Healthcare Analytics 4 (2023) 100275

A
2

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Healthcare Analytics

journal homepage: www.elsevier.com/locate/health

A deterministic compartmental model for investigating the impact of
escapees on the transmission dynamics of COVID-19
Josiah Mushanyu a,b, Chidozie Williams Chukwu c,∗, Chinwendu Emilian Madubueze d,e,
Zviiteyi Chazuka f, Chisara Peace Ogbogbo g

a Department of Computing, Mathematical, and Statistical Sciences, University of Namibia, Windhoek 13301, Windhoek, Namibia
b Department of Mathematics and Computational Sciences, University of Zimbabwe, Box MP 167 Mount Pleasant, Harare, Zimbabwe
c Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA
d Department of Mathematics, Joseph Sarwuan Tarkaa University Makurdi, Nigeria
e Department of Mathematics and Statistics, York University, Toronto, Canada
f Department of Decision Sciences, College of Economics and Management Sciences, University of South Africa, South Africa
g Department of Mathematics, University of Ghana, Ghana

A R T I C L E I N F O

Keywords:
Deterministic model
Correlation
COVID-19
Numerical simulations
Quarantine
Escapees

A B S T R A C T

The recent outbreak of the novel coronavirus (COVID-19) pandemic has devastated many parts of the globe.
Non-pharmaceutical interventions are the widely available measures to combat and control the COVID-19
pandemic. There is great concern over the rampant unaccounted cases of individuals skipping the border
during this critical period in time. We develop a deterministic compartmental model to investigate the impact
of escapees (individuals who evade mandatory quarantine) on the transmission dynamics of COVID-19. A
suitable Lyapunov function has shown that the disease-free equilibrium is globally asymptotically stable,
provided 0 < 1. We performed a global sensitivity analysis using the Latin-hyper cube sampling method
and partial rank correlation coefficients to determine the most influential model parameters on the short and
long-term dynamics of the pandemic to minimize uncertainties associated with our variables and parameters.
Results confirm a positive correlation between the number of escapees and the reported COVID-19 cases. It is
shown that escapees are primarily responsible for the rapid increase in local transmissions. Also, the results
from sensitivity analysis show that an increase in governmental role actions and a reduction in the illegal
immigration rate will help to control and contain the disease spread.
1. Introduction

The novel Coronavirus (COVID-19), which has taken the world
by surprise since its first case in Wuhan, China, in December 2019,
has ravaged Africa, and Zimbabwe has felt its negative impact. As
of the 14th of July 2020, there were 13,141,440 confirmed cases
worldwide and 573,349 deaths. The United States of America was
the most affected country in the world, with 3,481,677 confirmed
COVID-19 infections and 138,291 deaths [1]. In Africa, South Africa
is currently the epicenter of the coronavirus, with 287,796 confirmed
cases and 4172 deaths [2]. Zimbabwe recorded its first COVID-19 case
on the 20th of March 2020 and its first death was recorded on the 23rd
of March 2020 [3]. Thereafter, the government of Zimbabwe imposed
a 21-day nationwide lockdown on the 30th of March 2020 [3]. When
this manuscript was drafted, Zimbabwe recorded 1034 confirmed cases
with 19 deaths [3].

∗ Corresponding author.
E-mail address: wiliam.chukwu@gmail.com (C.W. Chukwu).

Recently, Zimbabwe experienced a sharp spike in the number of
confirmed COVID-19 cases despite the lockdown. The sharp spike
was suspected to be due to the increase in the number of returnees
from other countries. As a measure to control the spread of the in-
fection, Zimbabwe closed her borders to normal traffic except that
of returnees and essential services. Also, the government imposed a
mandatory 21-day quarantine on all returnees, a measure that helped
identify quarantined returnees who tested positive early. These were
immediately placed under mandatory isolation within the country’s
isolation healthcare centers. However, some of the returnees escaped
from the quarantine centers either after they had tested positive or
simply because they did not want to stay at the quarantine centers.
This led to escapees exposing individuals in the community to COVID-
19. Escapees in this study refer to those individuals who decided to
evade the mandatory quarantine on all returnees (Zimbabwean citizens
vailable online 31 October 2023
772-4425/Published by Elsevier Inc. This is an open access article under the CC BY

https://doi.org/10.1016/j.health.2023.100275
Received 28 May 2023; Received in revised form 29 September 2023; Accepted 21
-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

October 2023

https://www.elsevier.com/locate/health
http://www.elsevier.com/locate/health
mailto:wiliam.chukwu@gmail.com
https://doi.org/10.1016/j.health.2023.100275
https://doi.org/10.1016/j.health.2023.100275
http://creativecommons.org/licenses/by-nc-nd/4.0/


Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.

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who had moved out of the country for various reasons such as work,
business, or tourism, and other foreign nationals who were visiting
the country) upon being frustrated by the length of stay in quarantine
imposed by the Zimbabwean government. We use escapees in this
context to refer to this act of evading a mandatory and monitored
quarantine as reported in private and government-owned media. As
of 10 July 2020, there were 209 escapees from quarantine centers;
the police arrested only 28 of them, leaving 181 escapees unaccounted
for [4]. Among these escapees, it was also reported that some had tested
positive, implying that there was a high possibility of spreading the
infection.

The global economic impact of the novel COVID-19 has brought
researchers from across the globe together to fight the virus through
various research works. These works include the mathematical model-
ing of the impact of COVID-19 within countries such as South Africa,
China, the United States and India, to name a few [5–17], control
strategies [18–20] and the impact assessment of the control strategies
and their sensitivity [21–25] on COVID-19. Most of the mathematical
models developed used an 𝑆𝐸𝐼𝑅 modeling framework, which only
aried in the descriptions of the compartments used and the results
btained from the studies. The main thrust of the mathematical mod-
ling approaches was to predict the future trend of the COVID-19 virus
nd provide recommendations to policymakers on the way forward.
he role of governmental action in controlling the spread of COVID-
9 within communities was modeled by Mushayabasa et al. [5]. In
heir work, they presented a model that looked at the effects of actions
uch as imposed social distancing measures, travel restrictions, quaran-
ine, sanitizing and hospitalization on the control of the transmission
ynamics of COVID-19. Results from the study indicated that inter-
ention methods by the government positively impact the reduction
f infection, and the authors also suggested important thresholds by
hich these intervention strategies must be bound. Nyabadza et al. [26]
lso modeled the effect of one governmental action (social distancing)
n the South African population. Results of their study indicated that
f social distancing is relaxed by as much as 2%, it could have an
mpact of 23% rise in infections while an increase in social distance
dherence by 2% could effectively reduce infections by 18% [26].
mbikapathy et al. [27] assessed the effect of the imposed lockdown on

he transmission dynamics of COVID-19 in India, and results from the
odel suggested that a 21-day lockdown was effective in the reduction

f infections in India and if the government could increase lockdown to
2 days the infections would further be reduced. Other mathematical
odels of note on the subject matter include [28–31], just to mention
few.

