# Modelling COVID-19 Transmission Dynamics in Ghana

Edward Acheampong  Edward.Acheampong1<EMAIL_ADDRESS>Eric Okyere  Samuel Iddi  Joseph H. K. Bonney  Jonathan A. D. Wattis
<EMAIL_ADDRESS>Rachel L. Gomes
<EMAIL_ADDRESS>School of Mathematical Sciences, University of
Nottingham,
University Park, Nottingham, NG7 2RD, UK  Food Water Waste Research Group,
Faculty of Engineering,
University of Nottingham, University Park, Nottingham, NG7 2RD, UK  Department
of Mathematics and Statistics University of Energy
and Natural Resources, P.O. Box 214, Sunyani, B/A Ghana  Department of
Statistics and Actuarial Science University of Ghana,
P.O. Box LG 115, Legon, Ghana  Virology Department, Noguchi Memorial Institute
For Medical Research,
University of Ghana, P.O. Box LG 581, Legon, Ghana

###### Abstract

In late 2019, a novel coronavirus, the SARS-CoV-2 outbreak was identified in
Wuhan, China and later spread to every corner of the globe. Whilst the number
of infection-induced deaths in Ghana, West Africa are minimal when compared
with the rest of the world, the impact on the local health service is still
significant. Compartmental models are a useful framework for investigating
transmission of diseases in societies. To understand how the infection will
spread and how to limit the outbreak. We have developed a modified SEIR
compartmental model with nine compartments (CoVCom9) to describe the dynamics
of SARS-CoV-2 transmission in Ghana. We have carried out a detailed
mathematical analysis of the CoVCom9, including the derivation of the basic
reproduction number, $\mathcal{R}_{0}$. In particular, we have shown that the
disease-free equilibrium is globally asymptotically stable when
$\mathcal{R}_{0}<1$ via a candidate Lyapunov function. Using the SARS-CoV-2
reported data for confirmed-positive cases and deaths from March 13 to August
10, 2020, we have parametrised the CoVCom9 model. The results of this fit show
good agreement with data. We used Latin hypercube sampling-rank correlation
coefficient (LHS-PRCC) to investigate the uncertainty and sensitivity of
$\mathcal{R}_{0}$ since the results derived are significant in controlling the
spread of SARS-CoV-2. We estimate that over this five month period, the basic
reproduction number is given by $\mathcal{R}_{0}=3.110$, with the 95%
confidence interval being $2.042\leq\mathcal{R}_{0}\leq 3.240$, and the mean
value being $\mathcal{R}_{0}=2.623$. Of the 32 parameters in the model, we
find that just six have a significant influence on $\mathcal{R}_{0}$, these
include the rate of testing, where an increasing testing rate contributes to
the reduction of $\mathcal{R}_{0}$.

###### keywords:

Transmission model , SARS-CoV-2 , Uncertainty , Sensitivity , Mathematical
analysis , Monte Carlo-least squares.

††journal: *

##  1 Introduction

The recent COVID-19 pandemic has caused a devastating burden on the global
economy. Since there are currently no widely-available vaccines to bring down
or reduce the infection levels on the susceptible human population, many
governmental decision-makers worldwide have resorted to intensive non-
pharmaceutical interventions such as wearing of face-masks, social distancing,
cleaning of suspected infected surfaces, avoiding crowded places, the use of
hand sanitizers. These non-pharmaceutical interventions have significantly
helped to reduce the risk of transmission of COVID-19.

Mathematical and statistical modelling tools are important in providing key
epidemiological parameters of infectious diseases such as infection or
transmission rate, recovery rate, incubation period, isolation and
hospitalization rate, quarantine rate, disease-induced death rate, vaccination
rate (with other factors depending on the model formulation)(Chowell et al.,
2009). Using mathematical models, parametrised to confirmed reported cases of
infection, helps estimate the basic reproduction number, $\mathcal{R}_{0}$
which is a crucial epidemiological parameter that determines whether the
infection persists in the population or dies out (Li et al., 2020b; Dietz,
1993; Ma, 2020; Roberts and Heesterbeek, 2007; Chowell et al., 2004).

Nonlinear mathematical models have significantly contributed to the
understanding of transmission dynamics of infectious diseases, see, e.g.,
(Hethcote, 2000; Chowell and Hyman, 2016; Brauer et al., 2008; Khan et al.,
2015), and the recent COVID-19 pandemic is of no exception (Blyuss and
Kyrychko, 2021; Giordano et al., 2020; Ali et al., 2020; Asamoah et al., 2020;
Mushayabasa et al., 2020; Ndairou et al., 2020; Gevertz et al., 2020; Carcione
et al., 2020). Qianying et al. (2020) have proposed and studied a data-driven
SEIR type epidemic for the recent COVID-19 outbreak in Wuhan which captures
the effects of governmental actions and individuals’ behaviour. This
literature is growing rapidly; Abou-Ismail (2020) has reviewed the
fundamentals in SIR/SEIR modelling of the recent COVID-19 outbreak; here we
give a brief overview of literature relevant to our work.

