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SIRTEM: Spatially Informed Rapid Testing for Epidemic Modeling and Response to COVID-19

Authors:

Gerardo Chowell-PuenteAuthors Info & Claims

ACM Transactions on Spatial Algorithms and Systems, Volume 8, Issue 4

Article No.: 29, Pages 1 - 43

https://doi.org/10.1145/3555310

Published: 02 November 2022 Publication History

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Abstract

COVID-19 outbreak was declared a pandemic by the World Health Organization on March 11, 2020. To minimize casualties and the impact on the economy, various mitigation measures have being employed with the purpose to slow the spread of the infection, such as complete lockdown, social distancing, and random testing. The key contribution of this article is twofold. First, we present a novel extended spatially informed epidemic model, SIRTEM, Spatially Informed Rapid Testing for Epidemic Modeling and Response to COVID-19, that integrates a multi-modal testing strategy considering test accuracies. Our second contribution is an optimization model to provide a cost-effective testing strategy when multiple test types are available. The developed optimization model incorporates realistic spatially based constraints, such as testing capacity and hospital bed limitation as well.

1 Introduction

On December 31, 2019, the first COVID-19 outbreak was reported in Wuhan, China. Since then, the infection has spread rapidly and a worldwide pandemic was declared by the World Health Organization (WHO) on March 11, 2020. As of April 19, 2021, about $\sim$140 million cases and $\sim$3 million deaths have been reported [87]. Although tragic, the loss of human life is not the only cost of this pandemic. Economists forecast that the world economy will plunge about 2.4%$\sim$3% in GDP (86.6 trillion dollars) [79]. To minimize casualties and the impact on the economy, various mitigation measures are being employed with the purpose to slow the spread of the infection, such as complete lockdown, social distancing, and random testing. Governments worldwide have chosen to employ these mitigation methods in different combinations and levels depending on their cultural background, political systems, and social consensus [4]. Screening of the infected individuals from the uninfected susceptible population through isolation and quarantine is often enabled by diagnostic testing. Empirical data has shown this to be an effective form of prevention [85].

Any effective action plan for stopping the spread of a virus requires quantitative understanding of the dynamical evolution of the disease, and of the impact of policy measures on such dynamics. In fact, it is key to develop effective models that enable the analysis of the rate of transmission of the disease, the spread at different spatial scales, and the assessment of the effect of travel restrictions, school closures, therapeutics to manage the disease. Such models require foundational epidemiological understanding and, importantly, a possibly large volume of diverse data.

In particular, testing can be seen as an important, and controllable, source of data at different fidelities. Diagnostic tests can be developed with different costs and accuracy (sensitivity and specificity). Testing is fundamental to identify outbreaks, and, if performed at reasonable scale, it can provide a snapshot of the evolution of the epidemic. Testing becomes even more crucial in presence of a disease that has important asymptomatic spread and is highly infectious. This was the case for Sars-CoV2. Summarizing, testing strategies should be designed with three, core, objectives:

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Objective 1: Obtain a faithful picture of the COVID-19 model as well as epidemic trajectory;

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Objective 2: Identify individuals and populations who are at risk of exposure or are already sick;

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Objective 3: Guide intervention efforts, such as isolation of infectious individuals, quarantine of the suspected contacts, and minimization of the contact rates without completely disrupting the society.

Nonetheless, testing is expensive, difficult, and creating a timely and accurate test against a new and growing epidemic is not always feasible. CDC, for example, reportedly shunned the WHO test guidelines for COVID-19 and set out to create a more advanced test. But the test failed in the field as it correctly identified COVID-19, but it falsely flagged other harmless viruses in the samples [62]. The testing program in Italy also created controversy: experts at the WHO and the Italian Health Ministry argued that Lombardy had created an inflated perception of the threat by including in case totals people who tested positive for the virus but who had not gotten sick. Other experts argue that tracking even mild cases of the virus is essential to containing its spread [93]. Research indicates that an individual infected with COVID-19 in the absence of symptoms could spread the virus to a susceptible population [93]. This potentially necessitates us to test individuals randomly to discriminate infected individuals without symptoms in addition to the testing for the individuals showing COVID-19 like symptoms.

“Surveillance testing” of hundreds of people in possible hotspots helped epidemiologists in several countries tracking the spread of the epidemics before large numbers of people turn up at hospitals [80].

Yet, the decision on the daily testing rate of random tests and the symptomatic test is an essential issue for policymakers to balance virus spreading and testing costs. When policymakers implement a particular testing strategy, they need to consider multi-faceted aspects and practical limitations in employing a particular testing policy. In fact, different data collection and testing modalities and strategies available to help generating models and predicting spread/severity of a disease, have varying costs, response times, and accuracies.

For instance, accuracy of available tests can have significant impact on the epidemic progress: When test sensitivity accuracy is low, we will not be able to separate the infected population efficiently from the healthy population, resulting in a spike in disease spread. However, if the specificity accuracy is low, then we falsely quarantine healthy individuals. Consequently, widespread testing of asymptomatic people has the potential of disrupting the economy and affect the care infrastructures in the presence of tests with high rates of false positives. However, the testing strategy should account for several practical constraints, such as the daily testing capacity, limitations for different testing types and the potential of the epidemic to disrupt the healthcare infrastructure if we fail in identifying dangerous outbreaks.

For this, we need to account for (1) the spatial distribution and mobility of the susceptible population, (2) the sensitivity and specificity of the tests, (3) how quickly results are obtained, (4) the maximum number of tests administered per day, and others. Given the above challenges, several critical questions arise:

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“What is the value of testing?”

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“Should we only test sick people for virus detection?”

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“How should we handle limited testing capacity?”

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“How much resources should be devoted to the development of highly accurate tests (low false positives, low false negatives)?”

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“ Should we only use one type of test aiming at the best cost/effectiveness trade off or should we rather adopt a non-homogeneous testing policy?”

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“How do we account for spatial distribution of the populations and their movements in developing testing strategies?”

Motivated by the aforementioned questions, in this article, we develop a spatially informed model, SIRTEM, which couples the spatio-temporal dynamics of the COVID-19 epidemic with multiple testing modalities to solve the optimal cost-effective multi-modal COVID-19 testing strategy problem while satisfying practical constraints.

1.1 Contributions

In this article, we extend the SEIR model into SIRTEM, Spatially Informed Rapid Testing for Epidemic Modeling and Response to COVID-19, an epidemic spread compartment model that takes into account isolation, quarantine, and hospitalization processes, and, most prominently, multiple tests are considered with varying accuracy and costs. In particular, we consider the test sensitivity (i.e., ability to identify positive cases—error of Type 1), and specificity (ability to identify correctly negative cases—error of Type 2).

The SIRTEM model builds on the SEIR model, but is significantly extended to reflect testing, quarantine, and hospitalizations.

The model is designed to apply different testing rates for the symptomatic and asymptomatic individuals; for instance, a higher testing rate for the symptomatic individuals could be a more effective testing strategy than applying the same testing rate to both individuals.

SIRTEM model allows the positive and negative confirmed cases to help predict the disease’s propagation. Based on the SIRTEM model, we develop a nonlinear black-box optimization approach to help identify the best possible testing strategy, taking account daily testing capacities and hospitals physical bed limitations, testing, hospitalization, and quarantine costs. We separate asymptomatic and symptomatic testing, and consider multiple testing options with different levels of cost and accuracy. The SIRTEM model is spatially informed, since it takes into account the spatial distribution of populations as well as their non-homogeneous mixing patterns. In particular, we rely on multiple data sources, including [13, 29, 34], to consider intra- as well as inter-mixing and explicitly accounts for work-related vs. within-home and other interactions and movements.

1.2 Organization of the Article

The remainder of the article is organized as follows. We review the relevant literature in section 2. Section 3 presents the overall approach with relevant models and optimizers. Sections 3.1.1 and 3.1.2 present the single city and multiple city epidemiological models, respectively, while Section 3.2 presents the optimization model. Section 4 presents the numerical analysis and Section 5 draws the conclusions.

2 Relevant Literature

Epidemic studies generally focus on (a) predicting the progress of an epidemiological phenomenon over time (Section 2.1); (b) analyzing the effects of mitigation policies over the spread of the disease; and (c) evaluating the effectiveness of diagnostic testing in reducing the spread of the disease in early stages or when no vaccines are available [83].

In this section, we will briefly review the related literature within these three areas with particular focus on COVID-19.

2.1 Epidemiological Modeling

After Bernoulli devised the first mathematical epidemiology model to estimate the mortality rate due to smallpox in 1766, a plethora of epidemiological models have been proposed with increasing presence of data driven techniques in place of or embedded with the more traditional differential equations typically used to describe the evolution of a disease across time and space. The review presented in Wei et al. [24] classifies epidemiological models into three categories: (i) Mathematical Model, (ii) Complex network model, (iii) Agent-based model.

Analytical Evolutionary Models. These models are implemented as systems of differential equations and can be deterministic or stochastic [24, 32]. The most commonly used deterministic models to date are the SEIR, SIR, and SIS approaches and their extensions [1, 2, 3, 10, 25, 40, 41, 50, 55, 60]. COVID-19 has also been studied through this lens. As an example, quarantine and hospitalization augmented SEIR model have been proposed to investigate the dynamics of the COVID-19 outbreak in Hubei (China) [38], while a fractional order SEIR model was proposed for analyzing the spread within the U.S. [89].

For the class of stochastic models, an common example is the Reed-Frost method [5, 11], which uses a binomial process to describe the transmission of the virus between two individuals. Such approach has been proposed to extend the SEIR model in Reference [53], focusing on the spread of Cholera. A similar approach was used in Reference [48] to study the Ebola outbreak in Congo (1995). More recently, several SEIR variants were proposed to investigate the progress of COVID-19 and the effectiveness of mitigation strategies [6, 7, 15, 27, 28, 37, 54, 59, 65, 91].

Network-based Models. This class of models is traditionally adopted to understand the effect of person-to-person interaction dynamics over the spread of the disease [17]. According to this approach, the evolution of the disease is encoded as a space-time graph, where the nodes represent individuals, and edges the interactions [24]. Due to the level of detail, network models typically are not used for large-scale studies due to the computational complexity.

Agent-based Models. Similar to the case of networked models, agent-based approaches focus on individual to individual interaction and behavior (e.g., mobility, choice). This allows us to analyze transmission of the virus in high resolution and embedding the effect of individual level actions on the epidemic development. Several epidemic models of this type have been investigated for Dengue, Syphilis, AIDS, and Ebola [14, 19, 30, 33, 36, 57, 73, 77]. In the scope of the COVID-19 pandemic, various studies have been conducted with agent-based modeling [21, 39, 42, 44, 70, 74, 78, 82, 88]. However, also in this case, only small case studies can be conducted due to the high computational demand.

2.2 Effects of Mitigation Policies

In the early phases of a new disease, mitigation strategies are the key to attempt controlling the increase of cases [75]. Strategies normally considered include individual recommendations, community, and state-level mandates, such as lockdown, social distancing, and mass testing. Different combinations of such policies have been implemented amid the COVID-19, thus motivating research in the understanding of the efficacy of such measures in reducing the spread of the disease [46, 67, 72]. In this regard, Chu et al. [18] investigated the effect that measures such as mask wearing, eyes protection, and social distancing had on the reduction in cases. As part of the analysis, the authors estimated the effect of wearing face masks, eye protection, and distancing over 1[m], was a reduction of 14.3%, 10.6%, and 10.4% in number of infections, respectively. Eikenberry et al. [26] incorporated the effect of wearing a face mask in the SEIR-based epidemic model, by modulating the time-varying infection rate within the SEIR model. Similarly, Ngonghala et al. [61] embedded the effect of the face mask in the SEIR-based model, not only using a time varying infection rate but also using control variable to represent the the level of mask wearing thus impacting the contact rate of the SEIR model. Milne et al. [58] devised an agent-based model to simulate person to person virus transmission, thus allowing the detailed analysis of small-scale effects of social distancing. However. as previously mentioned, such approach cannot be used in the context of large-scale analysis. Kim et al. [45] investigated the effect of school closure in South Korea using a SEIR model with population stratified by age. In this study, the authors use the confirmed cases as a proxy for the number of infected individuals in the model. However, confirmed cases are a result of testing of a minimal percentage of the population, while the actual count of the infected population is censored. Greenstone and Nigam [35] investigated the nationwide social distancing effect on the mortality rate. The authors estimate the value of social distancing to $8 trillion as the value the American population is willing to pay to decrease of COVID-19 casualties.

