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A SIRD model applied to COVID-19 dynamics and
intervention strategies during the first wave in Kenya
Wandera Ogana1, Victor Ogesa Juma1, and Wallace D. Bulimo2
1
School of Mathematics, University of Nairobi, Nairobi, Kenya
Department of Biochemistry, University of Nairobi, Nairobi, Kenya
2
Abstract
The first case of COVID-19 was reported in Kenya in March 2020 and soon after nonpharmaceutical interventions (NPIs) were established to control the spread of the disease.
The NPIs consisted, and continue to consist, of mitigation measures followed by a period of
relaxation of some of the measures. In this paper, we use a deterministic mathematical
model to analyze the dynamics of the disease, during the first wave, and relate it to the
intervention measures. In the process, we develop a new method for estimating the disease
parameters. Our solutions yield a basic reproduction number, R 0 = 2.76, which is consistent
with other solutions. The results further show that the initial mitigation reduced disease
transmission by 40% while the subsequent relaxation increased transmission by 25%. We
also propose a mathematical model on how interventions of known magnitudes collectively
affect disease transmission rates. The modelled positivity rate curve compares well with
observations. If interventions of unknown magnitudes have occurred, and data is available
on the positivity rate, we use the method of planar envelopes around a curve to deduce the
modelled positivity rate and the magnitudes of the interventions. Our solutions deduce
mitigation and relaxation effects of 42.5% and 26%, respectively; these percentages are
close to values obtained by the solution of the SIRD system. Our methods so far apply to a
single wave; there is a need to investigate the possibility of extending them to handle
multiple waves.
1 Introduction
Coronavirus Disease of 2019 (COVID-19) is the disease caused by the novel coronavirus
that appeared in Wuhan, China, in December 2019. The disease has since spread to all
parts of the world and resulted in 103,513,141 confirmed infections with 2,237,247 deaths by
31st January 2021 [1]. In Africa, the first reported case of COVID-19 was on 14th February
2020, in Egypt [2]. It has since afflicted 47all countries in the continent and led to 3,585,676
confirmed infections with 91,079 deaths to 31st January 2021 [1]. To control the spread of
the disease, countries have introduced several non-pharmaceutical interventions (NPIs) in
addition to strengthening health facilities and treatment regimes. The disease has wreaked
havoc on the economies of most countries and transformed people’s lifestyles permanently.
We briefly present pertinent biological information about COVID-19 and a related disease,
influenza, popularly known as flu. Our reference to influenza here will mean seasonal and
not pandemic influenza. In the early days of COVID-19 many people confused the disease
with flu since they both yield almost identical symptoms. On 11th February 2020, the virus
NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.
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which causes COVID-19 was named “severe acute respiratory syndrome coronavirus 2
(SARS-CoV-2” by the International Committee on Taxonomy of Viruses (ICTV) [3]. The flu is
caused by influenza virus types A, B and C. SARS-CoV-2 and influenza viruses use
different receptors to enter the host cell; the former uses the spike (S) protein for entry [4]
while the latter uses the hemagglutinin (HA) protein [5]. Both viruses are spread through
droplets released from the nose and mouth of an infected individual as they cough or sneeze
[6]. Nevertheless, unlike influenza, it has been shown that COVID-19 may also be spread by
the long-range airborne route at greater distances [7]. A critical determinant of the infectivity
of these viruses is the concept of reproduction number, R0 , which represents the degree of
transmissibility of the virus and provides a representation of how many people can be
infected by one person infected with the virus, in a population where everyone is susceptible
to the disease. In the early stages of the epidemic, the R0 for COVID-19 was estimated to be
2.2 – 2.5 [8]; since then most of the published estimates for R0 lie in the range 2 – 3, with
some as high as 6 and others as low as 1.9 [9,10]. For seasonal influenza, studies
yielded R0 in the range 1.1 – 1.5 [11], suggesting that COVID-19 is more easily spread than
seasonal influenza. For both flu and COVID-19, it is possible to spread the virus at least one
day before experiencing any symptoms. Once an individual has flu, the person may be
contagious for 5 – 7 days while for COVID-19 the person could be contagious for 10 – 14
days; the number of days of staying contagious will depend on the age of the patient,
severity of the disease and whether there are other underlying medical conditions [12, 13].
Comparison of mortality from flu and COVID-19 is generally problematic due to the
differences in data collection: in most countries deaths from COVID-19 must be recorded
while deaths from influenza do not have to be recorded and are usually estimated from
prevalence [14]. Research is currently going on to determine more accurate ways of
estimating mortality from COVID-19 but it is believed to be much higher than for seasonal flu
[13, 14].
Mathematical models can be used to inform and provide health decisions during a disease
outbreak; besides they can be used to predict and perform peak detection of infected cases
in a particular country [15]. A variety of models have been applied towards understanding
the dynamics of COVID-19 [16]. . These models can be broadly categorised into; Stochastic
type models [17-19] and deterministic type [15, 20]. In the deterministic model, the
population is usually divided into various compartments, namely Susceptible (S), infectious
(I), Recovered (R) and Dead (D). Some models use all the compartments and are labelled
SEIRD [21, 22]] while other models omit the Dead compartment and are SEIR [15, 23] or
omit the exposed compartment and are SIRD [24 - 26] or end up with the basic SIR
formulation by omitting the Exposed and Dead compartments [20, 27]. Irrespective of the
number of compartments used above, models can be modified to include additional
compartments like hospitalised, symptomatic, asymptomatic, among others [28 - 33]. In the
context of deterministic models, are the so-called meta-population models which are capable
of capturing the inherent heterogeneity of the populations, an aspect which cannot be done
by compartmental models [16, 34]. In the early days of COVID-19 experts wondered if the
disease would follow other virus pandemics and exhibit a second wave and possibly
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subsequent waves. Since then second waves have appeared in some countries on all
continents; a few countries have even experienced third waves. This development has
helped to promote mathematical modelling of the genesis and dynamics of second and
subsequent COVID-19 waves [35 - 38]. In this paper we will use the SIRD model applied to
a single wave.
In addition to mathematical models for the analysis of COVID-19 dynamics, there have
emerged mathematical models that address intervention measures designed to control the
rate of spread of the disease. As a starting point for these models, it is important to have an
understanding of the baseline dynamics and hence estimates of the parameters associated
with the unmitigated disease, since they are essential to our planning for initial interventions.