Buonomo (2020) describes a susceptible-infected-recovered-infected
compartmental model to investigate the effects of information-dependent
vaccination behavior on COVID-19 infections. A simple SEIR COVID-19 epidemic
model with nonlinear incidence rates that capture governmental control has
been designed by Rohith and Devika (2020) to examine the dynamics of the
infectious disease in India. Pang et al. (2020) parametrise a nonlinear SEIHR
model to estimate the value and sensitivity of $\mathcal{R}_{0}$ using data
from Wuhan from December 31st, 2019. A classic SEIR epidemic is used to study
the spreading dynamics of the 2019 coronavirus disease in Indonesia (Annas et
al., 2020) using vaccination and isolation as model parameters. They
constructed a Lyapunov function to conduct global stability of the disease-
free equilibrium point. A data-driven epidemiological model that examines the
effect of delay in the diagnosis of the deadly COVID-19 disease is formulated
and studied by Rong et al. (2020), who estimate parameters and performed a out
global sensitivity analysis of their model parameters on $\mathcal{R}_{0}$.

A nonlinear SEIQR COVID-19 epidemic model is introduced by Zeb et al. (2020)
who present a local and global stability analysis for their model. The spread
of Covid19 in China due to undetected infections in is examined by Ivorra et
al. (2020). Chen et al. (2020) propose a model based on dividing the total
population into five non-overlapping classes: susceptible, exposed, infected
(symptomatic infection), asymptomatic infected, and recovered. Sardar et al.
(2020), investigate the effects of lockdown using an SEIR model. Using
reported cases of this highly infectious disease in some cities and the whole
of India, they performed a global sensitivity analysis on the computed
$\mathcal{R}_{0}$.

The exposed and infectious epidemiological classes used in formulating
infectious diseases models mentioned above have been left as abstract
concepts. In reality, especially regarding SARS-CoV-2, it is hard to
distinguish between individuals exposed to or infected with SARS-CoV-2, due to
the presence of asymptomatic carriers. In this present study, we introduce two
epidemiological classes, which are: (1) an identified group of exposed
individuals suspected to be carriers of SARS-CoV-2 (denoted by $Q$); and, (2)
individuals who have been clinically confirmed-positive for SARS-CoV-2
(denoted by $P$). Those identified as suspected exposed individuals are
denoted by $Q$ because they are quarantined as required by the COVID-19
protocols in Ghana. Likewise, confirmed-positives ($P$) are treated as
infectious individuals who have clinically tested positive for SARS-CoV-2.
Introducing these distinctions in the epidemiological classes for SARS-CoV-2
is vital for gaining an understanding of its transmission dynamics within the
Ghanaian population. Using published data from March 13 to August 10, 2020
(Ritchie, 2020), we have parametrised our model using a Monte Carlo-least
squares technique together with information derived from literature.

The purpose of this research is to investigate the transmission dynamics of
SARS-CoV-2 in Ghana using these more specific epidemiological classes to
estimate the basic reproduction number, $\mathcal{R}_{0}$. We have used Latin-
Hypercube Sampling-Partial Rank Correlation Coefficient (LHS-PRCC) technique
to quantify the uncertainty in $\mathcal{R}_{0}$ as well as to identify those
parameters to which $\mathcal{R}_{0}$ is most sensitive. We have organised the
subsequent sections of the paper as follows: in Section 2 we present a
detailed formulation of an epidemiological model of SARS-CoV-2 transmission in
Ghana, together with corresponding mathematical analysis of the positivity and
boundedness of solutions, a derivation of the basic reproduction number, and
global stability analysis of the disease-free equilibrium, which are given in
Section 3. Section 4 is dedicated to parameter estimation and numerical
simulation. The uncertainty and sensitivity analysis of $\mathcal{R}_{0}$ and
its implications are presented in Section 5, together with some simulations
predicting possible future dynamics of the epidemic. Finally, we give a brief
discussion and conclusion of the study in Section 6.

