Computational Epidemiology From Disease Transmission Modeling To Vaccination Decision Making 1st Ed

Health Information Science
Jiming Liu
Shang Xia
Computational
Epidemiology
From Disease Transmission Modeling
to Vaccination Decision Making
Series editor
Yanchun Zhang, Victoria University, Melbourne, Victoria, Australia
Editorial Board
Riccardo Bellazzi, University of Pavia, Italy
Leonard Goldschmidt, Stanford University Medical School, USA
Frank Hsu, Fordham University, USA
Guangyan Huang, Victoria University, Australia
Frank Klawonn, Helmholtz Centre for Infection Research, Germany
Jiming Liu , Hong Kong Baptist University, Hong Kong
Zhijun Liu, Hebei University of Engineering, China
Gang Luo, University of Utah, USA
Jianhua Ma, Hosei University, Japan
Vincent Tseng, National Cheng Kung University, Taiwan
Dana Zhang, Google, USA
Fengfeng Zhou, Shenzhen Institutes of Advanced Technology, Chinese Academy
of Sciences, China
With the development of database systems and networking technologies, Hospital
Information Management Systems (HIMS) and web-based clinical or medical
systems (such as the Medical Director, a generic GP clinical system) are widely
used in health and clinical practices. Healthcare and medical service are more data-
intensive and evidence-based since electronic health records are now used to track
individuals’ and communities’ health information. These highlights substantially
motivate and advance the emergence and the progress of health informatics research
and practice. Health Informatics continues to gain interest from both academia and
health industries. The significant initiatives of using information, knowledge and
communication technologies in health industries ensures patient safety, improve
population health and facilitate the delivery of government healthcare services.
Books in the series will reflect technology’s cross-disciplinary research in IT and
health/medical science to assist in disease diagnoses, treatment, prediction and mon-
itoring through the modeling, design, development, visualization, integration and
management of health related information. These technologies include information
systems, web technologies, data mining, image processing, user interaction and
interfaces, sensors and wireless networking, and are applicable to a wide range of
health-related information such as medical data, biomedical data, bioinformatics
data, and public health data.
Series Editor: Yanchun Zhang, Victoria University, Australia;
Editorial Board: Riccardo Bellazzi, University of Pavia, Italy; Leonard Gold-
schmidt, Stanford University Medical School, USA; Frank Hsu, Fordham Uni-
versity, USA; Guangyan Huang, Victoria University, Australia; Frank Klawonn,
Helmholtz Centre for Infection Research, Germany; Jiming Liu, Hong Kong Baptist
University, Hong Kong, China; Zhijun Liu, Hebei University of Engineering, China;
Gang Luo, University of Utah, USA; Jianhua Ma, Hosei University, Japan; Vincent
Tseng, National Cheng Kung University, Taiwan; Dana Zhang, Google, USA;
Fengfeng Zhou, Shenzhen Institutes of Advanced Technology, Chinese Academy
of Sciences, China.
More information about this series at http://www.springer.com/series/11944

Jiming Liu • Shang Xia
Computational Epidemiology
From Disease Transmission Modeling
to Vaccination Decision Making
Jiming Liu Shang Xia
Department of Computer Science Department of Computer Science
Hong Kong Baptist University Hong Kong Baptist University
Kowloon, Hong Kong Kowloon, Hong Kong
ISSN 2366-0988 ISSN 2366-0996 (electronic)

ISBN 978-3-030-52108-0 ISBN 978-3-030-52109-7 (eBook)
https://doi.org/10.1007/978-3-030-52109-7
© Springer Nature Switzerland AG 2020

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This book is dedicated to all the people
around the world, who came together to fight
against the novel coronavirus (COVID-19)
pandemic.
Jiming Liu and Shang Xia
Preface
What Can We Learn from COVID-19?
To see a World in a Grain of Sand

And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour
William Blake (1757–1827) Auguries of Innocence
The title of this preface may look a bit unusual for a research monograph.
Nevertheless, this was indeed the kind of questions that came to our mind when
we were writing it.
This book was born in a special time. As it was being written and published, the
world was enduring one of its greatest challenges in decades, if not in centuries. The
novel coronavirus, known as COVID-19, had rapidly spread to around 200 countries
and territories in 6 continents (with only Antarctica untouched) within a few months,
resulting in more than 5 million people infected and over 300,000 deaths (as of
May 2020). All the people, no matter where they were and who they were, found
themselves caught right amid this most unprecedented global crisis, with devastating
casualties, country lockdowns, service/business shutdowns, and possible economic
meltdown.
The world is truly in a state of emergency, a time of great uncertainty and anxiety.
Yet, as in the history of human civilization, we all should be hopeful that
humankind will be able to learn and prevail in the end. There will be no exception
this time. One of the important lessons that we can probably learn from the
Mother Nature in this global fight against COVID-19 is that only by being
united as one, as humanity, working together to remove the barriers of races,
vii
viii Preface
nation-states, political ideologies, religions, and special interests, and coexisting

harmoniously in an increasingly interconnected and interdependent world, can
human beings be saved. This also calls upon scientists to rethink their roles and
social responsibilities, to rediscover the world, and to advance sciences beyond
the usual disciplinary boundaries. Under such a unique circumstance, the theme
of this book becomes particularly appropriate, as it attempts to show how disciplines
such as computer science, systems science, and epidemiology can converge and
address some of the most pressing, socially relevant issues in eradicating diseases.
The contents presented in this book reflect part of our ongoing initiatives at Hong
Kong Baptist University (HKBU), which are aimed to address several important
problems in infectious disease epidemiology and to solve them in a systematic
way through the developed computational models, methods, tools, and case studies.
Some examples of the problems are as follows:
• How has the field of epidemiology evolved (Chap. 1)? How can data-centric
technologies be incorporated? (Chaps. 1 and 7)
• How can the heterogeneous nature of disease transmission be modeled and
characterized? (Chap. 2)
• How can we strategically plan and achieve disease interventions (Chap. 3)?
• How can we take into consideration the human (individual and social) aspects of
decision-making in disease interventions? (Chaps. 4–6)
• How can the epidemiological challenges be best addressed from a systems
perspective? (Chap. 7)
• What promises does systems epidemiology hold? What is the best way to pursue
it? (Chap. 7)
Solutions to the above problems can help governments, public health policy-
makers, scientists, and front-line practitioners in seeing the current and future global
health challenges, such as COVID-19, from a systematic, data-driven computational
modeling perspective, and hence developing the corresponding effective interven-
tion strategies. For instance, the solutions provided in this book can help respond
to the following questions in the case of COVID-19: Once a coronavirus vaccine
becomes available, what will be the best (scientifically sound and yet practically
acceptable) way to administer the limited supplies? Who will have the priorities?
Will there be enough people to take the vaccine, so that the target coverage (herd
immunity) can be achieved? How will people make their vaccination decisions?
The book is intended to serve as a reference book for researchers and practition-
ers in the fields of computer science and epidemiology, who may read Chaps. 1 and 7
of the book first, to gain a holistic view of the domain, prior to reading Chaps. 2–6
for further studies on the specific problems and issues involved.
Together with the provided references for the key concepts, methods, and
examples being introduced, the book can readily be adopted as an introductory text
for undergraduate and graduate courses in computational epidemiology as well as
systems epidemiology and as training materials for practitioners and field workers,
Preface ix
who may study the book in the regular order of Chaps. 1–7 and then revisit Chaps. 2–
6 to extend some of the topics and problems.
Hong Kong Jiming Liu

Hong Kong Shang Xia
May 2020
Acknowledgements
Jiming Liu is extremely grateful for the rigorous foundation development in

Physics that he acquired from East China Normal University in Shanghai in the
late 1970s and early 1980s and for the mind-opening education and enriched
inquiries in Philosophy, Cybernetics, and Psychology that he gained through the
most inspirational teachings of David Mitchell, Gary Boyd, and Gordon Past, as
well as other thinkers and visionaries, from Concordia University in Montreal in
the mid-1980s. Both periods have profoundly impacted him throughout his career
and life. He would like to acknowledge the amazing collegiality and friendships that
he has enjoyed in more than three decades from many of his mentors, colleagues,
collaborators, and students in Montreal before 1994, in Windsor in 2006–2007, and
in Hong Kong since 1994, who have not only accompanied, but also enlightened,
him throughout his odyssey of intellectual discovery, exploration, and wonder. For
the past ten years, he has made special efforts in developing solutions to address
real-world problems, such as global health and infectious disease epidemiology
in particular, from the novel perspectives of complex systems, network science,
machine learning, and autonomy-oriented computing. For this and other rewarding
journeys, he would like to express his heartfelt gratitude to: Xiao-Nong Zhou (as
well as dedicated colleagues) of National Institute of Parasitic Diseases (NIPD) at
Chinese Center for Disease Control and Prevention (China CDC), with whom he
co-established the Joint Research Laboratory for Intelligent Disease Surveillance
and Control; his long-time colleagues as well as collaborators and supporters at
Hong Kong Baptist University (HKBU), William Cheung, Pong Chi Yuen, Yiu-
ming Cheung, Yang Liu, among so many others; his previous postdoctoral fellows
and research collaborators, Bo Yang, Zhiwen Yu, Xiaofeng Xie, Qing Cai, Zhanwei
Du, etc.; his earlier research students, Shang Xia, Benyun Shi, Chao Gao, Li Tao,
Xiaolong Jin, Hongbing Pei, Hechang Chen, Xiaofei Yang, Shiwu Zhang, Hongjun
Qiu, Jianbing Wu, Qi Tan, Jinfu Ren, and many more. Also, he would like to thank
HKBU as a whole for the trust and opportunities to shape and contribute to the
university environment in the capacities of Chair Professor in Computer Science,
Head of Computer Science Department, Associate Dean (Research) of Faculty
of Science, Dean of Faculty of Science, and Associate Vice-President (Research)
xi
xii Acknowledgements
of the university, to make it the most conducive place for scholarship. He would
like to thank Hong Kong Research Grants Council (RGC) for the funding support
over the years; a number of grants have been awarded to specifically support his
team’s research on understanding and solving epidemiological problems through the
exciting routes of computer science, machine learning, and artificial intelligence.
Last but the foremost, he would like to express his deepest thanks to his wife
M.L. and his daughters I.Y.Y. and B.Y.X. for their long-lasting love and the most
wonderful time.
Shang Xia would like to express his sincere gratitude to Prof. Jiming Liu for his
enlightening, patience, motivation, enthusiasm, and profound knowledge. Without
his encouragement and persistence, this book could not be accomplished. He would
like to express his sincere gratefulness to Computer Science Department at Hong
Kong Baptist University (HKBU), where he acquired his PhD degree, benefited a lot
from the most inspirational guidance, and enjoyed a fulfilling campus life. For this
rewarding journey in Hong Kong, he would sincerely express his heartfelt gratitude
to Dr. Benyun Shi, Dr. Li Tao, and Dr. Yang Liu, from whom he benefited their
collaboration and support. The sincere thanks also go to Prof. Xiao-Nong Zhou and
the National Institute of Parasitic Diseases at Chinese Center for Disease Control
and Prevention for the great support for his academic career and research. Last but
not least, he would like to thank his family: his wife Yao Q.Q. and his daughters
Yoyo and Xiuxiu for their caring, love, and support in this wonderful life.
Both authors wish to express their special thanks to Dr. Yang Liu for his great
efforts in proofreading the manuscript and offering excellent editorial suggestions
and help.
Hong Kong Jiming Liu

Hong Kong Shang Xia
May 2020
Contents
1 Paradigms in Epidemiology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Methodological Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Infectious Diseases and Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Objectives and Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Modeling Infectious Disease Dynamics . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.2 Modeling Vaccine Allocation Strategies. . . . . . . . . . . . . . . . . . . . . . . 8
1.4.3 Modeling Vaccination Decision-Making . . . . . . . . . . . . . . . . . . . . . . 9
1.4.4 Modeling Subjective Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Computational Modeling in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Modeling Infectious Disease Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Infectious Disease Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Age-Specific Disease Transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Modeling Contact Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Empirical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Hong Kong H1N1 Influenza Epidemic . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Age-Specific Contact Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Strategizing Vaccine Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Herd Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 Vaccine Allocation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Vaccination Priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Age-Specific Intervention Priorities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Modeling Prioritized Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
xiii
xiv Contents
3.3.2 Effects of Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.3 Effects of Contact Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.4 Integrated Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Hong Kong HSI Vaccination Programme . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Effects of Prioritized Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Explaining Individuals’ Vaccination Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Costs and Benefits for Decision-Making. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Game-Theoretic Modeling of Vaccination Decision-Making . . . . . . . . . 51
4.3 Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.1 Hong Kong HSI Vaccination Programme . . . . . . . . . . . . . . . . . . . . . 53
4.3.2 Vaccination Coverage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Characterizing Socially Influenced Vaccination Decisions . . . . . . . . . . . . . . 57
5.1 Social Influences on Vaccination Decision-Making . . . . . . . . . . . . . . . . . . . 57
5.2 Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Vaccination Coverage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6 Understanding the Effect of Social Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.1 Modeling Subjective Perception. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Subjective Perception in Vaccination Decision-Making . . . . . . . . . . . . . . 74
6.2.1 Dempster-Shafer Theory (DST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.2 Spread of Social Awareness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3.1 Vaccination Decision-Making in an Online Community . . . . . 78
6.3.2 Interplay of Two Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Welcome to the Era of Systems Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1 Systems Thinking in Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Systems Modeling in Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3 Systems Modeling in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.4 Toward Systems Epidemiology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Abbreviations
ACIP Advisory Committee on Immunization Policy

AEFI Adverse events following immunization
BPA Basic probability assignment
CHP Centre for Health Protection
DST Dempster–Shafer theory
H1N1 Influenza A virus (H1N1)
H7N9 Avian influenza A virus (H7N9)
HSI Human swine influenza
HSIVP Human Swine Influenza Vaccination Programme
MMR Measles–mumps–rubella
NVAC National Vaccine Advisory Committee
SARS Severe acute respiratory syndrome
SEIR Susceptible–exposed–infectious–recovered
SIR Susceptible–infectious–recovered
SIS Susceptible–infectious–susceptible
SIT Social impact theory
STD Sexually transmitted disease
xv
Notation
S Population in susceptible compartment

I Population in infectious compartment
R Population in recovered compartment
N Overall population
Si Susceptible subpopulation i
Ii Infectious subpopulation i
Ri Recovered subpopulation i
Ni Overall subpopulation i
α Infectivity
β Susceptibility
λ Infection rate
μ Transmission rate
γ Recovery rate
cij Contact frequency between two subpopulations i and j
R0 Basic reproduction number
Rt Effective reproduction number
CH Contact matrix for household setting
CS Contact matrix for school setting
CW Contact matrix for workplace setting
CG Contact matrix for general community setting
C Contact matrix for overall social setting
Φ Social settings (H, S, W, G)
rH Household contact coefficient
rS School contact coefficient
rW Workplace contact coefficient
rG General community contact coefficient
K Disease reproduction matrix or next-generation matrix
A Infectivity matrix, diag (α1 , . . . , αN )
B Susceptibility matrix, diag (β1 , . . . , βN )
S Susceptible population matrix, diag (S1 , . . . , SN )
I Infectious population vector, [I1 . . . IN ]T
xvii
xviii Notation
ρ(K) Top eigenvalue of K

x1 Top left eigenvector of K
y1 Top right eigenvector of K
Nivac Number of vaccinated neighbors
Ninon Number of unvaccinated neighbors
wij Social closeness between two connected individuals i and j
λ̂i Perceived infection rate for subpopulation i
β̂ Perceived susceptibility
θ Herd immunity threshold
ζ Cost of disease infection
ξ Cost of vaccination
rc Cost ratio rc = ξ/ζ
σi Vaccination decision
σˆi Cost-minimized choice
σ˜i Social opinion from connected neighbors
ιvac
i Social influence for vaccination
ιnon
i Social influence against vaccination
ιi Influence discrepancy
ν Responsiveness to influence discrepancy in Fermi function
P (ιi ) Probability generated from Fermi function
rf Conformity rate
G Social network, G = V , L
V Network nodes (individuals)
L Network links (interactions)
Θ Universal set of vaccination decision, {Yes, No}
φ Empty set
2Θ Power set, {φ, {Y es}, {No}, Θ}
m(·) Basic probability assignment
m(Y es) Belief value of vaccination
m(No) Belief value of non-vaccination
m(Θ) Belief value of no decision (uncertainty)
mi Set of belief values
mi Updated belief values
mei Obtained awareness about negative events
medis Belief values generated by a severe disease infection
mevac Belief values generated by a vaccine adverse event
f Coefficient of awareness fading
Reporting rate of severe disease infections
κ Reporting rate of vaccine adverse effects
Chapter 1
Paradigms in Epidemiology
Epidemiology deals with “the study of the occurrence and distribution of health-
related states or events in specified populations, including the study of the deter-
minants influencing such states, and the application of this knowledge to control
the health problems” [1]. As defined by MacMahon et al. [2], epidemiology is
interdisciplinary by nature, concerning the sciences of etiology, genetics, biology,
pharmacy, geography, ecology, as well sociology and human behavior. Epidemio-
logical studies motivated by combating infectious diseases mainly focus on four
aspects of challenges, as follows: (1) pattern analysis, by investigating the spatio-
temporal distributions of the observed disease occurrences; (2) causal inference, by
identifying and evaluating associated impact factors; (3) forecasting and prediction,
by evaluating the dynamics of infectious diseases with reference to different sce-
narios; and (4) policy analytics, by exploring and conducting effective intervention
measures.
Toward these ends, the pioneers in epidemiology have provided much useful
knowledge to guide efforts in infectious disease control. As pointed out by Merrill
[3], epidemiology has evolved from supernatural practices to research based on
scientific foundations, from ad hoc reports to systematic investigations of public
health events and problems, from ignorance of the causes of diseases to a scientific
understanding of their hidden factors, determinants, and outcomes, and from lacking
feasible means for solving public health problems to having effective approaches to
disease intervention.
Developmental milestones in infectious disease epidemiology can be dated
back to the work of Hippocrates (460–377 BC), who examined the influence of
environments and attempted to explain how diseases transmit and cause infection
in a group of host individuals [3]. Other early studies include the work of John
Graunt (1620–1674), who described disease mortality rates by applying statistical
and census methods [4], and Thomas Sydenham (1624–1689), who studied disease
distribution patterns, moving from an observational to an analytical perspective [5].
In the nineteenth century, John Snow (1813–1858) traced the sources of disease
© Springer Nature Switzerland AG 2020 1

J. Liu, S. Xia, Computational Epidemiology, Health Information Science,
https://doi.org/10.1007/978-3-030-52109-7_1
2 1 Paradigms in Epidemiology
outbreaks (e.g., cholera in Soho, London, in 1854) and thereafter pointed out the
associations of disease outbreaks with social and natural environments [6]. To
more formally describe the dynamics of disease transmission, Ross (in 1911) and
MacDonald (in 1957) developed a set of mathematical equations and proposed
a threshold indicator, named the basic reproduction number, to quantitatively
characterize the extent of disease transmission [7].
1.1 Methodological Paradigms
Various methodologies have been developed to address a wide range of challenges

in infectious disease control and prevention, and these methods have been applied in
epidemiological studies in the past several decades. As stated by Zadoks [8], based
on the observation of disease occurrences, descriptive methods, such as clustering
and hot spot analysis, have been used to analyze the patterns of infectious diseases
in terms of their temporal, spatial, and demographic distributions in a population,
i.e., to answer the questions of when, where, and who. Statistical methods, such
as regression or Bayesian inference, can be used to further explore the causal
relationships between disease occurrence and the possible impact factors, i.e., to
answer the questions of why and how. Predictive methods, such as mathematical
modeling or computer-based simulation, have been developed to forecast the
dynamics of infectious diseases during an epidemic, and identify the most suitable
indicators for representing such a dynamic process. Based on these, prescriptive
methods, such as optimization, or scenario and sensitivity analyses, can be used by
public health authorities to decide how to implement the most effective intervention
strategies, such as the allocation of pharmaceutical resources (e.g., vaccines and
antivirals) and social distancing (e.g., segregation and school closures).
Infectious disease epidemiology has undergone a number of methodological
paradigm shifts throughout its development, as highlighted in Fig. 1.1. The typical
methods mentioned in the preceding paragraph, i.e., descriptive, predictive, and pre-
scriptive methods, correspond to three of those paradigms (the fourth is introduced
in the next subsection and discussed in detail in the final chapter of this book).
These three paradigms are (1) empirical investigation, (2) theoretical modeling, and
(3) computational modeling. Accordingly, we refer to the epidemiological method-
ologies based on these paradigms as (1) empirical epidemiology, (2) theoretical
epidemiology, and (3) computational epidemiology, respectively.
• Empirical Methods
The paradigm of empirical observation and investigation is well suited to the
early stage of epidemiological studies. As mentioned by Rothman et al. [9], it
typically involves (1) collecting observational data about disease transmission,
i.e., when, where, and who, and associated impact factors, e.g., the characteristics
of disease pathogens and host individuals at the microscopic scale, and of etiolog-
ical and meteorological environments at the macroscopic scale; (2) qualitatively
describing or quantitatively analyzing observational data to establish associative
1.1 Methodological Paradigms 3
Fig. 1.1 Major methodological paradigms in infectious disease epidemiology
or causal relationships between impact factors and disease transmission; and (3)
conducting further experiments or field investigations to test epidemiological
hypotheses, usually relating the proposed causes to the observed effects, the
findings of which may serve as the foundation for planning and implementing
disease intervention.
• Theoretical Methods
The theoretical paradigm in epidemiological studies involves the use of mathe-
matical tools, and is focused on generalizing and characterizing the processes of
disease transmission and their interrelationships with various impact factors [10].
Mathematical equations or models are typically constructed to quantitatively
describe the dynamics of disease transmission and estimate possible outcomes.
By evaluating different conditions under which the models reach convergent,
stable, or equilibrium states, public health authorities can potentially make long-
term projections and informed decisions on disease intervention. Theoretical
epidemiology sometimes draws on certain assumptions and simplifications about
the real processes of disease transmission. Meanwhile, it may also require
mathematical operations to derive model constructs of the behaviors of various
diseases, and use these to infer the disease dynamics and the corresponding
intervention measures.
• Computational Methods
With the developments in artificial intelligence, machine learning, data analytics,
data mining, and geographic science and information systems, the computational
paradigm has rapidly emerged in epidemiological studies. Computational meth-
ods are aimed to better characterize and understand the real processes of disease
transmission, by modeling and analyzing the patterns of transmission and quan-