Two of these parameters are particularly useful, namely, the transmission rate, β(t), which is
the number of contacts per person per unit time, and the basic reproduction number, R0,
already defined earlier. In most situations, intervention commences almost immediately
COVID-19 emerges and the baseline parameters are estimated later, using data collected
from the period preceding any major mitigation measures. The main purpose of the
intervention is to reduce the contact rate, hence the reproduction number, so that the peak of
infection reaches a level that can be managed by the available healthcare facilities and
personnel. The mathematical models can broadly be classified into two categories. In the
first category, variables and parameters associated with interventions are incorporated into
the system of differential equations and hence they directly influence disease variables and
parameters [28 – 29, 31 – 33, 39 - 40]. Although they are mathematically rigorous and
elegant, they lead to more parameters that must be estimated; in addition, they tend to
account for the effect of only a few intervention measures, whereas the impacts of
interventions arise from all the measures taken together. In the second category, the
intervention measures collectively, or a subset of them, are deemed to affect the
transmission rate only, yielding expressions for the transmission rate which are piecewise
continuous functions involving exponential, logistic, linear or constant functions [21, 26, 37,
42]. In this paper, we use the methods in the second category, since they are flexible and
can easily be applied to develop scenarios, pending more rigorous investigation on the effect
of the interventions.
We present the paper according to the following outline.
In the next section we briefly describe the COVID-19 situation in Kenya and
present the major NPIs and the timelines in which they were proposed.
In Section 3, the SIRD model equations and initial conditions are given.
In Section 4, we introduce a new method of estimating the parameters associated
with the disease dynamics.
A description is given in Section 5 of a mathematical model for interventions that
takes into consideration the fact that interventions lead to a reduction of
transmission rate, in the case of mitigation, or increase in the transmission rate,
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when some mitigation measures are lifted or when society violates prescribed
mitigation regulations.
In Section 6.1 we present solutions of the SIRD model, taking into consideration
the mitigation and relaxation timelines.
Section 6.2.1 presents the results of the effect of intervention measures of known
magnitudes on the observed and predicted positivity rates.
Using the observed positivity rates, in Section 6.2.2, we apply the method of
planar envelopes around curves to deduce the model positivity trajectory,
together with the mitigation and relaxation magnitudes that lead to the observed
positivity rates.
Finally, we give a few concluding remarks and recommendations in Section 7.
2 COVID-19 in Kenya
The rapidly spreading outbreak of the novel coronavirus in the African continent prompted
the Kenya government to establish the National Emergency Response Committee on
Coronavirus (NERC) on 28th February 2020 by Executive order. About 2 weeks later, the
first case of COVID-19 in Kenya was confirmed on 13th March, 2020 [43] and has resulted in
100,773 confirmed infections and 1,763 deaths to 31st January 2021 [44]. Due to the rapid
increase in cases, the Government of Kenya instituted several measures designed to curb
the spread of the disease, while, at the same time, providing economic support to individuals
identified as vulnerable. COVID-19 has hurt the Kenyan economy and the livelihood of the
residents, despite the commendable steps taken by the government to alleviate the suffering
of citizens. Concern has been raised, however, on the declaration by public and private
insurers requiring COVID-19 patients to share in the cost of their diagnosis and treatment.
Given that the costs are beyond the reach of many Kenyans, there have been reports of
patients not seeking medical help at health centres for fear of being burdened with large
bills, which they and their families would be unable to pay.
For our modelling, the strategies pursued can be divided into three periods, each with its
distinct characteristics as outlined hereunder and also shown in Table 1.
Period 1 (13 March to 8 April): Since COVID-19 was a novel disease, this period was
spent in formulating policies and protocols on how to respond to it. A major decision was to
immediately close learning institutions. Other mitigation measures were put in place, for
instance: no overcrowding in public transport; social distancing and mask-wearing in public
places; handwashing and sanitization at malls and supermarkets. There was low compliance
and so a lot of time was spent appealing to residents to comply with the measures. In
anticipation of the potential impact of the pandemic, the government introduced a tax break
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CATEGORIES
PERIOD 1
13 March to 8 April
ACTIONS
PERIOD 2
9 April to 8 June
ACTIONS
PERIOD 3
9 June to 8 August
ACTIONS
1. CONGREGATIONS
Bars and Clubs
Closed on 22 March
Closed
Closed
Places of worship
Closed on 22 March
Closed
Open from 6 July for 100 attendants
Funerals and Family
gatherings
Limited numbers; social
distancing and hygiene
Limited numbers; social distancing
and hygiene
Limited numbers; social distancing
and hygiene
Political and Social
gatherings
Banned
Banned
Banned
Restaurants and Eateries
Open for limited hours for takeaway Operate with social distancing and
Limited numbers; no alcohol from 31
Work from home
meals
Compliance encouraged
hygiene
Compliance encouraged
July
Compliance encouraged
Closed
Closed
Closed
Closed
Closed
Closed
None
For Nairobi, Mombasa, Kwale, Kilifi
Lifted in 2 stages: 7 June and 7 July
per session
2.
LEARNING
INSTITUTIONS
Schools
Tertiary Institutions
3.
RESTRICTION OF
MOBILITY
Cessation of Movement
from 6 April
Curfew
Lockdown
Country-wide overnight curfew from Country-wide overnight curfew still in Country-wide overnight curfew still in
27 March
force
None
For Eastleigh, Old Town Mombasa & Lockdown lifted on 7 June
force
Mandera
4.
PREVENTION
Covid-19 Regulations
None
Published; criminal offence to
contravene
Criminal offence to contravene
Public social distancing
Public mask-wearing
Public Hygiene
Compliance encouraged
Compliance encouraged
Compliance encouraged
Compliance mandatory
Compliance mandatory
Compliance mandatory
Compliance mandatory
Compliance mandatory
Compliance mandatory
International air travel
A few allowed initially, later all
suspended
Suspended
To resume on 1st August
Local air travel
Suspended 2 April
Suspended
Resumed on 7th July
Public transport (within the
county)
To operate with social distancing & To operate with social distancing &
To operate with social distancing &
hygiene
hygiene
5.
TRAVEL
Public transport (intercounty)
hygiene
To operate with social distancing & None to/from counties on cessation of To operate with social distancing &
hygiene
movement
hygiene
Support for Vulnerable
families
Plans to support the vulnerable
Money sent directly to vulnerable
families
Money sent directly to vulnerable
families
National Hygiene
Programme (Kazi Mtaani)
Plans to support youth
Payment to youth for restoring public
Payment to youth for restoring public
hygiene
hygiene
Implemented
Implemented
6.
ECONOMIC
INCENTIVES
General economic stimulus Announced
Table 1: Strategies for mitigating COVID-19 in Kenya
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to provide some relief to residents. Despite the enacted mitigation measures, in most places
it was business as usual. This period, therefore, can safely be regarded as the time when
covid-19 was still unmitigated. Hence our model treats this period as the baseline, since no
significant impact of mitigation measures had been realised.
Period 2 (9 April to 8 June). Due to the rapid increase in cases, and the public’s relaxed
attitude to COVID-19, the government took a move to enforce the mitigation measures by
enacting COVID-19 regulations whose contravention was a criminal offence [45]. It also
published proto-cols to govern the operations of restaurants and eateries. To provide an
economic incentive to a category of residents the government implemented two programmes
to help the vulnerable and youth. The period can be regarded as one of the application of
mitigation measures, despite attempts by a cross-section of society to flout the rules.