2.3 Effects of Diagnostic Testing

Individuating the infected individuals from the susceptible population is crucial to slow disease spread. Social distancing, as well as lockdown measures are among the most effective, and also expensive (economically as well as socially), ways to perform such separation. Testing can represent a less socially expensive alternative by allowing to target populations that require isolation [51, 63, 76].

In fact, the effect of testing during the pandemic has become the focus of several contributions. As an example, Toshikazu et al. [47] investigated the efficacy of mass testing and social distancing. The authors analysis reveals that 80% of diagnostic testing on the population combined with 30% reduction in contact rates (attainable by implementing social distancing) could halt the pandemic in Japan. The authors do not differentiate symptomatic from randomized testing, and the work focuses on understanding the ideal extent of testing efforts. Facundo et al. [68] studied the efficacy of quarantine and testing in terms of overall resulting economical cost. Unlike the previous contribution, the authors consider symptomatic and randomized tests for a single testing method. Also, they incorporate hospitalization and immunization into the simulation model. However, they do not consider the test accuracy. However, test accuracy rose as a key aspect in evaluating effectiveness of testing-based policies and analyses [22]. Incorrect diagnoses will increase economical costs (false positives) or incidence of the disease (false negatives). David et al. [64] proposed a cost-effective screening strategy for a 5,000 student college campus. They showed that testing every two days with sensitivity less than 70% could keep the infected population under control. However, extending such testing rates to a larger population could be not realistic primarily due to the lack of analysis capabilities for accurate test whose results are produced by labs with finite capacity. Similarly, Panovska et al. [66] provided the optimal testing strategy and analysis to reopen schools in the U.K. The authors proposed a stochastic model that simulates the disease transmission among individuals. In addition, Wells et al. [86] investigated the possibility of reducing the quarantine period using different testing strategies. The authors suggested that shortened quarantine period with exit testing could be an effective alternative for 14 days full quarantine.

In Table 1, we compare SIRTEM, with the two most related models, proposed by Berger et al. [68] and Piguillem et al. [9]. As we see in this table, which highlights the majors differences among these models, the three models make somewhat different assumptions about the quarantine, hospitalization, testing, and immunity processes. Mathematically speaking, however, the major difference lies in SIRTEM’s use of a mixture of rate-based and delayed differential equations, instead of a purely rate-based mathematical formulation. SIRTEM also considers the spatial context as described in Section 3.1.2. We experimentally compare SIRTEM against References [9, 68] in Section 4.5.

Table 1.

	Berger’s Model [68]	Piguillem’s Model [9]	SIRTEM
Temporal resolution	This model uses a rate-based transition from one state to the other.	This model also uses a rate-based transition from one state to the other.	SIRTEM uses delay differential equations, in addition to rate-based transitions, to move people one state to the other.
Hospitalization (Unknown Quarantine)	No hospitalization. They call it known quarantine. People come out of this state with a recovery rate.	No separate hospitalization state. Death rate depends on the hospital capacity. If the number of symptomatic individuals fills hospital capacity, then the additional people have a higher death rate.	SIRTEM has a separate hospitalization state. People with critical conditions go to this state and either recover or die from this state.
Imperfect Quarantine	This model has imperfect quarantine. Unknown cases are put into quarantine with a rate and unknown cases come out of quarantine with a rate (Temporary Asymptomatic, Permanent Asymptomatic, Symptomatic)	This model includes unknown/ indiscriminate quarantine (S, E, Ru). They create a separate quarantine state when they add testing.	We do not explicitly model imperfect quarantine; instead, we let the $\beta$ values vary over time to take into account changes in interaction patterns, including the lockdown periods.
Testing	This model uses virological and serological testing to identify known infected and known recovered cases, respectively.	First, antigenic testing is done. Those who are tested positive, take Molecular test.	SIRTEM has the two different kinds (virological and serological) testing.
Immunity	This model assumes that people from known recovered state never get infected again.	The model assumes that people from known or unknown recovered state never get infected again.	SIRTEM assumes that recovered people can get reinfected but that happens after 90 days after the first infection as stated on the published papers.
Exposed state	The exposed state is modeled as in the vanilla SEIR Model.	This model has a concept called critical mass If the exposed population goes above the critical mass, then the model takes on an additional mixing parameter.	SIRTEM does not introduce critical mass; instead, our approach relies on the $\beta$ values varying over time to take into account changes in interaction patterns
Objective Function	The objective function considers the productivity difference between quarantined and non-quarantined population.	The objective function represents the daily production/output.	In SIRTEM, the objective function takes into account the costs of quarantine, hospitalization, testing. The quarantine and hospitalization costs include reductions in economic production.

Table 1. SIRTEM vs. Berger’s [68] and Piguillem’s [9] Models

3 Proposed Approach

In this article, we take a three-step approach to testing policy development (Figure 1): (i) we develop a spatially informed and coupled SIRTEM model that includes multi-modal diagnostic testing policies combined with isolation, quarantine, hospitalization, and immunization; (ii) we propose a custom estimation algorithm to calibrate the simulation parameters of the model—this approach exploits available information on hospitalizations, deaths, and positive and negative cases daily; and (iii) we formulate and solve the problem of optimal testing considering the economical cost of quarantine, hospitalization, death, and testing, under constrained resources (e.g., hospital bed, testing).

Fig. 1.

3.1 SIRTEM Model

The equations that make up the SIRTEM model, presented in this section, extend the basic SEIR model to account for asymptomatic and symptomatic testing processes, quarantine, and hospitalization, recovery and immunization, and immunity loss processes (Figure 2). We first introduce the model for single city (Section 3.1.1) and then extend the model to multiple cities (Section 3.1.2).

Fig. 2.

3.1.1 Coupled Epidemic-Testing Single City Model.

We first expand the standard four-state SEIR model to reflect reality better by considering different population groups (or compartments).

Population Groups/Compartments. The model distinguishes individuals into five population groups: susceptible, infected symptomatic, infected asymptomatic, and symptomatic but not COVID-19 infected (flu or general sickness), and falsely presumed susceptible:

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We define susceptible as the population that is non-infected and displaying no symptoms associated with COVID-19.

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Infected symptomatic and asymptomatic are the sections of the population infected with COVID-19 who are showing symptoms and not exhibiting symptoms, respectively.

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Symptomatic but not infected are the people who are displaying symptoms (flu, general sickness) similar to COVID-19, but are not infected.

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Falsely presumed susceptible population consists of people who have immunity from natural recovery, but erroneously test negative to COVID-19 antibodies.

As with the standard SEIR model, an individual enters and leaves these compartments according to the relevant transition rates and other parameters (Table 2). Different from the traditional set of dynamical equations, these transition rates are governed by four different processes: (a) asymptomatic testing, (b) symptomatic testing, (c) immunization, and (d) falsely presumed susceptible process (Figure 2).

Table 2.

Parameters	Description
$tp_{i}$	Sensitivity of diagnostic test i
$tn_{i}$	Specificity of diagnostic test i
$\tau _{i}$	Time to obtain the result for diag. test i (days)
$\phi _{i}$	Testing rate for diagnostic test i the symptomatic population (ratio per day)
$\phi _{ai}$	Testing rate for diagnostic test i the non-symptomatic population (ratio per day)
$tp_{se}$	Sensitivity of the serology test
$tn_{se}$	Specificity of the serology test
$\tau _{se}$	Time to obtain the result for the serology test (days)
$\phi _{se}$	Testing rate for the serology test (ratio of the relevant population per day)
$\beta$	Infection rate for the susceptible population (ratio)
	= transmission rate $\times$ contact rate for the susceptible population
$\beta ^{\prime }$	Infection rate for the population of individuals who are falsely presumed immune (ratio)
	= transmission rate $\times$ contact rate for the presumed immune population
r	The ratio of the transmission rate of asymptomatic individuals to the transmission rate of symptomatic individuals
$per_{a}$	Percentage of individuals with COVID-19 who are asymptomatic
$per_{s}$	Percentage of individuals with COVID-19 who are symptomatic
$\eta$	Incubation length (days)
$\lambda _{a}$	Length of recovery for asymptomatic individuals (days)
$\lambda _{s}$	Length of recovery for symptomatic individuals (days)
$\lambda _{q}$	Length of quarantine (days)
$\lambda _{h}$	Hospitalization length (days)
h	Hospitalization rate (ratio of the quarantined population, per day)
$\kappa$	Mortality rate for symptomatic population (per day)
$\kappa _{h}$	Mortality rate for the hospitalized individuals (per day)
g	Ratio of the susceptible individuals who has fever and cough for non-COVID infections (ratio, per day)

Table 2. SIRTEM Model Parameters

Exposure Dynamics. The exposure process is governed by the following equations:

\begin{equation} \frac{dE}{dt} = \Big (\beta \cdot S(t) + \beta ^{^{\prime }}\cdot FPI(t) \Big) \cdot \frac{Infected(t)}{N} - \Big (per_{a} +per_{s}\Big) \cdot E(t), \end{equation}

(1)

where the total number of infected individuals at time t is

\[\begin{eqnarray} Infected(t) = r\cdot \Big (PS(t) + PA(t) + IA(t) + ATN(t)\Big) +IS(t) + STN(t). \end{eqnarray}\]

(2)

Note that here the term $Infected(t)$ is the sum of all compartments that account for all infections and, thus, is the aggregate of all known and unknown infected individuals at time t. Above, the infection rate parameter, $\beta$, represents the number of new daily infection produced by a single infectious individual and it is equivalent to the product of contact rate and disease transmission rate. Equivalently, the total number of new infected individuals will be $\beta \cdot Infected(t) \cdot \frac{S(t)}{N}$, and this population will move from susceptible to exposed, where N denotes the total population. The parameter, $\beta ^{^{\prime }}$, represents a higher infection rate parameter, which we apply to the falsely presumed immune (FPI) individuals (i.e., individuals that falsely test positive to antibodies). More specifically, FPI is a recovered individuals that falsely tests positive for the serology, his/her risk is higher, since the individual will likely behave in the assumption of larger than actual protection. Finally, Equation (2) shows that asymptomatic individuals (IA(t) and ATN(t)) and individuals before symptom onset (PA(t), PS(t)) infect at r times the rate compared to symptomatic individuals.

Asymptomatic Testing Process. The asymptomatic testing process emulates random testing to discriminate infected asymptomatic from the susceptible population. The asymptomatic population is divided into two groups, infected asymptomatic and not infected (susceptible). In practice, we cannot differentiate among these two groups of individuals without testing. Table 3 lists the sub-compartments relevant for the following equations that describe the asymptomatic testing process:

Table 3.