##  2 Formulation of the model

Compartmental models are useful means of qualitatively understanding the
dynamics of disease transmissions within a population (Martcheva, 2015;
Chowell et al., 2009). In formulating our compartmental model to gain insight
into COVID-19 transmission dynamics, the total human population is divided
into nine distinct epidemiological classes which are summarised in Table 1.
The numbers of individuals in each category is treated as a continuous
variable, the classes being: susceptible, $S(t)$, exposed, $E(t)$, infectious,
$I(t)$, quarantined suspects $Q(t)$, confirmed-positive $P(t)$, hospitalised
in the ordinary ward $H(t)$, hospitalised in the intensive care unit $C(t)$,
self-isolation $F(t)$ and recovered, $R(t)$. The total number of individuals
in the population is thus given by

$N(t)=S(t)+E(t)+I(t)+Q(t)+P(t)+H(t)+C(t)+F(t)+R(t).$ (2.1)

Table 1: Description of the variables of the CoVCom9 model. Variable | Description
---|---
$N$ | Total population
$S$ | Susceptible individuals
$E$ | Exposed individuals
$I$ | Infectious individuals
$Q$ | Quarantine suspected individuals
$P$ | Comfirmed-positive individuals
$H$ | Hospitalised at ordinary ward individuals
$C$ | Hospitalised at intensive care individuals
$F$ | Self-isolation individuals
$R$ | Recovered individuals
Figure 1: Transmission diagram for the model of COVID-19 involving ten
compartments. See Tables 1 and 2 for explanations of the parameters and
variables used in the model, respectively.

Figure 1 summarises the dynamic processes by which individuals pass from one
class to another. The susceptible class ($S$) represents individuals not
exposed to the SARS-CoV-2 virus, and the exposed class ($E$) represents
individuals that have recently been exposed to the SARS-CoV-2 virus so are
still in the incubation period and can infect others (that is, asymptomatic
individuals). An individual in an exposed class can infect another person but
with a probability lower than an individual in the infectious class ($I$).
This rate of infection is given by the nonlinear function $f$ which depends on
the parameters
$\varphi,\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2},\beta_{3}$.
Individuals in an infectious class show clear symptoms and have high
infectivity. These individuals have not yet been clinically confirmed-
positive, and thus can spread the disease to the susceptibles. Individuals in
class $Q$ are quarantined, that is, individuals identified to have had contact
with an exposed individual and so might be carrying the SARS-CoV-2 virus (but
this has not yet been confirmed), this class also includes individuals not
infected with SARS-CoV-2 but are quarantined as a result of enforcement of
COVID-19 protocols. These individuals may either enter the susceptible class
if test is confirmed negative or to the confirmed-positive class if confirmed
to be infected.

Individuals in the confirmed-positive class $P$ are carriers of the SARS-CoV-2
virus who have had clinical confirmation of this status. These individuals may
either enter the intensive care hospitalised class, or be admitted to the
ordinary hospitalised class or enter the self-isolated class after this
period. The rates of the these processes are governed by the parameters
$\gamma_{1},v_{1},\rho_{1},\rho_{2},\rho_{3}$. The individuals in the ordinary
Hospitalised class shows some level of sickness due to infection that need to
be cared for at the ordinary ward. Though there is chance of entering into
recovery class, these individuals’ conditions may deteriorate causing them to
enter the intensive care hospitalised class. Individuals move between these
categories with rates determined by $\kappa_{2},\kappa_{3},\delta_{2},\eta$.
These individuals can still infect other individuals who become exposed
through close contact. Individuals in intensive care ($C$) can still infect
other individuals and have a high risk of dying (rates $d_{j}$) although
improved care conditions may allow transfer to the ordinary ward ($H$, at rate
$\eta$).

Individuals in the self-isolated class ($F$) are on medication at home and can
still infect other individuals. These individuals ($F$) may either enter the
recovered class ($R$, at rate $\delta_{1}$) or enter the ordinary hospitalised
class (rate $\delta_{2}$). Individuals who have recovered from SARS-CoV-2
virus enter into the recovered class ($R$) but can be re-infected since there
is no life-long immunity, hence there is a flux from $R$ to $S$ with rate
parameter $\tau$. We assume that individuals in all the compartments can die
of COVID-19 (rates $d_{j}$) in addition to natural death (rate $\mu$) with the
exception of the susceptible compartment with only natural death. A summary of
all the parameter definitions is given in Table 2.