Consequently, our model treats this period as that of mitigation.
Period 3 (9 June to 8 August): Enforcement of the mitigation measures during Period 2
had a devastating effect on the country’s economy and people’s livelihoods. Many industries
and small businesses laid off workers or simply folded. Other establishments placed workers
on half salary or gave leave without pay while waiting for the situation to stabilize. To ease
the hardship being experienced, the government gradually relaxed some of the mitigation
measures. There was discussion about opening learning institutions in September but the
idea was shelved on based on the trend of the pandemic. Our model treats this period as
that of gradual relaxation of control measures.
Table 1 gives a summary of the major actions taken during each of the three periods. The
information in this table was obtained from the Ministry of Health, Kenya [44] and
Presidential Addresses on COVID-19 [46]. Most of the information was also available from
Academia Kenya [47] and in the dailies .
Figure 1 shows the trend of the 7-day moving averages of numbers and percentages of
three para-meters, that is, infections, deaths and recoveries, from 13th March 2020 for
numbers and 25th March 2020 for percentages, to 31st January 2021. The percentages
were of the numbers of daily observed variables relative to the daily number of people tested
on that day. It is argued, with some justification that more realistic percentages can be
obtained by computing the percentages relative to test numbers lagged by a week or two,
since people do not necessarily get infected, recover or die on the same day they are tested.
In Figure 1(a), the infections increase gradually and reach a maximum in mid-July, with a
positivity rate of just under 16%. The infections then decrease until early September when
they begin to increase thus indicating the onset of the second wave, which peaked in early
November with a positivity rate of about 18% and prompted fresh mitigation measures to be
put in place from 4th November 2020. Figure 1(b) shows that recoveries also follow the
wave pattern of the infections but there exist considerable fluctuations, with significant
spikes, possibly due to accumulated data not accounted for in previous days. The
fluctuations and spikes can also be partly attributed to uncertainty in obtaining data on
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recoveries from home-based care which was in effect from July 2020. Figure 1(c) shows
that the deaths also exhibit the wave pattern of the infections but they fluctuate considerably
but are generally on the low side, with the 7-day averages not exceeding 25 in numbers or
0.4 in percentages.
(a)
(b)
(c)
Figure 1: The 7-day moving averages of numbers and percentages of three variables, from 13th March
2020 for numbers and 25th March 2020 for percentages, to 31st January 2021.
3 SIRD model formulation
In this article, we consider a SIRD mathematical model. We assume a homogeneous mixing
in the population. At the time, t, the population is divided into four classes; Susceptible,
infectious, recovered and the dead, denoted respectively by, S(t), I(t), R(t) and D(t), as shown
in Figure 2 Since this is a new disease, there is no prior immunity, hence everybody is
susceptible to COVID-19. Upon being infected with the disease, susceptible individuals
move to the infectious class, from which they either recover or die from the infection.
R
I
S
SI
I
I
D
Figure 2: Compartmental SIRD model.
We assume that the total population, N is constant over time. For simplicity we assume that
the variables are already normalised on division by N such that
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S (t ) I (t ) R(t ) D(t ) 1
(1)
In the presentation of our results, the variables will be given as convenient in terms of actual
observed numbers or proportions or percentages or as fractions of the total population.
The mathematical equations describing the movement of individuals in different
compartments are given by:
dS / dt SI
(2a)
dI / dt SI ( ) I
(2b)
dR / dt I
(2c)
dD / dt I
(2d)
The system in Equation (2) is solved subject to the initial conditions: S(0) = S0, I(0) = I0, R(0)
= R0 and D(0) = D0, where S0, I0, R0 and D0, are the initial proportions of the Susceptible,
Infectious, Recovered and Dead, respectively. At the very start of the epidemic, there is one
infected individual so that I0 = 1/N. At this stage there are no recoveries or deaths so that
R0 = D0 = 0 and Equation (1) yields S 0 = 1-1/N. The SIRD model has been applied to
seasonal influenza [48 - 50] and COVID-19 [20, 24 – 26], among other publications.
4 Parameter estimation
To solve the system in Equation (2) subject to appropriate initial values, it is necessary to
compute the parameters γ, β and δ. This is usually by use of optimization software [51 –
53], preceded by specifying approximate values for the parameters, or the interval in which
the parameter values lie, or by use of Monte-Carlo methods to select initial values of the
parameters at random and facilitate computation of the optimum values using methods
based on diverse mathematical concepts. The process sometimes is long and may require
hundreds, if not thousands, of iterations. In this paper, we propose a new approach to
determining initial estimates of the parameters.
Observational or experimental values are usually available at discrete points in time
denoted, t0, t1, t2, · · · , tmax, where tmax is the maximum time for which the disease data is
available. Starting from the initial values, S(0) = S0, I(0) = I0, R(0) = R0 and D(0) = D0, we
find that at the current iteration, tk, Equation (1) yields
Sk I k Rk Dk 1, k 0,1,2, , kM
(3)
where S k S (t k ), I k I (t k ), Rk R (t k ) and Dk D (t k ) and kM is the maximum number of
time steps or nodes.
Given the significance of infection in understanding the dynamics of the disease, data on
the infections, Ik, is often available to a considerable degree of accuracy and reliability.
Whereas data on the deceased, Dk, and the recovered, Rk, may be available, it is often less
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reliable, particularly in countries with limited health facilities. In situations where only data
on the infections is given, it is possible to estimate the recovered and deceased by
integrating Equations (2c) and (2d) to yield:
Rk Rk 1 I k 1
(4a)
Dk Dk 1 I k 1
(4b)
where γ and δ are the recovery and death rates, respectively, and are to be estimated.
From Equation (4a) using k = 1 yields R1 I 0 , since R0 0 . Hence R1 is dependent on the
observed value I 0 . Using k = 2, we conclude that R2 is dependent on the observed values
I 0 and I1 . We can show that in general, Rk is dependent on the observed values
I 0 , I1 , I 2 , , I k 1 . Analogously, we can show that Dk is dependent on the observed
values I 0 , I1 , I 2 , , I k 1 . Although the quantities Rk and Dk are not observed values, they are
dependent on observed infections and, to meet our computational objectives, they are
regarded as “observed recovered fraction” and “observed dead fraction”, respectively.
Similarly, values of the susceptible fractions, Sk, are rarely availed, except in a highly
controlled environment, for instance, a school setting [54]. Despite this, by solving for Sk in
Equation (3), we obtain a value of the susceptible which is dependent on observed infections
and will also be regarded as “observed susceptible fraction”.