titatively evaluating the potential outcomes of disease intervention [11]. Primary
computational tools that are used comprise computational modeling, simulation,
prediction, and optimization, as well as data analytics and visualization, to
make the results accessible to public health authorities and epidemiologists. This
has further expanded the scope and capabilities of epidemiology for analyzing
and predicting the dynamics of disease transmission and the effects of disease
intervention in a given population. In addition, public health authorities are now
able to more effectively conduct scenario analysis, which facilitates their strategic
decision-making.
1.2 Recent Developments
The above-mentioned methods have been in vogue for several decades and have
been used to make great contributions to our understanding and ability to combat
infectious diseases. However, there remain a number of challenges. As schemati-
cally illustrated in Fig. 1.2, these challenges come from emerging and re-emerging
infectious diseases, which are significantly correlated with multiple impact factors
and their interacting effects, such as genetic mutation of disease pathogens/parasites
[12], human socio-economic and behavioral changes [13], and environmental and
ecological conditions [14, 15].
Fig. 1.2 Some interacting components (in circles) and their associated impact factors that can
affect the transmission of infectious diseases
1.2 Recent Developments 5
Now we examine influenza as an example. It has been shown that a wide range
of factors are involved in the dynamic processes of these outbreaks [16], which may
include the following: (1) pathogenic factors, such as viral genetic recombination
and the expression of pathogens; (2) host factors, such as the immunity of people
at different ages; (3) social and behavioral factors, such as people’s movement
or travel activities; and (4) policy factors, such as disease intervention measures.
Furthermore, these factors closely interact with each other. For instance, disease
pathogens are carried by humans as they travel, which accelerates the recombination
of different types of viruses. However, the implementation of disease intervention,
e.g., school closures, changes people’s contact behaviors, effectively cutting off the
route of disease transmission.
This highlights how various factors can interrelate and interact at various scales.
Crucially, the coupling and interactive relationships among those impact factors
can determine the intrinsic (yet possibly hidden) spatial, temporal, and social
mechanisms of disease transmission. These mechanisms can involve systemic
characteristics, such as feedback, saturation, bifurcation, and chaos, thus posing new
challenges for comprehensive epidemiological investigations [17].
Effective intervention measures rely on biomedical understanding of disease
pathogens/parasites, descriptive studies of spatio-temporal patterns of disease occur-
rences, and causal analysis of impact factors. In addition, predictive explorations
of the trends of disease transmission, i.e., the mechanistic interactions among
the components of the transmission process, are also key to understanding and
combating infectious diseases. For example, an early warning system for an
emerging infectious disease, like COVID-19, requires knowledge about the possible
geographic routes of disease transmission, such as human air-travel networks
[18]. The prevention of zoonotic and vector-borne diseases, like COVID-19 and
malaria, requires both environmental and ecological changes of animal/vector
species to be addressed [19], as well as human migrant and mobile behaviors
[20]. Furthermore, the effectiveness of disease intervention measures depends on
the efficacy of resource allocation, compliance of targeted host populations, and
responsive feedback to environmental modifications.
In addition to the above-mentioned challenges, epidemiological studies also
face new opportunities in the present and future data-centric era, enabled by the
confluence of data from various sources and the development of modeling and
analytical tools in data science [21]. For example, a global disease surveillance
system connects the health agencies of its member countries and partners at
different levels, via local, regional, national, and international organizations [22].
This surveillance system can be used for managing and sharing historical records
and reports on when and where specific people have been infected by certain kinds
of diseases.
Other data sources are also helpful for analyzing and modeling potential disease
transmission. For example, remote sensing data from satellites can readily be
utilized for mapping the meteorological and ecological conditions of local or global
environments [23, 24].
Another important source of data is Internet-based media, which can serve as

an informative channel for revealing individuals’ health-related behaviors and opin-
ions. For example, Google Flu Trends was earlier used to assess the transmission
of influenza virus [25], and the use of Internet search data was demonstrated to be
effective in predicting dengue fever [26].
In view of these challenges and opportunities, it is imperative that new method-
ologies and paradigms are developed that offer novel perspectives and methods for
comprehensive investigation of disease dynamics and associated impact factors,
thus expanding our capabilities to understand, predict, control, and prevent the
transmission of infectious diseases.
1.3 Infectious Diseases and Vaccination
Faced with the threat of infectious diseases, implementing timely and effective
disease intervention measures is critical for preventing mortality and debilitating
morbidity, and reducing the socio-economic losses. Various types of intervention
measures have been widely studied and adopted for these purposes. For example,
immediate isolation/quarantine can prevent transmission during an influenza-like
epidemic [27, 28]. The mass prophylactic use of antiviral drugs can reduce the
vulnerability of susceptible individuals exposed to infectious diseases [29]. Inter-
ventions by social distancing (e.g., school closures and workplace shutdowns) can
lower the frequency of contacts among the host population and, hence, reduce the
probability of transmitting diseases between susceptible and infectious individuals
[30, 31].
Besides the above-mentioned intervention measures, vaccination has been
regarded as one of the most effective methods for preventing infectious diseases,
due to the effect of vaccine-induced herd immunity (i.e., immunizing a certain
portion of the host population provides indirect protection for the unimmunized
individuals [32]). That is to say, to prevent a potential outbreak, the vaccination
coverage in a host population needs to be above a critical level for inducing the
effect of herd immunity, known as herd immunity threshold. In practice, it remains
a continual challenge for public health authorities to achieve such a threshold of
vaccination coverage for preventing disease outbreaks.
The task is challenging due to a series of reasons. For one thing, although
significant progress has been made over the years in vaccine development, the
capacity for providing adequate and timely vaccine doses remains a concern,
especially when encountering emerging infectious diseases, e.g., 2009 influenza A
(H1N1) [33]. Supply restrictions can arise due to many factors, including the time
needed for finalizing vaccine compositions, to respond to the constantly evolution
of new disease strains [34], the limited capacity for vaccine manufacturing and
logistics [35], and the difficulties in access to and uptake of vaccines due to
poor delivery infrastructures and economic constraints, especially in developing
countries [36]. In such situations, public health authorities in charge of vaccination
1.4 Objectives and Tasks 7
programs face the question of how to allocate a finite number of available vaccine
doses to most effectively prevent disease transmission. For example, the World
Health Organization (WHO) has strongly suggested that each country should
respond to a possible shortage of vaccine supplies by deciding in advance which
groups should have access [37].
Furthermore, the public acceptance of a vaccination program will crucially affect
the actual level of vaccine uptake: any loss of confidence in vaccine safety and
efficacy will lead to huge gaps between the level of public vaccination willingness
and the level needed to contain disease transmission. Historically, societies have
experienced several events of vaccine refusal, e.g., the pertussis vaccine scare in
the 1970s [38], the decline of measles-mumps-rubella (MMR) vaccine uptake in
the 1990s [39, 40], and the rise and popularity of anti-vaccination movements [41,
42]. The rejection of vaccination and the subsequent decline of vaccine uptake have
brought about outbreaks of certain vaccine-preventable diseases that were thought
to no longer be threats to humankind [43, 44].
In view of this, an in-depth understanding of individuals’ voluntary vaccination
compliance is urgently required. It has been found that public acceptance of
vaccination, which amounts to individuals’ decisions on whether or not to take
vaccines, are affected by a mixture of cultural, behavioral, and socio-economic
factors. For example, the public may have doubts about vaccine safety and efficacy
due to scare stories around the adverse effects of vaccination [45, 46]. Or, behaving
in their own self-interest, individuals may be inclined not to get vaccinated if
enough other people have been vaccinated [47, 48]. The affordability and convenient
accessibility of new vaccines are also of importance for individuals considering
vaccination, especially in developing countries [49, 50].
Furthermore, the rapid emergence of online social media, e.g., Facebook and
Twitter, allows opinions, whether for or against vaccination, to spread broadly
and immediately among the population [51]. Therefore, social influences play an
increasingly important role in individuals’ vaccination decisions. In this regard, an
individual’s decision on whether or not to vaccinate himself/herself is no longer a
personal affair, but will affect the decisions of others, and collectively determine the
final coverage of a vaccination program.
Clearly, there exists an urgent need for more systematic studies of vaccination
at both population and individual levels, and thereby improve the efficacy of
vaccination programs for preventing the outbreak of infectious diseases.
1.4 Objectives and Tasks
In this book, we examine the dynamics of disease transmission in a host population,

in which individuals’ contact relationships are inferred from the socio-demographic
data. Based on such a description of disease transmission, we address the problem
of vaccine allocation by developing a novel prioritization method that targets certain
subpopulations to most effectively reduce disease transmission. Furthermore, to
investigate individuals’ acceptance of vaccination, we present decision models to

characterize individuals’ voluntary vaccination and evaluate the impact of social
influences and individuals’ subjective perception on the effectiveness of disease
intervention by vaccination.
1.4.1 Modeling Infectious Disease Dynamics
As aforementioned, the dynamics of disease transmission depend on many disease-

and host population-related factors. To characterize the heterogeneity of a host pop-
ulation, we need to consider its age structure, and then construct a compartmental
model to describe disease dynamics with respect to individuals’ age-specific vari-
ations, i.e., the heterogeneity in terms of age-specific infectivity and susceptibility,
as well as cross-age contact relationships, as in the case of COVID-19 transmission
[52].
For the purpose of demonstration in this book, we consider the real-world
scenario of the 2009 Hong Kong H1N1 influenza epidemic, and calibrate our
demonstration parameters with reference to the epidemiological characteristics of
influenza A (H1N1). As detailed information about the actual contacts among age-
specific subpopulations is often unavailable, we exploit a computational method to
infer the contact relationships in terms of individuals’ cross-age contact frequencies
from the census data in Hong Kong. Specifically, we represent individuals’ actual
contacts as cross-age contact frequencies within four specific social settings, i.e.,
school, household, workplace, and general community. We then estimate the overall
contacts that account for disease transmission by incorporating four setting-specific
contact frequency matrices, which are weighted with the coefficients corresponding
to the proportions of individuals’ contacts within the considered social settings.
To computationally evaluate our model, we carry out a series of simulation-
based experiments to examine its predicted disease dynamics. That is, we validate
our epidemic model by comparing the model predictions with the real-world
observations, in terms of the daily new infection cases and the age-specific attack
rates, i.e., the proportion of infected individuals in each subpopulation. In essence,
we reproduce the dynamics of disease transmission based on the heterogeneity of
the age-structured host population. The results, as we describe later, can serve as
the basis for further discussions on vaccine allocation methods and on individuals’
voluntary vaccination.
1.4.2 Modeling Vaccine Allocation Strategies
The heterogeneity of the host population means that the disease-preventing effects
of vaccination in individuals of different ages can vary markedly. An immediately
related practical question is how to allocate a finite number of vaccine doses to
most effectively reduce disease transmission; crucially, this requires knowledge

of the effectiveness of this intervention. In this book, we focus on developing a
problem-solving method for answering this question. Specifically, we develop a
computational method for identifying the relative priority of each subpopulation,
by evaluating the effectiveness of age-specific vaccination in reducing disease
transmission. We examine the effects of disease intervention on containing disease
transmission by measuring the reproduction number corresponding to the age-
specific heterogeneity of a host population. By doing so, we identify subpopulations
whose vaccination will lead to the greatest reduction in disease transmission, by
considering the marginal effects of reducing the reproduction number in cases of
age-specific vaccine allocation.
Unlike the existing optimization-based methods, this proposed vaccine allocation
method has the following distinct characteristics:
• The method utilizes prior knowledge about individuals’ age-specific susceptibil-
ity and infectivity, real-time disease prevalence in each subpopulation, and the
basic patterns of individuals’ cross-age contact frequencies within each social
setting. Moreover, it does not rely on detailed information about individuals’
actual contacts, nor the potential changes in these contacts in response to disease
transmission, which would be difficult to rapidly and accurately determine in
practice.
• The method is designed to most effectively reduce disease transmission by
allocating a finite number of vaccine doses to certain target subpopulations.
The identified vaccination priorities can be adaptively regulated based on the
dynamics of disease transmission, i.e., the number of vaccine doses suggested to
be allocated to each subpopulation can be dynamically adjusted according to the
latest progress of disease transmission and vaccine supply.
• The method incorporates the effects of other disease intervention measures
being implemented simultaneously with vaccination, e.g., individuals’ contact
reduction. Therefore, in the situation of integrated disease intervention, the
method can provide more accurate and effective solutions for vaccine allocation.
We apply above-mentioned method to the real-world scenario of the 2009 Hong
Kong H1N1 influenza epidemic to identify the relative priorities of subpopulations
for disease intervention in Hong Kong. The results show that this method of
prioritizing age-specific subpopulations for vaccine allocation and social settings
for contact reduction can readily improve the effectiveness of disease transmission-
containing efforts.
1.4.3 Modeling Vaccination Decision-Making
In a voluntary vaccination program, individuals’ decisions on whether or not to

uptake vaccine crucially affect the level of vaccination coverage and, thus, the effec-
tiveness of disease intervention. In this regard, modeling and evaluating individuals’
vaccination decision-making would provide useful information for public health

authorities on how to improve the effectiveness of vaccination programs [53].
Researchers have typically utilized payoff-based approaches to characterize
individuals’ vaccination decision-making with respect to the perceived risks and
benefits of vaccination. Moving beyond that, we consider the fact that whether
an individual does or does not get vaccinated is also influenced by the decisions
of others, i.e., social influences. We thus view individuals’ voluntary vaccination
as an integrated decision-making process that incorporates both a cost analysis of
vaccination and the impact of social influences.
Our integrated decision model is an improvement over the existing models, and
has several interesting features, as follows:
• We model an individual’s vaccination decision-making as an integrated process
that balances his/her self-initiated cost minimization and the social influences
of others’ decisions. Moreover, this model introduces a parameter, called the
conformity rate, to modulate individuals’ tendency toward two decision-making
mechanisms: an individual will adopt his/her own cost-minimized decision, or
the social opinion of his/her interconnected neighbors.
• Based on the existing studies in which the social influences on the process of
opinion formation have been addressed, we further consider the heterogeneity of
individuals’ social relationships, i.e., how individuals are socially interconnected.
Computationally, we model and characterize the effect of networked social
influences on individuals’ vaccination decisions based on Social Impact Theory
(SIT).
• Based on this new model, we examine the effects of social influences on indi-
viduals’ decisions and on the effectiveness of disease intervention (vaccination
coverage), with respect to three determinants: (1) the relative costs of vaccination
and infection; (2) individuals’ conformity to social influences, i.e., conformity
rate; and (3) individuals’ initial level of vaccination willingness.
We parameterize the integrated decision model based on the real-world sce-
nario of the 2009 Hong Kong H1N1 influenza epidemic, and perform a series
of simulation-based experiments to infer the coverage of voluntary vaccination
programs as a result of individuals’ decision-making. The results indicate that
individuals’ vaccination decision-making can be affected by both the associated
costs and their conformity to social influences. Thus, it becomes necessary for public
health authorities to estimate the level of individuals’ acceptance of vaccination
prior to the start of a voluntary vaccination program, as well as to rapidly assess
and enhance the effectiveness of their adopted vaccination policies, e.g., providing
certain financial subsidies to reduce the cost of vaccination.
1.4.4 Modeling Subjective Perception
It has long been observed that the spread of awareness about an epidemic via social
media can affect individuals’ opinions and behaviors concerning an epidemic. In the
case of an emerging infectious disease, it can be difficult for individuals to become
informed about the disease and/or a newly developed vaccine prior to their decision-
making. In such a case, the spread of awareness about disease severity and vaccine
safety could affect individuals’ subjective perception about vaccination and, hence,
substantially affect their decisions [54].
To gain a better understanding of individuals’ voluntary vaccination, we develop
a belief-based decision model to evaluate the effect of the spread of awareness
on individuals’ decision-making and on the effectiveness of disease intervention.
Compared with the existing studies on modeling individual-level vaccination
decision-making, this belief-based model has the following unique properties:
• Unlike existing decision models that represent decision-making as a binary
problem, we consider the role of uncertainty in individuals’ vaccination decision-
making. Specifically, the situation in which an individual has made no firm
decision can be considered as a state of “yet to decide”, due to uncertainty. In this
regard, we introduce three belief variables to characterize the possible decision
response from an individual, namely that he/she will accept or reject the vaccine,
or has not yet decided.
• We further consider the fact that individuals’ decisions depend on their subjective
perception about whether or not vaccination is acceptable. Moreover, awareness
of disease severity and vaccine safety can spread from person to person—akin
to a disease itself—and will substantially affect their subjective perception of
vaccination.
• To model the spread of awareness, we utilize various real-world online social
networks to characterize the structure of individuals’ social relationships. There-
after, we further extend Dempster-Shafer Theory (DST) to computationally
model the propagation and evolution of individuals’ beliefs, as well as their
decision-making, having incorporated the awareness obtained from their socially
interconnected neighbors.
We investigate the effect of the spread of awareness on individuals’ vaccination
decision-making with respect to three considered impact factors, based on a series of
simulations of the 2009 Hong Kong H1N1 influenza epidemic. First, the reporting
rates of severe infection and adverse effects of vaccination are used to represent the
frequencies of these topics, which tend to draw public attention. Next, we consider
the coefficient of awareness fading, a parameter used to quantify the information
flows among individuals. Finally, we examine the effect of disease reproduction
number, which corresponds to the severity of an epidemic.
The simulation results show that the reporting rates will determine the number of
vaccinated individuals and the time at which they receive vaccination. A higher
fading coefficient will significantly reduce individuals’ vaccination willingness.
A larger value of disease reproduction number will enhance the proportion of

vaccinated individuals, although this cannot compensate for the growth of the
infected population size resulting from a more severe disease outbreak.
1.5 Summary
In this book, we develop a computational modeling approach to evaluate and guide

the implementation of different intervention measures for controlling infectious
diseases. We focus on the following topics in the remaining chapters:
In Chap. 2, we provide a general description of the concepts and related
computational models and tools for characterizing disease transmission dynamics
in a heterogeneous host population. Specifically, we introduce the concepts of
compartmental modeling for describing disease transmission in an age-structured
host population. Then, we present a computational method for inferring the cross-
age contact patterns of the population. Finally, we parameterize and validate the
epidemic model with a real-world epidemic scenario, which serves as the basis for
our further discussions on vaccine allocation and individuals’ voluntary vaccination.
In Chap. 3, we develop a prioritization method for identifying target subpop-
ulations for vaccine allocation that would enable us to most effectively reduce
disease transmission. We walk through a series of simulation-based experiments
that evaluate the performance of such a vaccine allocation method in improving the
effectiveness of disease intervention.
In Chap. 4, we examine vaccination decision-making from the perspective of
individuals. In particular, we show how to model individuals’ decision-making
processes in response to a voluntary vaccination program. We use computational
modeling to perform a game-theoretic analysis of the costs and benefits of vacci-
nation with respect to individuals’ social relationships. Then, we experimentally
examine the level of vaccination coverage, based on this game-theoretic model,
through a series of simulations of voluntary vaccination.
In Chap. 5, we introduce an extended decision model that additionally addresses
the effect of social influences on an individual’s decision whether to undergo
voluntary vaccination. In the extended decision model, we utilize the SIT to further
characterize social influences with respect to individuals’ social relationships. We
evaluate the effect of social influences by computing the level of vaccination
coverage through a series of simulations of voluntary vaccination based on such
an integrated decision model.
In Chap. 6, we present a more complete investigation of voluntary vaccination
by modeling and examining the effect of the spread of awareness on vaccination
decision-making. In doing so, we develop a belief-based decision model, in which
individuals’ decisions are affected by their subjective perception of vaccination. In
this model, we utilize and extend DST to characterize individuals’ belief updates and
changes in vaccination decisions accordingly. We evaluate the effect of the spread of
1.5 Summary 13
awareness by carrying out simulation-based experiments to examine the time course

of vaccine administration and disease transmission.
In Chap. 7, we offer a fresh outlook on the latest methodological paradigm
in infectious disease epidemiology, which is known as systems epidemiology.
Specifically, we introduce the fundamental ingredients of systems thinking, which
are essential for viewing and addressing complex epidemiological questions holisti-
cally. We then provide detailed systems modeling principles and practical steps that
can be followed in future systems epidemiological studies.
Finally, under “References” section, we provide a detailed list of references for
further reading and research.
Chapter 2
Computational Modeling in a Nutshell
There are many parallels between epidemiological studies and systems studies.
That is, we can view the problems of epidemiology from a systems perspective. In
systems studies, the goal of modeling is to develop representations or frameworks,
in mathematical or computational languages, that are abstracted from and yet
allow for characterization of certain real-world observations. In epidemiological
studies, the essence of modeling further entails two important aspects: (1) Problem-
driven conceptual modeling, which translates certain real-world problems in an
epidemiological domain into conceptual models in a theoretical or computational
domain. As an example, a metapopulation-based compartmental model may be
utilized to describe disease dynamics in an age-structured host population. (2) Data-
oriented real-world grounding, which requires us to discover ways to embody
the developed conceptual models, i.e., model parameterization, by obtaining and
utilizing real-world data and statistical analysis of the real-world observations. For
instance, the effect of age-specific population contact patterns may be inferred from
data by utilizing a computational method. Furthermore, we may parameterize and
validate the contact-based model with a real-world scenario, such as the 2009 Hong
Kong H1N1 influenza epidemic.
In this chapter, we provide an overview of the most essential models for
characterizing infectious disease transmission, namely, epidemic models and con-
tact relationship models. By doing so, we show how some of the questions in
infectious disease epidemiology can be approached by modeling and characterizing
the dynamics of infectious disease transmission [55].
2.1 Modeling Infectious Disease Dynamics
Accurate predictions about the dynamics of disease transmission play an essential

role in controlling infectious diseases. In this regard, mathematical modeling and

https://doi.org/10.1007/978-3-030-52109-7_2
16 2 Computational Modeling in a Nutshell
analysis offers important tools for characterizing disease transmission in a host

population and evaluating the effectiveness of intervention measures [56]. In this
section, we focus on the mathematical characterization of disease transmission
based on the contact relationships of individuals responsible for disease transmis-
sion. Specifically, we review (1) basic notions and concepts in disease transmission,
and (2) existing models for representing disease dynamics.
The spread of an infectious disease results from the transmission of the conta-
gious pathogen between hosts through their direct or indirect contact. During an
epidemic, the dynamics of disease transmission within a host population involve
multiple determinants, including disease-dependent biological factors and disease-
independent behavioral factors [57]. Disease-dependent biological factors refer to
the viral or bacterial features of the pathogen, such as the pathogen’s life cycle,
its dynamics of replication and development within the hosts, and its virulence
and sensitivity to drug treatments. Biological factors also include the physiological
conditions of a host population, e.g., individuals’ innate or acquired immunity.
Disease-independent behavioral factors that affect disease transmission include the
demographic and social structures of a host population, e.g., age distribution and
the contact relationships among different age groups. The contacts responsible for
disease transmission depend on the specific pathogen and the transmission route.
Taking influenza-like respiratory diseases as an example, individuals’ contacts
during informal gatherings, such as face-to-face conversations and handshakes, can
result in disease transmission [58].
In infectious disease epidemiology, one important indicator, called the basic
reproduction number R0 , has been widely used to evaluate the force of disease
transmission at the population level. It is defined as the number of secondary
infection cases generated by a typical infected individual during their entire period
of infectiousness in a completely susceptible population [59, 60]. R0 has been used
to indicate whether or not an infectious disease will cause an outbreak in a host
population. Specifically, if R0 is greater than one, outbreaks can occur. Conversely,
if R0 is less than one, disease transmission in a host population will naturally
disappear. Based on this property, the estimation of R0 provides a feasible way
of predicting disease outbreaks and evaluating the effectiveness of disease control.
That is to say, to prevent disease transmission in a host population, the adopted
disease interventions should ensure that R0 is less than the threshold of outbreak.
2.1.1 Infectious Disease Models
In studies of infectious diseases, mathematical modeling and analysis of disease

transmission is crucial for control and prevention. Infectious disease models are
formulated to characterize disease dynamics in terms of how susceptible individ-
uals will potentially be infected when exposed to infectious individuals, hence
offering a quantitative description of disease incidence and prevalence within a
host population. Therefore, developing infectious disease models helps reveal the
2.1 Modeling Infectious Disease Dynamics 17
Population level Individual level
(a) (b)
Metapopulation level
Susceptible
Infectious
Recovered
(c)
Fig. 2.1 Epidemic models: (a) population-level compartmental models, (b) individual-level
network-based models, and (c) metapopulation-level models
mechanisms of disease transmission and evaluate the outcomes of potential control

and prevention measures during an epidemic. In what follows, we look at three types
of models that have been used to represent disease transmission at different levels
of a host population. As shown in Fig. 2.1, the models are (1) population-level, (2)
individual-level, and (3) metapopulation-level models.
Population-Level Models
Early epidemic models, known as compartmental models, were developed based on
the assumption that individuals randomly have contact with each other (the homo-
mixing assumption as illustrated in Fig. 2.1a) [61, 62]. During the time course of
disease transmission, each individual belongs to one of the compartments, namely,
uninfected and susceptible (S), infected and not yet infectious (E, or exposed),
infected and infectious (I ), and recovered with immunity (R). Based on the homo-
mixing assumption, the dynamics of disease transmission can thus be modeled as
the population transitions among those different compartments. For example, the
susceptible-infectious-recovered (SIR) model has been widely used for describing