Sub-Compartment		Sub-Compartment
S	Susceptible population	E	Pop. exposed to the virus
PA	Pre-asymptomatic population	IA	Infected pop. who are asymptomatic
$AT_{i}$	Asymp. pop. receiving diagnostic test i	ATN	Asymptomatic pop. with negative test result
QAP	Asymp. pop. quarantined after a test	UR	Pop. with unknown immunity due to unknown infection
KR	Pop. of known recovered individuals	$NT_{i}$	Susceptible pop. receiving diagnostic test i
NTN	Pop. of non-infected indiv. who test negative	NTQ	Non-infected pop. quarantined due to testing error
FPI	Pop. of indiv. falsely presumed immune	PS	Infected pop. who are pre-symptomatic
IS	Infected pop. who are symptomatic	STN	Symptomatic pop. who test negative (by error)

Table 3. Sub-compartments Relevant for the Asymptomatic Testing Process

An individual exposed to the virus will proceed as an infected asymptomatic individual with $per_{a}$ rate; the virus will take $\eta$ days for incubation (during which the individual will be considered pre-asymptomatic) and after that period the individual will be considered infected asymptomatic (assuming that s/he does not show symptoms):

\begin{equation} \frac{dPA}{dt} = per_{a}\cdot E(t) - PA(t-\eta). \end{equation}

(3)

Infected asymptomatic population grows by pre-asymptomatic individuals becoming asymptomatic and shrinks by infected asymptomatic individuals recovering from the disease. Asymptomatic individuals testing negative also contribute to the infected asymptomatic population. If a government implements a random test for the asymptomatic population, then $\phi _{ai}$ rate of IA(t) individuals get tested with test type i:

\begin{equation} \frac{dIA}{dt} = PA(t-\eta) + ATN(t) - \sum _{i=1}^2 \phi _{ai}\cdot IA(t) -\lambda \cdot IA(t). \end{equation}

(4)

Above, we assume that the $\lambda$ is the rate of IA(t) individuals recover naturally; in other words, this parameter governs the rate with which individuals, symptomatic or asymptomatic, obtain immunity to the disease through infection:

\begin{equation} \frac{dUR}{dt} = \lambda \cdot IA(t) + \lambda \cdot IS(t) - UR(t). \end{equation}

(5)

Individuals who are asymptomatic may receive random testing of different types, with different degrees of accuracies and may take different amount of time to produce a result.

The following equation models the impact of the response time, $\tau _{i}$, of test type i:

\begin{equation} \frac{dAT_{i}}{dt} = \phi _{ai}\cdot IA(t) - AT(t-\tau _{i}). \end{equation}

(6)

Since we cannot distinguish an infected asymptomatic and a susceptible one, random tests are being employed not only for the infected asymptomatic but also for the susceptible ones:

\begin{equation} \frac{dNT_{i}}{dt} = \phi _{ai}\cdot S(t) - NT(t-\tau _{i}). \end{equation}

(7)

The specificity, $tn_{i}$, of the test impacts the true negativity rate for the test:

\begin{equation} \frac{dNTN}{dt} = \sum _{i=1}^2 tn_{i}\cdot NT_{i}(t-\tau _{i}) - NTN(t). \end{equation}

(8)

However, a portion of the asymptomatic individuals may also test negative:

\begin{equation} \frac{dATN}{dt} = \sum _{i=1}^2 (1-tp_{i})\cdot AT(t-\tau _{i})- ATN(t). \end{equation}

(9)

Once testing positive, the asymptomatic individual is quarantined for $\lambda _{q}$ days:

\begin{equation} \frac{dQAP}{dt} = \sum _{i=1}^2 tp_{i}\cdot AT(t-\tau _{i}) - QAP(t-\lambda _{q}). \end{equation}

(10)

If a test result is falsely positive, then a non-infected susceptible individual will be, falsely, quarantined for $\lambda _{q}$ days:

\begin{equation} \frac{dNTQ}{dt} =\sum _{i=1}^{2} (1-tn_{i})\cdot NT_{i}(t-\tau _{i}) - NTQ(t-\lambda _{q}). \end{equation}

(11)

At the end of the quarantine period, the individual is assumed to recover:

\begin{equation} \frac{dKR}{dt} = QAP(t-\lambda _{q}) + FSQ(t-\lambda _{q}) + QSP(t-\lambda _{q}) + \sum _{i=1}^2 tn_{1}\cdot HT_{i}(t-\tau _{i}) - KR({t}). \end{equation}

(12)

Note that the above equation accounts for populations other than asymptomatic individuals who leave the quarantine. In particular, FSQ denotes those individuals who are falsely presumed susceptible and thus wrongly quarantined (Table 6), QSP denotes symptomatic individuals who are quarantined with positive test results (Table 4), and HT$_i$ denotes portion of the population that received test i while hospitalized (Table 4).

Table 4.

Sub-Compartment		Sub-Compartment
E	Pop. exposed to the virus	PS	Pre-Symptomatic population
IS	Infected pop. who are symptomatic	$ST_{i}$	Symptomatic pop. receiving test i
STN	Symptomatic pop. with negative test result	QSP	Symp. pop. quarantined after a test
UR	Pop. with unknown immunity due to unknown infection	KR	Pop. of known recovered individuals
HBQ	Pop. need hospitalization before a quarantine	HDQ	Pop. need hospitalization during quarantine
HBT	Portion of HBQ tested while hospitalized	HDT	Portion of HDQ tested while hospitalized
FS	Pop. showing flu symptoms	$FT_{i}$	Pop. with flu symptom receiving test i
FTN	Pop. with flu symptom tested negative for COVID-19	FTQ	Pop. with flu symptom quarantined due to false positive
GS	Pop. with other COVID-like symptoms	$GT_{i}$	Pop. with other COVID-like symptoms reciving test i
GTN	Pop. with other symptoms tested negative for COVID-19	GTQ	Pop. with other symptoms quarantined due to false positive
D	Pop. who have not recovered from the infection (dead)

Table 4. Sub-Compartments Relevant for the Symptomatic Testing Process

Symptomatic Testing Process. The symptomatic testing process is designed to model the testing process for individuals who show COVID-19 like symptoms. We categorize symptomatic individuals into three populations: (a) COVID-infected symptomatic, (b) general sickness (fever, coughing), and (c) flu symptomatic. We assume that these populations can be distinguished through diagnostic testing.

Much like the asymptomatic process discussed above, the process consists of testing, isolation, and quarantine sub-processes. Unlike the asymptomatic process, however, the symptomatic process also includes hospitalization and death for severe cases. The compartments presented in Table 4 along with the following differential equations define the transitions between relevant states in the symptomatic process.

An individual exposed to the virus will proceed as an infected symptomatic individual with $per_s$ rate; the virus will take $\eta$ days for incubation (during which the patient is pre-symptomatic) and after that period the individual will be considered infected symptomatic (assuming that s/he does show symptoms):

\begin{equation} \frac{dPS}{dt} = per_{s}\cdot E(t) - PS(t-\eta). \end{equation}

(13)

Analogously to the asymptomatic population (Equation (4)), when the diagnostic tests are implemented for the symptomatic population, $\phi _{si}$ rate of infected symptomatic (IS(t)) individual get tested with test types i. As before, assume that $\lambda$ rate of IS(t) individuals recover naturally; in this case, however, we also take into account $\kappa$ and h rates of IS(t) individuals are dead and hospitalized with the severe cases, respectively:

\begin{equation} \frac{dIS}{dt} = PS(t-\eta) + STN(t) - \sum _{i=1}^2 \phi _{si}\cdot IS(t) -\lambda \cdot IS(t) -\kappa \cdot IS(t) -h\cdot IS(t). \end{equation}

(14)

As before, $\tau _{i}$, represents response time of test type i:

\begin{equation} \frac{dST_{i}}{dt} = \phi _{si}\cdot IS(t) - ST_{i}(t-\tau _{i}). \end{equation}

(15)

If testing positive, then the symptomatic individual is quarantined for $\lambda _{q}$ days until recovery; i.e., similarly to corresponding equation (Equation (10)) under asymptomatic testing process, we have

\begin{equation} \frac{dQSP}{dt} = \sum _{i=1}^2 tp_{i}\cdot ST_{i}(t-\tau _{i}) - QSP(t-\lambda _{q}). \end{equation}

(16)

Recovery process is governed by Equation (12) listed earlier, replicated below for quick reference:

\begin{equation*} \frac{dKR}{dt} = QAP(t-\lambda _{q}) + FSQ(t-\lambda _{q}) + QSP(t-\lambda _{q}) + \sum _{i=1}^2 tn_{i}\cdot HT_{i}(t-\tau _{i}). \end{equation*}

If a symptomatic individual falsely tests negative, then the individual will continue contributing to the spreading of the virus:

\begin{equation} \frac{dSTN}{dt} = \sum _{i=1}^2 (1-tp_{i})\cdot ST_{i}(t-\tau _{i}) - STN(t). \end{equation}

(17)

Equation (5), considered earlier for asymptomatic individuals and listed below for quick reference, also captures the rate, $\lambda$, with which symptomatic individuals recover from the disease and are (naturally) immunized:

\begin{equation*} \frac{dUR}{dt} = \lambda \cdot IA(t) + \lambda \cdot IS(t) - UR(t). \end{equation*}

Note that Equation (19) is the same as Equations (5) for the asymptomatic testing process, replicated here for completeness.

If their situation worsens, with rate h, then symptomatic individuals, before or during quarantine, may be admitted to a hospital:

\begin{equation} \frac{dHBQ}{dt} = h\cdot IS(t-\lambda _{h}) + (1-tn_{1}) \cdot HBT(t-\tau _{1}) - \kappa _{h}\cdot HBQ(t), \end{equation}

(18)

\begin{equation} \frac{dHDQ}{dt} = h\cdot QSP(t-\lambda _{h}) + (1-tn_{1}) \cdot HDT(t-\tau _{1}) - \kappa _{h}\cdot HDQ(t). \end{equation}

(19)

Individuals leave the hospital either through a negative test result (with more accurate test 1) or through death. After spending $\lambda _{h}$ days in the hospital, the hospitalized individuals will take a test to check if the virus is still active and they will keep being hospitalized when the result is positive:

\begin{equation} \frac{dHBT}{dt} = HBQ(t-\lambda _{h}) - HBT(t-\tau _{1}), \end{equation}

(20)

\begin{equation} \frac{dHDT}{dt} = HDQ(t-\lambda _{h}) - HDT(t-\tau _{1}). \end{equation}

(21)

Otherwise, they move to the recovered compartment as described in Equation (12).

The death rate for hospitalized individuals is $\kappa _{h}$,while the death rate for symptomatic individuals who are not hospitalized is, $\kappa$:

\begin{equation} \frac{dD}{dt} = \kappa _h \cdot \big (HBQ(t) + HDQ(t)\big) + \kappa \cdot QSP(t) + \kappa \cdot IS(t). \end{equation}

(22)

During the epidemic, portion of the non-infected population may also show COVID-19 like symptoms due to other illnesses and they may need to be tested. The following two equations leverage the flu and $g_{sick}$ rate parameters to describe the portion of the population that show flu-like and general sickness symptoms, such as fever and coughing, indistinguishable from COVID-19:

\begin{equation} \frac{dFS}{dt} = flu\cdot S(t) - FS(t), \end{equation}

(23)

\begin{equation} \frac{dGS}{dt} = g_{sick}\cdot S(t) - GS(t). \end{equation}

(24)

Since they show COVID-19 like symptoms, these individuals may be subject to symptomatic testing, analogous to Equation (15):

\begin{equation} \frac{dFT_{i}}{dt} = \phi _{si}\cdot FS(t) - FT_{i}(t-\tau _{i}), \end{equation}

(25)

\begin{equation} \frac{dGT_{i}}{dt} = \phi _{si}\cdot GS(t) - GT_{i}(t-\tau _{i}). \end{equation}

(26)

Even though these individuals are not COVID-19 infected, in the case of a false positive, they will be quarantined for $\lambda _{q}$ days:

\begin{equation} \frac{dFTQ}{dt} = \sum _{i=1}^2 (1-tn_{i})\cdot NT_{i}(t-\tau _{i}) - FTQ(t-\lambda _{q}), \end{equation}

(27)

\begin{equation} \frac{dGTQ}{dt} = \sum _{i=1}^2 (1-tn_{i})\cdot GT_{i}(t-\tau _{i}) - GTQ(t-\lambda _{q}). \end{equation}

(28)

those non-infected patients whose test are true negative return to the susceptible population:

\begin{equation} \frac{dGTN}{dt} =\sum _{i=1}^2 tn_{i}\cdot GT_{i}(t-\tau _{i}) - GTN(t), \end{equation}

(29)

\begin{equation} \frac{dFTN}{dt} = \sum _{i=1}^2 tn_{i}\cdot FT_{i}(t-\tau _{i}) - FTN(t). \end{equation}

(30)

Immunity Process. The role of the immunity process is to model how individuals who recover from COVID-19 obtain immunity against reinfection for a certain period of time. This model is governed with sub-compartments presented in Table 5 along with the following equations:

Table 5.