The model developed in this paper was designed to study the
ontribution of escapees on local COVID-19 transmissions during the
eriod of imposition of mandatory lockdown in Zimbabwe. This model
ives a first attempt to investigate escapees’ contribution to COVID-
9 disease spread. The model is applied to data on COVID-19 cases
n Zimbabwe during the period of mandatory lockdown and returnee
uarantine. It is essential to quantify the contribution of escapees to
ocal transmissions and assess the implications for public health and
olicy-making. Thus, we focus on the effects of quarantine measures
n the dynamics of COVID-19 infection within the Zimbabwean pop-
lation. Of particular note, the paper looks at the effects of those
ho escape from quarantine facilities on the dynamics of COVID-
9 in Zimbabwe. We employ advanced mathematical and statistical
echniques, such as the global sensitivity analysis using the Latin-
ypercube sampling method and partial rank correlation coefficients.
hese techniques allow the identification of the most influential model
arameters responsible for driving both short and long-term COVID-19
andemic dynamics. By reducing uncertainties surrounding influential
ariables and parameters, the results of this study will provide more
ccurate and actionable recommendations for policymakers and public
ealth officials. These results can also be applied in other countries
2

cross the globe with similar challenges in quarantining individuals.
The next section, Section 2, presents the model formulation, while Sec-
tion 3 presents the model analysis. Numerical simulations and global
uncertainty analysis are carried out in Section 4. The paper is concluded
in Section 5.

2. Model formulation

The human population is subdivided into 8 distinct classes namely;
the susceptible individuals, 𝑆(𝑡), returnees in quarantine who were
ot exposed to COVID-19, 𝑄1(𝑡), returnees in quarantine who were
xposed to COVID-19, 𝑄2(𝑡), locals who are exposed to COVID-19, 𝐸(𝑡),
ndetected infected individuals, 𝐼(𝑡), detected and isolated individu-
ls, 𝐼𝐷(𝑡), recovered individuals, 𝑅(𝑡) and deceased individuals, 𝐷(𝑡).
e assume that the population mixes homogeneously. The returnees

oming into the country are classified as 𝑄1(𝑡) and 𝑄2(𝑡), where 𝑄1(𝑡)
enotes returnees who undergo quarantine within the country’s centers
s prescribed by the government but are found to be COVID-19 negative
fter completing quarantine. Upon completion of quarantine, these
ndividuals are assumed to enter class 𝑆(𝑡) at a rate, 𝜃, as they remain
usceptible to local infections. The recruitment rate of returnees is given
y 𝛬. A proportion, 𝛱 , join the class 𝑄1(𝑡) and the remainder (1 −𝛱)
oin the class 𝑄2(𝑡). The model assumes that a proportion, 𝑞𝜙, of indi-
iduals in 𝑄2(𝑡) will escape quarantine while positive and hence join
lass 𝐼(𝑡) of undetected infected individuals, where 𝑞 is the probability
f escape and 𝜙 is the escape rate. The remaining proportion (1 − 𝑞)𝜙
rom 𝑄2(𝑡) undergo testing and may be found positive and proceed
o join class 𝐼𝐷(𝑡). The rationale here is that returnees are tested at
he beginning of quarantine, during quarantine, and towards the end
f quarantine. During these prescribed periods, an individual may be
ound positive and proceed to class 𝐼𝐷(𝑡) or escape while positive,
voiding further probation. It is assumed that the exposed individuals
ho are locals, 𝐸(𝑡), become undetected infected individuals at a rate,
because they are yet to be detected via testing while individuals

nfected with COVID-19 in class 𝐼(𝑡) are detected through testing at
rate, 𝛿1 and isolated. Since the beginning of the outbreak, scientists

ave suggested the possible mutation of the severe acute respiratory
yndrome coronavirus 2 (SARS-Cov-2) virus [32]. According to their
esearch, these mutations may cause the weakening of the strength of
he virus. At the same time, some individuals may develop resistance
nd can fight off the virus without seeking medical attention. We thus
ssume that the recovery of undetected infected individuals, 𝐼(𝑡), occurs
t a rate, 𝛿2 and moves to class 𝑅(𝑡). However, a certain proportion
f infected individuals will succumb to the infection at a rate 𝛿3 and
ove into class 𝐷(𝑡), which is the death class. A proportion, 𝑑, of

ndividuals that have been detected (and isolated) in 𝐼𝐷(𝑡) recover at
rate 𝜌 and join class 𝑅(𝑡) of recovered individuals. The remaining

roportion (1 − 𝑑)𝜌 succumb to the infection and join class 𝐷(𝑡). We
ssume permanent immunity of recovered individuals since the model
nly considers a short period covering lockdowns. The model has two
nfectious compartments, 𝐼(𝑡) and 𝐼𝐷(𝑡). Hence, we assume that the
orce of infection for the model is given by

=
𝛽(1 − 𝜅)

(

𝐼 + 𝜂𝐼𝐷
)

𝑁
, (1)

where 𝛽 is the transmission rate of COVID-19, 𝜂 is the modification
parameter, where 0 ≤ 𝜂 ≤ 1 and the term (1−𝜅) represents the effects of
governmental action such as wearing of face masks, sanitization, social
distancing etc., where 0 ≤ 𝜅 ≤ 1. A value of 𝜅 close to 1 indicates
that governmental action is effective while a value of 𝜅 close to zero
implies that there is little or no governmental action. The model does
not include the vital dynamics. The population total is given by

𝑁 = 𝑆(𝑡) + 𝐸(𝑡) +𝑄1(𝑡) +𝑄2(𝑡) + 𝐼(𝑡) + 𝐼𝐷(𝑡) + 𝑅(𝑡) +𝐷(𝑡).
The model dynamics can be summarized as in Fig. 1.



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 1. Model flow diagram. The dotted red oval indicates quarantined individuals.
The description of model variables, parameters and assumptions
combined with the model flow diagram (Fig. 1) leads to the following
set of non-linear ordinary differential equations:

𝑆′ = 𝜃𝑄1 − 𝜆𝑆,
𝐸′ = 𝜆𝑆 − 𝜎𝐸,
𝐼 ′ = 𝜎𝐸 + 𝑞𝜙𝑄2 − (𝛿1 + 𝛿2 + 𝛿3)𝐼,
𝐼 ′𝐷 = (1 − 𝑞)𝜙𝑄2 + 𝛿1𝐼 − 𝜌𝐼𝐷,
𝑄′

1 = 𝛱𝛬 − 𝜃𝑄1,
𝑄′

2 = (1 −𝛱)𝛬 − 𝜙𝑄2,
𝑅′ = 𝛿2𝐼 + 𝑑𝜌𝐼𝐷,
𝐷′ = 𝛿3𝐼 + (1 − 𝑑)𝜌𝐼𝐷,

⎫

⎪

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎪

⎭

(2)

with the initial conditions

𝑆(0) > 0, 𝐸(0) ≥ 0, 𝐼(0) ≥ 0, 𝐼𝐷(0) ≥ 0, 𝑄1(0) > 0, 𝑄2(0) ≥ 0, 𝑅(0) ≥ 0

and 𝐷(0) ≥ 0.