disease transmission, such as seasonal influenza and measles [63, 64].
Many variations have been proposed based on the basic framework of compart-
mental modeling. For example, the susceptible-infectious-susceptible (SIS) model
has been used to characterize the situation of secondary/repeated infections in
recovered individuals. The susceptible-exposed-infectious-recovered (SEIR) model
characterizes the important infections within the period of time when an individual
has been infected but is not yet infectious (i.e., in the exposed compartment E).
Furthermore, the S, I , and R compartments may be further divided into several
subgroups to reflect the demographic structure (e.g., age) of a host population
[56]. Thus, the dynamics of different subpopulations can be taken into account by
incorporating the transmission parameters for each subpopulation.
Although compartmental models can provide deterministic descriptions of dis-
ease dynamics at the population level, the assumption of homogeneous individuals
may not adequately capture the reality. In view of this, new approaches for
individual-level characterization of disease transmission have been developed.
Individual-Level Models
In attempts to explicitly characterize the heterogeneity of a host population, which
can affect the dynamics of disease transmission, new individual-level epidemic
models (also known as network-based models as illustrated in Fig. 2.1b) have been
developed to study the effects of the heterogeneity of pathogens, individuals, and
their interactions within a host population [65].
During an epidemic, disease transmissions depend on individuals’ contact
relationships. The contact patterns (frequency and structure of contacts in a host
population) can profoundly affect the population-level disease dynamics. Con-
tradicting the homo-mixing assumption that individuals uniformly and randomly
have contact with each other, each individual in reality has a different number of
relationships with contacts to whom he/she could potentially transmit a pathogen,
or from whom he/she could become infected with a pathogen. The structure
of individuals’ contacts in such a heterogeneous host population can readily be
described using a network model, in which nodes denote host individuals and links
represent the contact relationships among the interconnected nodes [66]. In a contact
network, the degree of a node (the number of links connected to an individual)
reflects the number of neighbors through which disease transmissions may take
place [67].
As mentioned, compartmental models utilize deterministic differential equations
to describe the mean dynamics of disease transmission at the population level,
but these cannot adequately characterize the effects of heterogeneity within a
host population. Network-based models overcome this shortcoming by modeling
individual-level disease transmission in a heterogeneous contact network. Here, the
transmission is characterized as a stochastic percolation process, in which each
individual may with a certain probability be infected by his/her network neighbors
[68]. This probability is computed based on the number of contact relationships
an individual may have. In other words, in a heterogeneous contact network,
2.1 Modeling Infectious Disease Dynamics 19
the number of contact relationships one has will determine his/her corresponding
chance of getting infected and, hence, the route of disease transmission through the
network.
Network-based models provide a way of incorporating individual-level het-
erogeneity. However, they also have limitations. For example, to construct a
contact network of disease transmissions, one must know in advance the detailed
characteristics of each individual and the disease-associated contact relationships.
In practice, it is difficult, if not impossible, to have such prior knowledge, sometimes
even for a small group of host individuals.
Thus far, existing studies have addressed the problem of representing individuals’
contact relationships primarily through statistical means, by collecting and analyz-
ing empirical data on individuals’ contact activities in certain places/regions for a
fixed time period [69]. These studies may provide static and empirical descriptions
of individuals’ contact relationships (who meets whom, when, where, and how
often). However, this approach also has limitations, as follows: (1) it is hard to define
the type of contacts responsible for disease transmission, because this depends on
the type of disease; (2) it is not always feasible to have such an empirical description
of contact relationships for a certain host population; and (3) individuals’ actual
contacts may change during the spread of an infectious disease, invalidating the
empirical knowledge basis on which disease transmissions are predicted.
In this regard, the challenge lies in the lack of accurate and reliable descrip-
tions of individuals’ contact relationships, without which the accuracy of model
predictions cannot be improved. As discussed in the following subsection, one
promising alternative is to develop a computational method to infer individuals’
contact relationships from the socio-demographic data of the population. Such
inferred contact relationships can serve as a good foundation for modeling age-
specific disease dynamics and the corresponding intervention strategies.
Metapopulation-Level Models
As aforementioned, population-level compartmental models based on differential
equations provide a deterministic description of the dynamics of disease transmis-
sion in a host population, and have the advantage of being tractable. However, the
assumption that individuals are homogeneously mixed neglects the variations in
the underlying disease transmission. In contrast, individual-level models based on
contact networks do address the host population’s heterogeneity, but the disease
dynamics resulting from a stochastic process are often intractable and sensitive to
the specific settings for simulations. In addition, it remains challenging to construct
realistic contact networks.
As a solution to bridge the gap between population-level compartmental mod-
els and individual-level network models, metapopulation-level models have been
developed, which modify the conventional compartmental models by further taking
into account the structure of a host population (Fig. 2.1c) [70, 71]. These models
address the heterogeneity of the host population by subdividing it into several
subpopulations according to certain characteristics of the individuals, such as their
age, occupation, and geographic location [72]. Once this is done, the infection
dynamics of the subpopulations are characterized, based on their distinct transmis-

sion characteristics and interrelationships.
There are some strong reasons for modeling the subpopulations in such a
manner. For example, in a disease outbreak, children and the elderly may be more
vulnerable to infection than younger adults [73]. School-age students may be more
inclined to have contact with individuals of similar ages, while adults of working
age tend to mostly associate with each other [74]. To address the challenge of
effective and timely infectious disease control, we need to investigate the role
of each subpopulation in disease transmission, corresponding to their specific
heterogeneous characteristics.
2.1.2 Age-Specific Disease Transmissions
The basic compartmental model (susceptible-infectious-recovered model, SIR for

short) deals with the dynamics of disease transmission within a single epidemic, in
which the birth and death rates of a host population are not taken into consideration
[56]. The number of individuals in each compartment is described by the following
set of ordinary differential equations:
dS
= −λS
dt
dI
= λS − γ I (2.1)
dt
dR
= γI
dt
where S, I , and R denote the numbers of individuals in the susceptible, infectious,
and recovered/immunized compartments, respectively. The parameter λ is the
infection rate, which denotes the proportion of the susceptible population that will
become infected during the present time step. The parameter γ is the recovery rate,
which describes the rate at which the infectious individuals recover and, therefore,
become immunized from secondary infection. In detail, λ can be viewed as a
composite of three factors:
I
λ=β· ·α (2.2)
N
where α describes the infectivity of infected individuals and β is the susceptibility
of uninfected individuals. N is the total size of the host population. In this case, the
basic reproduction number R0 can be computed as follows:
α·β
R0 = (2.3)
γ
2.2 Modeling Contact Relationships 21
We now introduce an age-specific compartmental model, in which individuals

are divided into n subpopulations with reference to their ages. Each individual in
age group i belongs to one of three infection-associated compartments: susceptible
(S), infectious (I ), and recovered/immunized (R). Correspondingly, the number
of individuals in each compartment is denoted by Si , Ii , and Ri , respectively.
We consider disease transmission within a single circulation season, such that the
natural birth and death rates of the population are not taken into account. The total
number of individuals in age group i, denoted by Ni = Si + Ii + Ri , is static.
The dynamics during the time course of disease transmission is described by the
following set of differential equations:
dSi
= −λi Si
dt
dIi
= λi Si − γi Ii (2.4)
dt
dRi
= γi Ii
dt
where γi represents the rate of recovery, corresponding to the duration of disease
infection. λi is the infection rate, which denotes the probability of being infected for
susceptible individuals in age group i. For each time step, λi can be calculated as
follows:
n

Ij
λi = μ · βi · cij · αj · (2.5)
Nj
j =1
where cij describes the contact frequency between individuals in age groups i and
j , αj measures the infectivity for individuals in group j , which is the probability
of disease transmission when an infectious individual has contacts with other
susceptible individuals, βi denotes the susceptibility for individuals in group i,
which represents the probability of being infected when a susceptible individual is
exposed to infectious contacts, and μ is a constant disease transmission rate for all
age groups and can be estimated from R0 in the initial stage of disease transmission.
2.2 Modeling Contact Relationships
To accurately predict the dynamics of disease transmission, it is crucial to know indi-

viduals’ contact relationships, as these determine the routes of disease transmission
within a host population. In what follows, we examine some of the frequently used
contact models. For the identified impact factors, the contact relationships among
the host population play an important role in determining the dynamics of disease
transmission. In practice, it remains a major challenge to accurately and reliably
describe individuals’ contact relationships during the course of disease transmission.

As a result, it is difficult to rapidly and reliably predict disease transmissions. For
this reason, efforts have been made to characterize the social contact relationships
based on the empirical data of individuals’ contact frequencies and durations [75].
In this section, we review some of the existing studies on characterizing individ-
uals’ contact patterns for modeling disease transmission within a host population.
Specifically, based on different ways of collecting contact-related information, we
consider two types of studies, namely (1) empirical methods that collect individuals’
actual contact activities, and (2) computational models that infer/simulate individu-
als’ contact behaviors from data.
2.2.1 Empirical Methods
Empirical methods involve collecting information about individuals’ actual contact

relationships to describe individuals’ contact patterns and thereby reveal how a
disease is transmitted from one person to another. As mentioned by Keeling [65],
we can use three types of empirical methods:
• Infection Tracing
Infection tracing is performed to determine the route of disease transmission
for each case of infection, i.e., for each infected individual, infection tracing
can identify by whom the individual was infected, and the group of persons
who have been infected by the individual. By doing so, infection tracing can
collect detailed information about the majority of infection cases and, hence,
reveal the probable routes of disease transmission. For example, Haydon et
al. constructed epidemic trees with respect to the 2001 U.K. foot-and-mouth
outbreak [76], while Riley et al. traced cases of the severe acute respiratory
syndrome (SARS) epidemic in Hong Kong [77]. Thus, by constructing a tree-like
network of disease transmissions, the corresponding spatial and temporal patterns
can be accurately depicted. Moreover, the epidemiological parameters, e.g.,
disease reproduction number R0 , can be directly estimated from the collected
infection cases. Therefore, infection tracing is an individual-based method that
can describe the network of disease transmissions within a relatively small
population, and provides a quick and simple analysis of disease dynamics.
• Contact Tracing
In contrast to infection tracing, which gathers information on the actual links
in the chain of disease transmissions, contact tracing focuses on identifying all
of the contact relationships for an infected individual. This method emphasizes
the potential routes of disease transmission, in terms of a group of individuals
who might be infected due to their contacts with infected individuals. In this
case, individuals in the network of contacts can be targeted for treatment or
quarantine [78]. Contact tracing has been successfully used to depict the network
of various disease transmissions, including that of sexually transmitted diseases
2.2 Modeling Contact Relationships 23
(STDs) [79, 80], as well as air-borne diseases [81, 82]. However, although both
infection tracing and contact tracing can provide a contact network to represent
the potential routes of disease transmission, such a network covers only a subset
of the host population, as it is focused on the contact relationships immediately
surrounding the infected individuals.
• Survey-Based Studies
Survey-based studies record individuals’ contact activities in detail for the
entire host population. For example, Mossong et al. collected diaries from
individuals in eight European countries that recorded the characteristics of those
individuals’ daily contacts with reference to their contactees’ age and sex, and
the location, duration, frequency, and occurrence of physical contacts [83]. Read
et al. launched a detailed diary-based survey of contact in terms of a group of
adults’ casual and close contact encounters [84]. Hens et al. carried out a 2-day
population survey in Belgium to mine social contact patterns for epidemic models
[85]. Such survey-based studies have provided comprehensive descriptions of
individuals’ contact patterns, which can be used to improve the accuracy of
epidemic models for infectious disease control.
The above-mentioned empirical studies can provide quantitative descriptions
of real networks of individuals’ contact relationships. However, there are several
limitations to epidemic modeling and disease control. First, it is difficult to define
the connections of a contact network in a way that generally represents the routes of
disease transmission, as the types of contact responsible for infection are disease-
dependent; e.g., influenza spreads through the air, while HIV is transmitted via
sexual contact. Next, it is time- and labor-intensive (and thus impractical) to
obtain detailed information from the entire population with respect to their contact
behaviors. Therefore, it is not always feasible to acquire such an empirical character-
ization of contact patterns within a specific host population. Further, survey-based
studies can only provide static and empirical descriptions of individuals’ contact
relationships. Finally, individuals’ potential behavioral changes, in response to
both disease transmission and adopted disease interventions, are not taken into
account by surveys, although these can significantly affect the dynamics of disease
transmission.
2.2.2 Computational Methods
The challenge for characterizing disease dynamics is to obtain sufficient realistic

data to represent the contact relationships among a host population. In this section,
we discuss how this challenge can be solved by applying computational models
that enable the representation and estimation of individuals’ contact relationships
and correspondingly characterize the effects of these relationships on disease
transmission.
• Contact Networks
Contact networks are essentially graphs, where a graph is a collection of nodes,
which are joined by a set of connections, called links. Each link denotes a
relationship or an interaction between the nodes it joins. Contact networks
can be further categorized as undirected or directed, and the directedness of a
network is highly epidemiologically relevant, as it indicates the possible paths of
propagation of a contagion.
• Contact Matrix
In a contact network of size N (i.e., with N nodes), a compact way to specify
all contact relationships is to utilize an adjacency matrix C, in which elements
cij = 1 if a link connects nodes i and j , and zero otherwise. Such a
matrix C is also known as a contact matrix, and can be used to describe the
individuals’ contact relationships, such as the occurrence of pairwise contacts or
the frequencies of their contacts [52]. Both contact networks and contact matrices
are computational characterizations of individuals’ contact relationships. Note
that contact networks that are undirected or directed correspond to contact
matrices that are symmetric or asymmetric, respectively.
Computational methods to infer contact networks from the demographic and
social characteristics of a host population are in practice very useful for infec-
tious disease modeling. For example, in studying the contact network of STDs,
simulation-generated networks have been used to characterize the patterns of
observed contact relationships, e.g., the role of hub individuals with a large number
of partners [86, 87]. Potterat et al. described a risk network for individuals with
HIV infection in Colorado Springs, U.S.A., by analyzing the community-wide
HIV/AIDS contact-tracing records during the 1980s and 1990s, i.e., sexual and
drug-injection partners [88].
In modeling outbreaks of air-borne diseases, simulation-generated contact net-
works have been used to capture the demographic and social characteristics of
the considered host population [89]. For example, Halloran et al. simulated the
stochastic spread of smallpox in a community of 2000 people, in which each
individual was generated from population’s age distribution and household size
that can be derived from census data [90]. Meyers et al. investigated the dynamics
of SARS by generating a contact network based on the social setting in the city
of Vancouver [91]. Eubank et al. explored dynamic bipartite graphs to model
the physical contact patterns corresponding to individuals’ movements between
different locations [92, 93].
2.3 Case Study
In this section, we examine a case study of the 2009 Hong Kong H1N1 influenza
epidemic. We discuss how to characterize the heterogeneous contact relationships
within an age-structured host population. Note that in this case, an empirical
2.3 Case Study 25
description of individuals’ contact relationships is not available and, thus, we need

a computational method to infer the cross-age contact frequency and structure from
the social-demographic data of the population. In addition, we examine how such an
age-structured contact characterization helps assess the dynamics of disease spread.
In doing so, we use a compartmental model that is parameterized and validated with
respect to the real-world epidemic scenario.
2.3.1 Hong Kong H1N1 Influenza Epidemic
In Hong Kong, the first H1N1 influenza (also known as human swine influenza,
HSI for short) infection case was an imported case confirmed on May 1, 2009
[94]. The first reported local case (i.e., the first indigenous HSI infection without
an epidemiological link to imported patients) was laboratory-confirmed on June 10,
2009. As of September 2010, there were over 36,000 laboratory-confirmed cases
of HSI in Hong Kong [95]. Figure 2.2a shows the daily number of newly reported
H1N1 infection cases over the period of 200 days since the disease onset in early
May 2009, as reported by the Centre for Health Protection (CHP) of Hong Kong
[96].
Figure 2.2b further gives the proportions of disease infections in different
age groups. In what follows, we present a data-driven computational model
for investigating the dynamics of disease transmission among those age-specific
subpopulations.
2.3.2 Age-Specific Contact Matrices
In this section, we discuss how to computationally characterize social contacts

based on individuals’ contacts in various social settings, namely, school, household,
workplace, and general community, which capture the patterns of interest or the
likelihoods of individuals having contact with each other. From such patterns, we
can further investigate ways of reducing social contact-based disease transmission,
as discussed in the next chapter, by enforcing changes to individuals’ contacts in
these social settings.
Here, we consider susceptible individuals being infected through their social
contacts with infectious individuals. The number of disease transmissions among
different age groups is therefore determined by the frequencies of contacts among
them. Individuals’ contact relationships can be represented using a contact matrix,
and it has been shown that there exist strong diagonal elements (indicating high
contact frequencies) among those aged 5–24 years [83]; this pattern reflects that
individuals tend to have contact with others of similar ages within places such
as schools. At the same time, there also appear parallel secondary diagonals that
(a)
(b)
Fig. 2.2 The 2009 Hong Kong H1N1 influenza epidemic. The confirmed cases of the H1N1
infection reported by the Centre for Health Protection (CHP) of Hong Kong for the first 200 days
since the disease onset. (a) The dynamics of disease transmission in terms of the daily number of
newly infected cases. (b) The proportions of reported infections in different age subpopulations
represent children mixing with adults, mainly in households, and a wider contact
“plateau” among adults, which accounts for contacts occurring in workplaces.
Individuals’ cross-age contacts exhibit specific patterns that correspond to the
likelihoods of individuals’ mixing together within certain social settings (school,
household, workplace, and general community), which in turn depend on the
socio-demographic structure of the population (age distribution, school attendance,
2.3 Case Study 27
household size, and working population). These patterns suggest that disease
transmissions through social contacts mainly occur in certain typical social contact
settings. In addition, individuals’ social contacts may change due to either their self-
initiated behaviors (e.g., avoidance of public places) or governmental compulsory
policies (e.g., school closures and workplace shutdowns).
Based on the above empirical findings, a reasonable strategy is to devise a
computational means for directly inferring individuals’ setting-specific contact
patterns from the demographic structure of the considered population. Then, these
setting-specific patterns can be used to estimate the overall social contacts that
account for disease transmissions, by combining the patterns via their respective
coefficients, to reveal the proportions of individuals’ contacts within different social
settings.
Specifically, we can define and compute the contact frequency between a pair of
individuals in age groups i and j , i.e., cij , as the total number of contacts between
two age groups, Cij = Cj i , divided by the product of their population sizes, Ni and
Nj :
Cij Cj i
cij = = = cj i (2.6)
Ni Nj Nj Ni
Note that matrix C with the elements of cij is the overall contact matrix,
which describes individuals’ cross-age contact frequencies. Based on this definition,
matrix C is symmetric for cij = cj i .
Next, we calculate the probability for individuals of different ages mixing
within certain social settings, i.e., individuals sharing the same places, namely
households, schools, workplaces, and general communities. Then, we generate four
matrices accounting for the specific patterns of individuals’ contacts within each
social setting, which are represented by CH for contacts within households, CS
for schools, CW for workplaces, and CG for general communities. Thus, we can
estimate the overall matrix of individuals’ cross-age contact frequencies as a linear
combination of the four setting-specific matrices:
C = r H CH + r S CS + r W CW + r G CG (2.7)
where the coefficients r H , r S , r W , and r G denote the proportions of individuals’

effective contacts occurring in the above-mentioned social settings, respectively, and
sum to one:
rH + rS + rW + rG = 1 (2.8)
For illustration, we infer individuals’ setting-specific contact patterns from the

available census data of Hong Kong by calculating the likelihoods of individuals’
mixing within different social settings [97]. The generated setting-specific contact
pattern matrices are shown in Fig. 2.3.
Household School
(a) (b)
Workplace
(c) (d)
(e) (f)
Fig. 2.3 Contact patterns inferred from the census data of Hong Kong. We consider disease
transmissions among individuals between 0 and 85+ years old and divide these into 18 age groups.
The contact matrices are generated corresponding to the likelihoods of individuals’ mixing within
respective social settings: household (CH ), school (CS ), workplace (CW ), and general community
(CG ). The overall contact matrix is calculated as the linear combination of the four setting-specific
contact matrices. The combination coefficient of each matrix denotes the proportion of effective
contacts occurring in that social setting
2.3 Case Study 29
Specifically, Fig. 2.3a describes the contacts in households, in which the main
diagonal and two secondary diagonals correspond to the contacts within families and
between parents and children. Figure 2.3b shows the pattern of contacts in schools,
in which the strong diagonal elements among individuals below 20 years old
indicate that students are more inclined to have contact with same-age individuals.
Figure 2.3c presents the pattern of contacts in workplaces, in which the contacts are
more frequent among individuals aged between 20 and 65. Figure 2.3d gives the
pattern of individuals’ random contacts with each other in general communities.
We normalize the elements of the four generated contact pattern matrices, such
that their total numbers of contacts are equal. For the overall contact matrix (see
Fig. 2.3e), the coefficients used for combining the setting-specific matrices, rΦ and
Φ ∈ {H, S, W, G}, can be approximately estimated as the fraction of disease
infections occurring in the respective social settings.
Based on the population size of each age group (Fig. 2.3f), it has been shown that
31% of infections during the 2009 Hong Kong H1N1 influenza epidemic occurred
in households [98]. In addition, we assume that the other three contact matrix
coefficients are identical to those empirical estimations for the respective social
settings [99, 100], as follows: 0.24 in schools, 0.16 in workplaces, and 0.29 in
general communities.
2.3.3 Validation
To examine our aforementioned computational method, we now revisit the real-

world scenario of the 2009 Hong Kong H1N1 influenza epidemic [101, 102]. Based
on the census data of Hong Kong, we divide the host population between 0 and
85+ years of age into 18 groups to represent the individuals’ age-specific contact
patterns. In addition, we estimate the reproduction number to be R0 = 1.5 in the
initial stage of disease transmission [103]. The infectivity is α = 1.0, which is
homogeneous for all age groups. The susceptibility is estimated to be β = 2.6
for individuals below 20 years old, who are more susceptible than the rest of the
population. The duration of H1N1 influenza infection is set to be 3.2 days [104].
Therefore, the recovery rate is calibrated as γ = 0.312 (i.e., 3.2−1 day−1 ) based on
an assumption that there is an exponential distribution of individuals’ recovery from
disease infection.
Based on the above parameterization, we now validate our epidemic model by
comparing the model predictions with the real-world observations in terms of the
daily new infection cases and the age-specific infection rates, as shown in Fig. 2.4.
In doing so, we use the confirmed daily cases of H1N1 infection, as reported by the
CHP of Hong Kong for the first 200 days since the disease onset in early May 2009,
in which the infected cases were classified into five age groups.
The results of the simulation over the specified period show that disease infection
peaked around 120 days after the disease onset, as shown in Fig. 2.4a. The young
(a)
(b)
Fig. 2.4 The baseline dynamics of disease transmission. We calibrate our epidemic model based
on the real-world scenario of the 2009 Hong Kong H1N1 influenza epidemic, as reported by the
Centre for Health Protection (CHP) of Hong Kong for the first 200 days since the disease onset.
(a) The dynamics of disease transmission in terms of the daily number of newly infected cases
reported as a proportion of the total number of infections. (b) A comparison of the observation and
model predictions in terms of disease infection rates by age
2.4 Further Remarks 31
and school-age students between 0 and 19 years old constituted a large proportion of
the infection cases, while that of adults was relatively small, as shown in Fig. 2.4b.
2.4 Further Remarks
In infectious disease control, the effectiveness of intervention relies heavily on

accurate predictions about where and when outbreaks may happen, and in what
specific population groups. Such spatial, temporal, and demographic patterns of dis-
ease transmission are determined by the biological properties of a pathogen, i.e., its
infectivity and transmissibility, as well as the demographic and social characteristics
of a host population, i.e., individuals’ vulnerability and their contact relationships.
In view of this, public health authorities increasingly turn to mathematical models
to characterize the dynamics of disease transmission and to evaluate the effect of
adopted intervention strategies.
For example, compartmental models utilize deterministic differential equations
to describe disease transmission in a group of randomly mixed individuals as a
process of population transitions among different compartments, i.e., susceptible,
infectious, and recovered. Social network models represent individuals’ contact
relationships using a network structure, and thereafter, model disease transmission
as a stochastic percolation process. Based on epidemic models, strategies for
controlling infectious diseases, e.g., antivirals [105], case isolation/quarantine [106],
vaccination [107], and school and workplace closures [108], can be computationally
simulated and the effectiveness of disease control further evaluated.
Accurate descriptions of individuals’ contact relationships are essential for the
utilization of epidemic models to characterize the dynamics of disease transmission,
as such relationships affect the routes of disease transmission. In previous studies,
individuals’ contact relationships have been mainly characterized by utilizing
statistical means to collect and analyze empirical data, e.g., individuals’ contact
activities in certain places/regions over a fixed time period [109]. For example,
survey-based studies may provide static and empirical descriptions of the routes of
disease transmission with respect to individuals’ contact relationships. In this case,
individuals’ behavioral changes, i.e., potential responses to disease transmission and
adopted interventions, are not taken into account, although they significantly affect
the disease transmission during an epidemic [110].
However, in the real world, it is difficult to collect real-time data to characterize
behavioral changes throughout the period of an epidemic. In fact, the lack of realistic
data to characterize the disease transmission in a host population has presented a
major challenge to effective disease intervention. To overcome this, we resort to
developing computational methods for inferring contact patterns that incorporate
individuals’ social, demographic, and behavioral characteristics, and accordingly
evaluate the effects of these patterns on the disease dynamics.
2.5 Summary
In this chapter, we introduced a number of key concepts and computational models

that are essential for characterizing infectious disease dynamics, while adequately
addressing the heterogeneity of a host population. In doing so, we addressed the host
population heterogeneity in terms of age-specific susceptibility, infectivity, and the
contact frequency among different population groups. Furthermore, we developed
a specific compartmental model to describe the dynamics of influenza-like disease
transmission in an age-structured host population.
In order to characterize individuals’ contact relationships, we decomposed indi-
viduals’ disease transmission-related contacts into those within four specific social
settings: school, household, workplace, and general community. Accordingly, the
coefficient for each contact matrix represents the proportion of individuals’ contacts
occurring in that social setting. In doing so, we utilized a computational method to
infer individuals’ setting-specific contact matrices from the socio-demographic data
of the population.
Finally, to demonstrate our computational method, we conducted simulation-
based experiments based on the real-world 2009 Hong Kong H1N1 influenza
epidemic. The results showed that individuals’ contacts, as inferred from the
considered host population, exhibited clear age-specific patterns. Based on the
analysis, we further validated our epidemic model by comparing the temporal-
demographic patterns of disease transmission, i.e., the daily new infection cases
and age-specific infection rates, between our model predictions and the real-world
observations. This work provides the basis for our further discussion of vaccination
programs in the following chapters.
Chapter 3
Strategizing Vaccine Allocation
In the control and prevention of infectious diseases, one of the most effective
measures is vaccination. A key question for public health authorities is the optimal
allocation of a finite number of available vaccine doses to most effectively reduce
disease incidence. This is closely related to the question of which age groups or
subpopulations will be most vulnerable and should be vaccinated first. Thus, if
we can answer this question, we will be in a strong position to combat infectious
diseases.
In this chapter we focus on the above questions. Building on the age-structured
compartmental model of infectious diseases introduced in the preceding chapter,
we discuss a computational means for identifying and prioritizing the target
subpopulations for effective vaccine allocation [55]. Through a series of simulation-
based experiments, we examine the performance of such a vaccine allocation
strategy by considering different epidemic scenarios as well as other intervention
strategies.
3.1 Vaccination
Vaccination has long been recognized as one of the most effective methods for the
control and prevention of infectious diseases. In this section, we review the basic
notions and existing methods of vaccination. Specifically, we discuss the principal
idea behind vaccination, in terms of achieving the herd immunity effect. We also
look at some of the existing vaccination strategies that have been designed for
disease control and prevention.