Sub-Compartment		Sub-Compartment
KR	Pop. of known recovered individuals	IM	Pop. of individuals with immunity
STI	Pop. of immune indiv. receiving serology (anti-body) test	SII	Pop. of not-immune indiv. receiving serology test
FPI	Pop. of individuals who are falsely presumed being immune

Table 5. Sub-compartments Relevant for the Process Through Which Individuals Obtain Immunity

Any person who recovers from the disease (KR(t)), will have immunity for $\gamma$ days. The recovered individuals are divided into two groups: (a) individuals having immunity and (b) those who do not get sufficient immunity. We assume that the government will administer serology tests at a predetermined rate, $\phi _{se}$, for both groups, IM(t) and FPI(t), since we cannot discriminate an individual having immunity from those who lost immunity without a serology test. The serology test takes $\tau _{se}$ days to provide the result:

\begin{equation} \frac{dIM}{dt} = KR(t) - IM(t - \gamma) - \phi _{se}\cdot IM(t) + tp_{se} \cdot STI(t- \tau _{se}). \end{equation}

(31)

Some of the individuals (STI) who receive serology tests are immune, while others may be FPI:

\begin{equation} \frac{dSTI}{dt} = \phi _{se}\cdot IM(t) - STI(t- \tau _{se}), \end{equation}

(32)

\begin{equation} \frac{dSII}{dt} = \phi _{se}\cdot FPI(t) - FPI(t- \tau _{se}). \end{equation}

(33)

Note that a group of individuals may be Falsely Presumed Immune (FPI(t)) for various reasons: (a) the individual may lose immunity (IM(t-$\gamma$)), (b, c) some individuals may be quarantined after false-positive result with flu symptoms (FSQ(t-$\lambda _{q}$), after a general sickness (GSQ(t-$\lambda _{q}$)), or (d) after a testing error ((NTQ(t-$\lambda _{q}$)):

\begin{equation} \frac{dFPI}{dt} = IM(t -\gamma) + GSQ(t-\lambda _{q}) + FSQ(t-\lambda _{q}) + NTQ(t-\lambda _{q})-\beta ^{^{\prime }}\cdot FPI(t). \end{equation}

(34)

For these individuals, we apply higher infection rate ($\beta ^{\prime } \gt \beta$) in Equation (1), because these individuals are likely to socialize more than pure susceptible ones.

Finally, like the diagnostic tests, the serology tests are also imperfect. We use accuracy of $tp_{se}$ and $tn_{se}$ for sensitivity and specificity, respectively.

Falsely Presumed Susceptible Process. The falsely presumed susceptible process is designed to consider those individuals recovered naturally from the COVID-19. These individuals have immunity against the disease, but they think themselves susceptible. To model this group of individuals, we consider the sub-compartments listed in Table 6, along with the following equations:

Table 6.

Sub-Compartment		Sub-Compartment
UR	Pop. with unknown immunity due to unknown infection	FPS	Pop. falsely presumed susceptible
$FST_i$	Pop. falsely presumed susceptible receiving diag. test i	FSTP	Pop. falsely presumed susceptible who test negative
FSQ	Pop. falsely presumed susceptible (wrongly) quarantined

Table 6. Sub-compartments Relevant for Individuals Who Are Falsely Presumed Susceptible

As formalized in Equation (5), unknown recovered individuals, who had COVID-19 but recovered naturally, have immunity against COVID-19. These individuals move to the falsely presumed susceptible (FPS), since they regard themselves susceptible. Other individuals who are falsely presumed susceptible include individuals who are already falsely presumed negative and test negative for a diagnostic test and others who are immune, but receive a false negative for a serology test:

\begin{equation} \frac{dFPS}{dt} = UR(t) + FSTN(t) +(1-tp_{se}) \cdot STI(t- \tau _{se}) -FPS(t-\gamma)-\sum _{i=1}^2 \phi _{ai}\cdot FPS(t). \end{equation}

(35)

Note that, as captured in the above equation, these individuals will lose immunity after $\gamma$ days and move to the pure susceptible compartment. The above equation also captures that, while they are falsely presumed susceptible, they may be subject to regular random testing strategies, with rate, $\phi _{ai}$. These tested individuals move to the corresponding FST compartment:

\begin{equation} \frac{dFST_{i}}{dt}=\phi _{ai}\cdot FPS(t)-FST_{i}(t-\tau _{i}). \end{equation}

(36)

Some of these falsely presumed susceptible individuals will test negative:

\begin{equation} \frac{FSTN}{dt}=\sum _{i=1}^2 tn_{i}\cdot FST_i(t-\tau _{i}) -FSTN(t). \end{equation}

(37)

Others, however, may test positive and get quarantined for $\lambda _{q}$ days, due to testing errors:

\begin{equation} \frac{dFSQ}{dt} = \sum _{i=1}^2 (1-tn_{i})\cdot FST_{i}(t-\tau _{i}) -FSQ(t-\lambda _{q}). \end{equation}

(38)

3.1.2 Spatially Informed SIRTEM Model.

We note that the coupled epidemic/testing model described so far is not spatially informed. In particular, it assumes one group of individuals, whose interactions are defined through a single mixing rate. Moreover, the model further ignores the possibility of different geographic locations applying different testing policies. To account for these, we therefore extend the model with spatially indexed compartments along with spatially informed parameters and mixing rates. Let ${\mathcal {S}} = \lbrace s_1, \ldots , s_M \rbrace$ be a set of spatial locations.

•

Spatially indexed model compartments: Tables 3–6 list the compartments of the base model. In the spatially informed model, each of these components are spatially indexed: for example, each spatial location $s_i$ has a corresponding susceptible population, $S_{(i)}$, and an exposed population $E_{(i)}$.

•

Spatially informed mixing rates: The infection rate parameter, $\beta$, in Table 2 includes two components: transmission rate, which (ignoring the local mutations of the disease) is a spatially insensitive parameter, and the contact/mixing rate, $\mu \in [0,1]$, which depends on the behaviors of local populations, and therefore is spatial-sensitive:

\begin{equation*} \forall _{s_{i} \in {\mathcal {S}}}\;\;\beta _{(i)} = transmission\_rate \times \mu _{(i)}. \end{equation*}

Here $\mu$ is the likelihood of each individual in the population to interact with other individuals in the same population.

The above equation, however, would fail to take into account potential mixing among spatially distributed population groups due to mobility patterns (such as daily commute between spatial regions). We, therefore, extend the above model as follows:

\begin{equation*} \forall _{s_{i}, s_{j} \in {\mathcal {S}}}\;\; \beta _{(i,j)} = transmission\_rate \times \mu _{(i,j)}, \end{equation*}

where $\mu _{(i,j)} \in [0,1]$ denotes the likelihood of an individual in population at spatial location $s_i$ to interact with an individuals in the spatial location $s_j$. Consequently, $\beta _{(i,j)}$ models the transmission of the disease from one population to another due to the underlying mobility patterns. We, therefore, revise the part of the model that governs the exposure process as follows:

\begin{equation*} \frac{dE_{(i)}}{dt} = \left(\sum _{s_j \in {\mathcal {S}}} \Big (\beta _{(i,j)}\cdot S_{(i)}(t) +\beta _{(i,j)}^{^{\prime }}\cdot FPI_{(i)}(t) \Big) \cdot Infected_{(j)}(t) \right) -(per_{a} + per_{s})\cdot E_{(i)}(t), \end{equation*}

where

\begin{equation*} Infected_{(i)}(t) = r \cdot \left(PS_{(i)}(t) + IA_{(i)}(t)+PA(t) + ATN_{(i)}(t)\right) +IS_{(i)}(t) + STN_{(i)}(t). \end{equation*}

Above, as described earlier, the parameter $\beta ^{^{\prime }}$ represents the larger infection coefficient that applies to the falsely presumed immune individuals who potentially have larger mixing rates compared to the susceptible population. The value of r, in contrast, takes into account the fact that the $transmission\_rate$ parameter is lower for asymptomatic and pre-symptomatic individuals than the fully symptomatic individuals.

In this article, we consider two alternative sources of data to compute mixing rates: the worker mobility data available at Reference [13], and the monthly mobility dataset from SafeGraph [17, 34]. In the following, we discuss the calculation of the mixing rates with these sources.

Table 7.

	Start	End	Work	Home	Other
Wave 0	22-Mar-20	9-Apr-20	0.25	2	0
Wave 1	10-Apr-20	4-May-20	0.5	2	1
–	5-May-20	16-Jun-20	0.625	2	1
Wave 2	17-Jun-20	23-Jun-20	0.75	2	1
–	24-Jun-20	10-Sep-20	1	2	1.5
Wave 3	11-Sep-20	26-Sep-20	1.25	2	2

Table 7. Average Daily Interactions Per Day Per Person in Different Contexts During Different Waves of the Epidemic [29]

Obtaining the Mixing Rates using Worker Mobility Data. The data source available at Reference [29] provides the interaction rate of people at home, work, and other activities during the COVID-19 pandemic. More specifically, this data source provides data about the number of per person daily interactions over time during different waves of the epidemic Table 7. While Reference [29] does not include any spatial differentiation, we combine this data source with the work commute data available at Reference [13] to partition the mixing rate into to three spatially differentiated components:

\begin{equation*} \mu {(i,j)}= h_{(i,j)}+w_{(i,j)}+o_{(i,j)}, \end{equation*}

where $h_{(i,j)}$ is the rate with which infected individuals in city i interact with susceptible people in city j in the home context, $w_{(i,j)}$ is the rate with which infected individuals in city i interact with susceptible people in city j in the work context, and $o_{(i,j)}$ is the rate with which infected individuals in city j interact with susceptible people in city i in all other contexts. In the following, we define the three interaction contexts:

•

At-home mixing: To compute the rate of interactions in the home context, we make the simplifying assumption that, at home, individuals only interact with those individuals living in the same city; i.e.,

\begin{equation*} h_{(i,j)}= Ø\quad if\quad i\ne j. \end{equation*}

Given this assumption, we can obtain the at-home mixing rate, $h_{(i,i)}$, at city i as

\begin{equation*} h_{(i,i)}= \frac{num\_home\_interaction}{pop_i}, \end{equation*}

where $num\_home\_interaction$ is the number of per day, per person at-home interactions reported in Table 7 and $pop_i$ is the population of the ith city.

•

At-work mixing: To compute the rate of interactions in the work context, we rely on the commute rate data available from Reference [13]. More specifically,

\begin{equation*} w_{(i,j)} = \left(\frac{num\_work\_interactions}{pop_j} \times \frac{W_{j\rightarrow i}}{W_{j}}\right)+ \left(\frac{num\_work\_interactions}{pop_i} \times \frac{W_{i\rightarrow j}}{\sum _{h}W_{h\rightarrow j}} \right). \end{equation*}

Above, the first term indicates the interaction rate due to individuals traveling from city j to city i for work-related reasons: more specifically, $W_{j}$ is the number of workers living in city j and $W_{j\rightarrow i}$ is the number of those that commute to city i for work.