Here, all the model parameters are considered to be non-negative.
Fig. 1 correctly captures the epidemiological status of the returnees.

However, for simplicity, we assume that all uninfected quarantined
individuals will eventually join the class 𝑆(𝑡) of susceptible, as shown
in Fig. 1. This basically means incorporating some people who are
in quarantine in the susceptible population. The error that results
from this consideration is negligible. Further, we assume that all the
remaining quarantined individuals will either escape to join the class
𝐼(𝑡) or move to the class of detected and isolated individuals upon
being screened through testing, which is a similar modification done for
(SARS-Cov-2) model by Gumel et al. [33]. This is related to this work
regarding a respiratory disease like COVID-19. They also considered
the impact of undetected entry of infected individuals on the SARS
transmission dynamics. Thus, the modified model can be represented
by Fig. 2.

We have the following set of nonlinear ordinary differential equa-
tions for the modified model:

𝑆′ = −𝜆𝑆,
𝐸′ = 𝜆𝑆 − 𝜎𝐸,
𝐼 ′ = 𝜎𝐸 + 𝑞𝜙𝑄2 − (𝛿1 + 𝛿2 + 𝛿3)𝐼,
𝐼 ′𝐷 = (1 − 𝑞)𝜙𝑄2 + 𝛿1𝐼 − 𝜌𝐼𝐷,
𝑄′

2 = 𝛬 − 𝜙𝑄2,
𝑅′ = 𝛿2𝐼 + 𝑑𝜌𝐼𝐷,
𝐷′ = 𝛿 𝐼 + (1 − 𝑑)𝜌𝐼 ,

⎫

⎪

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎪

(3)
3

3 𝐷
⎭

with the initial conditions

𝑆(0) > 0, 𝐸(0) ≥ 0, 𝐼(0) ≥ 0, 𝐼𝐷(0) ≥ 0, 𝑄2(0) ≥ 0, 𝑅(0) ≥ 0 and 𝐷(0) ≥ 0,

where all the model parameters are considered to be non-negative.
Here, 𝛬 is the recruitment rate of COVID-19-exposed individuals re-
turning to the country.

3. Model analysis

3.1. Positivity of solutions

We now consider the positivity of system (3). We show that all the
state variables remain non-negative, and the solutions of system (3)
with positive initial conditions remain positive for all 𝑡 > 0.

Theorem 1. Given that the initial conditions of system (3) are 𝑆(0) >
0, 𝐸(0) ≥ 0, 𝐼(0) ≥ 0, 𝐼𝐷(0) ≥ 0, 𝑄2(0) ≥ 0, 𝑅(0) ≥ 0 and 𝐷(0) ≥ 0, then for
all 𝑡 > 0, all solutions of model system (3) remain positive in R7

+.

Proof. Following the approach in Gweryina et al. [34], we assume that

𝑡1 = 𝑠𝑢𝑝
{

𝑡 > 0 ∶ 𝑆(0) > 0, 𝐸(0) ≥ 0, 𝐼(0) ≥ 0, 𝐼𝐷(0) ≥ 0, 𝑄2(0) ≥ 0,

𝑅(0) ≥ 0, 𝐷(0) ≥ 0 ∈ [0, 𝑡]
}

.

Thus, 𝑡1 > 0 and it follows from the first equation of system (3) that
𝑑𝑆
𝑑𝑡

= −𝜆𝑆. (4)

Solving this using separation of variables yields

ln𝑆 = −∫

𝑡1

0
𝜆(𝑡)𝑑𝑡 +𝐾. (5)

Hence,

𝑆(𝑡) ≥ 𝐴 exp
[

−∫

𝑡1

0
𝜆(𝑡)𝑑𝑡

]

, (6)

where 𝐴 is a constant to be determined. Therefore, applying initial
conditions 𝑆(0) = 𝑆0 yields 𝐴 = 𝑆0 such that

𝑆(𝑡) ≥ 𝑆0 exp
[

−∫

𝑡1

0
𝜆(𝑡)𝑑𝑡

]

> 0. (7)

Therefore, it can be shown in a similar manner that 𝐸(𝑡) ≥ 0, 𝐼(𝑡) ≥
0, 𝐼𝐷(𝑡) ≥ 0, 𝑄2(𝑡) ≥ 0, 𝑅(𝑡) ≥ 0 and 𝐷(𝑡) ≥ 0, for all 𝑡 > 0. This concludes
the proof. □



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 2. Model flow diagram. The dotted red oval indicates quarantined individuals.
3.2. Invariant region

Theorem 2. The region

𝛺 =
{

(𝑆,𝐸, 𝐼, 𝐼𝐷, 𝑄2, 𝑅,𝐷) ∈ R7
+ ∶ 𝑆(𝑡) + 𝐸(𝑡) + 𝐼(𝑡) + 𝐼𝐷(𝑡) +𝑄2(𝑡)

+𝑅(𝑡) +𝐷(𝑡) ≤ 𝑁0
}

is positively invariant for system (3), with non-negative initial conditions
where 𝑁0 is the initial population.

Proof. Adding all the equations of system (3) yields
𝑑𝑁
𝑑𝑡

= 0.

Then, lim sup𝑡→∞ 𝑁 ≤ 𝑁0. Thus, we have the feasible region for system
(3) defined by

𝛺 =
{

(𝑆,𝐸, 𝐼, 𝐼𝐷, 𝑄2, 𝑅,𝐷) ∈ R7
+|0 ≤ 𝑁 ≤ 𝑁0

}

.

Our analysis will be based on the dynamics in 𝛺. □

3.3. Disease-free equilibrium state and the basic reproduction number

The model has a disease-free equilibrium state

0 =
(

𝑆0, 𝐸0, 𝐼0, 𝐼0𝐷, 𝑄
0
2, 𝑅

0, 𝐷0) =
(

𝑆0, 0, 0, 0, 0, 0, 0
)

.

Following the next-generation matrix approach by van den Driessche
and Watmough [35], we have

 =

⎛

⎜

⎜

⎜

⎜

⎝

0 𝛽(1 − 𝜅) 𝛽𝜂(1 − 𝜅) 0
0 0 0 0
0 0 0 0
0 0 0 0

⎞

⎟

⎟

⎟

⎟

⎠

and

 =

⎛

⎜

⎜

⎜

⎜

⎝

𝜎 0 0 0
−𝜎 𝛿1 + 𝛿2 + 𝛿3 0 −𝑞𝜙
0 −𝛿1 𝜌 −(1 − 𝑞)𝜙
0 0 0 𝜙

⎞

⎟

⎟

⎟

⎟

⎠

, (8)

giving

0 = 𝐼 +𝐼𝐷 where 𝐼 =
𝛽(1 − 𝜅)

( ) and
4

𝛿1 + 𝛿2 + 𝛿3
𝐼𝐷 =
𝛽(1 − 𝜅)𝛿1𝜂

(

𝛿1 + 𝛿2 + 𝛿3
)

𝜌
. (9)

𝐼 is the reproduction number contributed by the undetected infected
class, 𝐼(𝑡) while 𝐼𝐷 is the reproduction number contributed by the
𝐼𝐷(𝑡) class. The term 𝛽(1−𝜅)