https://doi.org/10.1007/978-3-030-52109-7_3
34 3 Strategizing Vaccine Allocation
3.1.1 Herd Immunity
Generally, the effect of vaccination is judged in two respects: (1) how effectively
it can directly immunize vaccinated individuals; and (2) how effectively it can
indirectly protect unvaccinated individuals from getting infected by reducing disease
transmission. To discuss effectiveness, we need to further introduce the concept of
herd immunity, which is often mentioned in the literature although with several
slightly different meanings. Here, we adopt the definition as given by John et
al. [111], which is that herd immunity refers to “the proportion of subjects with
immunity in a given population,” which may be due to natural recovery from
infection, vaccine-induced immunization, or a combination of both. In essence, herd
immunity indicates the proportion of immunized individuals in a host population
who will be able to resist the spread of a disease. Due to herd immunity, it will not
be necessary for everyone in a host population to be vaccinated to prevent outbreaks,
because the size of the susceptible population has been reduced, and so has the
probability of disease transmission from the infectious to the susceptible. This
indirect protection of unvaccinated individuals, as a result of vaccine-induced herd
immunity, is often referred to as the herd immunity effect, which is schematically
illustrated in Fig. 3.1.
Next, we introduce the notion of a herd immunity threshold. This indicates the
critical portion of a host population that must be vaccinated/immunized to prevent
disease transmission. Studies on a herd immunity threshold have focused primarily
on how to determine the minimum vaccination coverage needed to prevent a poten-
tial outbreak. It should be noted that in epidemiology, the prevention of an outbreak
entails that the average number of secondary infections per infectious individual
should be less than one, which is equivalent to saying the basic reproduction number
R0 < 1.0 (see also Sect. 2.1.1). In this regard, the concept of a herd immunity
threshold reflects the vaccination coverage that leads to R0 = 1.0. In a standard
SIR model, which assumes that individuals of a host population have uniform and
random contacts with each other, the herd immunity threshold, denoted by θ , for
random vaccination (assuming 100% vaccine effectiveness), can readily be written
as follows:
Fig. 3.1 The effect of vaccine-induced herd immunity

3.1 Vaccination 35
1
θ =1− (3.1)
R0
As can be seen, the usefulness of the notion of the herd immunity threshold is that
it provides a way to examine the effectiveness of vaccination programs for disease
control, that is, to evaluate the level of vaccination coverage in a host population.
3.1.2 Vaccine Allocation Strategy
As aforementioned, vaccination coverage in a host population plays a significant

role in ensuring the effectiveness of a vaccination program. Due to limited vaccine
supply, providing adequate and timely vaccine doses remains an open challenge,
especially when encountering emerging infectious diseases. In this case, public
health authorities in charge of vaccination programs need to address the problem
of how to allocate a finite number of available vaccine doses for most effectively
controlling disease transmission.
Public health authorities in different countries may adopt different vaccination
prioritization plans in situations of limited vaccine supply. The WHO generally
recommends that the priority for vaccine-based immunization should be given to
essential service providers and individuals at a high risk of death and severe com-
plications [37]. In addition to front-line workers and high-risk individuals, another
important group of the susceptible population is healthy adults and children, which
can be very large in size [112–114]. Effectively vaccinating ordinary susceptible
individuals can help build up the population-level herd immunity and hence can
directly contribute to decreasing disease transmission [115, 116]. In this regard,
we focus on vaccine allocation strategies for specifically targeting the ordinary
susceptible population to enhance population-level herd immunity.
Previous studies on modeling vaccine allocation have been focused on mathe-
matical optimization approaches, and aimed at optimizing the predicted outcomes
of disease control. In this context, the results of optimal vaccine allocation depend
on the considered outcome measures and the projected time intervals. For example,
some researchers have suggested that vaccine doses should be allocated to high-risk
groups, thereby reducing influenza-attributed morbidity and mortality [117, 118].
Meanwhile, other researchers suggest that disease transmission could be more
effectively reduced by allocating vaccine doses to the population groups with a
higher possibility for disease transmission, e.g., school children, instead of the high-
risk groups [119, 120].
Regarding vaccination during the different stages of disease transmission, Med-
lock et al. pointed out that optimal vaccine allocation should take into account
disease dynamics and the vaccine availability [121]. Matrajt et al. observed that the
optimal allocation of vaccines, e.g., with 25% vaccination coverage, might involve
switching between allocating vaccine doses to transmitting groups to allocating
vaccine doses to vulnerable groups at the approximate peak of an outbreak [122].
Similar results for vaccination in different stages were also reported by Myliusa et
al. [123] and Bansal et al. [124].
Generally, the methods outlined in the preceding chapter adopt a fixed objective
function to be optimized and an optimal strategy to be computed for the given period
of time. To function, these methods make several assumptions. Specifically, they
assume we are able to make accurate predictions about disease dynamics, and that
we know in advance the number of available vaccine doses (needed to set up the
constraints) and the timing of vaccine release (needed to determine the optimization
time period). In real-life applications, such assumptions are evidently rather strong,
as we generally have limited knowledge about disease-associated epidemiological
features, e.g., infectivity and transmissibility, during the spread of an emerging
disease. In addition, we may not know the quantity and timing of vaccine supply.
As a result, the optimization-based methods may not be practically feasible.
Several recent studies have been performed to address the vaccine allocation
problem by focusing on disease transmission rather than on future predictions.
Importantly, the authors of these studies allowed vaccine allocation to be dynam-
ically/adaptively adjusted in relation to the dynamics of disease transmission,
e.g., based on the time-varying, age-specific incidences of infection revealed from
surveillance data [125], or the real-time monitoring of hospitalization and infection-
induced death [126]. The findings of these studies have partly solved the problem
of obtaining detailed knowledge of an emerging disease (i.e., the parameters
used in an epidemic model). For example, proxy indicators may be used for
determining vaccine allocation, e.g., the group-specific confirmations of infection,
or the hospitalization and mortality rates [127].
So far, in this section we have reviewed some of the key concepts and methods
used in planning and evaluating the effectiveness of vaccination programs in terms
of vaccination coverage in a host population. Another important issue is that
the actual coverage of a vaccination program could be affected by individuals’
willingness to be vaccinated (for more details, see Chaps. 4–6).
3.2 Vaccination Priorities
In a real-world situation, the vaccine supply is usually limited. This is partly because
the quantity of available vaccine doses may be insufficient to meet the actual
demand, and partly because the vaccine’s release after the disease onset may be
delayed. Thus, an important concern for public health authorities is to achieve the
goals of vaccination by making the best use of the finite number of available vaccine
doses. For example, in the United States, the National Vaccine Advisory Committee
(NVAC) and the Advisory Committee on Immunization Policy (ACIP) have set the
goals of vaccination as being to weaken health effects, including severe morbidity
and death, and minimize socio-economic effects [128]. Accordingly, the NVAC and
ACIP have recommended a priority vaccination for vaccine workers, health-care
providers, and the ill elderly, followed by healthy people aged 2–64 [129].
3.3 Age-Specific Intervention Priorities 37
In this book, we consider vaccine allocation for the purpose of most effectively
minimizing disease transmission in the whole population. In this case, the key
problem is how to determine the relative vaccine allocation priority for each of the
subpopulations corresponding to their respective roles in disease transmission. That
is, the challenge for effectively allocating a finite number of vaccine doses is herein
translated into the question of how to adaptively adjust vaccine allocation to various
subpopulations with reference to the immediate situation of disease transmission,
such as the disease incidence and prevalence rates in different subpopulations,
the possible changes of individuals’ contact relationships, and the total number of
available vaccine doses and the time of vaccine release.
3.3 Age-Specific Intervention Priorities
During the 2009 Hong Kong H1N1 influenza epidemic, the government announced
the immediate closure of all primary schools and kindergartens when the first non-
imported case was confirmed, and at the same time, targeted children between
the ages of 6 months and 6 years as the priority groups for vaccination [130].
Relevant to this, a pertinent question is how we can systematically determine the
relative priorities of subpopulations for disease interventions during the time course
of disease transmission, where such intervention measures may include vaccine
allocation, contact reduction, or a combination of both.
The aim of our work is to provide a method to enable optimal prioritization
of subpopulations for disease interventions, using a combination of age-specific
vaccine allocation with setting-specific contact reduction. We evaluate the effects
of disease interventions for containing disease transmission by measuring the
reduction in the reproduction number. By doing so, we show which subpopulations
should be prioritized for vaccination, so as to generate the greatest reduction in the
number of disease transmissions. We do so by considering the marginal effects of
reducing the reproduction number for different cases of vaccine allocation by age,
and reducing contacts by social setting.
For demonstration, we use a compartmental model to describe the dynamics
of an influenza-like disease transmission in an age-structured host population. We
parameterize the epidemic model with the epidemiological data from the 2009 Hong
Kong H1N1 influenza epidemic, and further implement our method to identify
the relative priorities of subpopulations for disease interventions in Hong Kong.
Additionally, we carry out a series of simulation-based experiments with different
settings of disease intervention.
Compared with existing optimization-based approaches, our method has the
following features:
• The strategy utilizes prior knowledge about individuals’ age-specific susceptibil-
ity and infectivity, real-time disease prevalence in each of the subpopulations, and
the basic patterns of individuals’ cross-age contact frequency within each social
setting. Moreover, it does not rely on detailed information about individuals’

actual contacts, nor their potential behavioral changes in response to disease
transmission, which would be difficult to determine rapidly and accurately.
• The strategy is designed to most effectively reduce disease transmission by
optimizing the allocation of a finite number of vaccine doses to certain target
subpopulations. Crucially, these identified vaccination priorities can be adap-
tively regulated based on the dynamics of disease transmission. That is to say,
the number of vaccine doses suggested to be allocated to each subpopulation can
be dynamically adjusted according to the latest progress of disease transmission
and vaccine supply.
• The developed vaccine allocation method also incorporates the effects of other
disease intervention measures being implemented simultaneously with vaccina-
tion, e.g., individuals’ contact reductions. Therefore, in a situation of integrated
disease interventions, our method can provide more accurate and effective
solutions for vaccine allocation.
3.3.1 Modeling Prioritized Interventions
We now introduce the standard epidemic model and our detailed computational
method for identifying the relative priorities of age-specific subpopulations for
intervention measures.
First, we use vector I(k) = [I1 (k), . . . , IN (k)]T to denote the number of
infectious individuals in each of N age groups at the kth generation of disease
infection. Then, we characterize the dynamics of disease infection from generation
k to generation k + 1 as follows:
I(k + 1) = KI(k) (3.2)
where K is the reproduction matrix, also known as next generation matrix (NGM)
[131].
For the earlier-mentioned SIR model, the reproduction matrix K can be written
as follows:
K = (μγ −1 )SBCA (3.3)
where matrix S describes the size of the susceptible population in each age group;
it has elements S1 , S2 , . . . , SN in the diagonal and zeros elsewhere. Matrix B
summarizes individuals’ age-specific susceptibility in the diagonal elements of
β1 , β2 , . . . , βN , and zeros elsewhere. Matrix A gives the age-specific infectivity
of infected individuals with the diagonal elements of α1 , α2 , . . . , αN , and zeros
elsewhere. Matrix C, known as the contact matrix (see Eq. (2.7) in Sect. 2.3.2),
describes the frequency of contacts between two age groups. Moreover, during the
course of disease transmission, the susceptible populations will decrease in size over
time, and therefore matrices S and K will change dynamically.
In epidemiology, the effective reproduction number Rt refers to the number of
new infection cases caused by a typical infectious individual in a completely suscep-
tible population. By constructing the NGM, the effective reproduction number Rt in
the context of an age-structured host population can be approximately estimated as
the dominant eigenvalue of reproduction matrix K:
Rt = ρ (K) (3.4)
Given the condition that matrices S, B, C, and A are all symmetric, Rt can be
approximately calculated as:
Rt = x T1 Ky 1 (3.5)
where x 1 and y 1 are the top left and right eigenvectors of reproduction matrix K (the
corresponding top eigenvalue is Rt ). Specifically, we choose a normalized format of
each eigenvector in which the elements are positive and sum to one. As proposed
by Wallinga et al. [106], y 1 and x 1 approximately correlate to the number of new
infections in each age group, I:
y1 ∝ I
(3.6)
x 1 ∝ S−1 B−1 AI
Therefore, we can describe disease transmission (effective reproduction number

Rt ) with reference to the current disease prevalence, the susceptible population
sizes, and their cross-age contact frequencies. In the following subsections, we
examine the effects of disease interventions (vaccination and contact reduction)
on containing disease transmission, by measuring these interventions’ marginal
reductions of Rt .
3.3.2 Effects of Vaccination
As aforementioned, disease transmission can be estimated based on Rt , which

correlates to reproduction matrix K. Thus, the change of the effective reproduction
number, dRt , can be calculated from dK as follows:
dRt = x T1 dKy 1 (3.7)
As vaccination that immunizes susceptible individuals can reduce the sizes

of susceptible populations in different age groups, the reduction of Rt due to
vaccination will be proportional to the following term:
dRt ∝ x T1 (dS)BCAy 1 (3.8)
Specifically, when targeting a susceptible population in age group i, the effects

of vaccination can be calculated as follows:

dRt dS
∝ x T1 BCAy 1 (3.9)
dSi dSi
By combining the elements of each matrix, we arrive at an indicator that evaluates

the marginal reduction of Rt by vaccinating a unit of susceptible individuals in age
group i:
2
dRt αi Ii
∝ (3.10)
dSi βi Si
Due to the lack of knowledge about the reporting rate of disease infections, i.e.,
the ratio of confirmed cases to the overall infections, we approximate the susceptible
population size Si using the population size Ni , based on the assumption that the
number of infections is relatively small in the host population. Therefore, we can
prioritize each age group as follows:
2
dRt αi Ii
∝ (3.11)
dSi βi Ni
Therefore, with the above equation, we can determine the relative priorities
of age groups for vaccine allocation with respect to their age-specific infectivity,
susceptibility, population sizes, and current disease prevalence.
3.3.3 Effects of Contact Reduction
To contain disease transmission by reducing the effective contacts in a host

population, we examine the change of the effective reproduction number, dRt , with
respect to the reduction of individuals’ contact frequencies, dC:
dRt ∝ x T1 SB(dC)Ay 1 (3.12)
We focus on the effects of contact reduction in terms of reducing the number of

individuals’ contacts within a social setting, ψ ∈ {H, S, W, G}:
dRt
∝ x T1 SBCψ Ay 1 (3.13)
dr ψ
Then, based on the inferred setting-specific contact matrices, the relative priority
for contact reduction that targets social setting ψ can be computed as follows:
dRt ψ
∝ αi Ii cij αj Ij (3.14)
dr ψ
Thus, with the above equation, we will be able to further estimate the relative
priorities of social settings for contact reduction, having taken into consideration
individuals’ age-specific infectivity, current disease prevalence, and setting-specific
contact patterns.
3.3.4 Integrated Measures
Next, we consider the case of a campaign against the spread of an infectious disease,
in which multiple intervention measures will be implemented at the same time. In
view of this, we are interested in evaluating the effect of vaccination and contact
reduction being implemented simultaneously. We estimate the marginal reduction of
the effective reproduction number, d 2 Rt , corresponding to simultaneous vaccination
of the susceptible population dS and the reduction of effective social contacts dC:
d 2 Rt ∝ x T1 (dS)B(dC)Ay 1 (3.15)
Specifically, when selecting age group i for vaccination and social setting ψ
for contact reduction, the effect of implementing the two intervention measures on
reducing the effective reproduction number Rt can be evaluated as follows:

d 2 Rt dS
∝ x T1 BCψ Ay 1 (3.16)
dSi dr ψ dSi
That is,
d 2 Rt αi ψ
∝ c α I
ij j j (3.17)
dSi dr ψ Si
Hence, by evaluating the interplay of the two intervention measures, we can

identify the relative priorities of age groups and social settings when vaccine
allocation and contact reduction are implemented simultaneously. That is, the
number of vaccine doses allocated to each age group will be proportional to the
relative priority of that group, and the contact reduction will target the social setting
with the highest priority.
3.4 Case Study
In this section, we demonstrate our vaccine allocation strategy using the real-world
scenario of the 2009 Hong Kong H1N1 influenza epidemic, by determining the
relative priorities of the age-specific subpopulations for vaccination.
3.4.1 Hong Kong HSI Vaccination Programme
Figure 3.2 shows the H1N1 scenario in terms of the daily numbers of reported
infections in different age groups during the spread of the disease in Hong Kong.
Accordingly, Fig. 3.3 presents the relative priorities for vaccine allocation in those
age groups during the course of disease transmission.
Generally, individuals between 0 and 29 years old are the most important
subpopulation for containing disease transmission by vaccination. However, for
each specific age group, the identified priorities vary between different stages of
disease transmission. For the first month since the disease onset, up to day 25, we
can observe that individuals between 0 and 29 years of age are targeted as the
top priority subpopulation for vaccination. This describes the situation in which
outbreaks will appear among school-age students due to their high frequency of
contacts. Subsequently, on day 50, it can be observed that the relative priorities
of individuals aged between 0 and 9 and those between 20 and 29 are increased,
while the priorities of individuals between 10 and 19 are reduced. When disease
Fig. 3.2 The numbers of reported infections in different age groups during the 2009 Hong Kong
H1N1 influenza epidemic
3.4 Case Study 43
Fig. 3.3 Relative priorities of age-specific subpopulations for vaccine allocation during the course
of disease transmission
infection peaks near day 120, individuals between 0 and 19 are predicted to become
the dominant priority for vaccination, which agrees with the real-world observation
that children and school-age students accounted for a large proportion of the new
infections in this stage (Fig. 3.2). Finally, in the decay stage of the epidemic,
children between 0 and 9 will become the subpopulation with the highest priority
for vaccination. However, it should be pointed out that vaccination is more effective
in the initial stage of disease transmission than in the stage of decay.
Figure 3.4 shows the relative priorities of the social settings for disease interven-
tion by contact reduction among individuals. Overall, the reduction of individuals’
contacts within schools is identified as the key measure for containing disease trans-
mission throughout the whole period of disease transmission. Disease transmissions
within households and workplaces account for a relatively large proportion during
the initial and the decay stages of disease spread (between day 25 and day 100 and
between day 150 and day 200, respectively), however, when infection peaks around
day 120, interactions within these environments account for a relatively small
proportion of disease transmissions. As mentioned before, the estimated proportion
of infections occurring in schools was not the largest among the four considered
social settings (i.e., the empirical estimation for schools was approximately 24%,
and for households 31%). However, the disease infections that had already occurred
in schools played a significant role in the disease transmission among the host popu-
lation. Therefore, the reduction of individuals’ effective contacts in the social setting
of schools (through school closures as well as school sanitation and disinfection)
should be implemented immediately after the disease onset, and conducted for the
whole period of disease transmission.
Fig. 3.4 Relative priorities of social settings for contact reduction during the course of disease
transmission
Figure 3.5 further presents the relative priorities of the age groups and social
settings when vaccine allocation and contact reduction are implemented simultane-
ously. Generally, vaccination of individuals between 0 and 19 and contact reduction
in schools are the most important measures for containing disease transmission.
For people of older ages, contact reduction in households, workplaces, and general
communities is more important than contact reduction in schools. Specifically, as
shown in Fig. 3.5a, on day 1 of disease transmission, contact reduction in schools
and vaccination of individuals aged between 5 and 19 should be the top priority
to contain disease transmission. On day 60, as indicated in Fig. 3.5b, individuals
between 15 and 19 are identified as the target subpopulation for vaccination,
followed by individuals aged 10–14 and 5–9. At this stage, contact reduction in
the social setting of schools remains the top priority. When disease infection peaks
at approximately day 120, as illustrated in Fig. 3.5c, the age groups 5–9 and 10–14
are predicted to become the most important subpopulations for vaccination. In the
final stage of disease transmission on day 180, as shown in Fig. 3.5d, vaccinating
children becomes more important.
By comparing the results of disease intervention using only vaccination (Fig. 3.3)
with those using simultaneous vaccination and contact reduction (Fig. 3.5), we
can observe that the age group 20–29 has a higher priority for vaccine allocation
in the case of vaccination only than in the case of adopting both intervention
measures. This is mainly due to the interplay of the two intervention measures, in
that the effects of reducing effective contacts in schools can prevent or delay disease
transmission to other age groups.
3.4 Case Study 45
(a) (b)
(c) (d)
Fig. 3.5 Relative priorities of subpopulations and social settings for simultaneous implementation
of age-specific vaccination and setting-specific contact reduction during the course of disease
transmission
3.4.2 Effects of Prioritized Interventions
Next, to demonstrate the performance of our method, we carry out several

simulation-based experiments to examine infectious disease transmission under
different settings of vaccination and contact reduction. In the Hong Kong H1N1
influenza epidemic, the government closed all primary schools, kindergartens,
and special schools immediately after the first local case was confirmed on June
10, 2009. Also, the Human Swine Influenza Vaccination Programme (HSIVP)
was launched on December 21, 2009. As of early February 2010 (50 days after
the vaccine became available), approximately 180,000 individuals had been
administered vaccine doses, which accounted for approximately 2.5% of the
overall population. For demonstration, we examine disease transmission under
various intervention measures, i.e., vaccination with a coverage of 2.5% of the host
population and contact reduction within different social settings. The intervention
measures are set to be implemented on day 75.
The results in Fig. 3.6 show that vaccination and contact reduction can ameliorate
disease prevalence by reducing the incidence rate at the peak of an outbreak. In the
case of disease intervention by contact reduction only, we can clearly observe from
the time course of the infectious population sizes that reducing individuals’ contacts
in different social settings can lead to distinctly different results. Contact reduction
in schools (the blue solid curve) outperforms those in the other three social settings,
in terms of preventing the occurrence of an infection outbreak and lowering the
incidence rate at the peak of disease prevalence. Contact reduction in households
(the red solid curve) and workplaces (the yellow solid curve) each has a similar
effect on disease control and a better performance than that in general communities
(the green solid curve). These results agree well with our previous prioritization
of social settings for contact reduction, in which schools are identified as the top
priority, followed by households and workplaces.
For the implementation of vaccination and contact reduction simultaneously,
the results are shown as the dashed curves in Fig. 3.6. With contact reduction in
schools, disease transmission would be almost eliminated (the blue dashed curve).
Vaccination when combined with contact reduction in households and workplaces
performs better than contact reduction in general communities in reducing disease
prevalence, which, in turn, is still better than vaccination only. In addition, the
Effects of prioritized disease interventions