To illustrate our approach for estimating the parameters, we assume that the deceased
values, Dk are given and the death rate, δ, is approximated by the Case Fatality Rate (CFR)
or any other suitable method. Consequently, we consider Equations (2) and (4a) only and
estimate γ and β. Every disease has associated with it a number of days of recovery, i.e. the
days for which the patient remains contagious after being diagnosed with the disease; the
inverse of those days is the recovery rate, γ. For instance, the recovery days from influenza
are on average 5 – 7 days [12 – 13]; hence for influenza is on average 1/7 to 1/5. The
recovery days from mild to moderate COVID-19 are on average 10-14 days [12 - 13]; hence
for this type of COVID-19 is on average 1/14 to 1/10. To begin our computation, we
assume that the recovery days for COVID-19 do not exceed dM days, where dM is an integer
that is large enough to preclude any medical evidence that a patient of COVID-19 could still
be contagious beyond d M days, for instance 100. Hence the recovery rate, , will be in the
interval (1/dM , 1). We now establish grid point, or nodes,
i , in the interval (d M ,1)
such
that
i (1 / d M ,1),
for i 1, 2, , iM ,
where iM is the maximum number of node pertaining to
(5)
.
At the time step k, insert γi in Equation (4a) and then Equation (4a) into Equation (3) to find
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Sk I k Rk Dk 1 ki , k 1,2,3, , kM
(6)
where ϵki is the error, at the grid point tk , associated with observation and also with the value
γi If ϵki is close to zero then γi is close to the recovery rate for the disease.
The transmission rate, β(t), for COVID-19 is a finite non-negative number, and must
therefore be such that 0 < β(t) < βM , where βM is a positive number and is chosen large
enough to include the highest possible transmission rate of COVID-19. We then establish
grid points, or nodes, βj , in the interval (0, βM) such that
j (0, M )
where
for
j 1,2, , jM
(7)
jM is the maximum number of nodes pertaining to .
We now use γi from Equation (5) and βj from Equation (7) together with the value of δ
estimated from CFR, to solve the system (2), at the time step k. This will yield values which
we denote Sc, Ic, Rc and Dc, where the superscript signifies values from computation to
distinguish them from observed values S k , I k , Rk and Dk . Analogous to Equation (6), the
values from computation satisfy
Skc I kc Rkc Dkc 1 kij , k 0,1,2, , kM
(8)
where ϵkij is the error, at the grid point tk , associated with observation together with the
values γi and βj . If ϵkij is close to zero then γi and βj are close to the recovery and
transmission rates, respectively.
Subtracting Equation (6) from (8) yields
(Skc Sk ) ( I kc I k ) ( Rkc Rk ) ( Dkc Dk ) k , k 0,1,2, , kM
(9)
where ϵk is the total error at the time step tk. We note that each of the terms in brackets on
the LHS is the error in the corresponding variable.
The disease variables are nonlinear functions of the parameters γi and βj , hence it is not
advisable to compute the correlation coefficients between the computed (predicted) and the
observed variables to determine how closely the computed and observed values agree.
Hence we use an error metric based on the time average of the individual errors and we
choose to minimize the error metric. There are a number of possible error metrics but we
choose to use the Root Mean Square Error (RMSE), although we could also have used the
Mean Absolute error (MAE) [55]. For RMSE we determine i and j to obtain
min RMSE (S RMSE ( I ) RMSE ( R) RMSE ( D)
( , )
where for the variable X,
10 | P a g e
(10)
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k
RMSE ( X ) SQRT k 1 ( X kc X k ) 2 / kM
Starting from initial values of
M
(11)
i and it is possible to institute search procedures that will
j
lead to the minimum in Equation (10), for instance [51 – 53]. The amount of effort in
reaching this minimum will depend on how close the initial values are to the actual solutions.
To narrow down the intervals of the search, we use a different approach based on the fact
that the disease parameters, as used in the formulations leading to Equation (2), are spatial
and temporal averages. We begin by determining the maximum sum inside the curly
brackets in Equation (10). We then express the sum at each point as a percentage of this
maximum sum; consequently, the minimum values will be associated with the least
percentages. Thereafter, we average the disease parameters within regions bounded by
concentric circles whose radii are the percentages. The regions are expanded from the
innermost circle, associated with the least percentage, to larger circles associated with
higher percentages, while keeping track of the sum in Equation (10). Initially this sum is large
but decreases as we expand the percentage of averaging; eventually the sum reaches a
minimum and begins to increase again as the percentage of averaging continues to
increase. The minimum in Equation (10) is at values of the parameters associated with the
turning point. To ensure that we identify all the minima, it is important to start from the least
percentage and proceed till 100%, or close to it. This way it will be possible to identify which
minima are local and which one is global. The best that can be done with this averaging
technique is to obtain intervals within which lie isolated values of γi and βj that lead to the
suspected minimum of the error matrices in Equation (10). Thereafter, other search and
optimization procedures can be applied to obtain more accurate parameter values, if need
be [51 – 53]. Here again, we propose a slightly different approach: simply divide the interval
so identified into smaller subintervals to establish a denser network of grid points. We then
go through the computations leading to the sum in Equation (10) for all paired values of i
and j . This time, we use a procedure that searches for a minimum value from an array and
identifies the associated
i and . The new subinterval, within which lie the appropriate i
j
and j can again be divided into smaller subintervals and the process repeated until
required accuracy in
i and is obtained.
j
5 Modelling intervention
In this paper we formulate an intervention model which leads to piecewise exponential
functions for the transmission rate. We assume that the recovery rate, , and the death
rate, , do not change during the interval of the intervention. Our model takes into account
the fact that intervention not only leads to a reduction of the transmission rate, through
mitigation, but can lead to a surge in the transmission rate, through relaxation of, or noncompliance with, mitigation measures.
Let the daily events be at the time nodes denoted t0, t1, t2, · · · . Suppose intervention is
initiated at the time node tk then there will be a difference in the transmission rate before and
after tk . Let βb(t) be the transmission rate before, and up to, the time tk; the quantity βb(t)
could be the result of baseline dynamics or it could be due to the dynamics from some
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immediately preceding intervention. Associated with βb(t) is a reproduction number which we
label Rb(t). If this Rb(t) is due to baseline dynamics then it is the basic reproduction number
R0 ; otherwise it is an effective number, Re,
We now assume that for any time t > tk , the rate of decrease of the transmission rate, as a
result of intervention measures, is proportional to the transmission rate at that time. This
yields the following general solution for the effective transmission rate;
(t ) Aeb (t t ) , t tk
k
(12)
The main objective is to gradually change the transmission rate at the time of intervention,
namely, βb(tk), by a fraction c so that the effective transmission rate at a future time, say tk+m,
where m > 0, becomes (1 − c)βb(tk). To determine the constants A and b in Equation (12), we
impose the conditions
(tk ) b (tk ) and
(tk m ) (1 c) b (tk )
(13)
We need to consider what happens after the objective of the intervention has been met,
namely for t > tk+m . It is reasonable to assume that after the objective of a particular
intervention has been met, the transmission rate will remain constant at the level already
achieved, until another intervention takes place. The choice of m which enables this to be
achieved will depend on the implementation goals. When an intervention takes place on day
tk, the optimum transformation of the transmission rate, hence of the reproduction number
also, does not occur instantly but takes place say m days later, that is, at the time tk+m, where
m > 0. In fact it is a common observation that following an event that spreads COVID-19, an
increase in infections is usually observed about 7 to 14 days later. It is, therefore reasonable
to select m in, or close to, the interval 7 to 14 days. Using Equation (13) and the explanation
for what happens beyond tk+m, we obtain the following expression for the transmission rate,
before and after the intervention.