(𝛿1+𝛿2+𝛿3)
means that the individuals in the un-

detected infected class 𝐼(𝑡) have contact with the susceptible class at
a rate, 𝛽(1 − 𝜅), and they will spend a mean time of 1

(𝛿1+𝛿2+𝛿3)
within

the undetected Infected class, 𝐼(𝑡). It can be noted that governmental
action 𝜅 will reduce the transmission rate, 𝛽. The term 𝛽𝜂𝛿1(1−𝜅)

𝜌(𝛿1+𝛿2+𝛿3)
means

that some individuals within the undetected Infected class, 𝐼(𝑡) will be
detected positive and isolated, at a testing rate of 𝛿1 and spend a mean
time 1

(𝛿1+𝛿2+𝛿3)
within, 𝐼(𝑡) class. Thereafter, they will progress to the

𝐼𝐷(𝑡) class and spend an average infectious time of 1
𝜌 within the 𝐼𝐷(𝑡)

class. The contact rate of such individuals is 𝛽𝜂𝛿1(1 − 𝜅), where 𝜂 is the
modification parameter for the 𝐼𝐷(𝑡) class.

The following result follows from van den Driessche and Wat-
mough [35].

Theorem 3. The disease-free equilibrium for system (3) is locally asymp-
totically stable provided that 0 < 1.

3.3.1. Global stability of the disease-free equilibrium state
We shall now prove the global stability of the disease-free equilib-

rium point, 0, whenever the reproduction number is less than unity.

Theorem 4. The disease-free equilibrium for system (3) is globally asymp-
totically stable provided that 0 < 1.

Proof. Using the approach in Shuai and van den Driessche [36], we
construct the following Lyapunov function given by

 =
(

1
𝑓

+
𝛿1𝜂
𝑓𝜌

)

[𝐸 + 𝐼] +
𝜂
𝜌
𝐼𝐷 +𝑄2

[

𝑞
𝑓

+
𝜂𝑓 (1 − 𝑞) + 𝑞𝛿1

𝑓𝜌

]

, (10)

where 𝑓 = 𝛿1 + 𝛿2 + 𝛿3. Differentiating  yields

̇ =
(

1 +
𝛿1𝜂

)

[𝐸 + 𝐼]′ +
𝜂
𝐼 ′ +𝑄′

[

𝑞
+

𝜂𝑓 (1 − 𝑞) + 𝑞𝛿1
]

. (11)

𝑓 𝑓𝜌 𝜌 𝐷 2 𝑓 𝑓𝜌



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.

w



r



p


H
(
g
a
c
w
P
w
d
i
s

Substituting expressions for 𝐸′, 𝐼 ′, 𝐼 ′𝐷, 𝑄
′
2 leads to

̇ =
(

1
𝑓

+
𝛿1𝜂
𝑓𝜌

)

[𝜆𝑆 + 𝑞𝜙𝑄2 − 𝑓𝐼] +
𝜂
𝜌
[(1 − 𝑞)𝜙𝑄2 + 𝛿1𝐼 − 𝜌𝐼𝐷]

+
[

𝑞
𝑓

+
𝜂𝑓 (1 − 𝑞) + 𝑞𝛿1

𝑓𝜌

]

[𝛬 − 𝜙𝑄2]

= 𝜆𝑆
[

1
𝑓

+
𝛿1𝜂
𝑓𝜌

]

− (𝐼 + 𝜂𝐼𝐷) + 𝛬
[

𝑞
𝑓

+
𝜂𝑓 (1 − 𝑞) + 𝑞𝛿1

𝑓𝜌

]

.

(12)

Recall that 𝜆 =
𝛽(1 − 𝜅)

(

𝐼 + 𝜂𝐼𝐷
)

𝑁
and 0 = 𝛽(1 − 𝜅)

[

1
𝑓 + 𝛿1𝜂

𝑓𝜌

]

. Hence,
e obtain

̇ =
[

0𝑆
𝑁

− 1
]

(𝐼 + 𝜂𝐼𝐷) + 𝛬
[

𝑞
𝑓

+
𝜂𝑓 (1 − 𝑞) + 𝑞𝛿1

𝑓𝜌

]

. (13)

At the disease-free equilibrium 𝛬 = 0 (there are no returnees being
ecruited into quarantine centers in Zimbabwe), then

̇ =
[

0𝑆
𝑁

− 1
]

(𝐼 + 𝜂𝐼𝐷). (14)

Then, it follows that

̇ ≤ (0 − 1)(𝐼 + 𝜂𝐼𝐷), (15)

since 𝑆
𝑁 ≤ 1. This implies that ̇ ≤ 0 provided that 0 ≤ 1 and ̇ = 0

iff 𝐼 = 𝐼𝐷 = 𝐸 = 𝑄2 = 0. Hence, by applying LaSalle’s invariance
rinciple [37] it suffices to conclude that the disease-free equilibrium,
0 is globally asymptotically stable provided 0 < 1 and unstable

otherwise. This concludes the proof. □

3.4. Sensitivity analysis of 0.

We perform sensitivity analysis to establish which parameters have
a positive or negative effect on the 0 expression given in (9). We use
the following normalized forward sensitivity index formula as given
in [38].

𝛤0
𝜔 =

𝜕0
𝜕𝜔

× 𝜔
0

, (16)

where 𝜔 is the parameter of interest. Below are sensitivity indices of
0 with respect to each of its parameters.

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

𝛤0
𝛽 = 1, 𝛤0

𝜂 =
𝛿1𝜂

𝛿1𝜂 + 𝜌
, 𝛤0

𝜅 = − 𝜅
1 − 𝜅

,

𝛤0
𝜌 = −1 +

𝜌
𝛿1𝜂 + 𝜌

,

𝛤0
𝛿1

= 𝛿1

(

𝜂
𝛿1𝜂 + 𝜌

− 1
𝛿1 + 𝛿2 + 𝛿3

)

, 𝛤0
𝛿2

= −
𝛿2

𝛿1 + 𝛿2 + 𝛿3
,

𝛤0
𝛿3

= −
𝛿3

𝛿1 + 𝛿2 + 𝛿3
.