0.012
No intervention
Household only
School only
Infectious population
Workplace only
0.008 Community only
Vaccine only
Vaccine & household
Vaccine & school
Vaccine & workplace
Vaccine & community
0.004
0
0 50 100 150 200
Days
Fig. 3.6 Disease dynamics under the disease interventions of age-specific vaccination and setting-
specific contact reduction. The baseline dynamics of disease transmission without any intervention
(black solid curve); contact reduction only in schools (blue solid curve), households (red solid
curve), workplaces (yellow solid curve), and general communities (green solid curve); vaccination
only (black dashed curve); vaccination and contact reduction in schools (blue dashed curve),
households (red dashed curve), workplaces (yellow dashed curve), and general communities (green
dashed curve) (Color figure online)
implementation of contact reduction in schools only (the blue solid curve) leads
to a lower incidence rate at the peak of disease outbreak than the simultaneous
implementation of vaccination and contact reduction in general communities (the
green solid curve), and also delays the time of disease outbreak.
3.5 Further Remarks
The two disease intervention measures studied here are age-specific vaccine allo-
cation and social setting-specific contact reduction. Unlike previous studies in
which statistical means are utilized to describe individuals’ contact relationships,
in this chapter we have discussed how to infer setting-specific contact matrices
by decomposing individuals’ actual disease transmission-related contacts into their
contacts within four specific social settings, i.e., school, household, workplace,
and the general community. Therefore, the contact-reduction method of disease
intervention can be interpreted as the reduction of the proportion of individuals’
contacts in a certain social setting. Furthermore, the changes of individuals’ overall
contacts can be interpreted as the changes of contact proportions within different
social settings. We have also examined the reproduction number to evaluate the
effects of implementing different intervention measures on containing disease
transmission. Therefore, we can identify the priority subpopulations (in terms of
age groups and social settings) based on the marginal reduction of the reproduction
number caused by changes of the susceptible population sizes by age (vaccination)
and of the contact proportions by social setting (contact reduction).
We have evaluated our method with respect to the real-world scenario of the
2009 Hong Kong H1N1 influenza epidemic and, thereafter, examined the relative
priorities of subpopulations for age-specific vaccination and setting-specific contact
reduction in Hong Kong. Our study has practical value for public health authorities
in preparing and assessing their intervention measures for controlling an infectious
disease. First, the age distribution of new infections will always be available from
an epidemic surveillance system, e.g., the CHP in Hong Kong. Next, the basic
patterns of individuals’ contacts in different social settings will depend mainly on
the socio-demographic characteristics of the population, which can be derived either
through statistical means or by computational methods from the census data for
the host population. Finally, disease control will be more effective when multiple
intervention measures are implemented simultaneously.
It should be pointed out that the results from our method partly depend on
the accuracy of the age distribution of new infections reported by the surveillance
system. In addition, other potential factors that also affect the results may include the
reporting rates of infection, which may vary for individuals in different age groups
due to their physical and biological conditions, and the time needed for the case
confirmations, which may lead to a delayed response to disease transmission.
3.6 Summary
In this chapter, we developed a solution to the problem of how to allocate a finite

number of vaccine doses to an age-structured host population. To achieve this, we
considered two measures: vaccination, which immunizes susceptible individuals
in different age subpopulations, and contact reduction, which reduces individuals’
effective contacts in different social settings. We presented a prioritization method
to identify the target subpopulations that will lead to the greatest reduction in the
number of disease transmissions when subjected to age-specific vaccination, or to
vaccination integrated with contact reduction. We computed the relative priorities
of subpopulations by considering the marginal effects of reducing the reproduction
number in the cases of vaccine allocation by age and contact reduction by social
setting. We demonstrated the performance of our method, based on the real-world
2009 Hong Kong H1N1 influenza epidemic, through a series of simulation-based
experiments on disease transmission under different interventional settings.
Chapter 4
Explaining Individuals’ Vaccination
Decisions
Although vaccination has long been regarded as one of the most effective methods
in controlling infectious diseases, public concerns about the safety and efficacy of
vaccines will significantly affect the effective coverage of a vaccination program.
In the real world, it remains highly challenging to achieve adequate and lasting
vaccination coverage in a host population, especially for a voluntary vaccination
program. An example is the decline of MMR vaccine uptake in Britain [132],
subsequent to a controversial study associating the vaccine with the development
of autism. Even though other studies contested this association, the public remained
doubtful about the safety of the MMR vaccine, leading to a decrease in vaccine
uptake, which was in turn followed by outbreaks of measles [132]. Similarly, public
losses of confidence in the safety of pertussis vaccine have led to declines in
its uptake in many countries, subsequently resulting in a series of large pertussis
outbreaks [133].
In the remaining three chapters, we aim to explain how individuals make their
decisions once a voluntary vaccination program becomes available, and what could
be the dominant factors in their decision-making [53, 54]. Specifically, we focus our
discussions on the following three key aspects:
• Risks and Benefits of Vaccination
Vaccination immunizes susceptible individuals to prevent infection. However,
vaccinated individuals also face the potential risk of adverse effects of the
vaccine, which may sometimes lead to severe complications. In this regard,
individuals making vaccination decisions typically focus on several determinants
associated with the risks and benefits of vaccination, such as the perceived risk
of disease infection, the perceived safety and efficacy of the vaccine, and the
socio-economic costs associated with vaccination and disease infection (e.g.,
the monetary cost of vaccination, medical expenses for treatment in the case of
infection, and possible absence from work[134]).

https://doi.org/10.1007/978-3-030-52109-7_4
50 4 Explaining Individuals’ Vaccination Decisions
• Impact of Social Influences

In a socially connected population, an individual’s behaviors or opinions are also
subjected to the influence of his/her connected neighbors, i.e., social influences
[135, 136]. That is to say, in a voluntary vaccination program, an individual’s
decisions are affected by the decisions of others. The social influences on
individuals’ vaccination decisions can derive from multiple sources, such as
recommendations of friends or family members [137], suggestions from health
professionals [138], and advice given by trusted colleagues [139].
• Individuals’ Subjective Perception
Individuals inevitably lack prior knowledge when balancing the risks and
benefits of vaccination with a newly developed vaccine against an emerging
infectious disease. In this case, vaccination decision-making relies on individ-
uals’ subjective perception of disease severity and vaccine safety. Specifically,
individuals who are aware of the dangers of severe infectious diseases will tend
to seek protection through vaccination. However, those who are conscious of the
potential adverse effects of vaccination will be less willing to be vaccinated.
In this regard, there are two challenges for understanding the effectiveness of a
voluntary vaccination program, as follows: (1) how to characterize individuals’ vac-
cination decision-making, taking into consideration the above-mentioned factors;
and (2) how to estimate the final population vaccination coverage that results from
individuals’ decision-making.
In what follows, we develop computational models to characterize individual-
level vaccination decision-making corresponding to (1) the risks and benefits of
vaccination; (2) the effect of social influences; and (3) individuals’ subjective
perception. We further examine the effect of individuals’ vaccination decisions on
ensuring adequate population vaccination coverage for effective infectious-disease
control.
4.1 Costs and Benefits for Decision-Making
The perceived risks affecting individuals’ decision-making are twofold: the risk of
being infected for unvaccinated individuals, and the risk of a vaccine’s adverse
effects for vaccinated individuals. It has been found that individuals who worry
about being infected or believe themselves vulnerable to infection will be more
inclined to accept vaccination, and vice versa [140, 141]. Similar patterns have also
been found for individuals’ intention to be vaccinated when they felt that a pan-
demic would be severe and long-lasting [142]. Moreover, studies have shown that
individuals with positive attitudes to vaccination, and who believe that vaccination
can reduce their risk of infection, were motivated to be vaccinated [143, 144]. In
contrast, individuals who worried about the vaccine’s potential adverse effects and
doubted its efficacy were found to have a lower level of vaccine uptake [145, 146].
4.2 Game-Theoretic Modeling of Vaccination Decision-Making 51
Several mathematical models have been proposed to model individuals’ vac-

cination decision-making, with these models utilizing payoff-based methods to
characterize decision-making based on individuals’ perceptions of the risks and
benefits of vaccination. Bauch et al. characterized individuals’ vaccination decisions
as a modified minority game by exploring the herd immunity effect [147, 148].
In a related manner, game theory has been used to describe individuals’ decision-
making in favor of optimizing personal payoffs [149, 150]. Cojocaru extended the
game-theoretic model by considering a finite number of heterogeneous population
groups [151]. Perisic et al. further incorporated individuals’ contact networks
into the vaccination game-theoretic analysis [152, 153]. Moreover, some studies
have considered social and psychological aspects of decision-making (e.g., social
learning processes [154] and imitation behaviors [155–157]). Elsewhere, others
have considered the problem of incomplete information, by addressing either the
potential discrepancy between individuals’ perception and real situations (e.g.,
perceived disease prevalence and adverse effects of vaccination [158]) or different
sources of information [159, 160].
4.2 Game-Theoretic Modeling of Vaccination

Decision-Making
Game theory provides a useful tool for characterizing the dilemma associated with
voluntary vaccination. That is, for each individual, it would be better not to take
the risk of vaccination, while benefiting from the herd immunity generated from
the rest of the population keeping the vaccination coverage high. Game-theoretic
analysis assumes that individuals possess perfect rationality to maximize their own
gains by adjusting their decision of whether to vaccinate. If applied to a vaccination
program, game theory implies that the host population’s individuals will adjust their
willingness to vaccinate by balancing the costs and benefits associated with the
status of a spreading epidemic and the safety and efficacy of vaccines.
There are two types of costs associated with an individual’s vaccination decision:
(1) the cost of vaccination (e.g., the potential risk of a vaccine’s adverse effects or
the expense of vaccine administration); and (2) the cost of infection if not vaccinated
(e.g., disease complications, expenses for treatment, or absence from work). We let
ξ and ζ denote the costs associated with vaccination and infection, respectively, and
use λ̂i to represent the perceived risk of being infected by a susceptible individual
i (see the definition of λi in Sect. 2.1.2). Then, we can introduce a cost function for
individual i with a decision σi , as follows:
Fi (σi ) = (1 + σi ) · ξ + (1 − σi ) · λ̂i · ζ (4.1)
where ξ denotes the cost associated with being vaccinated, and λ̂i · ζ denotes the
cost associated with rejecting vaccination.
Vaccination cost-benefit analysis

Social network Decision costs
Nodes V : individuals Vaccination:
Edges L : social closeness Disease infection:
Status : decision-making
Game-theoretic analysis
Risk of infection: ˆ
Decision equilibrium
Cost ratio: rc
• Cost function Cost-minimized choice

F( ) F , rc , ˆ 1, if rc ˆ
• Cost minimization ˆ 1, if rc ˆ
ˆ min F ( ) unchanged, if rc ˆ
1
Fig. 4.1 A decision process for characterizing individuals’ voluntary vaccination. A group of
socially interconnected individuals can make decisions by minimizing all of the associated costs
Next, without loss of generality, we let rc = ξ/ζ describe the ratio of ξ and ζ .
Thus, we can further transform the cost function Fi (·) into the following:
Fi (σi ) = (1 + σi ) · rc + (1 − σi ) · λ̂i (4.2)
As illustrated in Fig. 4.1, we assume that individuals estimate the risk of disease
infection based on their perception of disease severity, as reflected in the perceived
disease transmission rate, β̂, as well as their neighbors’ vaccination decisions, as
represented by Nivac and Ninon for the numbers of neighbors who make the decision
to vaccinate or not, respectively. In addition, vaccinated individuals are assumed to
be successfully immunized from disease infection, while unvaccinated individuals
can be infected and thus transmit the disease. Therefore, the perceived infection risk,
λ̂i , can be computed corresponding to the proportion of unvaccinated neighbors as
follows:

Ninon
λ̂i = β̂ · (4.3)
Nivac + Ninon
Based on the above formulation, an individual can arrive at an optimal choice

by minimizing the cost function in Eq. (4.2). In our model, individual i will accept
4.3 Case Study 53
vaccination (σi = 1) if rc < λ̂i , reject vaccination (σi = −1) if rc > λ̂i , and keep
his/her decision unchanged from the previous step if rc = λ̂i . We can express this
cost-minimized choice of individual i, σ̂i , in the following form:
⎧
⎨ +1, if rc < λ̂i
σ̂i = −1, if rc > λ̂i (4.4)
⎩
unchanged, if rc = λ̂i
If all individuals follow the same strategy of minimizing their cost functions, after
a certain number of iterations of decision-making a steady state will be reached, in
which all individuals will have no incentive to change their decision in the next step.
4.3 Case Study
In the preceding section, we have shown how individuals’ vaccination decision-

making can be computationally modeled as an integrated decision process that
incorporates their cost minimization with the effect of social influences. Next, we
examine the individual-level decision-making process with respect to voluntary
vaccination, and examine its potential effect on disease control using a real-world
influenza epidemic as an example.
4.3.1 Hong Kong HSI Vaccination Programme
First, for our simulations, we calibrate the parameters of individuals’ vaccination

decision-making based on the 2009 H1N1 influenza epidemic (see also Sect. 2.3.1).
To focus on the effect of social influences, we assume that the perceived disease
transmission rate is equal to that of the actual disease transmission, i.e., β̂ = β. In
addition, we construct a social network based on data of individuals’ close proximity
interactions (i.e., at distances less than 3 m) at a high school [69], where the social
closeness wij between individuals i and j corresponds to the frequency of their
interactions (the sum of all interactions between the two individuals during the day).
The total number of nodes is N = 788 and the average node degree (the number
of connected neighbors) is 35. The average link weight (social closeness) is 115
units. Based on our model parameterization, we carry out Monte Carlo simulations
to experimentally study individuals’ vaccination decision-making and the effect of
the resulting vaccination coverage on disease control.
4.3.2 Vaccination Coverage
To investigate the impact of individuals’ decision-making on the coverage, we

first consider each individual’s vaccination decision-making, with respect to three
cases: (1) R0 = 1.2; (2) R0 = 1.6; and (3) R0 = 2.0. Figure 4.2 shows
the vaccination thresholds (see also Sect. 3.1.1) for eliminating the epidemic with
respect to different basic reproduction numbers R0 .
Based on our decision model, we perform several simulations of vaccination
dynamics to estimate the vaccination coverage at the steady state of individuals’
decision-making. As shown in Fig. 4.3, we investigate the effect of individuals’
decision-making on vaccination coverage, in which each individual tries to mini-
mize his/her perceived cost. The level of vaccine uptake is determined by the cost
ratio rc and the perceived severity of disease infection, i.e., reproduction number
R0 . Specifically, the simulation results show that the cost of vaccination (cost ratio
rc ) fundamentally determines the resulting vaccination coverage, in that increasing
the cost of vaccination will lower individuals’ vaccination willingness (at the steady
state of individuals’ decision-making). In our considered situation, the vaccination
coverage is greater than 70% when cost ratio rc = 0.1, which means that the
vaccination coverage is above the threshold for eliminating disease transmission.
Gradually, as rc increases and approaches 1.0, the vaccination coverage falls to
20%, which means voluntary vaccination may not be adequate to prevent disease
outbreaks. In our model, when an individual perceives a sufficiently high severity
of disease transmission (i.e., R0 is increased from 1.2 to 2.0), the individual will
Fig. 4.2 The vaccination thresholds for preventing disease outbreaks, with respect to: (1) R0 =
1.2; (2) R0 = 1.6; and (3) R0 = 2.0
8 .0
4 6
Reproduction number
Fig. 4.3 Vaccination coverage at the steady state of individuals’ decision-making. We investigate
the effect of individuals’ decision-making being motivated by cost minimization, with this effect
measured by the resulting vaccination coverage, through varying the values of cost ratio rc between
0 and 1.0. The disease severity (basic reproduction number R0 ) is set as: (1) R0 = 1.2; (2) R0 =
1.4; (3) R0 = 1.6; (4) R0 = 1.8; and (5) R0 = 2.0
change his/her previous choice of non-vaccination even if the cost of vaccination is

high, due to his/her consideration of the risk of disease transmission.
4.4 Further Remarks
Game-theoretic analysis has been used in various studies to investigate the collective
effect of individual-level vaccination choices on the population-level vaccination
coverage, that is, the coverage of vaccination achieved at the equilibrium state of
individuals’ strategy adjustment. It has been found that the vaccination coverage
resulting from individuals’ collective self-interested choices will always be below
the optimal level of vaccination for the whole population (i.e., the threshold
of vaccination for the eradication and elimination of disease infection). Specific
examples are the analysis of vaccination campaigns against smallpox [47], seasonal
influenza [48], and more recently the H1N1 epidemic [161]. In the real world,
individuals’ choices deviate greatly from those that would result from perfect
rationality.
4.5 Summary
In this chapter, we investigated individuals’ decision-making regarding whether or

not to be vaccinated in a voluntary vaccination program. Specifically, we considered
individuals as rational decision-makers who make decisions by evaluating the
risks and benefits associated with disease infection and vaccination. In this regard,
we presented a decision model driven by game-theoretic analysis, in which each
individual arrived at his/her vaccination decisions by minimizing their costs. By
doing so, we provided an in-depth study of vaccination decision-making that
integrates cost minimization. We used individuals’ social networks to represent the
structure of their interaction relationships and carried out a series of simulations of
voluntary vaccination against an influenza-like disease. By evaluating the resulting
coverage of voluntary vaccination, we examined the effect of individuals’ cost-
minimization motive on their decisions and on the effectiveness of disease control
for the entire population (vaccination coverage), where individuals’ choices were
driven by the relative cost of the decision to vaccinate, i.e., cost ratio rc . Based
on the results in this chapter, we conclude that individuals’ self-interested cost
minimization will negatively affect the control of an infectious disease, by reducing
the vaccination coverage. In this respect, our work can provide a means for
estimating the effectiveness of a voluntary vaccination program.
Chapter 5
Characterizing Socially Influenced
Vaccination Decisions
In addition to the self-estimated costs and benefits of vaccination against disease,

as discussed in the previous chapter, individuals’ vaccination decision-making is
also subject to social influences, i.e., the vaccination decisions of their family
members, friends, or colleagues. To better understand individuals’ vaccination
decision-making, in this chapter we further examine how people embark on an
integrated decision-making process that incorporates both the cost-benefit analysis
of vaccination decisions and the effect of social influences [53].
5.1 Social Influences on Vaccination Decision-Making
As illustrated in Fig. 5.1, we consider a group of individuals that make their

vaccination decisions by both minimizing the associated costs and evaluating the
decisions of others (social influences). Specifically, we focus on the social settings
of individuals, which are structured with reference to their interaction relationships
(interconnected individuals and their social closeness). Therefore, the effect of
social influences among them will be heterogeneous with respect to the structure
of their interactions. In addition, when individuals interact with those making
similar choices, their decisions may be further affirmed; in the opposite case, their
confidence in their decisions may be weakened [162]. In such a case, Social Impact
Theory (SIT) provides a computational approach to characterize the effect of social
influences corresponding to individuals’ interactive relationships [163].
In outline, SIT describes how individuals change their attitudes or decisions in
a structured social environment, and further suggests that the strength of the social
effect is determined by the characteristics of the source (e.g., various attitudes or
decisions), the closeness of their social relationships, and the number of sources
holding similar attitudes or decisions. In this work, we provide an in-depth modeling
framework for describing individuals’ vaccination decision-making by integrating

https://doi.org/10.1007/978-3-030-52109-7_5
58 5 Characterizing Socially Influenced Vaccination Decisions
Fig. 5.1 An integrated decision process for characterizing individuals’ voluntary vaccination. We
extend the existing game-theoretic analysis by incorporating the effect of social influences. By
doing so, we can investigate the steady state of individuals’ decision-making and examine the
effect of social influences on voluntary vaccination and, hence, the effectiveness of disease control
5.1 Social Influences on Vaccination Decision-Making 59
an extended SIT-based characterization of social influences with a game-theoretic

analysis of cost minimization. In this model, we use conformity to describe the
effect of social influences on vaccination decision-making, in terms of individuals’
tendency to be affected by the social influences of others. Additionally, we represent
individuals’ interaction relationships with reference to a social network structure,
in which individuals are heterogeneously interconnected with different numbers
of connected neighbors and where there is variable social closeness in their
interactions. We parameterize our model with an influenza-like disease and a real-
world social network.
Through a series of simulations of voluntary vaccination, we examine the steady
state of individuals’ decision-making and evaluate the vaccination dynamics and the
effect of disease control, in terms of vaccination coverage and the resulting infection
rate, respectively. By doing so, we investigate the interplay of cost minimization
and social influences in individuals’ vaccination decision-making, and examine the
effect of different levels of individuals’ conformity on the strength of social influ-
ences. Furthermore, we provide a new modeling framework that incorporates the
effect of social influences to investigate the effectiveness of voluntary vaccination
for infectious disease control.
In addition to cost minimization, an individual may be affected by the decisions
of others under the effect of social influences, and thus convert his/her cost-based
choice to the social opinion of his/her neighbors, as illustrated in Fig. 5.2. According
to SIT, the strength of such social influences is subject to the structure of individuals’
interactions, e.g., the types of opinions (acceptance or rejection σi ), interaction
relationships (social closeness wij ), and the number of opinion sources (the numbers
of vaccinated and unvaccinated neighbors, Nivac and Ninon , respectively). In our
social network, for individual i, the strengths of social influences for two opposite
opinions (vaccination acceptance and rejection), described by ιvac i and ιnon
i , can be
accordingly computed as follows:

= Nivac / ·
12 2
ιvac
i wij (5.1)
j ∈Nivac

= Ninon / ·
12 2
ιnon
i wij (5.2)
j ∈Ninon
We use σ˜i to denote the formalized social opinion resulting from the social influ-
ences of individual i’s neighbors. As a modification of the standard SIT definition
(where σ˜i corresponds to the opinion backed by stronger social influences), σ˜i , being
either acceptance or rejection of vaccination, will be determined by comparing the
influences of the two opposite opinions. We let Διi denote the discrepancy between
ιvac
i and ιnon
i . Then, we normalize Διi as follows:
ιvac − ιnon
Διi = i i
. (5.3)
ιi + ιnon
vac
i
Fig. 5.2 A decision process for characterizing individuals’ voluntary vaccination. We extend the
existing decision analysis by incorporating the effect of social influences. A group of socially
interconnected individuals can make decisions by evaluating the decisions of others. We utilize
SIT to characterize the effect of social influences on individuals’ decision-making with reference
to their interaction relationships
5.1 Social Influences on Vaccination Decision-Making 61
Δι
0 0.5
Δ
Fig. 5.3 Social opinion based on the influence of two opposite opinions. For individual i, σ̃i
is formalized as acceptance of vaccination with the probability P (Διi ) or otherwise with the
probability 1 − P (Διi ). In the Fermi function, ν denotes individuals’ responsiveness to the
discrepancy Διi
Therefore, we can write σ˜i in the following form:
+1, with probability P (Διi )