The second term, however, is the interaction rate due to individuals traveling from city i to city j for work: $W_{i\rightarrow j}$ is the number of people that commute from city i to city j for work, while $\sum _{h}W_{h\rightarrow j}$ is the number of individuals working in city j.

•

Other mixing: To obtain non-home, non-work mixing rates, we assume that the rate interaction among individuals is inversely proportional with the square of their distance; i.e.,

\begin{equation*} o_{(i,j)}= \frac{num\_other\_interactions}{pop_i} \times \frac{Z_{(i,j)}}{\sum _h Z_{(h,j)}}, \end{equation*}

where $num\_other\_interaction$ is the number of non-home, non-work-related interactions reported in Table 7 and $Z_{(i,j)}$ is the population of city i normalized with the square of the distance between cities i and j:

\begin{equation*} Z_{(i,j)}= \frac{pop_i}{\delta _{(i,j)}^2}. \end{equation*}

We obtained the distance $\delta _{(i,j)}$ between cities i and j from Reference [56] and approximated $\delta _{(i,i)}$ as $min_h\left(\delta _{(i,h)}\right)/2$.

We note that the probability of getting infected upon each contact may vary with scenes – for example, the risk of getting infected is larger in closed environments. This is apparent in winter surges of the COVID-19 epidemic. Our model does not explicitly account for the in-door and out-door interactions; however, we treat the parameter $\beta$ as a learnable parameter, whose values changing across time can indicate differences in the way individuals interact at a particular location.

When working with this data set, as explained in detail in Section 4, we considered 11 cities of the Maricopa County of Arizona to compute inter- and intra-city mixing rates.

Obtaining the Mixing Rates Using SafeGraph Mobility Data. As an alternative to the above mixing rate computations, we also computed a second set of mixing rates using the monthly mobility dataset from SafeGraph [34]. SafeGraph aggregates anonymized mobility data using various mobile applications. This dataset captures the mobility data of the general population between Census Block Groups (CBGs) and Points of Interest (POIs). CBGs are geographical regions with a population size generally between 600 and 3,000. POIs can be restaurants, grocery or any kind of stores, and religious establishments. We used the same 11 cities of the Maricopa County of Arizona for inter- and intra-city mixing rates. Since SafeGraph data provides raw data based on 5%–10% of the actual data, we have normalized the data using a location specific multiplier. The mixing rate was partitioned into three spatially differentiated components as follows:

•

At-home mixing: At-home mixing, $h_{(i,j})$ was computed using the same calculations as the previous mixing rate computation. Since the SafeGraph dataset includes the general mobility data, we have used the ratio of worker to non-workers using Table 7.

•

At-work mixing: At-work mixing, $w_{(i,j)}$ was computed using the same equation as the previous mixing rate calculation.

•

Other mixing:

\begin{equation*} o_{(i,j)} = \left(\frac{num\_other\_interactions}{pop_j} \times \frac{O_{j\rightarrow i}}{O_{j}}\right)+ \left(\frac{num\_other\_interactions}{pop_i} \times \frac{O_{i\rightarrow j}}{\sum _{h}O_{h\rightarrow j}} \right). \end{equation*}

Here, the first term indicates the interaction rate due to individuals traveling from city j to city i for reasons other than work: more specifically, $O_{j}$ is the number of non-workers living in city j and $O_{j\rightarrow i}$ is the number of those that commute to city i for non-work-related reasons. The second term, however, is the interaction rate due to individuals traveling from city i to city j for reasons other than work: $O_{i\rightarrow j}$ is the number of people that commute from city i to city j for reasons other than work, while $\sum _{h}O_{h\rightarrow j}$ is the number of non-workers in city j.

We note, in general, that accurate mobility data is hard to obtain in practice [90] proposes how mobile phone data can be used to characterize mobility pattern in developing countries; but, in any technically based solution to the problem, quality of data across regions are likely to vary as the reviewer suggested. To accommodate this, the model and the optimization framework can be extended to allow noise and values ranges as in our prior works [23, 43, 49, 52, 92]; but this is outside of the scope of this work.

3.2 Optimal Testing

We use the epidemic/testing model, SIRTEM, to develop an optimization problem for identifying optimal testing strategies: The objective of the problem P is to minimize the total economic cost, consisting of testing ($f_{1}$), hospitalization ($f_{2}$), and quarantine ($f_{3}$) costs, of the COVID-19, subject to resource (testing and hospital capacity) limitations. At each time frame t (indicated in the equations as a subscript), we apply rolling horizon optimization formulation(one time frame look ahead) to prevent locally optimized solution at each time frame as follows:

\[\begin{eqnarray} P_{t} : & \min _{x=\left(\omega _t,\omega _{t+1}\right)} z^{o}_{t}= & \sum _{k=1}^{3} (f_{k,t}(\omega _t)+f_{k,t+1}(\omega _{t+1})), \end{eqnarray}\]

(39)

\[\begin{eqnarray} \mbox{subject to}\nonumber \nonumber\\ Z^{c}_{1,d}: & g_{d,T1}(\omega _t)\le T1 & \forall d\in \mathcal {D}\left(t\right)\!, \end{eqnarray}\]

(40)

\[\begin{eqnarray} Z^{c}_{2,d}: & g_{d,T1}(\omega _{t+1}) \le T1 & \forall d\in \mathcal {D}\left(t+1\right)\!, \end{eqnarray}\]

(41)

\[\begin{eqnarray} Z^{c}_{3,d}: & g_{d,T2}(\omega _t) \le T2 & \forall d\in \mathcal {D}\left(t\right)\!, \end{eqnarray}\]

(42)

\[\begin{eqnarray} Z^{c}_{4,d}: & g_{d,T2}(\omega _{t+1}) \le T2 & \forall d\in \mathcal {D}\left(t+1\right)\!, \end{eqnarray}\]

(43)

\[\begin{eqnarray} Z^{c}_{5,d}: & H_{d}(\omega _t) \le H &\forall d\in \mathcal {D}\left(t\right)\!, \end{eqnarray}\]

(44)

\[\begin{eqnarray} Z^{c}_{6,d}: & H_{d}(\omega _{t+1}) \le H & \forall d\in \mathcal {D}\left(t+1\right)\!. \end{eqnarray}\]

(45)

We note that, in practice, a decision maker needs to set the objective function appropriately for her goals. In References [9, 68], for example, the objective functions are set in a way to quantify the economic cost in terms of the total loss in economic output. In this article, we model the economic cost as the total cost of testing, quarantine, and hospitalization. A higher testing rate potentially reduces the number of hospitalization and quarantine, which are generally more expensive than testing, not only because of their direct cost but also because of their indirect cost in terms of their negative economic impacts. In general, however, the objective function is orthogonal to the epidemic model. While References [9, 68] and the experiments reported in our model did not associate an explicit cost to the number of deaths, the objective function used in SIRTEM can very easily accommodate the number of deaths as an optimization parameter—it would, of course, be the role of the decision maker to assign a specific cost to the human life.

The optimization problem $P_{t}$, is solved in a rolling horizon manner for $t=1,\ldots ,T-1$ using a lookahead of one time period. Herein, a time period t refers to a month, while d is day, and the set $\mathcal {D}\left(t\right)$ represents the number of days in month t. At each time period t, we choose the testing rates to apply. Therefore, the decision variables $\omega _t= (\phi _{a1,t},\phi _{a2,t},\phi _{s1,t},\phi _{s2,t})$, where $(\phi _{a1,t},\phi _{a2,t},\phi _{s1,t},\phi _{s2,t})$ represent daily asymptomatic and symptomatic testing rate for test type 1 and 2 in time period t. These decision variables are real numbers in the range of [0,1]. The objective in Equation (39), together with the constraints in Equations (40)–(45) are blackbox functions and, therefore, they require simulation to be evaluated. Specifically, $g_{d,T1},g_{d,T2}$, represent the daily total number of tests of type 1 and 2 administered, respectively, to be compared to the total daily testing capacity $T_{1},T_{2}$. There are $\mathcal {D}\left(t\right)$ of these constraints for each month t, and for each test type. in constraints (44) and (45), ${H}_{d}$ denotes the hospital beds used daily, to be compared to the daily available hospital bed capacity, H. There are $\mathcal {D}\left(t\right)$ of these constraints for each time period t. The values, $f_2$ and $f_3$, are hospitalization cost and quarantine cost, respectively. They are calculated by multiplying the daily per person costs of hospitalization or quarantine with the number of people hospitalized or quarantined per day, respectively.

We used a Constrained Bayesian optimization with rolling horizon approach to solve the given problem. The sampling algorithm was taken from Reference [31], and it is reported as Step 1 in Algorithm 1, below.

4 Evaluation

In the proposed model, the transmission rate is a key learnable parameter of the disease. However, the contact rates also impact the disease propagation and can also be treated as a time variable unknown and may need to be learned from the data. In the experiments considering the single city model reported below, we treated both the transmission rate and the contact rate as unknowns, and we sought to learn a time varying aggregate infection rate $\beta$ that captures not only the transmission rate but also major changes in contact rates due to closures and other interventions and behavioral changes. When considering the multi-city model, we calculated the contact rates for the individual cities within the state to get finer grain information that captures how individuals within a city interact. In short, in SIRTEM, we not only treat the disease parameters as targets for learning but also behavioral parameters, such as inter-city and intra-city contact rates, as learnable parameters.

Note that, in this section, in addition to $\beta$, we also consider testing rate ($\phi _{s}$) as a learnable parameter. This is because one of the key goals of the SIRTEM model is to develop testing policies—therefore, in many of our experiments, we treat $\phi$ as an optimizable parameter subject to test availability constraints. Having said this, there is nothing that prevents us from constraining the value of $\phi$ with published values when the daily testing rates are available.

4.1 Single City Model Validation

4.1.1 Parameter Calibration.

For a given geographic location (this could be county, state, city), we calibrate the SIRTEM model by estimating three critical parameters, infection rate ($\beta$), testing rate ($\phi _{s}$), and the general symptomatic rate (g) using published data of confirmed positive and negative cases (Table 8). We made the following assumptions:

Table 8.

Parameter	Description	Values	Reference
$tp_{1}$	Sensitivity of diagnostic test	0.75	[22, 81]
$tn_{1}$	Specificity of the diagnostic test	0.95	[81]
$tp_{se}$	Sensitivity of the serology test	0.84	[8]
$tn_{se}$	Specificity of the serology test	0.97	[8]
$\tau _{1}$	Time to result for diagnostic test	3 days	Assumption
$\tau _{se}$	Time to result for the serology test	5 days	Assumption
$\phi _{1}$	Diagnostic testing rate for symptomatic individuals	Estimated
$\phi _{a1}$	Diagnostic testing rate for non-symptomatic individuals	0	Assumption
$\phi _{se}$	Serology Test rate	0.01	Assumption
$\beta$	Infection rate for the susceptible pop.	Estimated
$\beta ^{\prime }$	Inf. rate for falsely presumed immune pop. (ratio)	1.2$\cdot \beta$	Assumption
r	Ratio of transmission rates for asympt. population against sympt. population	0.51	[12]
$per_{a}$	Percentage of ind. with COVID-19 who are asymptomatic	0.16	[12]
$per_{s}$	Percentage of ind. with COVID-19 who are symptomatic	0.84	[12]
$\eta$	Incubation length (days)	3.2 days	[12]
$\lambda _{a}$	Length of recovery for asymptomatic ind. (days)	3.5 days	[12]
$\lambda _{s}$	Length of recovery for symptomatic ind. (days)	7 days	[12]
$\lambda _{q}$	Length of quarantine (days)	14 days	[16]
h	hospitalization rate (ratio of quarantined pop, per day)	0.06	[20]
$\lambda _{h}$	Hospitalization length (days)	6 days	[16]
$\kappa$	Mortality rate for symptomatic pop. (per day)	0.0088	[12]
$\kappa _{h}$	Mortality rate for hospitalized individuals (per day)	0.074	[20]
g	Ratio of susc. who have fever for non-COVID infections (ratio, per day)	Estimated

Table 8. Default Values for Various SIRTEM Model Parameters

•

There is only one diagnostic test and one serology test available;

•

Flu and general symptomatic-related compartments are collapsed into one, and the underlying rate (g) is assumed to be constant over time.