Table 2: Description of the CoVCom9 model parameters. Parameters | Description
---|---
$\Lambda$ |  Recruitment rate
$\mu$ |  Natural death rate
$\varphi$ |  Transmission rate of infectious individuals ($I$)
$\alpha_{1}$ |  Probability of transmission of exposed individuals ($E$)
$\alpha_{2}$ |  Probability of transmission of quarantine suspected infectious individuals ($Q$)
$\alpha_{3}$ |  Probability of transmission of confirmed-positive infectious individuals ($P$)
$\beta_{1}$ |  Probability of transmission of hospitalised at ordinary ward individuals ($H$)
$\beta_{2}$ |  Probability of transmission of hospitalised at intensive care individuals ($C$)
$\beta_{3}$ |  Probability of transmission of self-isolation at home individuals ($F$)
$\epsilon_{1}$ |  Progression rate of exposed individuals to infectious class per day
$\epsilon_{2}$ |  Progression rate of exposed individuals to quarantine suspected per day
$\gamma_{1}$ |  Progression rate of infectious individuals to confirmed-positive per day
$\gamma_{2}$ |  Recovery rate of infectious individuals per day
$\upsilon_{1}$ |  Progression rate of quarantined ($Q$) to confirmed cases ($P$) per day
$\upsilon_{2}$ |  Progression rate of quarantined cases to susceptible cases per day
$\rho_{1}$ |  Progression rate of confirmed-positive infectives to hospital class per day
$\rho_{2}$ |  Progression rate of confirmed-positive to intensive care class per day
$\rho_{3}$ |  Progression rate of confirmed-positive ($P$) to self-isolation
|  at home ($F$) class per day
$\kappa_{1}$ |  Recovery rate of hospitalised ($H$) individual per day
$\kappa_{2}$ |  Progression rate of hospitalised (ordinary, $H$) to intensive care ($C$) per day
$\kappa_{3}$ |  Progression rate of hospitalised (ordinary ward, $H$) to
|  self-isolation at home ($F$) class per day
$\delta_{1}$ |  Recovery rate of self-isolation at home individual per day
$\delta_{2}$ |  Progression rate of self-isolation at home
|  to hospitalised at ordinary ward ($H$) class per day
$\eta$ |  Progression rate of intensive care to ordinary ward class per day
$\tau$ |  Progression rate of recovery individuals to susceptible class per day
$d_{1}$ |  Disease-induced death rate of exposed individuals per day
$d_{2}$ |  Disease-induced death rate of infectious individuals per day
$d_{3}$ |  Disease-induced death rate of quarantine suspected infectives per day
$d_{4}$ |  Disease-induced death rate of confirmed-positive individuals per day
$\sigma_{5}$ |  Disease-induced death rate of intensive care individuals per day
$\delta_{6}$ |  Disease-induced death rate of self-isolated ($F$) cases per day
$d_{7}$ |  Disease-induced death rate of hospitalised individuals per day

The standard form of incidence which is formulated from the basic principles
that effective transmission rates are independent of the population size $N$
for human diseases is used in this study (Martcheva, 2015; Hethcote, 2000).
This principle has been shown in many studies to be a plausible assumption
(Hethcote, 2000). If $\alpha$ is the average number of sufficient contacts for
transmission of an individual per unit time, then $\alpha I/N$ is the average
number of contacts with infectives per unit time of one susceptible, and
$(\alpha I/N)S$ is the incidence. That is, the number of new cases per unit
time at time $t$ due to susceptibles $S(t)$ becoming infected (Hethcote,
2000). We use $\varphi$ to denote the effective transmission rate from an
infectious individual while $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$,
$\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ denote the transmission
probabilities, of exposed individuals, quarantine suspected exposed
individuals, confirmed-positive individuals, ordinary hospitalised
individuals, intensive care hospitalised individuals, and self-isolated
individual, respectively. All these probabilities lie between zero and one.
The incidence is therefore given by

$f(S,E,I,Q,P,H,C,F)=\varphi\bigg{(}\frac{\alpha_{1}E+I+\alpha_{2}Q+\alpha_{3}P+\beta_{1}H+\beta_{2}C+\beta_{3}F}{N}\bigg{)}S.$
(2.2)

Our COVID-19 model (CoVCom9) is obtained by ‘translating’ the compartmental
model summarised in Figure 1 into nine coupled ordinary differential equations