(17)

ere, the sensitivity indices for the given parameters as presented in
17) only indicate the direction of influence without quantifying the
iven parameter’s influence. We note that the parameters 𝛽 and 𝜂 have
direct proportional relationship with 0, with 0 most sensitive to

hanges in 𝛽. An increase (or decrease) in either of these parameters
ill result in an equivalent increase (or decrease) in the value of 0.
arameters 𝜅, 𝜌, 𝛿2 and 𝛿3 have an inversely proportional relationship
ith 0, an increase in the values of these parameters will lead to a
ecrease in the value of 0. However, putting down measures that will
ncrease the value of 𝛿3 is unethical. Thus, measures such as increased
urveillance and isolation of infected individuals (increase in 𝜌) and

enforcing governmental actions such as wearing of face masks, use of
sanitizers, social distancing, etc. (increase in 𝜅) will be of great help
in curtailing the spread of COVID-19. Interestingly, 𝛤0

𝛿1
< 0 when

𝜌 >
(

2𝛿1 + 𝛿2 + 𝛿3
)

𝜂 and 𝛤0
𝛿1

> 0 when 𝜌 <
(

2𝛿1 + 𝛿2 + 𝛿3
)

𝜂. This
entails that the detection rate on infected individuals will reduce the
value of 0 provided there is an adequate amount of time spent in
5

isolation for infected individuals.
4. Numerical simulations

In this section, we perform numerical simulations of system (3). We
consider a case study of COVID-19 transmission dynamics in Zimbabwe.
The model in the present study has many parameters whose values
must be ascertained to properly capture the COVID-19 transmission
dynamics in Zimbabwe for the period 30th of March 2020 up to the
30th of June 2020. To capture the COVID-19 dynamics in Zimbabwe
for this period, we fit the model to the observed COVID-19 data. We
assign reasonable ranges from which parameter values are chosen and
we obtain parameter values that give the best fit. These parameters are
used to perform our numerical simulations. We also carry out some
forecasting of the disease dynamics under certain conditions within the
framework of the objectives of the study. For instance, we investigate
the long-term impact of governmental actions such as lockdowns, social
distancing, closure of schools and universities, etc.

We consider COVID-19 data starting from the 30th of March 2020
(day 1), when the first lockdown was instituted in Zimbabwe, up to the
30th of June 2020 (day 93). Important timelines for the management
and control of COVID-19 in Zimbabwe, as given by the government
authorities, are as follows:

1. declaration of state of disaster on the 17th of March 2020 [39],
2. 1st lockdown instituted on the 30th of March 2020 [39],
3. 2nd lockdown instituted on the 6th of May 2020 [40],
4. 3rd lockdown instituted on the 20th of May 2020 for an indefi-

nite period covering the drafting of this manuscript [41].

We tabulate the data for COVID-19 cases in three parts, namely: (i) the
period covering the 1st lockdown (see Table 1), (ii) the period before
the massive surge of returnees (see Table 2. This period covers all of
the 1st lockdown period, all of the 2nd lockdown period and part of
the initial stages of the 3rd indefinite lockdown period), and (iii) the
period after the massive surge of returnees (see Table 3. This period
covers most of the 3rd indefinite lockdown period).

To estimate our unknown model parameters and unpack the un-
derlying dynamics of the COVID-19 pandemic in Zimbabwe, we use
the curve-fitting process to quantify the trend of the outcomes of
this pandemic. Here, we fit the equations of approximating curves
to the available raw field data on COVID-19 dynamics in Zimbabwe.
Generally, it will be observed that the fitting curves for any given data
set are not unique. Thus, a curve with minimum possible deviation
from all the data points will be desired. The least squares curve fit
routine (lsqcurvefit) in Matlab with optimization is used to estimate our
unknown model parameters. The procedure requires that a lower and
an upper bound be set from which estimates of the unknown parameter
values are obtained. Thus, using the available data on cumulative cases
for COVID-19 in Zimbabwe over a defined time frame, 𝑡𝑖−1 ≤ 𝑡 ≤ 𝑡𝑖
(where 𝑡𝑖−1 and 𝑡𝑖 indicate the beginning and end of the time interval,
respectively), we estimate using the function

𝐶(𝑡) = ∫

𝑡𝑖

𝑡𝑖−1

[

(1 − 𝑞)𝜙𝑄2(𝑡) + 𝛿1𝐼(𝑡)
]

𝑑𝑡. (18)

We applied the ordinary least squares (OLS), which is a standard
approach in regression analysis, to approximate the solution of overde-
termined systems by minimizing the sum of the squares of the residuals
made in the results of each equation while fitting our Model to Eq. (18).
The OLS is advantageous as it provides a rationale for placing the line
of best fit, taking into account the input data and the model under
parameterization of its validation process while giving the smallest
sum of squares of variance [10,43]. Also, the OLS algorithms allow
researchers to estimate unknown model parameter values by specifying
a lower bound and an upper bound from which the set of estimated
parameter values are generated once the algorithm obtains a line of best
fit. Applying OLS to our model, we validate the system (3) by fitting

it to the data, which yields Table 4 using the least square regression



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 3. Daily reported new cases in Zimbabwe starting on the 30th of March 2020 and ending on the 27th of May 2020.
Fig. 4. Model system (3) fitted to data for cumulative COVID-19 cases in Zimbabwe before the massive surge of returnees starting from the 30th of March 2020 and ending
on the 27th of May 2020. See Table 2. This period covers the whole of the 1st and 2nd lockdown period and partly covers the initial stages of the 3rd indefinite lockdown
period. The blue circles indicate the actual data, and the solid red line indicates the model fit to the data. The initial conditions obtained from data fitting are as follows:
𝑁(0) = 14 × 106; 𝑆(0) = 𝑁(0) − 𝐸(0) − 𝐼(0) − 𝐼𝐷(0) − 𝑄2(0) − 𝑅(0); 𝐸(0) = 0; 𝐼(0) = 0; 𝐼𝐷(0) = 7; 𝑄2(0) = 𝛬 = 61; 𝑅(0) = 0; 𝐷(0) = 1 where 𝑡 = 0 corresponds to the 30th of March
2020 for this case.
Fig. 5. Daily reported new cases in Zimbabwe starting on the 27th of May 2020 and ending on the 30th of June 2020.
Table 1
Confirmed COVID-19 cases in Zimbabwe before lockdown [42].

Date 20∕3∕20 21∕3∕20 22∕3∕20 23∕3∕20 24∕3∕20 25∕3∕20 26∕3∕20 27∕3∕20

Total cases 1 3 3 3 3 3 3 5

Date 28∕3∕20 29∕3∕20

Total cases 7 7
6



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 6. Model system (3) fitted to data for cumulative COVID-19 cases in Zimbabwe after the massive surge of returnees starting from the 27th of May 2020 and ending on the
30th of June 2020 (see Table 3). This period covers most of the 3rd indefinite lockdown period. The blue circles indicate the actual data, and the solid red line indicates the model
fit to the data. The initial conditions obtained from data fitting are as follows: 𝑁(0) = 14 × 106; 𝑆(0) = 𝑁(0) − 𝐸(0) − 𝐼(0) − 𝐼𝐷(0) − 𝑄2(0) − 𝑅(0); 𝐸(0) = 50; 𝐼(0) = 0; 𝐼𝐷(0) =
132; 𝑄2(0) = 𝛬 = 1000; 𝑅(0) = 25; 𝐷(0) = 4 where 𝑡 = 0 corresponds to the 27th of May 2020 for this case.
Table 2
Confirmed COVID-19 cases in Zimbabwe before massive surge of returnees. This period covers the whole of the 1st lockdown period, the whole
of the 2nd lockdown period and partly covers the initial stages of the 3rd indefinite lockdown period [42].