t tk
b (t ),
(t ) b (tk )e g ( t ) , tk t tk m
(1 c) (t ), t t
b k
k m
where
g (t )
(t t
k
)ln(1 c) / (tk tk m )
c 1
(14)
(15a)
(15b)
The last equation in Equation (14) gives the optimum value of the effective transmission rate
that will be achieved due to intervention. At any given time, t, the effective reproduction
number, Re, is computed from
Re (t ) / ( )
(16)
From the last equation in Equation (14) we note that when 0 < c < 1, then β(tk+m) < βb(tk) ;
this corresponds to the intervention being a mitigation, since it yields a smaller future
transmission rate which represents a reduction by a fraction c of the transmission rate at the
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time of intervention before mitigation occurred; we call the quantity 100c the “percent
mitigation”. On the other hand when c < 0 then β(tk+m) > βb(tk); this corresponds to the
intervention being a relaxation of the existing mitigation measures since it yields a larger
future transmission rate which represents an increase by a fraction
c
of the transmission
rate at the time of intervention; we call 100 c the “percent relaxation”. In computing Re
the values of γ and remain fixed and only the value of β changes. Consequently the
percentage changes in β, as indicated above, also apply to Re. Previous researchers
restricted c to the interval [0 , 1]; by Equation (15b), we extend c to negative values to
account for spikes in the dynamics that may occur as a consequence of relaxing mitigation
measures.
The parameter c is important for this type of intervention modelling and yet only two of its
values are obvious, namely, c = 0 implies the absence of any control and c = 1 is the unlikely
scenario of absolute control where there is no disease transmission. The other values of c
are more complicated to determine. The most thorough method is to identify all the
interventions that impact on the disease transmission, hence contribute to c, and assign
weights to their impacts. The parameter c can then be computed as the weighted average of
the impacts. Groups of the impacts could then be isolated to obtain their relative contribution
to the decrease or increase in the disease transmission. Table 1 lists some of the
intervention measures and related effects that could be taken into account in this exercise:
mask-wearing, closure of learning institutions, curfews, travel restrictions, limitations on
gatherings, restrictions on operations of bars and restaurants, economic activities, social
distancing, availability of PPEs and hospital space etc. To assess the impact of all these
factors on disease transmission requires a truly collaborative effort involving a
multidisciplinary team. Once the value of c is estimated, it can be described in simple terms
for public health implementation. For instance, two intervention scenarios are mentioned in
[23], namely: moderate lockdown, regarded as the intervention which reduces transmission
by 25% during lockdown followed by transmission at 90% of the pre-lockdown value; and
hard lockdown, regarded as the intervention which reduces transmission by 44% during
lockdown followed by transmission at 90% of the pre-lockdown value. In terms of our
formulation, moderate lockdown is equivalent to mitigation with c = 0.25 followed by
relaxation with c = −2.6 while hard lockdown is equivalent to mitigation with c = 0.44 followed
by relaxation with c = −1.05.
6 Results
The models were developed and computations carried out, per timelines associated with
government and public response to COVID-19, as specified in Section 2. We first present
results for solutions of the SIRD system using the new method described in Section 4; then
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we present results for the intervention model described in Section 5. COVID-19 data was
obtained from the following sources: Worldometer [1], Ministry of Health, Kenya [44], World
Health Organization [56], and Our World in Data [57].
6.1 Solution of SIRD system across different intervention periods
In this section we solve the system of equations using the methods described in Section 4
for the periods identified in Section 2: Period 1 (Baseline), Period 2 (Mitigation) and Period 3
(Relaxation). The objective is to determine whether there are any differences in the disease
parameters among the three periods. In Table 2 we list the values of parameters and initial
conditions used during various periods of computation.
Quantity
Symbol
Population
Initial Infections
Initial Recovered
Initial Deaths
Initial Susceptible
Number of days
Maximum time nodes
N
I(0)
R(0)
D(0)
S(0)
------
Maximum
days
of
recovery
Least value of (=1/dM)
No. of nodes for
Subintervals for
Baseline
Value
5586
0.1790190E-04
0
0
0.999982098
27
27
Mitigation
Value
100683
0.9932163E-05
0
0
0.999990067
88
88
Relaxation
Value
353727
0.2827039E-05
0
0
0.999997173
149
149
dM
100
100
100
M
0.01
0.01
0.01
------
1000
1000
1000
iM
1000
1000
1000
kM
Largest value of
M
1
1
1
Subintervals for
-----
750
750
750
jM
750
750
750
No. of nodes for
Table 2: Parameters and initial conditions at baseline, mitigation and relaxation periods
For all the periods, we assumed that covid-19 patients are unlikely to be contagious after
100 days, so that dM = 100, hence (0.01,1) . We also assumed that (0,1) for covid19, since the choice of a larger upper limit would not make any difference to the results. To
obtain the initial estimates of and , by the averaging method described in the last
paragraph of Section 5, the intervals (0.01 , 1), for , was divided into 1000 subintervals
while the interval and (0 , 1), for , was divided into 750 subintervals, and a node located
in the middle of each subinterval. On identification of the appropriate initial estimates, the
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subintervals in which they lay were each further divided into 400 subintervals, to establish
400 nodes, and the MIN function in MATLAB was used to solve Equation (10). Further
subdivision was not required, since the values obtained agreed to 5 significant figures.
(a) Baseline Solutions
For results involving the baseline, computation is carried out from 13th March to 8th April,
2020, with projections made till stability is reached. Use is made of the parameters and initial
conditions given in the baseline column of Table 2. Computed values of and
, are given,
to 5 significant figures, in the baseline row of Table 3. Also given are related parameters
and results. The basic reproduction number is 2.76 which is consistent with results from
other computations, e.g. [23].