σ̃i = (5.4)
−1, with probability 1 − P (Διi )
where P (Διi ) denotes the probability that social opinion Διi is to accept vaccina-
tion, and 1 − P (Διi ) to reject vaccination. Here P (Διi ) is computed from the Fermi
function as follows:
1
P (Διi ) = (5.5)
1 + exp (−ν · Διi )
The Fermi function is a sigmoid function that has been widely used for describing
individuals’ behavioral changes as a response to the payoff discrepancy between
two different choices. Here, ν describes individuals’ responsiveness to the effect
discrepancy, i.e., the difference in effect of the two opposite opinions. As shown
in Fig. 5.3, a larger value of ν means that the choice backed by a stronger
social influence will be more likely to dominate social opinion even if the effect
discrepancy, Διi , is relatively small.
Next, we introduce a probability, rf , termed the individual’s conformity rate, that
indicates the degree of individuals’ tendency to adopt the social opinion of his/her
interconnected neighbors, which corresponds to how likely individual i is to convert
his/her cost-minimized choice (σ̂i ) to the social influence-formalized opinion (σ̃i ).
Thus, rf = 0 corresponds to the case of an absolute cost-based decision maker,

whereas rf = 1.0 indicates that the individual is an absolute social follower (i.e.,
ignoring his/her own cost evaluation). In other words, the final decision of individual
i can be expressed as follows:
σ̃i , with rf
σi = (5.6)
σ̂i , with 1 − rf
5.2 Case Study
Further to the above description of the integrated decision-making model, we

now show several simulation-based experiments based on the real-world epidemic
scenario of the 2009 Hong Kong H1N1 influenza together with a social network
with a total of 788 nodes and the average node degree of 35 (see also Sect. 4.3.1).
5.2.1 Vaccination Coverage
To evaluate the effect of individuals’ vaccination decision-making on disease

control, we further construct a standard SIR model to describe the threshold of
vaccination coverage for mitigating an epidemic. The mathematical description of
the threshold of vaccination coverage can be found in Sect. 3.1.1.
Based on our decision model, we conduct a series of simulations on vaccination
dynamics to assess the vaccination coverage at the steady state of individuals’
decision-making. As shown in Fig. 5.4, we investigate the interplay of cost min-
imization and the effect of social influences on individuals’ vaccination decision-
making with reference to three initial levels of vaccination willingness: 30%, 45%,
and 60%. The level of vaccine uptake is subject to cost ratio rc and individuals’
initial level of vaccination willingness, where individuals’ conformity rate rf
takes different values. Specifically, the simulation results in Fig. 5.4a–c show that
when the effect of social influences is relatively weak (i.e., conformity rate rf is
relatively small), the cost of vaccination (cost ratio rc ) fundamentally determines the
resulting vaccination coverage, in that increasing the cost of vaccination will lower
individuals’ vaccination willingness (at the steady state of individuals’ decision-
making). In our considered situation, the vaccination coverage is approximately
31% when cost ratio rc = 1.0. Gradually, as rc decreases and approaches 0,
the vaccination coverage increases to 90%. Based on our model design, when an
individual perceives that all of his/her interconnected neighbors have decided to be
vaccinated, the individual will keep his/her previous choice of non-vaccination even
if the cost of vaccination is zero, due to the consideration that disease transmission
will no longer occur.
5.2 Case Study 63
Fig. 5.4 Vaccination

coverage at the steady state of
individuals’ decision-making.
We investigate the interplay
of cost minimization and the
effect of social influences on
individuals’ vaccination
decision-making, as measured
by the resulting vaccination
coverage, by varying the
values of cost ratio rc and
conformity rate rf between 0
and 1.0. Individuals’ initial (a)
level of vaccination
willingness is set as: (a) 30%;
(b) 45%; and (c) 60%
(b)
(c)
Furthermore, we can observe that the strength of social influences (conformity

rate rf ) moderates the aforementioned effect of cost ratio rc on individuals’
vaccination decisions. In the extreme case that individuals are purely cost-based
decision makers (rf = 0), the resulting vaccination coverage will be completely
determined by the relative cost of vaccination (cost ratio rc ). Conversely, in the
extreme case that individuals are absolute followers of social opinion (rf = 1.0),
the effect of social influences will promote an unvarying vaccination coverage,
the level of which depends on individuals’ initial level of willingness rather than
the associated costs. In this simulation setup, when rf = 1.0, the vaccination
coverage at the steady state of decision-making converges to approximately 2% for
individuals’ vaccination willingness at the initial level of 30% (Fig. 5.4a), 50% at
the level of 45% (Fig. 5.4b), and 97% at the level of 60% (Fig. 5.4c).
In addition, the effect of varying conformity rate rf (individuals’ tendency to
be swayed by social opinions) can also be observed from the changing vaccination
decisions depending on vaccination-associated costs (cost ratio rc ). When individ-
uals become more susceptible to social influences (gradually increasing conformity
rate rf ), as shown in Fig. 5.4a, the effect of such influences tends to increase
the vaccination coverage when the cost of vaccination is low (0 < rc ≤ 0.5).
Conversely, when the cost of vaccination is relatively high (0.5 < rc ≤ 1.0), the
effect of social influences will reduce the resulting vaccination coverage at the
steady state of individuals’ decision-making. Furthermore, when the conformity
rate rf approaches 1.0, the vaccination coverage will decrease/increase sharply
and finally converge to a fixed level that depends on individuals’ initial level of
vaccination willingness.
Based on the earlier-mentioned SIR model (see also Sect. 2.1.2), we investigate
the effect of social influences on disease control by evaluating disease infection rates
(the percentage of individuals being infected as a result of disease transmission) with
respect to different vaccination coverage levels resulting from individuals’ decision-
making.
Figure 5.5 shows the disease infection rates with respect to the interplay of
individuals’ cost minimization and the effect of social influences on vaccination
decision-making (the values of cost ratio rc and conformity rate rf each ranging
from 0 to 1.0). Regarding our considered epidemic scenario (basic reproduction
number R0 = 1.6), the simulation results in Fig. 5.5a–c show that disease infection
can be eliminated given a relatively low cost of vaccination (cost ratio 0 < rc ≤ 0.8)
and a moderate effect of social influences (conformity rate 0 < rf ≤ 0.6).
Specifically, when individuals are less susceptible to social influences (confor-
mity rate rf < 0.6), the effectiveness of disease control is determined by the
relative cost of vaccination (cost ratio rc ) in that a lower vaccination cost leads
to a reduction in the disease infection rate, due to higher vaccination coverage.
However, as individuals’ tendency to be affected by social influences increases
(conformity rate 0.8 ≤ rf < 1.0), the effect of vaccination cost on disease control
is weakened accordingly, while individuals’ initial level of vaccination willingness
matters more. In the extreme case of rf = 1.0 (i.e., individuals are absolute
followers of social influences), the disease infection rate increases to 46% for the
initial level of vaccination willingness of 30%, as shown in Fig. 5.5a. If the initial
level of vaccination willingness is set as 45%, as shown in Fig. 5.5b, the infection
rate will be higher than in the situation with the initial level of 60%, as shown in
Fig. 5.5c, where cost ratio rc > 0.8 and conformity rate 0.2 ≤ rf < 0.8.
In addition, we examine the steady-state vaccination coverage and the resulting
infection rate corresponding to different initial levels of individuals’ vaccination
willingness prior to their decision-making, the results of which are shown in Fig. 5.6.
We note that individuals’ initial level of willingness affects the converged level
5.2 Case Study 65
Fig. 5.5 Infection rates with

respect to different levels of
vaccination coverage
resulting from individuals’
decision-making. We
investigate the interplay of
cost minimization and the
effect of social influences on
the effectiveness of disease
control, as measured by the
percentage of individuals
being infected as a result of
disease transmission, through (a)
varying the values of cost
ratio rc and conformity rate
rf between 0 and 1.0.
Individuals’ initial level of
vaccination willingness is set
as: (a) 30%; (b) 45%; and (c)
60%
(b)
(c)
of the steady-state vaccination coverage as well as the effectiveness of disease

control when individuals are absolute followers of social opinion (conformity rate
rf ≈ 1.0). In our simulations, when the initial level of individuals’ vaccination
willingness is 30%, the converged steady-state vaccination coverage is approxi-
mately 2.4%, as shown in Fig. 5.6a. The vaccination coverage will reach 45% and
91% if the initial levels of vaccination willingness are 45% and 60%, respectively.
In addition, we observe a critical phase transition in vaccination coverage when
individuals’ initial level of vaccination willingness is between 0.4 and 0.5, as shown
Conformity rate rf =1
1
0.8
Vaccin tion coverage
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1.0
Initial vaccination willingness
(a)
Conformity rate rf =1
1
0.8
Disease attack rate
0.6
0.4
0.2
0
0.2 0.4 0.6 0.8 1.0
Initial vaccination willingness
(b)
Fig. 5.6 The effect of individuals’ initial levels of vaccination willingness when individuals are
all social followers (conformity rate rf = 1.0). (a) Vaccination coverage at the steady state of
individuals’ decision-making. (b) The resulting effects on epidemic control in terms of disease
attack rate
in Fig. 5.6a. That is, in the situation of individuals being absolute social followers,
there is a threshold value of individuals’ initial level of vaccination willingness,
which can be used to evaluate the effectiveness of a voluntary vaccination program
for eliminating the epidemic, as in Fig. 5.6b.
Figure 5.7 shows the vaccination coverage at the steady state of individuals’
decision-making with respect to different disease reproduction numbers. Here, we
observe a similar effect of social influences in all three considered situations: the
effect of social influences will increase the vaccination coverage when the relative
cost of vaccination rc is low (see Fig. 5.7a–c), reduce it when rc is relatively high
(see Fig. 5.7g–i), and bring the coverage to a certain level when individuals become
strong followers of social influences (conformity rate rf approaches 1.0). The
simulation results further show that when the effect of social influences is relatively
weak (conformity rate 0 < rf ≤ 0.6), the vaccination coverage will increase in the
case of relatively severe disease transmission, i.e., large reproduction numbers (e.g.,
R0 = 2.0). However, if the effect of social influences is strengthened (conformity
rate rf approaches 1.0), the vaccination coverage at the steady state of individuals’
decision-making is mainly determined by individuals’ initial level of vaccination
willingness, rather than the related costs and disease severity.
5.3 Further Remarks
The phenomenon of social influences, in which individuals’ behaviors or opinions

are affected by their social environment, has long been observed and studied. In
the context of vaccination, social influences can affect individuals’ vaccination
decisions and, thus, the effectiveness of disease control in terms of the resulting
population vaccination coverage. In this chapter, we addressed the effect of social
influences on individuals’ vaccination decision-making, vaccination coverage, and
disease control. By parameterizing our model based on a real-world contact network
and the 2009 Hong Kong H1N1 influenza epidemic, we carried out a series of
simulations on individuals’ voluntary vaccination. The simulation results confirmed
that the relative cost of vaccination (cost ratio rc ) is one of the determining factors of
the voluntary vaccination coverage. In our simulations, the cost ratio is particularly
decisive if individuals are relatively indifferent to social influences (conformity rate
rf is relatively small). However, if individuals become more susceptible to social
influences (rf is large), such influences increase the vaccination coverage when the
cost of vaccination is low and, conversely, reduce the vaccination coverage when
the cost is high. In the extreme case where individuals are absolute social followers
(conformity rate rf = 1.0), the vaccination coverage at the steady state converges
to a certain level that solely depends on individuals’ initial level of vaccination
willingness, and is independent of the vaccination-associated costs.
Several mathematical models have been proposed that utilize payoff-based
methods to characterize individuals’ vaccination decision-making based on their
perception of the risks and benefits of vaccination. As an improvement over the
1 R0=1.2 1 1
R =1.6 0.8
0.8 0
0.8
R0=2.0
0.6 0.6
0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
1.0 1.0 1.0
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
(c)
(a) (b)
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 1.0 0 0
0.8 1.0 1.0
0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
(d) (e) (f)
1 1 1
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
1.0 1.0 1.0
0.8 0.8 0.8
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
(g) (h) (i)
Fig. 5.7 Sensitivity analysis of the vaccination coverage at the steady state of individuals’
decision-making, with respect to: (1) R0 = 1.2; (2) R0 = 1.6; and (3) R0 = 2.0
existing models, we consider an individual’s vaccination decision as an integrated

process balancing his/her self-initiated cost minimization (e.g., behaviors that
exploit the herd immunity effect) against the social influences of neighbors’ deci-
sions (social conformity). Our model introduces the parameter rf (conformity rate)
to modulate individuals’ tendency toward the two decision-making mechanisms:
an individual will either adopt his/her cost-minimized decision, or convert to the
social opinion of his/her interconnected neighbors. Based on the existing studies that
address individuals’ vaccination decision-making as a process of independent opin-
ion formation [164], we further take into account the heterogeneity of individuals’
interaction relationships by an extended SIT-based characterization of the strength
5.4 Summary 69
of social influences. Additionally, by incorporating the effect of social influences,

we can investigate the effect of individuals’ initial level of vaccination willingness
on the vaccination coverage resulting from individuals’ final decision-making.
By computationally characterizing the effect of social influences, our work
has practical implications for understanding vaccination behaviors and improving
the effectiveness of vaccination policies. In recent years, the rapidly increasing
use of new communication tools e.g., Internet-based social media, has further
amplified social influences. For example, both the efficacy and the adverse effects
of vaccination are debated online, and opinions either for or against vaccination
spread fast among users. We have identified that individuals’ initial level of
vaccination willingness is an important factor determining the final vaccination
coverage, due to the effect of social influences (social conformity). Our results
show that when conformity rate rf approaches 1.0, the vaccination coverage at
the steady state of individuals’ decision-making will differ starkly, depending on
different initial levels of vaccination willingness. Moreover, the empirical studies
that have been performed to survey the determinants of individuals’ vaccination
decisions in a social environment readily provide us a practical means for measuring
and evaluating conformity to social influences [165]. As has been shown in our
study, individuals’ vaccination decisions can be affected by both the associated
costs and their conformity to social influences. Therefore, public health authorities
must estimate the level of individuals’ acceptance of a vaccine prior to the start
of a voluntary vaccination program, as well as rapidly assess and enhance the
effectiveness of their adopted vaccination policies, e.g., provide certain financial
subsidies to reduce the cost of vaccination.
So far, our study has provided a modeling framework for incorporating the
effect of social influences on individuals’ decision-making and disease control. We
note that the results demonstrated in this chapter may be reliant on the chosen
social network, which was constructed using data from students’ interactions
within an American high school. In our model, the social influences account only
for the localized interactions between an individual and his/her interconnected
neighbors. Additionally, by utilizing an SIT-based characterization of social effect,
we implicitly assume that individuals are passive recipients of social influences and
do not actively shape them.
5.4 Summary
In this chapter, we further addressed the characterization of individuals’ voluntary

vaccination rate. To achieve this, we considered a group of individuals making
their decisions with the aim of minimizing the associated costs while evaluating
the decisions of others. In this regard, we introduced an integrated process of
vaccination decision-making that incorporates both the cost analysis of vaccination
and the effect of social influences. In addition, we applied SIT to characterize
the effect of social influences on individuals’ changing vaccination decisions
corresponding to their interaction relationships. By doing so, we conducted an

in-depth study of vaccination decision-making that integrated an extended SIT-
based characterization of social influences with a game-theoretic analysis of cost
minimization. We adopted individuals’ social network to capture the structure of
their interaction relationships and carried out a series of simulations on voluntary
vaccination against an influenza-like disease. By evaluating the resulting coverage
of voluntary vaccination, we evaluated the effect of social influences on individuals’
decisions and, in turn, on the effectiveness of disease control (vaccination coverage),
subject to three determinants, as follows: (1) the relative cost of the decision to
vaccinate, i.e., cost ratio rc ; (2) individuals’ susceptibility to social influences, i.e.,
conformity rate rf ; and (3) individuals’ initial level of vaccination willingness.
Based on the results in this chapter, we conclude that social influences will have
an effect on the ability to control an infectious disease by indirectly affecting the
vaccination coverage. In this respect, our work can provide a means to model the
effect of social influences and estimate the effectiveness of a voluntary vaccination
program.
Chapter 6
Understanding the Effect of Social Media
It has long been observed that the spread of awareness about an epidemic affects
individuals’ behaviors or opinions. In the context of an emerging infectious disease,
it would be unrealistic for individuals to have prior knowledge about a newly
developed vaccine when deciding whether to be vaccinated. In this case, the spread
of awareness on social media about both disease severity and vaccine safety will
affect individuals’ subjective perception of vaccination and, hence, substantially
influence their decisions.
In this chapter, we address the further development of decision models enabling
us to characterize individual-level voluntary vaccination; the models should incor-
porate the factor of subjective perception, which is partly shaped by social media
[54]. In addition, we examine how such individuals’ vaccination decisions affect
efforts to ensure adequate vaccination coverage for infectious disease control.
6.1 Modeling Subjective Perception
In the preceding two chapters, voluntary vaccination decision-making was modeled

from the perspective of rational individuals who assess the socio-economic costs of
disease infection and vaccination (see also Sect. 4.1). In this regard, game-theoretic
analysis was used to describe individuals’ vaccination decisions by evaluating their
personal optimized payoffs based on the perceived risks and benefits of vaccination.
Payoff-based decision-making models make the assumption that individuals have
prior knowledge about the disease, the vaccine, and the associated costs, which
may be valid for routine vaccination programs against seasonal infectious diseases
(e.g., measles and chickenpox). However, for a newly developed vaccine against an
emerging infectious disease (e.g., the 2009 H1N1 influenza), there will be a lack
of such knowledge. Thus, the assumption of individuals’ rational decision-making
may not hold.

https://doi.org/10.1007/978-3-030-52109-7_6
72 6 Understanding the Effect of Social Media
As we have discussed, individuals’ vaccination decisions are influenced by their

perception of disease severity and vaccine safety. Such perception may be informed
by news of recent events relayed by others [166]. That is to say, individuals’
perception can be mediated by the spread of awareness in a host population,
which can potentially alter their vaccination decisions and, hence, affect disease
transmission and the effectiveness of vaccination programs.
In recent years, rapidly emerging social media, such as Facebook [167], Twitter
[168], and YouTube [169, 170], have served as new channels for the spread of public
health information. Specifically, the efficacy of vaccines is widely debated among
online social communities; reports of the adverse events of vaccination are shared
across the Internet, and opinions either for or against vaccination can be effectively
“transmitted” from person to person. In this case, the spread of awareness about
an infectious disease and a vaccine can be as fast as the spread of the disease
itself, so that individuals will respond immediately to negative events as they occur.
Therefore, the dynamics of individuals’ vaccination will be tightly coupled with
that of disease transmission. The interplay between the two dynamic processes can
have significant consequences for the resulting vaccination coverage for infectious
disease control.
In this chapter, we study individuals’ vaccination decisions under the influence
of the spreading awareness of disease- and vaccine-related negative events during
an epidemic. As illustrated in Fig. 6.1, we consider a group of individuals who
decide whether or not to take a vaccine based on their perceptions of disease
severity and vaccine safety. Specifically, individuals interact through their social
relationships (e.g., friendships on Facebook and follower relationships on Twitter).
In such a structured host population (represented by a social network), awareness
about negative events can spread from person to person and substantially affect
individuals’ perceptions of disease severity and vaccine safety. On the one hand,
reported cases of severe infection will cause individuals to perceive the disease as
more serious and, hence, increase their tendency to be vaccinated. On the other hand,
reported events of the vaccine’s adverse effects will weaken the public confidence
in vaccine safety, leading to lower acceptance of vaccination.
In this case, we develop a novel decision model for characterizing individuals’
voluntary vaccination rate, in which we suppose that an individual will voluntarily
decide to accept or reject the vaccine based on his/her beliefs on whether or not
vaccination is acceptable. If the individual is not sufficiently confident in his/her
view for or against vaccination, he/she will not make any firm decision but wait
to see future developments. This situation in which an individual makes no firm
decision can be considered as a state of “yet to decide”, due to uncertainty. In
this regard, we introduce three belief variables in the form of Yes, No, and No
decision to characterize the possible decision of an individual to accept or reject the
vaccine, or to have not yet decided. Due to the spread of awareness, the individual
will update his/her beliefs about vaccination by collecting information from his/her
social neighbors, which may either reinforce his/her own perception, or bring about
conflicting perception.
6.1 Modeling Subjective Perception 73
Fig. 6.1 A schematic

illustration of the spread of
awareness affecting
individuals’ vaccination
decision-making. We
consider a group of
individuals whose vaccination
decisions depend on their
perception of disease severity
and vaccine safety. Awareness
about negative events can
spread from person to person
through their interaction
relationships, which will
substantially affect their
perceptions of disease
severity and vaccine safety
and, thus, their vaccination
decisions
To characterize individuals’ evolving belief values based on the spread of

awareness and their subsequent vaccination decision-making in the presence of
uncertainty, we develop a new belief-based decision model by extending and
utilizing the framework of the Dempster-Shafer Theory (DST) [171]. DST, also
known as the theory of beliefs, was originally proposed as a generalization of the
Bayesian theory of subjective probability to characterize how individuals update
their beliefs by combining new pieces of evidence from multiple sources in the
presence of uncertainty [172]. In our DST-based model, individuals can update
their beliefs (with respect to perceived disease severity and vaccine safety) by
combining the new evidence collected from social neighbors (obtained awareness
about negative events). Furthermore, we extend the conventional DST framework by
incorporating the effect of awareness spreading and fading, in which the awareness
about a negative event (a piece of new evidence) will ripple through individuals’
social networks, but the certainty of belief regarding that event (the belief values
of both Yes and No) will decay gradually. In this regard, individuals’ vaccination
decision-making is modeled as a process affected by the spread of awareness about
negative events and the subsequent updating of their beliefs.
We parameterize our model with an influenza-like disease and a social network

from a real-world online community. Through a series of simulations of voluntary
vaccination and infectious disease transmission, we evaluate the effect of the spread
of awareness on individuals’ vaccination decisions and the consequences for disease
dynamics corresponding to three impact factors: (1) the reporting rates of negative
events, which denote the probabilities of an infected or vaccinated individual being
reported as a case of severe infection or vaccine adverse effects; (2) the coefficient
of awareness fading, which describes the effect of certainty decay as awareness
spreads from one person to another; and (3) the disease reproduction number, which
represents the infectiousness of a disease. Furthermore, we provide a new modeling
framework that extends the existing studies on voluntary vaccination for infectious
disease control.
6.2 Subjective Perception in Vaccination Decision-Making
We develop a belief-based decision model to evaluate the effect of the spread of

awareness on individuals’ decision-making as well as on the effectiveness of disease
control. Compared with the existing studies modeling individual-level vaccination
decision-making, our model has the following properties:
• In contrast to the decision models in preceding chapters, which represent
decision-making as a binary problem, we consider the uncertainty in individuals’
vaccination decisions. Specifically, the situation in which an individual makes no
firm decision may be considered as a state of “yet to decide”, due to uncertainty.
In this regard, we introduce three belief variables to characterize the possible
decisions of an individual to accept or reject the vaccine, or to have not yet
decided.
• We further consider the dependence of individuals’ decisions on their subjective
perception about the acceptability of vaccination. Moreover, awareness about
disease severity and vaccine safety can spread from person to person and
substantially affect personal perceptions of vaccination.
• To characterize the spread of awareness, we utilize real-world online social
networks to characterize the structure of individuals’ interactions. By doing so,
we further utilize and extend DST to characterize individuals’ belief updating and
their decision-making, having incorporated the awareness obtained from their
socially interconnected neighbors.
We consider a voluntary vaccination program for preventing the outbreak of
an emerging infectious disease, e.g., 2009 H1N1 influenza, in which individuals
can decide whether or not to take the vaccine based on their perception of disease
severity and vaccine safety. It is assumed that individuals do not possess any prior
knowledge about the disease or vaccine, while they can receive information about
related events (reported cases of severe infection and vaccine adverse effects).
In this case, awareness about these negative events will spread among the host
6.2 Subjective Perception in Vaccination Decision-Making 75
individuals, rippling through their interaction relationships, which will in turn affect
their vaccination decisions.
For such a situation, we construct a new individual-level model to characterize
vaccination decision-making. At the same time, we use an epidemic model to
describe the dynamics of disease transmission as a result of individuals’ voluntary
vaccination. Based on our model, we investigate how the spread of awareness
affects the changes of individuals’ vaccination decisions during an infectious disease
outbreak.
6.2.1 Dempster-Shafer Theory (DST)
We represent an individual’s willingness to accept or reject vaccination by using a