$\displaystyle\dfrac{dS}{dt}$ $\displaystyle=\Lambda+\upsilon_{2}U+\tau
R-f(S,E,I,Q,P,H,C,F)-\mu S,$ (2.3a) $\displaystyle\dfrac{dE}{dt}$
$\displaystyle=f(S,E,I,Q,P,H,C,F)-(\epsilon_{1}+\epsilon_{2}+\mu+d_{1})E,$
(2.3b) $\displaystyle\dfrac{dI}{dt}$
$\displaystyle=\epsilon_{1}E-(\gamma_{1}+\gamma_{2}+\mu+d_{2})I,$ (2.3c)
$\displaystyle\dfrac{dQ}{dt}$
$\displaystyle=\epsilon_{2}E-(\upsilon_{1}+\upsilon_{2}+\mu+d_{3})Q,$ (2.3d)
$\displaystyle\dfrac{dP}{dt}$
$\displaystyle=\gamma_{1}I+\upsilon_{1}Q-(\rho_{1}+\rho_{2}+\rho_{3}+\mu+d_{4})P,$
(2.3e) $\displaystyle\dfrac{dH}{dt}$ $\displaystyle=\rho_{1}P+\eta
C+\delta_{2}Q-(\kappa_{1}+\kappa_{2}+\kappa_{3}+\mu+d_{5})H,$ (2.3f)
$\displaystyle\dfrac{dC}{dt}$
$\displaystyle=\rho_{2}P+\kappa_{2}H-(\eta+\mu+d_{6})C,$ (2.3g)
$\displaystyle\dfrac{dF}{dt}$
$\displaystyle=\rho_{3}P+\kappa_{3}H-(\delta_{1}+\delta_{2}+\mu+d_{7})F,$
(2.3h) $\displaystyle\dfrac{dR}{dt}$
$\displaystyle=\gamma_{2}I+\kappa_{1}H+\delta_{1}F-(\tau+\mu)R,$ (2.3i)

with $t>0$. These are solved subject to the initial conditions

$\displaystyle S(0)=S_{0}\geq 0,\quad E(0)=E_{0}\geq 0,\quad I(0)=I_{0}\geq
0,$ (2.4) $\displaystyle Q(0)=Q_{0}\geq 0,\quad P(0)=P_{0}\geq 0,\quad
H(0)=H_{0}\geq 0,$ $\displaystyle C(0)=C_{0}\geq 0,\quad F(0)=F_{0}\geq
0,\quad R(0)=R_{0}\geq 0.$

In this paper, we will use the acronym CoVCom9 to indicate the nine
compartments of the model of SARS-CoV-2 transmission pattern in Ghana given by
Equation (2.3). The epidemiologically feasible region of interest of the model
(2.3) is the domain defined by

$\displaystyle\Omega=\biggl{\\{}(S(t),E(t),I(t),Q(t),P(t),H(t),C(t),F(t),R(t))\in\mathbb{R}_{+}^{9}:$
$\displaystyle S+E+I+Q+P+H+C+F+R\leq\frac{\Lambda}{\mu}\biggr{\\}}.$ (2.5)

In the following sections we present a mathematical analysis of the model with
respect to positivity and boundedness of the feasible region, $\Omega$, as
well as various stability results and the epidemiological threshold of
interest. In the subsequent sections, we discuss a theorem demonstrating that
solutions of Equation (2.3) with initial condidtions (2.4) in $\Omega$ remain
in $\Omega$.

##  3 Mathematical analysis of CoVCom9 model

###  3.1 Positivity, boundedness and invariant region

The CoVCom9 model (2.3) depicts COVID-19 transmission dynamics in the human
population, so it is vital to show that variables in (2.3) remain nonnegative
and bounded for all time $t\geq 0$ and do not leave the epidemiologically
feasible region of interest, $\Omega$.

######  Lemma 1 (Positivity and Boundedness).

For any given nonnegative initial conditions in Eq. (2.4), the CoVCom9 model
(2.3) has a nonnegative solution
$\\{S(t),E(t),I(t),Q(t),P(t),H(t),C(t),F(t),R(t)\\}$ of the system (2.3) for
all time $t\geq 0$ whenever the parameters are non-negative. Moreover

$\lim_{t\rightarrow\infty}\sup N(t)\leq\frac{\Lambda}{\mu}.$ (3.1)

###### Proof.

Considering the first equation of the CoVCom9 model (2.3), one can clearly see
that

$\dfrac{dS}{dt}\geq-(\lambda+\mu)S,$ (3.2)

where

$\lambda=\varphi\bigg{(}\frac{\alpha_{1}E+I+\alpha_{3}Q+\alpha_{4}P+\beta_{1}H+\beta_{2}C+\beta_{3}F}{N}\bigg{)}$

Next, integrating Eq. (3.2), we find

$S(t)\geq S_{0}\exp\biggl{[}-\int_{o}^{t}(\lambda(\zeta)+\mu)d\zeta\biggr{]}$
(3.3)

Therefore $S(t)\geq 0$ for all $t\geq 0$.

Following a similar argument, it can be shown that the rest of the model
variables have nonnegative solutions for all time $t\geq 0$. That is,
$E(t)\geq 0$, $I(t)\geq 0$, $Q(t)\geq 0$, $P(t)\geq 0$, $H(t)\geq 0$,
$C(t)\geq 0$, $F(t)\geq 0$, $R(t)\geq 0$, $\forall t\geq 0$.