Date 30∕3∕20 31∕3∕20 1∕4∕20 2∕4∕20 3∕4∕20 4∕4∕20 5∕4∕20 6∕4∕20

Total cases 7 8 8 9 9 9 9 10

Date 7∕4∕20 8∕4∕20 9∕4∕20 10∕4∕20 11∕4∕20 12∕4∕20 13∕4∕20 14∕4∕20

Total cases 11 11 11 13 14 14 17 17

Date 15∕4∕20 16∕4∕20 17∕4∕20 18∕4∕20 19∕4∕20 20∕4∕20 21∕4∕20 22∕4∕20

Total cases 23 23 24 25 25 25 28 29

Date 23∕4∕20 24∕4∕20 25∕4∕20 26∕4∕20 27∕4∕20 28∕4∕20 29∕4∕20 30∕4∕20

Total cases 29 30 32 32 32 32 32 34

Date 1∕5∕20 2∕5∕20 3∕5∕20 4∕5∕20 5∕5∕20 6∕5∕20 7∕5∕20 8∕5∕20

Total cases 34 34 34 34 34 34 34 34

Date 9∕5∕20 10∕5∕20 11∕5∕20 12∕5∕20 13∕5∕20 14∕5∕20 15∕5∕20 16∕5∕20

Total cases 35 36 36 36 37 37 42 42

Date 17∕5∕20 18∕5∕20 19∕5∕20 20∕5∕20 21∕5∕20 22∕5∕20 23∕5∕20 24∕5∕20

Total cases 44 46 46 48 51 51 56 56

Date 25∕5∕20 26∕5∕20

Total cases 56 56
technique. This technique minimizes the sum of the squared residuals
given by:

(𝛹 ) =
𝑛
∑

𝑖=1

(

𝑝𝑡𝑖 (𝛹 ) − �̃�𝑡𝑖
)2

.

Here, 𝛹 represents all the model parameters in Table 4, 𝑛 is the total
number of data points used for the fitting process, 𝑝𝑡𝑖 (𝛹 ) and �̃�𝑡𝑖 are
the Zimbabwe cumulative COVID-19 cases of infected individuals by
our model predictions and the actual reported data for the time frame
under investigation respectively (see Figs. 3–6).

4.1. Epidemiological parameter estimation

Under this section, we ascertain epidemiological parameter val-
ues used for conducting numerical simulations. We use the available
expanding literature on COVID-19 dynamics and the Zimbabwe COVID-
19 data fitting. Immigration of exposed individuals (𝛬) is estimated for
the periods before and after the massive surge of returnees. The manda-
tory quarantine period for returnees is 21 days, with some individuals
escaping from quarantine before this period lapses. Thus, we choose a
range of between 14 to 21 days and set the mean duration of quarantine
to be 𝜙−1 = 14 days [44]. The proportion, 𝑞, of those escaping from
quarantine facilities is chosen to fall within the range [0.00138, 0.025]
7

per day [44,45]. The proportion of the detected and isolated individuals
who progress into the recovered class 𝑅(𝑡) is considered to be within
the range 𝑑 = [0.26, 0.5], drawn from the work done in [44,46].
Following [5,47], we assign the range for the governmental action
parameter 𝜅 as 𝜅 ∈ [0.4239, 0.8478]. Further, the remaining parameter
values are estimated from the COVID-19 data fit, which are the disease
transmission rate; 𝛽 = 0.5643, modification parameter; 𝜂 = 0.7015,
detection rate of infected individuals; 𝛿1 = 4.0816 × 10−7 and the mean
duration period for isolation of detected individuals is 𝜌−1 = 10 days.
We provide a summary of the description of parameters, ranges, and
values used in Table 4.

4.2. Sensitivity analysis

According to Marino [53], we note that in any mathematical model-
ing exercise, the model variables and parameters are uncertain, making
it difficult to quantify. Also, sensitivity analysis can be carried out
over time intervals with no particular time frame under investigation
for exploratory purposes. For instance, the exact number of exposed
returnees remains difficult to accurately measure considering some
unaccounted cases of individuals who skip the border. More generally,
describing a phenomenon using models is often linked with incomplete
available knowledge due to a lack of analogous experimental measures.



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 7. Plots of PRCC values showing the effects of long-term dynamics before surging for parameter values on COVID-19 disease in Zimbabwe over time for (a) parameters
𝜅, 𝜎, 𝜙, 𝑞, 𝛿1 against 𝐼 . (b) parameters 𝜅, 𝜙, 𝑞, 𝛿1 , 𝜌 against 𝐼𝐷 . (c) parameters 𝛬, 𝜅, 𝜙, 𝑞 against 𝑄2.
Table 3
Confirmed COVID-19 cases in Zimbabwe after massive surge of returnees. This period covers most of the 3rd indefinite lockdown period [42].

Date 27∕5∕20 28∕5∕20 29∕5∕20 30∕5∕20 31∕5∕20 1∕6∕20 2∕6∕20 3∕6∕20

Total cases 132 149 149 174 178 203 206 222

Date 4∕6∕20 5∕6∕20 6∕6∕20 7∕6∕20 8∕6∕20 9∕6∕20 10∕6∕20 11∕6∕20

Total cases 237 265 279 282 287 314 320 332

Date 12∕6∕20 13∕6∕20 14∕6∕20 15∕6∕20 16∕6∕20 17∕6∕20 18∕6∕20 19∕6∕20

Total cases 343 356 383 387 391 401 463 479

Date 20∕6∕20 21∕6∕20 22∕6∕20 23∕6∕20 24∕6∕20 25∕6∕20 26∕6∕20 27∕6∕20

Total cases 486 489 512 525 530 551 561 567

Date 28∕6∕20 29∕6∕20 30∕6∕20

Total cases 567 574 591
Hence, we chose only the most important parameters and/or state
variables in relation to the subject/aim of the study. It is important
to note that as indicated in [53], all parameters that have partial rank
correlation coefficients (PRCCs) between −0.2 to 0.2 are considered to
be less sensitive, though this can be further justified by the calculation
of the associated P-values in further study. Sensitivity analyses were
done using the state variables: the undetected individuals 𝐼 , detected
and isolated individuals 𝐼𝐷 and the exposed individuals in quarantine
𝑄2 to determine the most influential parameters; 𝛬, 𝜌, 𝛿1, 𝑞, 𝜙, 𝜅
and 𝜎 on the increase/decrease of the escapee individuals on COVID-
19 disease transmission dynamics. We aim to determine the impact of
8

these parameters against the state variable of choice for their short and
long-term transmission dynamics for our model by considering their
Partial Rank Correlation Coefficients (PRCC) values over the modeling
time. This will help us to quantify the role of escapees on SARS-CoV-
2 transmissions before and after the surge of the disease. Here, we
consider the sensitivity analysis of these parameters before and after
a surge of COVID-19, as presented in the subsections below.