set of belief variables. Based on the Dempster-Shafer Theory (DST), we suppose
that the decision on whether or not to be vaccinated is a binary problem, which
is represented as Θ = {Y es, No}. Here Θ is the frame of discernment for the
vaccination decisions (a universal set). Individuals’ possible vaccination-decision
responses can be modeled as subsets of Θ, i.e., belonging to a power set,
2Θ = {φ, {Y es}, {No}, Θ}. Next, we use a function m(·) to assign a belief
mass (probability) to each element of the power set 2Θ , which is called the basic
probability assignment (BPA). The mass m(A) (A ∈ 2Θ ) denotes the proportion
of support for the particular subset A based on the current available evidence or
knowledge. The BPA has the following two properties: (1) the mass of empty set φ
is zero; and (2) the masses of the power set sum to one:
m : 2Θ → [0, 1]
m(φ) = 0 (6.1)
m(A) = 1
A⊆2Θ
Accordingly, the belief functions for an individual’s vaccination-decision

responses can be expressed as follows:
m(φ) = 0
m(Y es) ∈ [0, 1]
(6.2)
m(No) ∈ [0, 1]
m(Θ) = 1 − m(Y es) − m(No)
where m(Y es) describes an individual’s belief that he/she should be vaccinated to
prevent infection, m(No) represents the belief that he/she should reject vaccination
having considered the potential adverse effects, and m(Θ) denotes the belief that
he/she is yet to decide whether or not to be vaccinated (due to uncertainty about the
disease and vaccine). Based on the above formulation, an individual will decide to
be vaccinated with the probability of m(Y es), reject the vaccine with the probability
of m(No), and have no firm decision with the probability of m(Θ). In addition, we
assume individuals who decide to be vaccinated will be vaccinated immediately
and, therefore, will either be successfully immunized or suffer from vaccine adverse
effects. Those with no decision will review their decision state in the next time step.
6.2.2 Spread of Social Awareness
During the spread of an emerging infectious disease and the implementation of a

vaccination program, individuals’ perceptions of disease severity and vaccine safety
will be affected by their awareness of negative events caused by both the disease
and vaccine. On the one hand, a case of severe infection can be naturally regarded
as evidence that an individual should be vaccinated. On the other hand, news of
the vaccine’s adverse effects can be viewed as evidence that vaccination should be
rejected.
Here, we use a belief value me to characterize a piece of evidence of a reported
negative event that spreads through the host population. We suppose that a group
of individuals are interconnected through their social network, denoted by G =
V
, L, where V = {v1 , v2 , · · · , vN } is the set of nodes (individuals), and L =
vi , vj | 1 ≤ i, j ≤ N, i = j is the set of links (social interaction relationships).
N is the total number of individuals. During an epidemic, each reported negative
event will be treated as a piece of triggering evidence with a belief value of me =
{me (Y es), me (N o)}, where me = {1.0, 0} and me = {0, 1.0} for a reported case
of severe infection and vaccine adverse effects, respectively.
Individuals can learn of emerging evidence by interacting with their social
neighbors, update their belief values accordingly, and further spread the news to
others through their social networks. However, the certainty of belief regarding
a piece of evidence will decay as it is transmitted from person to person, which
is referred to as awareness fading. Here, we introduce a fading coefficient, f , to
indicate how fast the certainty will decay when a piece of evidence is transmitted
between two individuals. A larger value of f corresponds to a faster decay
(awareness fading). Therefore, the evidence that is transmitted (i.e., the spread of
awareness) from individual j to his/her socially interconnected neighbor i can be
computed as follows:
mei (Y es) = (1 − f ) · mej (Y es)

mei (N o) = (1 − f ) · mej (N o) (6.3)

mei (Θ) = mej (Θ) + f · mej (Y es) + mej (N o)
In the course of disease transmission and vaccine implementation, newly

reported negative events will constitute new sources of evidence at different
time steps. The spread of awareness about the events will cause an individual
6.3 Case Study 77
to continuously update his/her perception of the disease and vaccine and then make
his/her vaccination decision. Based on the obtained awareness, the individual will
update his/her belief values (denoted by mi ) by combining the present belief values
(denoted by mi ) with the newly received evidence mei . This can be expressed in the
following form (with ⊕ denoting the combination operation):
mi = mi ⊕ mei (6.4)
Specifically, based on the assumption that the multiple sources are independent,
the belief-value update corresponding to the extended Dempster rule of combination
is as follows:
mi (SB ) · mei (SC )

SB ∩SC =SA
mi (SA ) = , and SA , SB , SC ∈ 2Θ (6.5)
1− mi (SB ) · mei (SC )
SB ∩SC =φ
where mi (SB ) · mei (SC ) represents the basic belief mass associated with
SB ∩SC =φ
the conflict between present beliefs and the newly received evidence. In the
Dempster combination rule, the denominator, 1 − mi (SB ) · mei (SC ), is a
SB ∩SC =φ
normalization factor, which attributes the conflict probability mass to the universal
set m(Θ).
By doing so, we utilize DST to develop a belief-based decision model for
characterizing individuals’ vaccination decisions in the presence of uncertainty.
Furthermore, we extend the classical DST framework by incorporating the spread of
awareness in a structured host population, in which the certainty regarding a piece
of evidence will decay as it is transmitted from person to person.
For illustration, Fig. 6.2 shows individuals’ belief-value updates calculated with
respect to the spread of awareness about two independently reported negative events
on a synthetic lattice network, in which the fading coefficients are set as: 0.1
(Fig. 6.2a) and 0.2 (Fig. 6.2b), respectively.
6.3 Case Study
We carry out a series of simulations to examine the coupled dynamics of individuals’

voluntary vaccination and disease spread based on the real-world 2009 Hong Kong
H1N1 influenza epidemic.
Fig. 6.2 Illustrations of f

awareness spreading and
fading on a synthetic lattice
network (30 × 30 nodes).
Individuals can interact with
their socially interconnected
neighbors. The awareness of
two reported negative events
will spread independently in
the structured host
population, and affect
individuals’ belief values.
Here, the color of a cell
denotes an individual’s belief
value in terms of the
probability mass
(mi ∈ (0, 1.0)). At the source
of a reported negative event,
the probability mass is set as
mi = 1.0. Parameter f
denotes the coefficient of
awareness fading. We f
demonstrate the effects of
awareness fading
corresponding to two
coefficient values: (a)
f = 0.1 and (b) f = 0.2
6.3.1 Vaccination Decision-Making in an Online Community
We construct an infectious disease model to characterize the spread of an emerging

infectious disease in a host population, in which negative events (e.g., severe
infections and vaccine adverse effects) will be reported and, hence, awareness of
them will spread. In doing so, we adopt a standard SIR model, as mathematically
described in Sect. 2.1.2. In addition, for the sake of example, we assume that a
vaccine is available and adequately supplied at the time of disease onset. Only
6.3 Case Study 79
susceptible individuals can decide whether and when to be vaccinated. Once an

individual is vaccinated, it is assumed that he/she will be completely immunized and
move from the susceptible compartment to the recovered/immunized compartment.
For our simulations, we calibrate the parameters in our epidemic model with
reference to the 2009 Hong Kong H1N1 influenza epidemic (see also Sect. 2.3.1).
During that outbreak, there were more than 36,000 laboratory-confirmed cases (as
of September 2010), among which approximately 290 were identified as severe
(i.e., the reporting rate of severe infections was estimated as 0.805%). The Human
Swine Influenza Vaccination Programme (HSIVP) was launched on December 1,
2009. The numbers of vaccinated individuals and reported negative events since
that date are shown in Fig. 6.3. As of March 13, 2010, more than 180,000 doses
of HSI vaccine were administered to persons of various groups (Fig. 6.3a) [173].
Throughout HSIVP as a whole, a total of 34 cases of adverse events following
immunization (AEFI) were reported (Fig. 6.3b). The rate of AEFI was evaluated as
17.8 per 100,000 vaccinated individuals (i.e., the reporting rate of vaccine adverse
effects was estimated as 0.0178%) [95].
We further construct a social network to characterize individuals’ interaction
relationships based on the data of a Facebook-like online community [174], in which
registered users can communicate through personal blogs and forum postings. In this
network, there are in total 1899 nodes and 13, 838 undirected links among them. As
shown in the partial network snapshot of Fig. 6.4, the nodes denote the registered
users and the links among them represent their interaction relationships, in terms
of sending and receiving at least one message. Based on this network structure,
we carry out a series of Monte Carlo simulations and experimentally examine the
above-mentioned belief-based characterization of individuals’ vaccination decision-
making.
The simulation results in Fig. 6.5 show the dynamics of disease transmission
and individuals’ voluntary vaccination for the first 50 days. Figure 6.5a indicates
the dynamics of disease transmission with the impact of individuals’ voluntary
vaccination decision-making. Figure 6.5b presents the distributions of simulated
vaccine-induced negative events and severe disease infections over time. Figure 6.5c
shows the changes of individuals’ belief values in the three states. In this case,
we reproduce the observed patterns of the real vaccination program, as shown
in Fig. 6.5d, in terms of daily number of vaccinated individuals. We observe
that the number of vaccinated individuals increases steadily in the earlier days of
the vaccination program, as individuals’ uncertainty about vaccination decreases.
However, the reported cases of vaccine adverse effects significantly strengthen
individuals’ belief that non-vaccination is preferable, which causes the daily number
of vaccinated individuals to fall sharply when this belief peaks on day 12.
(a)
(b)
Fig. 6.3 The Human Swine Influenza Vaccination Programme (HSIVP) in Hong Kong. This
program was launched on December 1, 2009. As of March 13, 2010, more than 180,000 doses of
HSI vaccine were administered to persons of various groups. The rate of adverse events following
immunization (AEFI) was evaluated as 17.8 per 100,000 vaccinated individuals, i.e., the reporting
rate of AEFI was estimated as 0.0178%. (a) The daily number of vaccinated individuals since the
beginning of HSIVP. (b) The reported cases of infection and vaccine adverse effects
6.3 Case Study 81
Fig. 6.4 A partial snapshot of individuals’ social network. We utilize a network structure to
represent individuals’ interaction relationships, based on the data of a Facebook-like online
community. In this network, the nodes denote individuals and the links represent their interactions
in terms of sending and receiving messages
6.3.2 Interplay of Two Dynamics
We run our model with the aforementioned parameterization under various settings
to reveal the interplay between the dynamics of disease transmission and individ-
uals’ vaccination. In doing so, we investigate the effect of the spread of awareness
of disease severity and vaccine safety in a host population by investigating various
settings of the reporting rates of negative events and κ, the coefficient of awareness
fading f , and the disease basic reproduction number R0 .
As shown in Fig. 6.6, we investigate the reporting rates for the negative events of
severe infections () and vaccine adverse effects (κ) with respect to two levels: 1%
and 0.1%. Here, we set = 0.01 and κ = 0.001 for the situation of “disease scare”,
and similarity, = 0.001 and κ = 0.01 for the situation of “vaccine scare”. A higher
reporting rate of severe disease infections will prompt individuals to be vaccinated
(Fig. 6.6a, dash curve), which will in turn reduce disease transmission (Fig. 6.6b,
dash curve). Moreover, vaccination in the early stage will be more effective than in
the later stages. We can observe that when = 0.01, the difference in the number of
vaccinated individuals between the situations of κ = 0.001 and κ = 0.01 (Fig. 6.6a,
dashed curve and solid curve, respectively) is relatively small in the early stage of
disease transmission (before day 10). After that, the vaccination dynamics when
Severe disease infection

Vaccine adverse effects
Susceptible
Infectious
Recovered
Vaccinated
(a) (b)
Daily vaccinated individuals
(c) (d)
Fig. 6.5 Monte Carlo simulations of disease transmission and voluntary vaccination. (a) The
dynamics of disease transmission in terms of the sizes of susceptible, infectious, recovered, and
vaccinated populations. (b) The reported cases of severe disease infection and vaccine adverse
effects. (c) The average belief values about vaccination in a host population. (d) The dynamics of
voluntary vaccination in terms of the daily number of vaccinated individuals
= 0.01 and κ = 0.001 will peak on day 15 with more than 4% individuals
being vaccinated, while that of = 0.01 and κ = 0.01 will peak on day 11 at
2%. Accordingly, we can observe that the disease dynamics in the situations of
= 0.01, κ = 0.001 and = 0.01, κ = 0.01 (Fig. 6.6b, dashed curve and solid
curve, respectively) have a similar incidence rate at the peak of disease infection,
but the durations of the disease transmission period are different.
In addition, we investigate the effect of awareness fading corresponding to
different fading coefficients f , the results of which are shown in Fig. 6.7. We
note that awareness fading can affect the dynamics of individuals’ vaccination in
terms of the number of vaccinated individuals and the time of individuals’ being
vaccinated. In our simulation, when the fading coefficient f = 0.1, the daily
6.3 Case Study 83
(a)
(b)
Fig. 6.6 The effect of reporting rates of negative events (severe disease infections and vaccine
adverse effects κ). (a) The dynamics of voluntary vaccination (daily number of vaccinated
individuals). (b) The dynamics of disease transmission (daily number of infectious individuals)
Vaccination dynamics
0.1
f =0.1
f =0.4
Population size (100%)
0.08
f =0.7
0.06
0.04
0.02
0
0 10 20 30 40 50
Days
Disease dynamics
0.1
f =0.1
f =0.4
Population size (100%)
0.08
f =0.7
0.06
0.04
0.02
0
0 10 20 30 40 50
Days
Fig. 6.7 The effect of fading coefficient (f ). (a) The dynamics of voluntary vaccination (daily
number of vaccinated individuals). (b) The dynamics of disease transmissions (daily number of
infectious individuals)
number of vaccinated individuals peaks on day 10 at a rate of approximately 4%

(Fig. 6.7a, solid curve). In contrast, the vaccination rates will be approximately 2%
and peak on day 12 and day 20 if the fading coefficients are set as: f = 0.4 and
f = 0.7, respectively (Fig. 6.7a, dashed curve and dotted curve). In this case, we
can observe that when awareness spreads with only a weak fading effect (a smaller
fading coefficient), individuals will be prompted to be vaccinated and, thus, disease
transmission will be effectively prevented (Fig. 6.7b, solid curve when f = 0.1).
To investigate the sensitivity of our results, we consider individuals’ vaccination
decisions with respect to different situations of disease transmission. In Fig. 6.8,
we vary the disease basic reproduction number over 1.2, 1.6, and 2.0 to represent
different levels of disease infectiousness. Here, we can observe that a larger disease
reproduction number will enhance individuals’ vaccination rate (Fig. 6.8a, dotted
curve when R0 = 2.0). However, the increased levels of vaccination cannot provide
adequate population-level immunity for containing disease transmission in the cases
where R0 is relatively small. The simulation results in Fig. 6.8b show that the
reproduction number R0 = 2.0 results in disease dynamics with the largest peak-
incidence rate, i.e., nearly 6% as shown in Fig. 6.8b, dotted curve, when compared
with the situation of 3% for R0 = 1.6 (Fig. 6.8b, dashed curve) and 1% for R0 = 1.2
(Fig. 6.8b, solid curve). These results can be explained by considering the interplay
of the dynamics of disease transmission and vaccine implementation: an increased
number of vaccinated individuals will trigger more reporting of the vaccine’s
adverse effects, which will in turn reduce individuals’ vaccination willingness.
6.4 Further Remarks
It is well known that the spread of awareness about an epidemic will affect
individuals’ health-related behavior. For example, individuals who are aware of
the risk of infection may take measures to reduce their susceptibility or distance
their social contacts to protect themselves [175, 176]. In the context of vaccination,
the spread of awareness about severe infections and vaccine adverse effects will
affect individuals’ perception of disease severity and vaccine safety and, hence,
substantially change their vaccination decisions.
As a further step from the studies in the preceding chapters, here we have con-
sidered a belief-based characterization of individuals’ vaccination decisions. In our
model, we have correlated individuals’ subjective assessment of disease severity and
vaccine safety with the dynamics of disease transmission and voluntary vaccination
by exploring the awareness of reported negative events. Unlike existing belief-based
studies, e.g., that of Coelho et al. [177], we have characterized individuals’ belief-
value updates as a result of the spread of awareness in a structured host population
(a Facebook-like online community). In this case, we can represent the situation
in which individuals collect health-related information from online social media,
and make vaccination decisions according to their obtained awareness from socially
interconnected neighbors. Additionally, instead of a binary polarization of decisions,
(a)
(b)
Fig. 6.8 The effect of disease reproduction number R0 . (a) The dynamics of voluntary vaccination
(daily number of vaccinated individuals). (b) The dynamics of disease transmissions (daily number
of infectious individuals)
6.5 Summary 87
i.e., either to vaccinate or not, we have introduced a third decision response, “yet to
decide”, and associated it with the belief value of uncertainty. By doing so, we have
utilized an extended DST to characterize individuals’ belief-value updates in the
presence of uncertainty.
By computationally characterizing the effect of the spread of awareness, this
work has some practical implications for understanding individuals’ vaccination
decision-making and for improving the effectiveness of a vaccination program. A
growing number of individuals use Internet-based communication services to obtain
and share health-related information [178]. This represents the growing power of
data on users’ online communication to help track events in real time during an
epidemic, e.g., detecting a pandemic of influenza by monitoring related queries to
online search engines [179]. Salathe et al. collected individuals’ Twitter updates to
assess public sentiments toward a novel vaccine [180]. Henrich et al. used online
comments as obtained from the websites of CBC, Vancouver Sun, and Global
and Mail to capture public attitudes about the H1N1 vaccine [181]. Thus, online
social media provide effective tools to access real-time data to evaluate the public
perception of diseases and vaccines. These can be used to estimate public acceptance
of a prospective vaccination program, which will enable public health authorities to
make plans in advance to improve the effectiveness of vaccination strategies.
So far, our study has provided a modeling framework that incorporates the spread
of awareness with the belief-based characterization of decision-making. It should
be pointed out that the simulation results may depend on the social network used
in our example (a Facebook-like online community). In our model, the spread of
awareness only mediates individuals’ localized interactions (i.e., between socially
interconnected neighbors), while the global effect of public media is not taken into
account. For future work, it will be interesting to extend the current model by adding
a globalized spread of awareness, wherein each individual will become aware of
a reported negative event with a certain probability. Additionally, by utilizing a
DST-based characterization of vaccination decisions, the combination rule used to
describe individuals’ belief-value updates will determine the resulting decisions.
The above-mentioned issues are worthy of further investigation.
6.5 Summary
In this chapter, to gain a more complete understanding of individuals’ vaccination

decision-making, we discussed how to model and evaluate the effect of the spread of
vaccination and disease awareness on decisions. In doing so, we focused on a group
of individuals opting for or against vaccination, in which their awareness of negative
events spread via their social interactions and, thus, affected their perception of
disease severity and vaccine safety. We characterized vaccination decisions in the
form of individuals accepting, rejecting, or having not yet decided whether to be
vaccinated, and further associated them with a set of belief values denoted Yes, No,
and Uncertain, respectively. Furthermore, we extended DST to model individuals’
belief updates after incorporating their awareness received from interconnected

neighbors, and their corresponding changes in vaccination decisions. We studied
the situation of voluntary vaccination against an influenza-like disease through a
series of simulations. We examined three factors that affect individuals’ vaccination
decisions, namely (1) the reporting rates of negative events, (2) the coefficient
of awareness fading, and (3) the disease reproduction number. By doing so, we
evaluated the effect of the spread of awareness on their decisions by assessing
the vaccination dynamics through the time course of disease transmission, i.e., the
number of vaccinated individuals and the time point of vaccine administration.
Chapter 7
Welcome to the Era of Systems
Epidemiology
The preceding chapters have, in many aspects, alluded to a new way of viewing
and pursuing epidemiological studies in the modern context. Starting from cer-
tain empirical observations of a real-world infectious disease situation, beset by
unanswered questions, we then represent it using an abstract modeling language,
parameterize our models with the real-world data, and thereafter analytically and
quantitatively examine the underlying associated risk factors and the implications
for evidence-based policies and practice.
In this chapter, we continue to build on the above-mentioned discussions, and
further generalize them through the prism of systems epidemiology, a term coined
here to encompass the latest methodological development in epidemiology [182].
7.1 Systems Thinking in Epidemiology
We must begin by asking precisely what systems epidemiology is, and how the
notion of systems is relevant to epidemiology. These key questions will be answered
in the following discussions on systems thinking.
Systems thinking is a philosophical and methodological perspective that draws
on the fundamental notions of systems theory, which views a system as an
integration of components with the interacting relationships among them and the
environments in which they reside [183, 184]. As mentioned by Maani et al.
[185], systems thinking emphasizes two fundamental concepts, i.e., complexity
and entirety. Systems complexity is generated from the structure of integrated
components, i.e., how the constituent components are organized and interact with
each other and with the environment. Systems entirety is derived from the dynamic
behaviors of a system as a whole, which is to say, how a complex system of
interacting components behaves and exhibits emergent properties at the system
level, rather than being a simple behavioral aggregation of its basic components.