Baseline
Mitigation
Relaxation
γ
CFR (δ)
0.051750
0.015
0.055294
0.0344
0.015099
0.0297
β
R
0.18448 2.7638
0.14852 1.6558
0.092675 2.0687
1/γ
19.3
18.0
66.2
Peak
infection
0.270
0.0916
0.1655
RMSE sum
0.00255
0.000570
0.000804
Table 3: Parameters and related quantities for different disease periods. Recovery rate is denoted γ,δ
denoted case-fatality proportions(CFR), β, transmission rate and 1/γ denotes recovery days, R=R0,
basic reproduction number (for Baseline) and R=Re, effective reproduction number (for mitigation and
relaxation).
Figure 3(a) shows the observed and computed cumulative infection numbers and
percentages during the baseline period; the agreement is quite good. Although results are
available for the parameters S, I, R and D, we present results only for I, since it is the most
important variable we use in our subsequent analyses.
(a)
(b)
(c)
Figure 3: Observed and computed cumulative infected proportions and numbers at baseline (3(a)),
observed and computed cumulative infected proportions and numbers up to mitigation period (3(b))
and Observed and computed cumulative infected proportions and numbers up to relaxation period
(3(c)).
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Figure 4 shows the graphs for infection percentages obtained using data up to the end of the
baseline, mitigation and relaxation periods. The graph labelled baseline dynamics shows the
infection percentages associated with the baseline data. It achieves a peak of approximately
27% in late May 2020, thus indicating the way the dynamics would have proceeded if
mitigation measures were not put in place on 8th April 2020.
Figure 4: Infection proportions during baseline, mitigation and relaxation periods.
(b) Mitigation Solutions
For results involving mitigation, computation is carried out from 13th March to 8th June. The
objective of the computation is to find out whether the inclusion of data during the mitigation
period made any difference in the rate of transmission of the disease and the dynamics of
the infection. Use is made of the parameters and initial conditions given in the mitigation
column of Table 2. Computed values of and
, are given, to 5 significant figures, in the
mitigation row of Table 3. Also given are related parameters and results. The results
indicate that the reproduction number was reduced by about 40% from 2.76 at the baseline
to 1.66 at mitigation. These results indicate that the mitigation measures that were put in
place on 8th April helped to reduce the rate of spread of the disease. Other assessments
showed, however, that the gains in controlling the disease were met at a considerable
economic and social impact on the country. In Figure 4 the graph labelled mitigation
dynamics shows the infection percentages associated with data up to the end of the
mitigation period. This graph peaks at 9.2% thus indicating that the mitigation measures put
in place from 9th April to 8th June had reduced the peak infection to a level more
manageable for healthcare. Figure (3b) shows the observed and computed cumulative
infection numbers and percentages during the mitigation period; the agreement is quite
good.
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(c) Relaxation Solutions
For results involving the relaxation, computation is carried out from 13th March to 8th
August. The objective of the computation is to find out whether the inclusion of data during
the relaxation period made any difference in the rate of transmission of the disease and the
infection dynamics. Use is made of the parameters and initial conditions given in the
relaxation column of Table 2. Computed values of and
, are given, to 5 significant
figures, in the relaxation row of Table 3. Also given are related parameters and results. The
results indicate that the reproduction number increased by about 25% from 1.66 at mitigation
to 2.07 at relaxation, thus showing that relaxation of mitigation measures led to an increased
rate of transmission of the disease. Figure 3(c) shows the observed and computed
cumulative infection numbers and proportions during the relaxation period; the agreement is
good but not as well as in the previous two cases. In Figure 4 the graph labelled relaxation
shows the infection percentages, associated with the relaxation period, which peak at about
16.6%. The implication is that the lifting of mitigation measures on 8th June, increased the
peak infection to a level that could put more pressure on healthcare facilities, although not as
much as would have happened with unmitigated disease. These results indicate that lifting
the mitigation measures on June 8th subsequently placed an increased disease burden on
society despite the temporary relief from the adverse economic and social effects due to
mitigation. Figure 4 shows that if mitigation measures were not lifted, the disease would
virtually disappear by January 2021 but relaxation shifts the disease wave and makes it
disappear in mid-2021.
6.2 Results from modelling interventions
In the previous subsection, we presented results based on the solution of the SIRD system
of equations, in which interventions had taken place and we aimed to determine how much
they affected the transmission rate, reproduction number, infection rates and other variables.
In this subsection, we would like to solve the following two problems:
Given the magnitudes of the mitigation and relaxation measures, determine their
effects on the disease dynamics, especially the positivity rates.
Given the positivity rate curve, determine the magnitudes of the mitigation and
relaxation measures that could have resulted in the curve.
A solution to both problems requires the application of the methods described in Section 5.
It necessitates setting up scenarios hence the results are largely qualitative and yield only
broad outcomes for use in the initial planning and further investigations. We make the
following assumptions:
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(i) The largest portion of intervention measures is put in place at the beginning of the
intervention period, namely close to 8th April 2020 for mitigation and 8th June 2020 for
relaxation. Inevitably, there will subsequently be minor adjustments to the interventions
which, if they were significant enough, would be regarded as independent interventions in
their own right.
(ii) The effect of the intervention, resulting in the optimum values of the transmission rate
and effective reproduction number, is not achieved immediately but takes some days,
normally up to 21 days but mostly between 7 to 14 days.
(iii) Once the optimum values of the transmission rate and effective reproduction number
are achieved, they govern the disease dynamics till another intervention takes place, either a
mitigation or a relaxation.
6.2.1 Determination of positivity rates given intervention magnitudes
A solution to the problem here requires information on the magnitudes of the intervention
measures, hence appropriate values of the parameter c to be used in Equation (14). In case
there is a good accounting of the factors that contribute to the disease transmission rate, and
their relative effects, we can readily determine the parameter c as indicated in Section 5. If
necessary, the parameter c can be determined by solving the SIRD model as done in
Section 6.1. Ideally, this problem should be solved a priori, namely before any interventions
are put in place, so that the outcomes can be used to plan for the interventions. After
interventions have occurred, the problem can be solved to learn what action ought to have
been taken.
(a) Results from modelling mitigation
As pointed out earlier, the first incident of the disease in Kenya was on 13th March 2020. If
the disease was left unmitigated it would have spread as indicated in the blue baseline
dynamics curve in Figure 4. The infection would have peaked in late May 2020 and virtually
disappeared by early October 2020. The associated parameters and quantities are indicated
in the baseline row of Table 3. These values indicate that slightly over a quarter of the
population would have been infected at the peak of the disease, a fact which would have
placed a considerable burden on health facilities. Mitigation was consequently effected on
8th April 2020 and had the effect of slowing down the spread of the disease.