We obtain estimation of these three parameters by solving an equation. Given the complexity of the simulator, we cannot optimize the likelihood in closed form. Instead, we defined an algorithm to iteratively improve the discrepancy between the daily positive and negative cases predicted against the real data:

\[\begin{eqnarray} (P) : \min _{\omega \in \Omega } Z\left(\omega \right) =\frac{1}{2}\left[ \frac{\frac{1}{d}\sum _{t=1}^{d}(\hat{y}^{(+)}_{t}|{\omega })-y^{(+)}_{t})^2)}{\bar{y}^{(+)}_{t}} +\frac{\frac{1}{d}\sum _{t=1}^{d}(\hat{y}^{(-)}_{t}|{\omega })-y^{(-)}_{t})^2}{\bar{y}^{(-)}_{t}}\right]. \end{eqnarray}\]

(46)

In Equation (46), $\hat{y}^{(+)}_{t}$, $\hat{y}^{(-)}_{t}$ are the SIRTEM predictions for positive and negative cases at time t (day), respectively. $y^{(+)}_{t}$, $y^{(-)}_{t}$ denote confirmed positive and negative cases obtained from public sources [69]. Each error term is normalized using the average confirmed cases, $\bar{y}^{(+)}_{t}$ and $\bar{y}^{(-)}_{t}$, respectively.

The changes in the two key parameters, the relative infection rate $\beta$, and the diagnostic testing rate $\phi$ (assuming only one test), over time, follow a functional form. More specifically, as was also done in prior work, such as Reference [71], we assume that these two parameters are random temporal processes—precisely, we assume both to be autoregressive, and we use an AR(2) model for their calibration. This is because the parameter $\beta$ captures the transmissibility of the disease and the likelihood of interactions among individuals within the population; neither of which is memoryless processes. Similarly, in practice, the diagnostic testing rate cannot be assumed to be memoryless. Since autoregressive models are suited to capture temporal processes where observations at a given time instant depend on past time steps, they are suitable to model the evaluation of the $\beta$ and $\phi$ parameters over time. As a result, for each week k, we have

\[\begin{eqnarray} \beta _{k} &=& a_{1}\cdot \beta _{k-1} + a_{2}\cdot \beta _{k-2} + e^b_{k}, \end{eqnarray}\]

(47)

\[\begin{eqnarray} \phi _{{k}} &=& b_{1}\cdot \phi _{{k-1}} + b_{2}\cdot \phi _{{k-2}} + e^s_{k}, \end{eqnarray}\]

(48)

\begin{equation} \forall _k \quad g_{{k}} = g. \end{equation}

(50)

Note that, here $e^b_{k}$ is not a learnable parameter, but a noise term.

Under these assumptions, the decision vector $\omega =\left[\mathbf {a},\mathbf {b},g\right]$, requiring to estimate $\mathbf {a},\mathbf {b}$, both two-dimensional vectors, and the parameter g. This results in a five-dimensional decision problem, which we treat as a black box optimization, and use K weeks of data to calibrate. Specifically, we designed an iterative procedure for the optimization of the autoregressive parameters in Equation (47) and (48) and g. The approach is summarized in Algorithm 2.

Impact of the Model Parameters. Before considering specific U.S. states, we investigate the impact of the various key model parameters on the progression of the COVID-19 epidemic. For this purpose, we consider a state with total population of 7 million and start the epidemic with 10 symptomatic and 10 asymptomatic individuals. We simulate the epidemic for 180 days.

Impact of $\beta$ on COVID-19 Progression. Figure 3 shows how the infection rate, $\beta$ affects the positive and negative cases. As expected, a higher infection rate, $\beta$, results in a larger number of COVID-19 infected individuals as shown in Figure 3(a). This in turn results in larger numbers of positive and negative test results as shown in Figures 3(b) and 3(c). Here, the increase in the negative cases is due to the imperfection in the diagnostic test.

Fig. 3.

Impact of $\phi _{s1}$ on COVID-19 Progression. A higher testing rate, $\phi _{s1}$, would indicate that the authorities are actively trying to find COVID-19 infected individuals from the population, which is likely to increase positive test results. However, a strong testing campaign would decrease the infection itself, which in turn would push the positive test cases down. We study this complex relationship in Figure 4(a). As we see in Figure 4(b), as a result of this, a higher testing rate is able to slow the epidemic progress but does not significantly impact the peak positive test results. Additionally, a higher testing rate also increases the number of negative test results for a fixed general sick rate, g (Figure 4(c)).

Fig. 4.

Impact of g on COVID-19 Progression. General sick rate, g, contributes to increased negative cases, because a higher general sick rate means more individuals show symptoms (coughing, fever) but are not COVID-19 infected. As seen in Figure 5(c), the gap between the negative cases for different values of g diminishes after 110 days; this is largely due to significant drops in the false-negative cases.

Fig. 5.

4.1.2 Case Studies with Four U.S. States.

We next compare the prediction results of the proposed model with calibrated model parameters against the published confirmed case data for four U.S. states—Arizona, Florida, Wisconsin, and Minnesota. The model parameters are calibrated and estimated weekly (more specifically, for the $i\text{th}$ week, the parameters $\beta$ and $\phi$ phi are estimated based on the $(i-4){\text{th}}$ to $(i-1){\text{th}}$ weeks) relying on the parameter values presented in Table 8 obtained from various sources, including recent research results and CDC published data. These parameter values are then used to predict the behavior of the epidemic for the following week.

Table 9 and Figure 6 show the populations and the weekly moving averages of reported positive and negative cases of the four states considered in the case study [20].

Fig. 6.

Table 9.

State	Population
Arizona	7.3 million
Florida	21.5 million
Minnesota	5.6 million
Wisconsin	5.8 million

Table 9. Populations of the Four States Considered in the Case Study

Arizona and Florida have seen two waves of case growth during July and December. In contrast, Minnesota and Wisconsin kept the epidemic under control until September, but suffered from rapid growth of cases during the second wave of the pandemic in December 2020. We use these confirmed cases to calibrate the three key parameters of the epidemic for each state by solving the optimization problem reported in Section 4.1.1. The results are presented in Figure 7 and Table 10: as we see here, the model predicts the reported positive cases very accurately for the southern states, Arizona and Florida.

Fig. 7.

Table 10.

State	$\overline{y1}$	$\overline{y2}$	$\frac{\frac{1}{d}\sum _{t=1}^{d}(f1_{t}-y1_{t})^2}{\overline{y1}}$	$\frac{\frac{1}{d}\sum _{t=1}^{d}(f2_{t}-y2_{t})^2}{\overline{y2}}$	$\frac{1}{2}\left[ \frac{\frac{1}{d}\sum _{t=1}^{d}(f1_{t}-y1_{t})^2}{\overline{y1}} +\frac{\frac{1}{d}\sum _{t=1}^{d}(f2_{t}-y2_{t})^2}{\overline{y2}}\right]$
Arizona	1937.1	7975.8	0.295	0.264	0.2797
Florida	4427.4	24559.4	0.3227	0.2185	0.2706
Wisconsin	1752.9	7738.8	0.325	0.343	0.334
Minnesota	1417.7	9222.6	0.5201	0.2305	0.3753

Table 10. Normalized Mean-squared Error for the Four U.S States

The prediction, especially, for the negative test results is somewhat off for the northern states, Minnesota and Wisconsin. This is apparently because the general symptomatic rate was assumed to be constant throughout the simulation, which may not be less a valid assumption for the colder states in the north.

As we see in Figures 7(a) and 7(c), in Arizona and Florida, the first major hike of the infection rate, $\beta$, follows soon after the governments’ lifting of the stay-home orders in May. We see that while $\beta$ dropped fast as people in these States panicked with the rapid increase of the positive case numbers, soon after the value of the $\beta$ parameter started a slower but consistent creep up as the population started suffering from social distancing fatigue. The negative cases for these two states, reported in Figures 7(b) and 7(c), are also predicted quite accurately; in fact, they follow the the testing rates (reported in Figures 7(a) and 7(b) closely): the more testing done, the higher numbers of negative cases are reported.

In contrast to Arizona and Florida, Wisconsin and Minnesota managed the epidemic better until September. Unfortunately, the infection rate, $\beta$, in both of these states do see a rapid increase roughly 1 to 1.5 month after lifting of the stay home order, as seen in Figures 7(e) and 7(g). Similar to Arizona and Florida, the model negative cases (Figures 7(f) and 7(h)) follow the predicted testing rates; but the overall fit with the published negative cases is not as strong, presumably due to less stable general symptomatic rates in these two colder northern states.

4.2 Optimal Testing Policies

In this section, we present the optimization result with single area model (i.e., $c=1$) to investigate the relationship among testing policy, cost, and other parameters. In the results reported in this section, we consider various scenarios where two alternative tests, with different accuracies, are being deployed:

•

Test $\#1$: This is a relatively more accurate (high sensitivity, 0.75, and specificity, 0.95) test, with correspondingly high cost, $50.

•

Test $\#2$: This is a cheaper and less accurate test. In particular, we varied test sensitivity between 0.50 and 0.75, specificity between 0.70 and 0.95, and cost between 1 and 50,

For the test problem setting, we sampled 50 cases of test set with latin hypercube sampling. Detailed parameter values obtained through sampling are shown in the Appendix. We note that the optimization problem is solved Bayesian Optimization technique with inequality constraint [31] More specifically, we considered a 10-month period, with the first 2 months marked as the “early,” the last 2 months marked as the “late,” and the middle 6 months marked as the “active” period of the epidemic. For each scenario, we have computed the cost optimal deployment strategy and in the rest of this section, we present the key outcomes from this set of experiments.

Impact of Accuracy of the Alternative Test on the Testing Rates. In Figure 8, we plot four charts depicting the impact of the accuracy of the Test 2 on the testing rates for both Tests 1 and 2. As we see in Figures 8(a) and 8(b) as the accuracy of Test 2 increases, this translates into lesser use of Test 1—considering that Test 2 is cheaper, this is expected. We also see in Figures 8(c) and 8(d) that as the accuracy of Test 2 increases, a larger portion of the population is tested with the cheaper test, Test 2. This result is also confirmed in Figure 9, where we see that the percentage of test of Type 2 increases with increasing sensitivity and selectivity during all phases of the epidemic.

Fig. 8.

Fig. 9.

Impact of Accuracy of the Alternative Test on the Quarantine and Hospitalization Rates. In Figures 10 and 11, we consider the impact of accuracy of the alternative test on the quarantine and hospitalization costs. As we see in Figures 10(a) and 10(b), the sensitivity of the Test 2 has only minimal impact on the quarantine costs; in contrast, improved specificity of Test 2 has a strong impact on reductions on the quarantine rates. As we see in Figure 11, however, the accuracy of Test 2 have almost no impact on the hospitalization (while more accurate (especially specific) tests tend to reduce the quarantine costs, the trend is quite weak as indicated by a large p-value).

Fig. 10.

Fig. 11.

Impact of the Accuracy of the Alternative Test on the Percentage of Asymptomatic Tests Run. As we see in Figures 12(a)–12(c) if the cheaper test, Test 2, is highly sensitive, it raises the ratio of the symptomatic testing (as opposed to the asymptomatic testing) done during the early stages of the epidemic—the accuracy of the test, however, does not have an impact on the relative symptomatic versus asymptomatic testing rates during the active and late stages if the epidemic. The specificity of Test 2 also promotes symptomatic testing as opposed to asymptomatic testing, but this time during both early and late phases of the epidemic—see Figures 12(d)–12(f). Once again, during the active phase of the epidemic, the accuracy of the test does not have a significant impact on the balance of symptomatic versus asymptomatic testing rates.