Furthermore, we prove that the solutions are bounded. Adding the right-hand
side of the CoVCom9 model (2.3) yields

$\displaystyle\dfrac{dN}{dt}$ $\displaystyle=\Lambda-\mu N-dE-d_{1}I-dU-
d_{2}P-d_{3}H-d_{4}C-d_{5}F\;\leq\;\Lambda-\mu N,$ (3.4)

Since $dN/dt\leq\Lambda-\mu N$, it follows that

$\lim_{t\rightarrow\infty}\sup N(t)\leq\frac{\Lambda}{\mu}.$ (3.5)

∎

######  Lemma 2 (Positively Invariant Region).

The region defined by the closed set, $\Omega$ in Eq. (2) is positively
invariant for the model (2.3) with nonnegative initial conditions (2.4)
whenever the parameters are nonnegative.

###### Proof.

As in Lemma 1, it follows from the summation of all the equations of the
CoVCom9 model (2.3) that

$\displaystyle\dfrac{dN}{dt}$ $\displaystyle\leq\Lambda-\mu N.$ (3.6)

Using the initial condition $N(0)>0$ and an integrating factor, we have

$\displaystyle 0\leq N(t)$ $\displaystyle\leq\frac{\Lambda}{\mu}+N(0)\exp(-\mu
t),$ (3.7)

where $N(0)$ is the initial value of the total population. Thus
$N(t)\leq\Lambda/\mu$, as $t\rightarrow\infty$. Therefore all feasible
solutions of system (2.3) enter the region $\Omega$ defined by (2), which is a
positively invariant set of the system (2.3). This implies that all solutions
in $\Omega$ remain in $\Omega$ $\forall t\geq 0$. It is therefore sufficient
to study the dynamics of CoVCom9 model system (2.3) in $\Omega$.

∎

###  3.2 The basic reproduction number and existence of equilibria

The CoVCom9 model has a disease-free equilibrium point given by

$\mathcal{E}_{0}=(S_{0},0,0,0,0,0,0,0,0)\in\Omega,\qquad
S_{0}=\frac{\Lambda}{\mu}.$ (3.8)

The basic reproduction number is defined as the number of secondary infections
produced by a single infectious individual during the entire infectious period
(Van den Driessche and Watmough, 2002). In this study, the reproduction number
defined as the number of secondary SARS-CoV-2 infections generated by a single
active SARS-CoV-2 individual during the entire infectious period.
Mathematically, the basic reproduction number $\mathcal{R}_{0}$ is the
dominant eigenvalue of the next generation matrix (Diekmann et al., 2010; Van
den Driessche and Watmough, 2002). We apply the method formulation in (Van den
Driessche and Watmough, 2002) to obtain an expression of $\mathcal{R}_{0}$ for
the proposed CoVCom9 (2.3). Let ${\bf x}=\big{(}E,I,Q,P,H,C,F\big{)}^{T}$,
then the system (2.3) can be written in the form

$\frac{d\mathbf{x}}{dt}=\mathcal{F}(\mathbf{x})-\mathcal{V}(\mathbf{x}),$
(3.9)

where

$\mathcal{F}(\mathbf{x})=\begin{bmatrix}\dfrac{\varphi(\alpha_{1}E+I+\alpha_{2}Q+\alpha_{3}P+\beta_{1}H+\beta_{2}C+\beta_{3}F)S}{N}\\\
0\\\ 0\\\ 0\\\ 0\\\ 0\\\ 0\end{bmatrix},$ (3.10)
$\mathcal{V}(\mathbf{x})=\begin{bmatrix}\pi_{E}E\\\ -\epsilon_{1}E+\pi_{I}I\\\
-\epsilon_{2}E+\pi_{Q}Q\\\ -\gamma_{1}I-\nu_{1}Q+\pi_{P}P\\\ -\rho_{1}P-\eta
C-\delta_{2}F+\pi_{H}H\\\ -\rho_{2}P-\kappa_{2}H+\pi_{C}C\\\
-\rho_{3}P-\kappa_{3}H+\pi_{F}F\end{bmatrix}.$ (3.11)

and

$\displaystyle\pi_{E}=$
$\displaystyle\epsilon_{1}+\epsilon_{2}+\mu+d_{1};\qquad\pi_{I}=\gamma_{1}+\gamma_{2}+\mu+d_{2};$
(3.12) $\displaystyle\pi_{Q}=$
$\displaystyle\upsilon_{1}+\upsilon_{2}+\mu+d_{3};\qquad\pi_{P}=\rho_{1}+\rho_{2}+\rho_{3}+\mu+d_{4};$
$\displaystyle\pi_{H}=$
$\displaystyle\kappa_{1}+\kappa_{2}+\kappa_{3}+\mu+d_{5};\qquad\pi_{C}=\eta+\mu+d_{6};$
$\displaystyle\pi_{F}=$ $\displaystyle\delta_{1}+\delta_{2}+\mu+d_{7}.$