To study the effect of different mitigation strategies during the period 8th April to 8th June
2020, we can develop scenarios by varying the parameter c in Equation (15). Since the
mitigation is on the baseline dynamics, the transmission rate up to the time of the mitigation,
βb(t), must be the baseline transmission given in the baseline row of Table 3, namely, βb(t) =
0.1845. Using this value of βb(t) in Equation (14), with m = 15, and drawing the curves
associated with different values of c, we obtain Figure 5. For a given value of c the curve
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shows the trajectory of the infection as a result of a one-time mitigation force on 8th April
2020, with no other subsequent interventions. It can be seen that
Figure 5: Infection percentages for various mitigation scenarios.
the infection peaks reduce as the value of c increases and they tend to occur later, meaning
that the more stringent the mitigation measures, the lower will be the peak infections but they
will occur at a later time.
Table 4 shows how varying the parameter c affects the transmission rate, the effective
reproduction number, from Equation (16), and the infection. It is noted that as c increases,
the peak infection and the effective reproduction number decrease. Further increase shows
that for c = 0.8, the peak infection is 1.2% and Re = 0.55 < 1, meaning that there is no
disease spread. This situation represents taking mitigation action which is so drastic that the
disease is suppressed; the consequences to society for such action can be grave and so
disease suppression is not a practical option. Table 4 also gives the infection at the end of
the mitigation period, namely 8th June 2020. It enables the planner, who on 8th April 2020 is
proposing a 2-month mitigation action, to forecast the infection percent on 8th June 2020,
depending on the force of the mitigation.
c
% change
in
b
Re
% peak
infection
% infection
8 June
0
0.1
0.2
0
-10
-20
0.1845
0.1660
0.1476
2.76
2.49
2.21
27.1
23.3
19.1
24.1
23.0
18.7
0.4
0.6
-40
-60
0.1107
0.0738
1.166
1.11
9.6
1.3
6.6
1.2
0.8
-80
0.0369
0.55
0.71
0.16
Table 4: Scenarios involving changes in the parameter c under mitigation from baseline. The following
values are used βb = 0.1845, Rb = 2.76, γ = 0.0518 and δ = 0.015
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A question of interest is: Can we quantify the impact of the mitigation measures effected
from 8th April to 8th June? To answer this question a priori we would have to estimate,
before 8th April 2020, the value of the parameter c as described in Section 5. We can also
answer this question a posteriori, namely, after 8th June 2020, by making use of the results
in Section 6.1 where we solved the system of equations from the onset of the disease up to
the end of the mitigation, namely, from 13th Mach 2020 to 8th June 2020. Table 2 shows
that from the baseline period to the mitigation period the basic reproduction number
decreased from 2.76 to the effective reproduction number of 1.66. This change represents a
decrease of approximately 40% between the two periods. It is logical, therefore, to take c =
0.4 as the mitigation parameter in Equation (14) and the trajectory of the infection would be
the curve identified by 40% mitigation in Figure 5.
(b) Results from modelling relaxation
As a result of the adverse effects of mitigation, the government decided to relax some of the
mitigation measures from 8th June to 8th August 2020. This is the period we regard as the
“relaxation period” in our model; it is characterised by an increase in the transmission rate,
reproduction number, and infection. If we have a good accounting of the relaxation factors
that contribute to the disease transmission rate and can estimate their relative effects, we
can determine the parameter c to be used in Equation (14), as described in Section 5; the
values of c will be negative, since they imply an increase in transmission rate above
reference values. The scenarios cannot, however, be based on the baseline dynamics; they
must start from an appropriate mitigation scenario which ended on 8th June 2020. We have
established in the previous paragraph that such a scenario corresponds to mitigation with c
= 0.4. Consequently, to study the effect of different relaxation strategies from 9th June to 8th
August 2020, we develop scenarios by varying the parameter c, taking into consideration the
fact that the scenarios are dependent upon 40% mitigation , as shown in Figure 6.
Since the relaxation is on the 40% mitigation of the baseline dynamics, the transmission rate
up to the time of the relaxation, βb(t), must be the mitigation transmission corresponding to c
= 0.4 in Table 4, namely, βb(t) = 0.1107. Using this value of βb(t) in Equation (14), with m =
15, and drawing the curves associated with different values of c, we obtain Figure 6. For a
given value of c the curve shows the trajectory of the infection as a result of a one-time
relaxation force imposed, on 8 th June 2020, upon a 40% mitigation force, with no other
subsequent interventions. It is seen that the infection peak increases with decrease in c, or
increase in its magnitude, and relaxation percentage.
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Figure 6: Infection percentages for various relaxation scenarios on 40% mitigation.
Table 5 shows how varying the parameter c affects the transmission rate, the effective
reproduction number, from Equation (16), and the infection. It shows that continued
decrease in c, or increase in relaxation percentage, leads to a situation where the effective
reproduction number becomes 2.99 (for c = -0.8 or 80% relaxation) and thus exceeds the
basic reproduction number of 2.76. This reflects the fact that rapid lifting of mitigation
measures can result in an outbreak of faster spreading covid-19, as has been reported in
several countries since the outbreak of the disease. Public health advice is that mitigation
measures should not be relaxed too rapidly. Table 5 also gives the infection at the end of the
relaxation period, namely 8th August 2020; it enables the planner, who on 8th June 2020 is
proposing a 2-month relaxation action, to forecast the infection percent on 8th August 2020,
depending on the force of the relaxation.
% change
c
0
-0.1
-0.25
-0.4
-0.6
-0.8
in
b
0
10
25
40
60
80
0.1107
0.1218
0.1384
0.1550
0.1771
0.1993
Re
1.66
1.82
2.07
2.32
2.65
2.99
% peak
infection
% infection
8 August
9.6
11.1
13.6
16.2
19.5
22.6
6.6
7.7
8.8
9.2
9.1
8.5
Table 5: Scenarios involving changes in the parameter c under relaxation. The following values are
used βb = 0.1107, Rb = 1.660, γ = 0.0518 and δ = 0.015.
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A question similar to what was raised in the mitigation assessment is the following: Can we
quantify the impact of the relaxation measures effected from 8th June to 8th August? To
answer this question a priori we would have to estimate, before 8th June 2020, the value of
the parameter c as described in Section 5. We can also answer this question a posteriori,
after 8th August 2020, by making use of the results in Section 6.1 (c), where we solved the
system in Equation (14) from the onset of the disease, namely 13 th March 2020, through the
mitigation period, till the end of the relaxation period, namely, 8 th August 2020. Table 2
shows that from the end of the mitigation period to the end of the relaxation period, the
effective reproduction number increases from 1.66 to 2.07. This change represents an
increase of approximately 25% between the two periods and hence it is logical to take c =
−0.25 as the relaxation parameter in Equation (14) and the trajectory of the infection would
be the curve identified by 25% relaxation in Figure 6.
(c ) Combination of mitigation and relaxation effects
To assess the effect of interventions during the first wave of COVID-19 in Kenya, we trace
the trajectory of infection, taking into consideration the impact of the mitigation and relaxation
measures implemented by the Kenya government. The trajectory consists of three distinct
parts as indicated hereafter.