4.2.1. Sensitivity analysis before surge
Some parameters from Table 1 are used to carry out global uncer-

tainty sensitivity analysis of model system (3) with the combination



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 8. Plots of PRCC value showing the effects of long-term dynamics before surging for parameter values on COVID-19 disease in Zimbabwe over time for (a) parameters
𝜅, 𝜎, 𝜙, 𝑞, 𝛿1 against 𝐼 . (b) parameters 𝜅, 𝜙, 𝑞, 𝛿1 , 𝜌 against 𝐼𝐷 . (c) parameters 𝛬, 𝜅, 𝜙, 𝑞 against 𝑄2.
Table 4
Description of parameters in system (3) using data for Table 2.

Description Symbol Range Baseline value Source

Disease transmission rate 𝛽 [0.4, 1] 0.5643 Data fit
Governmental action 𝜅 [0.4239, 0.8478] 0.55 [5,47]
Modification factor 𝜂 [0.4, 1] 0.7015 Data fit
Mean incubation period 𝜎−1 5–6 days 5 [48,49]
Mean duration in quarantine 𝜙−1 14 days 14 [46]
Proportion of escapees 𝑞 [0.00138, 0.025] day−1 0.00138 [44,45]
Detection rate of infected individuals 𝛿1 [0, 0.7] day−1 4.0861 × 10−7 Data fit
Recovery rate of undetected individuals 𝛿2 [0.0752, 0.1370] day−1 0.0795 [50,51]
Disease-related death for infected individuals 𝛿3 [0.03, 0.04] day−1 0.04 [52]
Mean duration in isolation 𝜌−1 [0.03, 0.2] days 0.1 Data fit
Proportion of isolated individuals who recovers 𝑑 [0.26, 0.5] day−1 0.26 [44,45]
Immigration of exposed individuals 𝛬 [50, 100] day−1 61 Data fit
of Latin Hypercube Sampling (LHS) and the partial rank correlation
coefficients (PRCC) as in [53]. A sample size of 𝑁 = 500 with a unit step
of 1 is used to implement the simulations in this subsection for both the
short-term and long-term dynamics of the disease and their results are
depicted in Figs. 7 and 8 respectively. The basic reproduction number
for this case was found to be 0 = 2.1.

Short term predictions dynamics
In Fig. 7(a), we observe that the parameter, 𝜅 being the govern-

mental role actions during the pandemic, is negatively correlated at
the onset of the disease and becomes positively correlated from 𝑡 = 10
9

to 𝑡 = 300 days. Meanwhile, the parameter 𝜎 is negatively correlated
over the modeling time with respect to the undetected individuals, 𝐼(𝑡).
Also, Fig. 7(b) gives a similar trend to what we observe in Fig. 7(a),
only that the parameter 𝜙 becomes positively correlated at the onset
of the epidemic and diminishes as time progresses against the exposed
individuals in quarantine, 𝐼𝐷(𝑡). On the other hand, we observe in
Fig. 7(c) that the immigration rate, 𝛬, has a perfect strong correlation,
while the parameter, 𝜙, is positive and diminishes with time. The
parameter, 𝜅, has a negative correlation from the time 𝑡 = 0 to time
𝑡 = 100 and a positive correlation from 𝑡 > 100 and becomes less
significant after 𝑡 > 170.



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 9. Plots of PRCC values showing the effects of short-term dynamics after surging for parameter values on COVID-19 disease in Zimbabwe over time for (a) parameters
𝜅, 𝜎, 𝜙, 𝑞, 𝛿1 against 𝐼 . (b) parameters 𝜅, 𝜙, 𝑞, 𝛿1 , 𝜌 against 𝐼𝐷 . (c) parameters 𝛬, 𝜅, 𝜙, 𝑞 against 𝑄2.
Long term predictions dynamics
We observe in Figs. 8(a) and 8(b) that at the onset of the disease,

the parameter 𝜅 has a negative correlation and eventually becomes
positive after time 𝑡 = 80 for the undetected and exposed individuals
in quarantine. Further, in Fig. 8(c) 𝜅 is negatively correlated to the 𝑄2
individuals and becomes positive after 90 days. The parameters, 𝜎 and
𝛬 are strongly negative and positive correlated respectively against 𝐼
and 𝑄2 as observed in Figs. 8(a) and 8(c) respectively. Biologically, this
signifies that an increase in the immigration rate, 𝛬, will increase the
number of quarantined individuals, while a decrease in 𝜎 will lead to
a decrease in the number of undetected individuals before a surge.

4.2.2. Sensitivity analysis after surge
We also use Table 3, and a sample size of 𝑁 = 500 runs with a

unit step to perform sensitivity analysis under this subsection. Short-
term results are depicted in Fig. 9 while long-term results are depicted
in Fig. 10. The basic reproduction number for this case is found to be
0 = 2.8.

Short term predictions dynamics
In Figs. 9(a)–9(c), we observe that the parameter, 𝜅, has a negative

correlation from time 𝑡 = 0 to 𝑡 = 50 and thereafter it becomes positively
correlated against the state variables 𝐼, 𝐼𝐷 and 𝑄2. This implies that
an increase in governmental role action, such as implementing lock-
downs and wearing masks, needs to be continued after the first surge
10
to control the disease effectively. On the other hand, the immigration
rate, 𝛬, has a strong positive correlation over time as seen in Fig. 9(c)
while the parameter 𝜎 has a strong perfect negative correlation against
the state variable, 𝐼𝐷. Also, as seen before a surge, an increase in the
immigration rate during COVID-19 will result in more infections in a
wholly susceptible population. On the other hand, the mean duration
of quarantine parameter 𝜙 does not strongly impact the class, 𝐼𝐷, as
time increases as it becomes positive and diminishes.

Long term predictions dynamics
A similar trend is also seen in Figs. 10(a) and 10(c) for the parameter

𝜅. It has negative PRCC values from the first 30 days and positive PRCC
values for 𝑡 > 40. Also, in Figs. 10(a)–10(c), we observe that the mean
duration in quarantine, 𝜙, has a positive PRCC value and diminishes as
time progresses. The mean duration of incubation is not significant at
the onset after the surge but becomes prominently positively correlated
after time 𝑡 > 200. In addition, the immigration rate, 𝛬, has a perfect
positive correlation PRCC value over the modeling time.

4.3. Effects of varying parameters 𝑞 and 𝜅 on 𝐼(𝑡)

We investigate the impact of parameters 𝑞 and 𝜅 on the number
of undetected infected individuals, 𝐼(𝑡). The impact of 𝑞 is considered
for the period before and after the massive surge of returnees, whereas
the impact of 𝜅 is considered for the period after the massive surge of
returnees.



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 10. Plots of PRCC value showing the effects of long-term dynamics after surging for parameter values on COVID-19 disease in Zimbabwe over time for (a) parameters
𝜅, 𝜎, 𝜙, 𝑞, 𝛿1 against 𝐼 . (b) parameters 𝜅, 𝜙, 𝑞, 𝛿1 , 𝜌 against 𝐼𝐷 . (c) parameters 𝛬, 𝜅, 𝜙, 𝑞 against 𝑄2.
Fig. 11. Effect of varying 𝑞 on the number of undetected individuals.
We observe from Fig. 11 that the variation of 𝑞 before the massive
surge of returnees has less impact than the period after the massive
surge of returnees. We note that an increase in the value of 𝑞 fuels
the presence of undetected infected individuals. A 20% increase in the
value of 𝑞 increases 10 more daily undetected infected cases, which will
be responsible for any undetected local transmissions. Fig. 12 illustrates
that increased governmental action results in fewer undetected infected
individuals. We observe that a 20% increase in the intensity of 𝜅 will
11
result in approximately a maximum drop of 15 daily undetected cases.
This reflects the importance of ensuring adherence to measures put in
place by the government to help control this pandemic.