The Jacobians of $\mathcal{F}(\mathbf{x})$ and $\mathcal{V}(\mathbf{x})$
evaluated at the disease free equilibrium $E_{0}$ are, respectively,

$J_{F}=\begin{bmatrix}\varphi\alpha_{1}&\varphi&\varphi\alpha_{2}&\varphi\alpha_{3}&\varphi\beta_{1}&\varphi\beta_{2}&\varphi\beta_{3}\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ \end{bmatrix},$ (3.13)
$J_{V}=\begin{bmatrix}\pi_{E}&0&0&0&0&0&0\\\
-\epsilon_{1}&\pi_{I}&0&0&0&0&0\\\ -\epsilon_{2}&0&\pi_{Q}&0&0&0&0\\\
0&-\gamma_{1}&-\nu_{1}&\pi_{P}&0&0&0\\\
0&0&0&-\rho_{1}&\pi_{H}&-\eta&-\delta_{2}\\\
0&0&0&-\rho_{2}&-\kappa_{2}&\pi_{C}&0\\\
0&0&0&-\rho_{3}&-\kappa_{3}&0&\pi_{F}\end{bmatrix}.$ (3.14)

The basic reproduction number, $\mathcal{R}_{0}$ is given by the dominant
eigenvalue of $J_{F}J_{V}^{-1}$

$\displaystyle\mathcal{R}_{0}=$
$\displaystyle\;\varphi\biggl{\\{}\dfrac{\alpha_{1}}{\pi_{E}}+\dfrac{\epsilon_{1}}{\pi_{E}\pi_{I}}+\dfrac{\alpha_{2}\epsilon_{2}}{\pi_{E}\pi_{Q}}+\dfrac{\alpha_{3}}{\pi_{P}}\biggl{(}\dfrac{\epsilon_{1}\gamma_{1}}{\pi_{E}\pi_{I}}+\dfrac{\epsilon_{2}\upsilon_{1}}{\pi_{E}\pi_{Q}}\biggr{)}$
$\displaystyle+\dfrac{\beta_{1}}{\pi_{P}}\biggl{(}\dfrac{\rho_{1}\pi_{C}\pi_{F}+\delta_{2}\rho_{3}\pi_{C}+\eta\rho_{2}\pi_{F}}{\pi_{H}\pi_{C}\pi_{F}-\delta_{2}\kappa_{3}\pi_{C}-\eta\kappa_{2}\pi_{F}}\biggr{)}\biggl{(}\dfrac{\epsilon_{1}}{\pi_{E}}\dfrac{\gamma_{1}}{\pi_{I}}+\dfrac{\epsilon_{2}}{\pi_{E}}\dfrac{\upsilon_{1}}{\pi_{Q}}\biggr{)}$
$\displaystyle+\dfrac{\beta_{2}}{\pi_{P}}\biggl{(}\dfrac{\rho_{2}\pi_{H}\pi_{F}+\kappa_{2}\rho_{1}\pi_{F}+\delta_{2}(\kappa_{2}\rho_{3}\\!-\\!\kappa_{3}\rho_{2})}{\pi_{H}\pi_{C}\pi_{F}-\delta_{2}\kappa_{3}\pi_{C}-\eta\kappa_{2}\pi_{F}}\biggr{)}\biggl{(}\dfrac{\epsilon_{1}\gamma_{1}}{\pi_{E}\pi_{I}}+\dfrac{\epsilon_{2}\upsilon_{1}}{\pi_{E}\pi_{Q}}\biggr{)}$
$\displaystyle+\dfrac{\beta_{3}}{\pi_{P}}\biggl{(}\dfrac{\rho_{3}\pi_{H}\pi_{C}+\kappa_{3}\rho_{1}\pi_{C}+\eta(\kappa_{3}\rho_{2}\\!-\\!\kappa_{2}\rho_{3})}{\pi_{H}\pi_{C}\pi_{F}-\delta_{2}\kappa_{3}\pi_{C}-\eta\kappa_{2}\pi_{F}}\biggr{)}\biggl{(}\dfrac{\epsilon_{1}\gamma_{1}}{\pi_{E}\pi_{I}}+\dfrac{\epsilon_{2}\upsilon_{1}}{\pi_{E}\pi_{Q}}\biggr{)}\biggr{\\}},$
(3.15)

which can be written as

$\displaystyle\mathcal{R}_{0}=$
$\displaystyle\;\mathcal{R}_{0E}+\mathcal{R}_{0I}+\mathcal{R}_{0Q}+\mathcal{R}_{0P}+\mathcal{R}_{0H}+\mathcal{R}_{0C}+\mathcal{R}_{0F},$
(3.16)