Fig. 12.

Impact of the Unit Quarantine and Hospitalization Costs on the use of Test 2. In Figure 13, we consider the impact of the unit quarantine and hospitalization costs on the use of the cheaper Test 2, as opposed to the more expensive Test 1. As we see in the figure, during the early and late phases of the epidemic, larger unit quarantine and hospitalization costs favor the use of more expensive and accurate Test 1. The interesting observation, however, is that during the active phase of the epidemic, the quarantine and hospitalization unit costs have no discernible impact on the choice of accurate and expensive Test 1 versus less-accurate and cheaper Test 2.

Fig. 13.

4.3 Multi-city Epidemic Dynamics

We next use SIRTEM to investigate the multi-city dynamics, with intra- and inter-city mixing. For these experiments, we rely on two distinct mixing rate calculations as described in Section 3.1.2: (a) worker movement data available at Reference [13] and (b) mobility dataset from SafeGraph [34]. For both data sets, we consider 11 cities/regions¹ in Maricopa county, Arizona, under home, work, and other mixing as described in Section 3.1.2.

As we have seen earlier in Table 7, the worker movement data set in References [13] and [29] together provide mobility data from early Spring (March 22) to Fall (September 26) 2020 in multiple waves. Table 11 shows the inter- and intra-city mixing rates during Spring (22 March–9 April 2020) and Fall (11 September–26 September 2020) periods per these data. As we see here, during the initial phase of the epidemic, mixing was largely confined to within individual cities. By Fall 2020, however, inter-city mixing across most city-pairs became relatively significant.

Table 11.

Figures 14 and 15 show the intra-city mixing rates obtained from the two worker movement data set and the SafeGraph data set (inter-city mixing rates are omitted in these figures, due to space limitations). As we see in Figure 15, for the time period for which both data sets do overlap, the two data sources do not perfectly agree on the predicted mixing rates, with the SafeGuard data yielding higher intra-city mixing rate predictions across the board.

Fig. 14.

Fig. 15.

4.3.1 Results with Mixing Rates Based on Worker Mobility Data.

Figure 14 provides an overview of the average intra-city mixing rates for all phases of the epidemic reported in Table 7 against the size of the population, based on the worker mobility data available at Reference [13] (inter-city mixing rates are omitted due to space constraints). As expected, the intra-city mixing rate is generally inversely correlated with the population size of the city—the larger the city, the less likelihood for any pair of individuals to meet.

Once again, we consider a scenario where there are two diagnostic tests and a serology test. The high accuracy test has sensitivity 0.75 and specificity 0.95. The low accuracy test has sensitivity 0.65 and specificity 0.85. For both tests, we consider the testing rate, $\phi$, to be 0.01 for asymptomatic population and 0.2 for symptomatic population. The model also includes an accurate serology test with the test rate, $\phi$, value of 0.01.

In Figures 16 and 17, we investigate the impact of various key parameters on the multi-city epidemic dynamics. In particular, we consider two distinct per-contact transmission rates 0.05 and 0.1 for COVID-19 (roughly covering the range of values reported in the literature) and consider initial infected population sizes of $\sim \!80$ (the reported cases on 03/22/2020) [84] and $\sim \!400$, assuming that the number of real cases in the population is $5\times$ those of the reported infections. The Y-axis in the charts denote the level of exposed individuals at the point when the population reaches herd immunity; i.e., when the per-day exposure rate crests in the population and the epidemic starts to decline. We denote this as the exposure at herd immunity or $\tt EHI$. Note that, in the SIRTEM model, exposed individuals develop symptomatic or asymptomatic COVID-19—therefore, this is also the level of the infected individuals in the population before the epidemic starts to decline.

Fig. 16.

Fig. 17.

As we see in Figure 16, the $\tt EHI$ rate is generally between $35\%$ and $65\%$, with the population size being highly correlated with $\tt EHI$. We see in the charts that, in general, the larger the population, the higher the $\tt EHI$; the correlation, however, is not perfect due to variations in work-related inter-city mixing. In these charts, we also see that, as expected, a lower per-contact transmission rate (Figure 16(b)) results in a lower $\tt EHI$, especially for smaller cities. As expected, an overall higher mixing rate also results in a larger $\tt EHI$ (Figure 16(c)). Interestingly, however, a higher initial population does not necessarily result in a larger $\tt EHI$ (Figure 16(d))—while the disease grows faster in the population, the $\tt EHI$ is not significantly impacted by the size of the initial infected population.

Figure 17(a) shows the average impact of work-related movement (as a ratio of the cities’ own populations) to the $\tt EHI$ rate. As we see here, some cities (such as Tempe) have relatively large work-related movement considering their own populations, whereas others (such as Mesa) have a much lower work-related movement as a ratio of the population. As we see in the figure, even during the lockdowns, the work-related movements within and across cities have a slight, but generally positive impact on the $\tt EHI$ of the epidemic. Finally, in Figure 17(b), we see the importance of accounting for inter-city movements—when the model is modified to ignore the inter-city movements, it becomes insensitive to population mixing (and, in fact, appears to predict a counter-intuitive negative correlation between cities with large work-related movement and their exposure rates).

4.3.2 Results with Mixing Rates Based on SafeGraph Mobility Data.

Figure 18 plots the number of exposed individuals for the first month of the epidemic based on mixing rates obtained from the worker commute [13] and SafeGraph [34] data sets, under different transmission rate and initial exposure configurations. As we see in this figure, the higher mixing rates implied by the SafeGraph data (see Figure 15) also leads to a significantly larger rates of growth in the exposed population. This charts highlights the importance of reliable mixing rate information align with the difficulty of relying on data obtained through mobile applications in epidemic predictions.

Fig. 18.

4.4 Ablation Studies

The proposed SIRTEM model has various sub-components modeling different processes, as depicted in Figure 2. In this section, we present an ablation study that focuses on two of the key processes testing and FPS population, as well as spatial modeling.

4.4.1 Impact of Testing.

As we can see from Figure 19, when we compare the results of the SIRTEM model with the results with no testing, we see a significant difference: Figure 19(a) shows that without testing (and the following quarantine) the susceptible population drops significantly, since a substantially large number of people get infected. This is confirmed by Figures 19(b), 19(c), and 19(d), which show that the number of infected symptomatic and infected asymptomatic individuals and the number of dead increase significantly when the testing and test-based quarantine are removed. For these experiments, the default parameters of the model from Table 8 were used.

Fig. 19.

4.4.2 Impact of FPS.

Figure 20 summarizes the results when considering the same setting as above, but this time eliminating the FPS process. As we see in Figure 20, at early stages of an epidemic, the impact of FPS is low—however, the difference becomes non-negligible at later stages of the epidemic, where individuals who recover from the disease start losing their immunities.

Fig. 20.

4.4.3 Impact of Spatial Modeling.

We have already seen in Figure 17, the importance of accounting for inter-city movements. In this ablation study, we investigate the impact of using sufficiently, fine-grain spatial context (e.g., city-wide versus regional)—even when we ignore inter-city interactions (such as during when there are travel restrictions and/or due to remote-work). Such scenarios are well documented (see Table 11). To illustrate that it is important that we consider individual cities separately from each other, the ablation study is constructed as follows:

•

We first consider two cities where there are intra-city interactions, but no inter-city interactions.

•

We then consider a third city, which has the same population as the sum of the first to cities and the same number of total interactions.

•

We then show that despite having the same number of individuals and the same number of overall interactions, these two scenarios result in vastly different outcomes, highlighting the importance of spatial modeling.

In Figure 21, we consider the impact of the spatial context on the model results. In particular, we are considering two cities, $city_1$ and $city_2$, each with 100K population suffering from the same epidemic. The populations in the two cities have different interaction patterns, leading to different $\beta$ parameters, 1 for $city_1$ and 1.4 for $city_2$. As stated above, we assume that the two cities have the same testing rate 0.6 and individuals across these two cities do not interact. As explained above, to see the impact of fine-grain vs. low-grain spatial context, we are also considering a third city, denoted as $city_{(1,2)} = city_1 \cup city_2$, which represents the union of the two cities, without considering their spatial delineation. Note that, to account for the same number of of interactions among individuals in $city_{(1,2)}$ as $city_1$ and $city_2$ combined, the beta parameter for the super-city, $city_{(1,2)}$, can be computed as $\beta = (100\text{K}\times 100\text{K} \times 1 + 100\text{K}\times 100\text{K} \times 1.4)/(200\text{K}\times 200\text{K}) = 0.6.$

Fig. 21.

As we see in Figure 21, the model results for $city_{(1,2)}$, where the spatial context is ignored, look very different from model results for $city_1$ and $city_2$ (and their combinations). In particular, the epidemic kicks-in much later than the two cities when considered individually and its overall impact, in terms of the infections and deaths, is relatively low. This highlights the importance of considering spatial context to account for differences in population characteristics in modeling epidemics—in this particular case, we see that multiple smaller populations are more conducive to the quick growth of the epidemic than a large, well mixed population, even when the total number of interactions are the same.

4.5 Comparisons Against Other Models

In this section, we compare SIRTEM, with the models proposed by Berger et al. [9] and Piguillem et al. [68]. The major differences between these models are summarized in Table 1. To ensure that this comparison is as fair as possible, we have modified the SIRTEM model to reflect the key characteristics of References [9, 68], by replacing the delayed differential equations with purely rate-based state transitions and by appropriately eliminating/replacing sub-processes, such as FPS.

Figure 22 compares observed data against the prediction made by the three different models. As we see here, SIRTEM is able to match the observed data very well, whereas the two other models fail to match the observations. To understand why, Figure 23 provides a deeper study of the how the numbers of Susceptible, Symptomatic Infected, Asymptomatic Infected, and Dead change over time based on different models.

Fig. 22.

Fig. 23.

As we see here, both References [9, 68] significantly overestimate infections relative to SIRTEM: this is primarily due to the fact that rate-based transitions have a tendency to “leak” a portion of the transitions among states too early, contributing to erroneous exponential growth, whereas the delayed differential equations hold timed transitions more accurately, preventing these erroneous growth in the number of infections.

Another major difference between the results is the number of late infections captured by SIRTEM: this is because SIRTEM properly models loss of immunity, whereas References [9, 68] do not have the corresponding processes in their models—preventing them from capturing the long term evolution of the epidemic.

5 Conclusions

Epidemic testing strategies are designed with three complementary objectives: (a) obtaining a faithful picture of the COVID-19 model as well as epidemic trajectory; (b) identifying individuals and populations who are at risk of exposure or are already sick; and (c) guiding intervention efforts, such as isolation of infectious individuals, quarantining of suspected contacts, and minimization go contact rates without completely disrupting the society. To achieve these goals, the testing strategy should account for several practical constraints, such as the daily testing capacity, limitations for different testing types, and the potential of the epidemic to disrupt the healthcare infrastructure if we fail in identifying dangerous outbreaks. For this, we need to account for the spatial distribution and mobility of the susceptible population as well as the sensitivity and specificity of the available tests. In particular, testing can be a costly process especially during the onset of an epidemic and accuracy of available tests can have significant impact on the epidemic progress: When test sensitivity accuracy is low, we will not be able to separate the infected population efficiently from the healthy population, resulting in a spike in disease spread. However, if the specificity accuracy is low, then we can falsely quarantine healthy individuals. In this article, we presented a novel spatially informed epidemic model, SIRTEM (for “Spatially Informed Rapid Testing for Epidemic Modeling”), that integrates multi-accuracy testing strategies, along with quarantine and hospitalization processes. The model is coupled with an optimization model that incorporates spatially based testing and hospitalization resource constraints. We presented extensive experiments that shows the utility of SIRTEM and the associated optimization model in both single-city and multi-city scenarios.