4.4. Effects of parameters on 0

We use contour plots to investigate the effects of some crucial
parameters on  . These results support the analytical findings of the
0



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
Fig. 12. Effect of varying 𝑘 after the 27th of May 2020 on the number of undetected individuals.
sensitivity analysis done on 0. Figs. 13(a) and 13(c) show that an
increase in the rate of implementation of governmental actions (𝜅)
results in a decrease in the value of 0. Also, we note that the instituted
governmental policy of isolating detected infected individuals reduces
0. Fig. 13(b) shows that an increase in the proportion of escapees,
𝑞, will increase the disease transmission rate (𝛽). This entails more
infectious individuals due to escapees, which will, in turn, increase
local COVID-19 transmission.

4.5. Effects of different lockdown measures

We investigate the impact of different lockdown scenarios on the
COVID-19 dynamics in Zimbabwe.

Figs. 14, 15, 16, and 17 illustrate the impact of different lockdown
strategies to curb the spread of the COVID-19 pandemic. Results show
that the longer the period the lockdown is in force, the more reduction
in the number of cases observed. However, this is usually accompanied
by an economic impact that may be difficult to endure. Thus, some
trade-offs between the economy and the infection would need to be
taken care of to ensure human life is preserved at a bearable economic
cost. Besides the length of the lockdown period, we also note that the
timing of the lockdown implementation is crucial in the fight against
the COVID-19 pandemic. A long delay in implementing the lockdown
can fail to minimize the rising number of infections. This also can lead
to the number of infections increasing above the threshold healthcare
capacity, resulting in an otherwise avoidable loss of life. Lastly, Fig. 18
estimates the projected time for reaching peak cases of COVID-19 in
the presence of the current dynamics of escapees. We observe that the
peak number of cases will be reached by approximately the beginning
of October 2020.

5. Conclusions

In this research paper, we investigate the impact of individuals
escaping from quarantine centers on the dynamics of COVID-19 trans-
mission in Zimbabwe. To analyze this, we employ a deterministic
compartmental model and subject it to a comprehensive examination
utilizing tools from the extensive field of mathematical epidemiology.
Our study includes determining and exploring the model’s reproduction
number, which is pivotal in assessing the model’s stability. We establish
that the disease-free equilibrium is locally and globally asymptotically
stable provided that 0 < 1. Global stability of the disease-free equi-
librium poses a more robust condition and suggests that even if there
are initial infections, COVID-19 is not expected to persist within the
population in the long run. Based on this, sufficient control measures
and interventions are needed to control and eliminate the pandemic.

Our numerical simulations yield several key findings. Firstly, we
observe that an increase in the number of escapees from quarantine
centers leads to a higher rate of local transmissions, resulting in more
12
individuals contracting the infection. Secondly, our research under-
scores the necessity of governmental actions exceeding a threshold
of 65% to effectively contain the spread of COVID-19 in Zimbabwe.
This highlights the critical importance of implementing measures that
enhance compliance with government-introduced intervention strate-
gies to combat the infection. Thirdly, our sensitivity analysis reveals
a strong positive correlation between the escapee parameter, denoted
as ‘‘𝑞’’, and the daily immigration rate, represented by ‘‘𝛬’’, with the
number of newly reported daily COVID-19 infections. This correlation
signifies that more escapees contribute to increased local transmissions
and prolonged disease duration. As a result, implementing mitigation
measures such as improving surveillance in quarantine centers becomes
crucial to reducing the number of escapees.

While our research offers valuable insights, it does have limita-
tions. Notably, we did not account for vital dynamics as the disease
approaches its peak. Incorporating vital dynamics in future studies is
imperative for a more comprehensive understanding of the disease’s
dynamics. Also, the study did not account for the potential evolution
of the COVID-19 virus or the emergence of new variants, which can
impact transmission dynamics and government regulations. Another
limitation of the present study involves the contribution of asymp-
tomatic infections. The current model was formulated at a time when
the contribution of asymptomatic infections was still being debated,
with many researchers conjecturing the possibility of a reasonable
proportion of new infections attributable to asymptomatic infected
individuals. However, the major difficulty was ascertaining the actual
contribution from asymptomatic individuals and very limited studies
had considered including asymptomatic individuals in their modeling
framework. Hence, the non-inclusion of this class in the current model.
However, the current model can be further improved by incorporating
a class of asymptomatically infected individuals, depending on the
availability of more information on this aspect. The results presented in
this paper are actionable and hold the potential to inform policy man-
agement and decision-making regarding the control of future outbreaks
of the COVID-19 pandemic in Zimbabwe and beyond.

Declaration of competing interest

The authors declare that we do not have any competing interests
regarding the production of the above-mentioned manuscript.

Data availability

Data will be made available on request.

Acknowledgments

The authors acknowledge, with thanks, the support of their respec-
tive departments towards the production of this manuscript.



Healthcare Analytics 4 (2023) 100275

13

J. Mushanyu et al.

Fig. 13. (a) 3D plot of parameters governmental action and mean duration in isolation versus the model basic reproduction number, 0. (b) Contour plots of the model parameters
proportion of escapees (𝑞) versus disease transmission rate, 𝛽. (c) Contour plots of the model parameters governmental role action (𝜅) versus detection rate of infected individuals,
𝛿1.

Fig. 14. Simulation of cumulative detected cases under different lockdown scenarios.



Healthcare Analytics 4 (2023) 100275

14

J. Mushanyu et al.

Fig. 15. Simulation of cumulative deaths under different lockdown scenarios.

Fig. 16. Simulation of daily detected cases for different delays in implementing lockdown measures.

Fig. 17. Simulation of daily undetected cases for different delays in implementing lockdown measures.

Fig. 18. Simulation of projected peak cases of COVID-19 in the presence of escapees.



Healthcare Analytics 4 (2023) 100275J. Mushanyu et al.
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	A deterministic compartmental model for investigating the impact of escapees on the transmission dynamics of COVID-19
	Introduction
	Model formulation
	Model analysis
	Positivity of solutions
	Invariant region
	Disease-free equilibrium state and the basic reproduction number
	Global stability of the disease-free equilibrium state

	Sensitivity analysis of R0.

	Numerical simulations
	Epidemiological parameter estimation
	Sensitivity analysis
	Sensitivity analysis before surge
	Short term predictions dynamics
	Long term predictions dynamics
	Sensitivity analysis after surge
	Short term predictions dynamics
	Long term predictions dynamics

	Effects of varying parameters q and κ on I(t)
	Effects of parameters on R0
	Effects of different lockdown measures

	Conclusions
	Declaration of competing interest
	Data availability
	Acknowledgments
	References