Baseline trajectory: This consists of the curve labelled “No mitigation (c=0)” in Figure 5.
This curve is reproduced in Figure 7 (black curve) and is shown in two parts: the continuous
portion, from 13th March to 8th April 2020 shows the percent infection trajectory during the
baseline period and the dotted portion, after 8th April, indicates the projected trajectory in the
absence
of further interventions. As we shall see later, this projected trajectory is not followed due to
subsequent mitigation and relaxation actions.
Mitigation trajectory: This consists of the curve labelled “40% mitigation (c=0.4)” in Figure
5. This curve is reproduced in Figure 7 (blue curve) and is shown in two parts: the
continuous portion, from 9th April to 8th June 2020 shows the percent infection trajectory
during the mitigation period and the dotted portion, after 8th June, indicates the projected
trajectory in the absence of further interventions. We see later that this projected trajectory is
also not followed due to subsequent relaxation action.
Relaxation trajectory: This consists of the curve labelled “25% relaxation (c=-0.25)” in
Figure 6. This curve is reproduced in Figure 7 (red curve) and is shown in two parts: the
continuous portion, from 9th June to 8th August 2020 shows the percent infection trajectory
during the relaxation period and the dotted portion, after 8th August, indicates the projected
trajectory in the absence of further interventions. We see later that this projected trajectory is
followed until the second COVID-19 wave emerges in early September 2020.
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perpetuity.
All rights reserved. No reuse allowed without permission.
Figure 7: Infection percentages for different mitigation or relaxation dynamics. The red curve shows
infection trajectory arising from 40% mitigation followed by 25% relaxation, the blue curve shows the
infection trajectory arising from mitigation dynamics while the black curve shows infection trajectory
arising from baseline dynamics.
Combined trajectory: Combining the continuous portions of the curves in Figure 7, yields
the trajectory of the infection from the onset of the disease to the end of the relaxation
period, namely 8th August 2020.. The result is in Figure 8, where the 7-day moving
averages are given and are compared with observed values. The projected trajectory is
Figure 8: The 7-day moving average of observed and computed percent infections combining
mitigation and relaxation strategies.
extended to the beginning of the 2nd wave in early September, for explanations given in the
preceding paragraph. It can be seen that the predicted percent infection trajectory
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reasonably matches the observed values during the first wave of COVID-19 till early
September when the second wave commences. The extension of the projected trajectory
beyond the beginning of the 2nd wave indicates that had there not been further events that
contributed to a second wave surge, COVID-19 infections would have reduced to
insignificant levels by January 2021.
6.2.2 Determination of intervention magnitudes given positivity rates
In this section, the observed positivity curve is given and we seek to estimate the mitigation
and relaxation values that could have yielded such a curve. The problem can be solved only
a posteriori, and it will enable assessment of the interaction between interventions and
disease dynamics. The approach involves generating a suitable surface envelope around
the curve and using the trajectory of the midpoint of the envelope to reflect the computed
positivity curve. The concept of envelopes is applied to numerous phenomena, including
robotics and kinematics, motion involving collision avoidance, gear transmission design [58 –
60]. To start the process, we require a suitable initial surface around the curve that is not too
large, for computational efficiency. The surface is bounded by curves defined by constant
values of mitigation and relaxation percentages. We have already seen that hard lockdown
was defined as mitigation of 44% and moderate lockdown as mitigation of 25% [23]; it is
therefore reasonable to start with mitigation percentages in the range (25 , 50). Relaxation
techniques applied after mitigation are designed to increase the transmission rate, preferably
to 80% - 90% of the pre-mitigation transmission. If we assume that the relaxation increases
the transmission to 90% of the pre-mitigation level then from Equation (14), we conclude that
relaxation percentages will be in the range (20, 80). To identify the initial width of the
envelope, we choose a fixed relaxation percentage and draw curves of interventions
consisting of mitigation values in the range (25, 50), or a suitable subset of it, followed by the
selected relaxation. The curves all start at the same point but, as time progresses, they
begin to diverge till two curves appear on either side of the observed positivity rate; this
enables us to define the initial left and right boundaries of the envelope. Further
configurations involving intermediate mitigation percentages and other relaxation
percentages can be made, if necessary, to enable identification of an initial envelope that is
closer to the positivity rate curve.
To generate the envelope, we considered the time series values of infection percentages
obtained from mitigation and relaxation percentages selected above. From these time
series, maximum and minimum infection percentages are obtained at each time point and
they are taken as the boundaries of the envelope, as shown in the final optimized envelope
in Figure 9. The midpoint of the envelope trajectory, shown in red, reflects the computed
positivity rate curve, with a mitigation of 42.5% and relaxation of 26.0% as averaged from
values at the midpoints of the family of envelopes. Solution of the SIRD model in Section 6.1
yielded mitigation of 40% and relaxation of 25%. Although the approach differed significantly
from the current one, the two methods yield intervention parameters with less than 10%
relative error from each other and illustrates significant consistency in the findings.
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Figure 9. Optimized envelope around the positivity rate curve, with computed positivity curves from
the envelope midpoint
7 Conclusions
We have developed a new numerical scheme for determining initial estimates of the COVID19 parameters in the SIRD model. The parameters can be refined by a simple search of the
minimum point, instead of more complex procedures. The results yield values of R0 which
are comparable with other numerical schemes, thus validating our approach. We focused
on the case where the death rate was known and there was a need to estimate only the
recovery rate; the method can, however, be extended readily to estimate the death and
recovery rates. By carrying out computations from the onset of the disease to the end of
intervals that coincide with major intervention measures, we can quantify the effect of the
interventions. There is a need for further analysis to determine the optimum grid outlay.
The mathematical model for interventions takes into account mitigation, which results in
transmission rate decrease and in relaxation, which results in a transmission rate increase,
hence in spikes. The model yields an infection curve, subject to intervention measures, that
closely follows the observed trends. The process depends on a parameter, whose value is
positive for mitigation and negative for relaxation. This parameter should be computed a
priori, if the method is to be used as a basis for decision-making in order to enact guidelines
commensurate with the level of planned interventions. The computation would require a
multidisciplinary team of researchers drawn from diverse backgrounds. In determining a
surge in the transmission, it should not be assumed that the spike is purely a result of the
government having lifted some mitigation measures. It is often a result of society violating
the laid down mitigation guidelines and, in some cases, openly protesting and against
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lockdowns as a result of “COVID-fatigue”. Further investigation should be carried out to
determine how the method here, and that in the previous paragraph, can be extended to
second and subsequent waves.
Acknowledgements: We acknowledge Alice Wangui Wachira, Anne Kinyua and Lucy
Nyanchama for their assistance with data collection. . Victor Juma acknowledges the support
from KEMRI-Wellcome Trust Research Programme (KEMRI-WTRP)
Data sources: All the data used is in the public domain [1, 44, 56, 57]
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