https://doi.org/10.1007/978-3-030-52109-7_7
90 7 Welcome to the Era of Systems Epidemiology
Systems thinking offers a novel comprehensive perspective that examines the

process of infectious disease transmission as a system, and thus having structural
complexity and behavioral entirety. In such a system, the components include
disease pathogens/parasites, animal/vector species, human populations, and their
natural, social, and behavioral environments. The interactions among components
are present, such as the ability of disease pathogens/parasites to infect and be
transmitted between and within animal/vector species and human populations. The
interacting relationships of components with their environments can be described
as those components’ responses to potential environmental changes, for example,
biomedical genetic mutations of pathogens or parasites as a result of drug resistance
selection, animal/vector population fluctuations due to climate changes, and behav-
ioral changes affecting people’s exposure due to their socio-economic conditions.
The emergent behaviors of such a system, i.e., the emerging and re-emerging
infectious diseases, depend on the integrated effects of all of the constituting
components, including evolving microbial pathogens, zoonotic vector exposures,
environmental changes, and human behaviors.
Based on the above-mentioned perspective of systems thinking, the study of
infectious diseases can go beyond the conventional methods that are usually
confined by their disciplinary boundaries, such as the statistical analysis of disease
occurrences or laboratory research on pathogens. Intervention measures to combat
infectious diseases will thus be designed to modify the emergent behaviors at the
system level, by exploiting interdisciplinary methods to address systems complexity.
Toward this end, we need a set of novel modeling and analytical tools drawing on
the concepts of systems thinking. This will enable epidemiologists to develop and
deploy more effective intervention measures.
7.2 Systems Modeling in Principle
The complex systems approach is a holistic approach intended to model, character-

ize, explain, and predict the emergent behaviors of a system with reference to its
constituting complexity, which is difficult to derive or compute using conventional
top-down reductionist approaches [186, 187]. As stated by Liu [188], such an
approach pays special attention to the following three objectives:
• Systems Modeling
The systems-modeling step provides a blueprint/framework that is abstracted
and replicated from real-world observations in the languages of mathemati-
cal/computational characterizations. To model a system in this way requires
identification, abstraction, and reproduction of certain observations, which is the
starting point for the following steps of systems exploration and problem solving.
In systems modeling, the basic components of a model, also known as entities,
are the basic constituents of a complex system, which directly or indirectly inter-
act among themselves and with their environments, based on certain predefined
7.2 Systems Modeling in Principle 91
or known mechanisms or principles. For example, in a network-based disease

model, nodes represent individual humans whereas links represent the routes
of disease transmission. Diseases can be transmitted from one node to another
due to the predefined contact interactions [189]. Interrelationships exist among
entities and their local and global environments, through which the complex
system, as a whole, exhibits structural and behavioral complexity at and across
various scales. Emergence, which is the dynamic transmission of infectious
diseases, is defined in terms of the system-level patterns and regularities arising
from the dynamics of a group of interacting entities, as generated from the
reciprocally coupled and dynamically changing interrelations of the entities at
multiple scales.
• Systems Exploration
Systems exploration presents a set of analytical tools to understand the operating
mechanisms underlying a complex system and, furthermore, find explanations
and make predictions about the observed systems’ dynamics. To uncover the
operating mechanisms behind the observations, we perform systems modeling to
characterize or simulate the real system. For example, we can use the SIR model
to characterize the dynamics of disease transmission in a human population.
Then, by comparing the real-world observations and the synthetic simulation, the
models and interaction mechanisms of the system can be fine-tuned, as reflected
in adjustments of the relevant model parameters, and the structure and behavior
of the interacting entities. For example, when the incubation period is taken
into consideration, the SIR model can be modified into the susceptible-exposed-
infectious-recovered (SEIR) model with an additional latency compartment of E
(exposed).
• Problem Solving
Problem solving emphasizes the ability of the complex systems approach to
independently find ways to achieve adaptive solutions that are well suited to
the current problem settings. The ultimate goal is to develop a set of analytical
algorithms that can adjust their own parameters for different application domains.
For example, adaptive evolutionary algorithms can be used to automatically tune
parameters related to the developed systems being modeled or the proposed
operating mechanisms. Constrained optimization algorithms are dedicated to
finding the optimal solutions for resource allocation.
The complex systems approach can be used not only to build a modeling
framework for mapping real-world observations/phenomena in analytical languages
(i.e., mathematical and computational models), but also to reveal the operating
mechanisms behind a complex system. Problem solving is the advance application
of systems modeling and exploration, with respect to the specific domain problem
set.
7.3 Systems Modeling in Practice
In combating infectious diseases, the complex systems approach can help us

understand how the systems of infectious diseases are organized, in terms of the
causal relationships and the factors affecting disease prevalence, and how such
systems behave in time and space, by revealing the spatio-temporal distributions
of disease occurrences, and how the diseases can be better mitigated and eradicated
by developing more effective solutions for infectious disease control.
Figure 7.1 provides a schematic framework outlining the four essential steps for
applying the complex systems approach to epidemiological studies of infectious
diseases.
Fig. 7.1 The four essential steps (in ovals) for performing the complex systems approach in
combating infectious diseases. The directional arrows show their functional interrelationships
7.3 Systems Modeling in Practice 93
• Problem-Driven Conceptual Modeling first translates the real-world problems

in an epidemiological domain into conceptual models in a theoretical or com-
putational domain, which aim to describe the systems components in infectious
disease transmission, their impact factors, and interaction relationships.
• Data-Oriented Real-World Grounding then concentrates on discovering ways
to embody conceptual models, through model parameterization, by obtaining and
utilizing real-world data and statistically analyzing the real-world observations.
• Goal-Directed Analytical Inference is devoted to further developing analytical
methods and solutions addressing specific real-world tasks of disease surveil-
lance and control; that is, to finding appropriate methods and solutions to meet
specific goals.
• Evidence-Based Practice proceeds to the implementation, validation, and
improvement of the developed analytical solutions, aiming to bridge the
theoretical or computational analysis and the real world.
Specifically, the goal of the conceptual modeling step is to build theoretical
or computational prototypes of infectious disease systems, which can be used to
represent the real-world problems. For example, networks, consisting of nodes and
links between nodes, have been utilized to characterize the dynamics of infectious
diseases with various diffusion paths [190–192].
Based on the existing understanding and theoretical/empirical knowledge about
infectious diseases and the related impact factors, mathematical and computational
models can be used as a conceptual framework to reproduce disease dynamics.
For example, the compartmental models are well suited to characterizing disease
dynamics in several host populations, such as in the case of influenza [193].
Meanwhile, the network models or metapopulation-based models are more suitable
for representing disease diffusion due to human movement, such as the importation
of malaria cases in remote or cross-border areas [71, 194, 195]. To achieve the above,
we need to perform model selection with reference to the specific characteristics of
the current epidemiological problems.
As can be seen, conceptual modeling depends on simplifications and abstractions
about the operating mechanisms of infectious diseases, which also necessitates the
use of hypotheses for data collection in the real-world grounding step (function
c in Fig. 7.1). This step also provides a theoretical or computational qualitative
framework for performing analytical inference methods (function b). The goal of
real-world grounding is to collect and analyze data from multiple sources, aiming at
a more comprehensive, multidisciplinary understanding of the structural interrela-
tionships and behavioral mechanisms of real-world infectious disease systems. For
example, in the case of influenza, the demographic profiles and contact structures of
a human host population can be used to model disease transmission among different
human groups [196, 197]. In the case of malaria, environmental factors, such as
rainfall and temperature, can be identified from various sources, and incorporated
into a causal model for examining the effect of disease vector population over time
[198].
The products of this data-oriented real-world grounding can in turn provide

empirical intuitions for conceptual modeling (function d), generate experience-
based rules or principles to guide the practical implementation of infectious
disease-control measures (function f), and parameterize variables in performing
inference algorithms (function i). The real-world grounding step involves multi-
disciplinary data-fusion and knowledge discovery from massively accumulated data.
For example, based on the reported infection cases from the 62 towns in Yunnan
province of China, the underlying transmission networks of P. vivax can be inferred
for public health policy makers to accurately predict the geographic patterns of
malaria spread [199]. In order to evaluate the potential risks of H7N9 infection, data
about bird migration and poultry distribution can be collected and utilized [200].
Based on the developed models and collected data, the analytical inference step
provides a series of specific problem-solving methods and solutions, which can
be used as analytical tools to address the real-world problems that are taken into
account in the conceptual modeling step. The gaps between the desired situations
(goals) and the current situations (status quo) in disease active surveillance and
control necessitate the use of inference methods to find an improved solution. Ana-
lytical inferences will provide a set of quantitative representations for conceptual
modeling (function a). Furthermore, the end products of this step can also inform
solutions for the practical realization of infectious disease control (function g)
and guide the data collection in the real-world grounding step (function j). For
example, network inference methods can be used to identify the spatio-temporal
patterns of dengue transmission [201] as well as to reveal the key bird species and
geographic hotspots of avian influenza A (H7N9) [202]. Poisson regression methods
can be developed for epidemic prediction by integrating both inter-regional and
external, environmental, and social impact factors [203]. The methods of group
sparse Bayesian learning and partially observable reinforcement learning can be
utilized to achieve active surveillance of infectious diseases [204, 205].
The fourth step of evidence-based practice concerns the application and vali-
dation of the developed solutions in the real-world practice of infectious disease
surveillance and control. The goal of this step is twofold: (1) guiding the practice
of disease control and prevention (function e); and (2) validating and improving the
applied analytical methods (function h). For example, intervention and surveillance
planning can help public health authorities to know how to distribute their very
limited resources to high-priority regions so as to maximize the outcomes of active
surveillance [55, 206]. Risk ranking can also help identify the relative risks of
malaria among various villages in remote or cross-border areas [198, 207]. At the
same time, as more data are accumulated, the results of risk ranking will become
more precise and reliable [208].
The feedback from field studies will help validate the analytical results and
determine if the selected models and adopted inference methods truly represent
the real-world scenario and, thus, address the real-world problems. In other words,
theoretical analysis and results will be used to guide the practice of infectious
disease control, which will in turn validate or improve the developed models and
inference methods [209].
7.4 Toward Systems Epidemiology 95
7.4 Toward Systems Epidemiology
Systems epidemiology will be an interdisciplinary effort that will help provide a

deeper understanding of diseases and mitigation strategies. It will enable us to
pursue comprehensive epidemiological inquiries, throughout the evolutionary life-
cycle of problem-driven conceptual modeling, data-oriented real-world grounding,
goal-directed analytical inference, and evidence-based practice, as detailed above.
Each step will offer either a sound validation of, or a new insight into, another step,
thus bringing our understanding of the study problem closer to reality. The emerging
computational modeling and analytics tools and artificial intelligence technologies
will further empower such an endeavor.
Let us make this paradigm shift now—Welcome to the new era!
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Index
A propagation and evolution of individuals’

Age-structured host population beliefs, 11
age-specific attack rates, sources of evidence, 76
age-specific contact matrices, 25–29 theory of beliefs, 73
age-specific heterogeneity, 9
age-specific infectivity, 8, 38, 40, 41
age-specific subpopulations, 9, 38, 42, 43 C
age-specific susceptibility, 9, 32, 37, 38 Causal model
age structure, 8, 12, 15, 24, 25, 32, 33, 37, causal relationships, 3, 92
39, 48 Centre for Health Protection (CHP) of Hong
Artificial intelligence, x, 95 Kong, 25, 26, 29, 30, 47
Chickenpox, 71
Compartmental model
B disease attack rate, 66
Basic reproduction number (R0 ), 2, 16, epidemiological parameters, 22
20–22, 29, 34, 35, 54, 55, 64, 67, homo-mixing assumption, 18
68, 81, 85, 86 infection rate, 20, 21, 29, 30, 32, 64, 65
See also Reproduction number metapopulation-based compartmental
Belief-based decision model model, 15 (see also
Dempster-Shafer theory (DST) Metapopulation-based
basic probability assignment (BPA), 75 models)
belief functions, 75 recovery rate, 20, 28
belief updates, 12, 73, 87 susceptible-exposed-infectious-recovered
belief values, 73, 76–79, 82, 85, 87 (SEIR) model, 18, 91
belief variables, 11, 72, 74, 75 susceptible-infectious-recovered (SIR)
Dempster rule of combination, 77 model, 18, 20, 34, 38, 62, 64, 78,
extended DST, 87, 88 91, 94
frame of discernment, 75 susceptible-infectious-susceptible (SIS)
probability mass, 77, 78 model, 18
extended decision model, 12 See also Mathematical modeling
fading coefficients (f ), 11, 76, 77, 82, 84, Complex system
85 complex epidemiological system, 8
negative events, 72–74, 76–79, 81, 83, 85, complexity, 89–91
87, 88 complex systems approach, 90–92

https://doi.org/10.1007/978-3-030-52109-7
110 Index
Complex system (cont.) Contact reduction

emergence setting-specific contact reduction, 37, 45,
emergent behaviors, 90 47
emergent properties, 89 Contact tracing, 22–24
entirety, 89, 90 COVID-19, vii, viii, 5, 8
interacting entities, 91
operating mechanisms, 91, 93
regularities, 91
D
Computational methods
Data-centric era, 5
adaptive solutions, 91
Data science, 5
analytical algorithms, 91
Descriptive methods, 2
constrained optimization algorithms, 91
Diffusion, 93
inference methods, 93, 94
Disease dynamics, 3, 6, 8, 15–23, 32, 35, 36,
knowledge discovery, 94
46, 82, 93
Monte Carlo simulations, 79, 82
See also Infection dynamics
prioritization method, 12, 48
Disease intervention
sensitivity analysis, 2, 68
effectiveness of disease intervention, 8,
simulation-based experiments, 10, 12, 13,
10–12
37, 45, 48, 62
Disease pathogens/parasites, 4, 90
stochastic percolation process, 18, 31
Disease prevalence
Computational modeling
real-time disease prevalence, 9, 37
computational models, viii, 22, 23, 25, 32,
Disease scare, 81
50, 91
Disease severity, 11, 50, 52, 55, 67, 71–74, 76,
data-driven computational model, viii, 25
81, 85, 87
data-fusion, 94
Disease surveillance
decision model, 8, 10–12, 54, 56, 62,
active surveillance, 94
71–74, 77
reporting rates, 40, 47, 79, 81
deterministic description, 19
Disease transmission
differential equations, 18, 20, 21, 31
disease transmission systems, 8
epidemic models, 8, 12, 15, 17, 23, 30–32,
transmissibility, 31, 36
36–38, 75, 79
transmission networks, 94
epidemic trees, 22
transmission parameters, 18
infectious disease models, 17–20, 78
Disease vector population, 93
integrated decision model, 10, 12, 53,
58, 62 (see also Integrated
decision-making process)
model parameterization, 15, 53, 93 E
See also Mathematical modeling Emerging infectious disease, 5, 6, 11, 35, 71,
Conceptual models, 15, 93 74, 76
Contact matrix Empirical methods, 2–3, 22–23
adjacency matrix, 24 Empirical observations, 2, 89
Contact network Epidemiology
contact relationships, 18, 19, 23, 24 computational epidemiology, viii, 2
directedness of a network, 24 empirical epidemiology, 2
Contact patterns epidemiological inquiries, 95
cross-age contact frequencies, 8, 9, 26, 39 epidemiological studies, 2, 3, 5, 13, 15, 89,
cross-age contact patterns, 12 92
setting-specific contact frequency matrices, infectious disease epidemiology, vi, 1–3,
28, 32, 40, 47 13, 15, 16, 21
setting-specific contact patterns, 27, 37 theoretical epidemiology, 2, 3, 93
survey-based studies, 23, 31 Experience-based rules, 94
Index 111
G M
Game theory Machine learning
game-theoretic analysis, 12, 51, 55, 56, 58, group sparse Bayesian learning, 94
70 Poisson regression, 94
game-theoretic model, 12, 51–53 reinforcement learning, 94
minority game, 51 Macroscopic scale, 2
Geographic hotspots, 94 Mathematical modeling, 2, 15, 16, 31, 51, 67
Google Flu Trends, 6 Measles, 18, 49, 71
Measles–mumps–rubella (MMR), 7, 49
Metapopulation-based models
H subpopulations, 20 (see also Age-structured
Herd immunity host population)
herd immunity threshold, 6, 34, 35 Methodological paradigm
population-level herd immunity, 85 computational paradigm, 3
vaccine-induced herd immunity, 6 philosophical and methodological
Heterogeneity perspective, 89
heterogeneity of a host population, 8, 9, 18, theoretical paradigm, 3
32 (see also Age-structured host top-down reductionist approaches, 90
population) Microscopic scale, 2
heterogeneity of pathogens, 18 Mobility
HIV, 23, 24 human air-travel networks, 5
Human population mobile behaviors, 5
heterogeneous host population, 18 Morbidity, 6, 35, 36
human host population, 93 Mortality, 1, 6, 35, 36
Multidisciplinary, 93
Multiple scales, 91
I
Impact factors, 1–6, 11, 74, 93, 94
Individual-level model, 18–19, 75 N
Infection dynamics, 1, 2, 15, 93 Network models
Infection tracing, 22, 23 network-based disease model, 91
See also Contact tracing network inference, 94
Infectious disease control, 1, 20, 23, 31, 59, 71, network structure, 31, 59, 79, 81
92, 94 Next generation matrix (NGM), 38, 39
Infectivity, 8, 9, 20, 21, 29, 31, 32, 36, 38, 40,
41
Influenza
P
H1N1
Peak-incidence rate, 82
2009 Hong Kong H1N1 influenza
Predictive methods, 2
epidemic, 8, 10, 11, 25, 26, 29,
Prescriptive methods, 2
30, 32, 37, 42, 45, 47, 62, 67, 77
Problem solving, 9, 90, 91, 94
H7N9, 94
human swine influenza, 25, 42–45, 80
influenza-like disease, 32, 37, 56, 59, 70,
74, 87 R
seasonal influenza, 18 Real-world epidemic scenario, 12, 25
Integrated decision-making process, 10, 57, 62 Real-world problems, 15, 93, 94
Interdisciplinary methods, 90 Remote sensing data, 5
Internet-based media, 6 Reproduction matrix, 38, 39
Intervention measures, 3, 5, 6, 9, 16, 37, 38, See also Next generation matrix (NGM)
41, 44, 45, 47, 90 Reproduction number, 9, 12, 16, 22, 29, 37,
Intervention strategies 39–41, 47, 54, 55, 64, 74, 85, 86,
optimal prioritization, 37 88
prioritized interventions, 38–39, 45–47 See also Basic reproduction number (R0 )
112 Index
Risk factors socio-demographic data, 19, 23

behavioral-level factors, 5 temporal-demographic patterns, 2, 31, 32
environmental factors, 93 Socio-economic costs, 49, 71, 90
host population-related factors, 8 Spatial and temporal patterns, 22
national-level factors, 5 See also Spatio-temporal patterns
See also Impact factors Spatio-temporal distributions, 1, 92
Risk ranking, 94 Spatio-temporal patterns, 5
Statistical analysis
statistical methods, 2
S Surveillance, 5, 36, 47, 94
Search engines, 87 See also Disease surveillance
Severe acute respiratory syndrome (SARS), Susceptibility, 8, 20, 21, 29, 32, 38, 40, 70, 85
22, 24 Systematic studies, 7
Sexually transmitted diseases (STDs), 23, 24 Systemic characteristics, 5
See also HIV Systems epidemiology
Smallpox, 24, 55 behavioral complexity, 91
Social awareness behavioral entirety, 90
awareness fading, 11, 74, 76, 78, 82, 88 behavioral mechanisms, 93
effect of the spread of awareness, 11, 12, functional interrelationships, 92
81, 85, 87 infectious disease system, 93
spread of awareness, 11, 12, 71–76, 81, structural complexity, 89–91
85–87 structural interrelationships, 91
Social contacts system-level patterns, 91
social setting systems epidemiological studies, 13
community, 25, 26, 78–81, 87 systems exploration, 90, 91
household, 25, 26, 43 systems modeling
school, 25, 26 systems-modeling step, 90
workplace, 25, 26 systems modeling principles and practical
Social distancing, 2 steps
Social influences analytical inference
conformity, 10, 59, 61–70 goal-directed analytical inference, 93, 95
effect of social influences, 12, 50, 53, complex-system-based problem-solving
57–58, 62–67, 69 method, 9
imitation behaviors, 51 complex-system-based vaccine
impact of social influences, 8, 10, 50 allocation method, 9
opinion formation, 10 conceptual modeling
opinions, 7, 50, 59, 61, 64, 65, 67 problem-driven conceptual modeling, 93, 95
social impact theory (SIT) evidence-based practice, 93–95
extended SIT-based characterization, parameterization, 29, 53, 81, 93
59, 68 real-world grounding
Social media data-oriented real-world grounding, 15, 93–95
Facebook, 7, 72, 79, 81, 87 systems theory, 89
online community, 74, 78, 81, 87 systems thinking, 89–90
Twitter, 7, 72, 87
YouTube, 72
See also Internet-based media T
Social network Theoretical modeling
socially interconnected neighbors, 74, 76, theoretical methods, 3
78
social relationships, 10–12, 57
Socio-demographic characteristics U
census data, 8, 24, 27, 28, 47 Uncertainty, v, 11, 72–75, 77, 79, 85
Index 113
V individual-level vaccination decision-

Vaccination making, 11, 55, 74
adverse effects, 74–76, 79–83, 85 individual-level voluntary vaccination, 71
adverse events following immunization individuals’ decision-making
(AEFI), 79, 80 cost-benefit analysis
Advisory Committee on Immunization cost minimization, 10, 53, 56, 59, 62–65, 68
Policy (ACIP), 36 cost of infection, 51
age-specific vaccination, 9, 45–48 cost of vaccination
efficacy, 7, 49–51, 72 vaccination-associated costs, 64, 69
2009 Hong Kong HSI Vaccination costs and benefits of vaccination, 12, 50–51, 57
Programme, 42–45, 53 payoff-based approaches
Human Swine Influenza Vaccination payoff-based methods, 51
Programme (HSIVP), 45, 79, 80 rational decision-makers
National Vaccine Advisory Committee rationality, 51, 55
(NVAC), 36 steady state of individuals’ decision-
public health policies, vi, 12, 94 making, 54, 55, 58, 62–64,
risks and benefits of vaccination, 49, 50, 66–69
56, 67, 71 subjective perception
safety, 7, 11, 50, 71–76, 81, 85, 87 perceived infection risk, 52
vaccination coverage perceived risks and benefits of vaccination, 71
optimal level of vaccination, 55 vaccination decision-making, 9, 10, 12,
phase transition in vaccination 50–54, 56–63, 67–71, 73–79
coverage, 65 vaccination prioritization, 9, 36–38
vaccination dynamics, 59, 81, 88 vaccination willingness, 7, 10, 11, 54,
vaccination policies, 10, 69 63–67, 69, 70, 85
vaccination priorities, 9, 36–38 vaccine allocation
vaccination program, 7, 9, 10, 12, 35, 36, optimal allocation of vaccines, 35
42–45, 49–51, 53, 56, 67, 71, 72, vaccine-allocation strategy, 33
74, 79–80, 87 Vector species, 5, 90
vaccine administration
time-course of vaccine administration,
13
W
vaccine-based immunization, 35
WHO, 7, 35
vaccine scare, 7, 81
voluntary vaccination, 7–12, 49–52, 54, 56,
58–60, 67, 69–72, 74, 75, 82–88
Vaccination decision Z
age-specific vaccine allocation, 47, 48 Zoonotic and vector-borne diseases, 5

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Health Information Science

More information about this series at http://www.springer.com/series/11944

ISSN 2366-0988 ISSN 2366-0996 (electronic)

© Springer Nature Switzerland AG 2020

What Can We Learn from COVID-19?

To see a World in a Grain of Sand

William Blake (1757–1827) Auguries of Innocence

nation-states, political ideologies, religions, and special interests, and coexisting

Hong Kong Jiming Liu

Jiming Liu is extremely grateful for the rigorous foundation development in

Hong Kong Jiming Liu

3.3.2 Effects of Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ACIP Advisory Committee on Immunization Policy

S Population in susceptible compartment

ρ(K) Top eigenvalue of K

© Springer Nature Switzerland AG 2020 1

1.1 Methodological Paradigms

Various methodologies have been developed to address a wide range of challenges

Fig. 1.1 Major methodological paradigms in infectious disease epidemiology

transmission, by modeling and analyzing the patterns of transmission and quan-

1.2 Recent Developments

Another important source of data is Internet-based media, which can serve as

1.3 Infectious Diseases and Vaccination

1.4 Objectives and Tasks

In this book, we examine the dynamics of disease transmission in a host population,

investigate individuals’ acceptance of vaccination, we present decision models to

1.4.1 Modeling Infectious Disease Dynamics

As aforementioned, the dynamics of disease transmission depend on many disease-

1.4.2 Modeling Vaccine Allocation Strategies

most effectively reduce disease transmission; crucially, this requires knowledge

1.4.3 Modeling Vaccination Decision-Making

In a voluntary vaccination program, individuals’ decisions on whether or not to

vaccination decision-making would provide useful information for public health

1.4.4 Modeling Subjective Perception

A larger value of disease reproduction number will enhance the proportion of

In this book, we develop a computational modeling approach to evaluate and guide

awareness by carrying out simulation-based experiments to examine the time course

2.1 Modeling Infectious Disease Dynamics

Accurate predictions about the dynamics of disease transmission play an essential

© Springer Nature Switzerland AG 2020 15

analysis offers important tools for characterizing disease transmission in a host

2.1.1 Infectious Disease Models

In studies of infectious diseases, mathematical modeling and analysis of disease

Population level Individual level

mechanisms of disease transmission and evaluate the outcomes of potential control

susceptible-infectious-recovered (SIR) model has been widely used for describing

dynamics of the subpopulations are characterized, based on their distinct transmis-

2.1.2 Age-Specific Disease Transmissions

The basic compartmental model (susceptible-infectious-recovered model, SIR for

We now introduce an age-specific compartmental model, in which individuals

2.2 Modeling Contact Relationships

To accurately predict the dynamics of disease transmission, it is crucial to know indi-

describe individuals’ contact relationships during the course of disease transmission.

2.2.1 Empirical Methods

Empirical methods involve collecting information about individuals’ actual contact

2.2.2 Computational Methods

The challenge for characterizing disease dynamics is to obtain sufficient realistic

2.3 Case Study

description of individuals’ contact relationships is not available and, thus, we need

Fi (σi ) = (1 + σi ) · rc + (1 − σi ) · λ̂i (4.2)