Footnote

We selected the 10 largest cities in Maricopa county, Arizona, with respect to the population size and worker movement and merged the remaining small cities under a single region label, “Others.” For mixing rate computations, the distance of this aggregate region to a given city has been approximated as half of the maximum distance of the city to all the cities in the aggregate region.

Appendix

Table 12.

Case	Cap.T1	Cap.T2	Max H.Bed	Sen.T1	Sen.T2	Spe. T1	Spe.T2	Co.T1	Co.T2	Q.Cost	H.Cost
Case 1	24,349	26,000	1,024	0.75	0.71	0.95	0.81	50	18	30	1,603
Case 2	12,050	26,894	1,143	0.75	0.67	0.95	0.82	50	7	163	4,268
Case 3	21,986	63,310	1,235	0.75	0.65	0.95	0.76	50	8	141	3,818
Case 4	20,431	47,689	1,657	0.75	0.75	0.95	0.72	50	50	199	4,975
Case 5	16,127	36,959	1,398	0.75	0.63	0.95	0.86	50	47	4	1,071
Case 6	17,978	25,328	1,256	0.75	0.61	0.95	0.71	50	47	16	1,321
Case 7	22,551	46,142	1,474	0.75	0.66	0.95	0.74	50	1	171	4,414
Case 8	12,889	24,399	1,699	0.75	0.71	0.95	0.73	50	23	104	3,081
Case 9	24,519	53,270	1,705	0.75	0.55	0.95	0.92	50	29	79	2,585
Case 10	20,811	40,944	1,815	0.75	0.53	0.95	0.87	50	40	127	3,536
Case 11	19,303	48,640	1,934	0.75	0.74	0.95	0.73	50	16	48	1,962
Case 12	16,879	37,890	1,854	0.75	0.72	0.95	0.75	50	35	64	2,289
Case 13	20,730	43,578	1,449	0.75	0.69	0.95	0.86	50	18	13	1,267
Case 14	15,017	24,136	1,210	0.75	0.58	0.95	0.79	50	32	193	4,868
Case 15	22,957	48,826	1,485	0.75	0.52	0.95	0.71	50	14	151	4,012
Case 16	15,407	16,719	1,861	0.75	0.52	0.95	0.75	50	41	25	1,504
Case 17	15,400	20,159	1,518	0.75	0.70	0.95	0.90	50	22	130	3,598
Case 18	22,158	45,124	1,324	0.75	0.59	0.95	0.77	50	16	115	3,300
Case 19	14,058	38,234	1,559	0.75	0.67	0.95	0.89	50	8	95	2,892
Case 20	11,461	15,159	1,400	0.75	0.70	0.95	0.78	50	33	91	2,826
Case 21	19,790	35,043	1,270	0.75	0.64	0.95	0.85	50	20	101	3,025
Case 22	11,655	15,934	1,997	0.75	0.62	0.95	0.92	50	4	183	4,665
Case 23	18,159	44,931	1,959	0.75	0.72	0.95	0.89	50	25	97	2,930
Case 24	18,495	26,854	1,835	0.75	0.60	0.95	0.80	50	1	7	1,132
Case 25	14,494	16,408	1,779	0.75	0.57	0.95	0.79	50	3	57	2,142
Case 26	20,160	24,698	1,784	0.75	0.50	0.95	0.70	50	30	117	3,332
Case 27	13,409	39,921	1,636	0.75	0.73	0.95	0.91	50	38	165	4,291
Case 28	14,675	43,221	1,172	0.75	0.66	0.95	0.78	50	11	60	2,204
Case 29	19,278	53,500	1,918	0.75	0.60	0.95	0.87	50	43	68	2,370
Case 30	16,940	42,896	1,284	0.75	0.64	0.95	0.76	50	14	22	1,448
Case 31	22,794	28,839	1,973	0.75	0.56	0.95	0.90	50	38	187	4,742
Case 32	10,340	29,210	1,362	0.75	0.55	0.95	0.83	50	10	110	3,202
Case 33	12,300	22,461	1,616	0.75	0.54	0.95	0.82	50	26	158	4,164
Case 34	10,091	26,331	1,669	0.75	0.74	0.95	0.84	50	10	146	3,922
Case 35	18,806	22,485	1,097	0.75	0.63	0.95	0.72	50	24	175	4,505

Table 12. Parameter Setting for Single City Model (Cases 1–35)

Table 13.

Case	Cap.T1	Cap.T2	Max H.Bed	Sen.T1	Sen.T2	Spe.T1	Spe.T2	Co.T1	Co.T2	Q.Cost	H.Cost
Case 36	10,731	16,884	1,056	0.75	0.59	0.95	0.85	50	24	134	3,690
Case 37	15,722	25,842	1,307	0.75	0.58	0.95	0.80	50	27	35	1,691
Case 38	17,323	25,779	1,101	0.75	0.53	0.95	0.88	50	46	82	2,646
Case 39	23,376	60,585	1,585	0.75	0.68	0.95	0.77	50	43	43	1,861
Case 40	21,544	41,739	1,340	0.75	0.73	0.95	0.81	50	32	9	1,190
Case 41	13,037	20,024	1,528	0.75	0.51	0.95	0.88	50	49	48	1,954
Case 42	17,662	32,822	1,008	0.75	0.68	0.95	0.94	50	35	53	2,050
Case 43	21,269	51,610	1,124	0.75	0.57	0.95	0.91	50	5	39	1,778
Case 44	12,522	21,463	1,732	0.75	0.62	0.95	0.95	50	4	190	4,800
Case 45	24,978	59,681	1,064	0.75	0.69	0.95	0.94	50	44	156	4,111
Case 46	23,789	41,232	1,434	0.75	0.56	0.95	0.83	50	20	177	4,547
Case 47	11,112	11,314	1,189	0.75	0.61	0.95	0.74	50	13	138	3,754
Case 48	23,873	69,407	1,742	0.75	0.51	0.95	0.93	50	40	87	2,749
Case 49	16,370	43,315	1,567	0.75	0.54	0.95	0.93	50	30	124	3,474
Case 50	13,873	38,073	1,898	0.75	0.65	0.95	0.84	50	36	74	2,480

Table 13. Parameter Setting for Single City Model (Cases 36–50)

References

[1]

Jacob B. Aguilar and Juan B. Gutierrez. 2020. An epidemiological model of malaria accounting for asymptomatic carriers. Bull. Math. Biol. 82, 3 (2020), 1–55.

Parameters	Description
\(tp_{i}\)	Sensitivity of diagnostic test i
\(tn_{i}\)	Specificity of diagnostic test i
\(\tau _{i}\)	Time to obtain the result for diag. test i (days)
\(\phi _{i}\)	Testing rate for diagnostic test i the symptomatic population (ratio per day)
\(\phi _{ai}\)	Testing rate for diagnostic test i the non-symptomatic population (ratio per day)
\(tp_{se}\)	Sensitivity of the serology test
\(tn_{se}\)	Specificity of the serology test
\(\tau _{se}\)	Time to obtain the result for the serology test (days)
\(\phi _{se}\)	Testing rate for the serology test (ratio of the relevant population per day)
\(\beta\)	Infection rate for the susceptible population (ratio)
	= transmission rate \(\times\) contact rate for the susceptible population
\(\beta ^{\prime }\)	Infection rate for the population of individuals who are falsely presumed immune (ratio)
	= transmission rate \(\times\) contact rate for the presumed immune population
r	The ratio of the transmission rate of asymptomatic individuals to the transmission rate of symptomatic individuals
\(per_{a}\)	Percentage of individuals with COVID-19 who are asymptomatic
\(per_{s}\)	Percentage of individuals with COVID-19 who are symptomatic
\(\eta\)	Incubation length (days)
\(\lambda _{a}\)	Length of recovery for asymptomatic individuals (days)
\(\lambda _{s}\)	Length of recovery for symptomatic individuals (days)
\(\lambda _{q}\)	Length of quarantine (days)
\(\lambda _{h}\)	Hospitalization length (days)
h	Hospitalization rate (ratio of the quarantined population, per day)
\(\kappa\)	Mortality rate for symptomatic population (per day)
\(\kappa _{h}\)	Mortality rate for the hospitalized individuals (per day)
g	Ratio of the susceptible individuals who has fever and cough for non-COVID infections (ratio, per day)

Parameter	Description	Values	Reference
\(tp_{1}\)	Sensitivity of diagnostic test	0.75	[22, 81]
\(tn_{1}\)	Specificity of the diagnostic test	0.95	[81]
\(tp_{se}\)	Sensitivity of the serology test	0.84	[8]
\(tn_{se}\)	Specificity of the serology test	0.97	[8]
\(\tau _{1}\)	Time to result for diagnostic test	3 days	Assumption
\(\tau _{se}\)	Time to result for the serology test	5 days	Assumption
\(\phi _{1}\)	Diagnostic testing rate for symptomatic individuals	Estimated
\(\phi _{a1}\)	Diagnostic testing rate for non-symptomatic individuals	0	Assumption
\(\phi _{se}\)	Serology Test rate	0.01	Assumption
\(\beta\)	Infection rate for the susceptible pop.	Estimated
\(\beta ^{\prime }\)	Inf. rate for falsely presumed immune pop. (ratio)	1.2\(\cdot \beta\)	Assumption
r	Ratio of transmission rates for asympt. population against sympt. population	0.51	[12]
\(per_{a}\)	Percentage of ind. with COVID-19 who are asymptomatic	0.16	[12]
\(per_{s}\)	Percentage of ind. with COVID-19 who are symptomatic	0.84	[12]
\(\eta\)	Incubation length (days)	3.2 days	[12]
\(\lambda _{a}\)	Length of recovery for asymptomatic ind. (days)	3.5 days	[12]
\(\lambda _{s}\)	Length of recovery for symptomatic ind. (days)	7 days	[12]
\(\lambda _{q}\)	Length of quarantine (days)	14 days	[16]
h	hospitalization rate (ratio of quarantined pop, per day)	0.06	[20]
\(\lambda _{h}\)	Hospitalization length (days)	6 days	[16]
\(\kappa\)	Mortality rate for symptomatic pop. (per day)	0.0088	[12]
\(\kappa _{h}\)	Mortality rate for hospitalized individuals (per day)	0.074	[20]
g	Ratio of susc. who have fever for non-COVID infections (ratio, per day)	Estimated

Abstract

1 Introduction

1.1 Contributions

1.2 Organization of the Article

2 Relevant Literature

2.1 Epidemiological Modeling

2.2 Effects of Mitigation Policies

2.3 Effects of Diagnostic Testing

3 Proposed Approach

3.1 SIRTEM Model

3.1.1 Coupled Epidemic-Testing Single City Model.

3.1.2 Spatially Informed SIRTEM Model.

3.2 Optimal Testing

4 Evaluation

4.1 Single City Model Validation

4.1.1 Parameter Calibration.

4.1.2 Case Studies with Four U.S. States.

4.2 Optimal Testing Policies

4.3 Multi-city Epidemic Dynamics

4.3.1 Results with Mixing Rates Based on Worker Mobility Data.

4.3.2 Results with Mixing Rates Based on SafeGraph Mobility Data.

4.4 Ablation Studies

4.4.1 Impact of Testing.

4.4.2 Impact of FPS.

4.4.3 Impact of Spatial Modeling.

4.5 Comparisons Against Other Models

5 Conclusions

Footnote

Appendix

References

Cited By

Index Terms

Recommendations

Efficient COVID-19 testing via contextual model based compressive sensing

Effective Crisis Response to COVID-19 in Tourism and Hospitality: The Intersection of Crisis Leadership and Crisis Decision-Making

Microarchitectural synthesis for rapid BIST testing

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