where the effective reproduction number, $\mathcal{R}_{0}$ is made up of
contributions from secondary infections from the exposed group $E$
($\mathcal{R}_{0E}$) generated by asymptomatic individuals; the infected
(symptomatic) group $I$ ($\mathcal{R}_{0I}$); asymptomatic quarantine
suspected individuals - class-$Q$ ($\mathcal{R}_{0Q}$); confirmed positive
individuals - class $P$ ($\mathcal{R}_{0P}$); hospitalised cases ($H$,
$\mathcal{R}_{0H}$); intensive care ($C$) cases, ($\mathcal{R}_{0C}$); and
those self-isolating at home ($F$, $\mathcal{R}_{0F}$). Equation (3.15)
implies that intervention strategies of SARS-CoV-2 infections should target
those in classes $E$, $I$, $Q$, $P$, $H$, $C$, and $F$.

According to Theorem 3.2 of Van den Driessche and Watmough (2002), the
disease-free steady state $E_{0}$ is locally asymptotically stable if
$\mathcal{R}_{0}<1$ and unstable if $\mathcal{R}_{0}>1$. In the next section
we provide stability results for the disease-free equilibrium state.

###  3.3 Stability of disease free equilibrium (DFE)

In this section, we prove global stability results for the CoVCom9 model
(2.3). The epidemiological implication of the local stability is that a small
number of the infected individuals will not generate large outbreaks so in the
long run, resulting in SARS-CoV-2 dying out provided $\mathcal{R}_{0}<1$. The
global stability result helps demonstrate that the disappearance of SARS-CoV-2
disease is independent of the size of the initial subpopulations in the model,
provided $\mathcal{R}_{0}<1$ (Ali et al., 2020). The global stability of the
disease-free equilibrium, $E_{0}$ is established using a candidate Lyapunov
function.

######  Theorem 1.

The disease-free equilibrium state, $E_{0}$ of the CoVCom9 model (2.3) is
globally asymptotically stable in $\Omega$ if $\mathcal{R}_{0}<1$ and unstable
if $\mathcal{R}_{0}>1$.

###### Proof.

We construct a candidate Lyapunov function (3.17) for the CoVCom9 model (2.3)
as

$V(E,I,Q,P,H,C,F)=\varPhi_{1}E+\varPhi_{2}I+\varPhi_{3}Q+\varPhi_{4}P+\varPhi_{5}H+\varPhi_{6}C+\varPhi_{7}F,$
(3.17)

where $\varPhi_{i},i=1,2,\cdots,7$ are (as yet unknown) non-negative
coefficients. Since all the variables are bounded below by zero, then so is
$V$. Assuming that the variables are solutions of the model (2.3), the
derivative of $V$ with respect to $t$ can be bounded by

$\displaystyle\dfrac{dV}{dt}=$
$\displaystyle\;\varPhi_{1}\biggl{(}\varphi\big{(}\alpha_{1}E+I+\alpha_{2}Q+\alpha_{3}P+\beta_{1}H+\beta_{2}C+\beta_{3}F\big{)}\big{(}\frac{S}{N}\big{)}-\pi_{E}E\biggr{)}$
(3.18)
$\displaystyle+\varPhi_{2}\biggl{(}\epsilon_{1}E-\pi_{I}I\biggr{)}+\varPhi_{3}\biggl{(}\epsilon_{2}E-\pi_{Q}Q\biggr{)}+\varPhi_{4}\biggl{(}\gamma_{1}I+\upsilon_{1}Q-\pi_{P}P\biggr{)}$
$\displaystyle+\varPhi_{5}\biggl{(}\rho_{1}P+\eta
C+\delta_{2}F-\pi_{H}H\biggr{)}+\varPhi_{6}\biggl{(}\rho_{2}P+\kappa_{2}H-\pi_{C}C\biggr{)}$
$\displaystyle+\varPhi_{7}\biggl{(}\rho_{3}P+\kappa_{3}H-\pi_{F}F\biggr{)}$
$\displaystyle\leq$
$\displaystyle\;\biggl{(}\varPhi_{1}\varphi\alpha_{1}+\varPhi_{2}\epsilon_{1}+\varPhi_{3}\epsilon_{2}-\varPhi_{1}\pi_{E}\biggr{)}E+\biggl{(}\varPhi_{1}\varphi+\varPhi_{4}\gamma_{1}-\varPhi_{2}\pi_{I}\biggr{)}I$
$\displaystyle\\!+\\!\biggl{(}\\!\varPhi_{1}\varphi\alpha_{3}\\!+\\!\varPhi_{4}\upsilon_{1}\\!-\\!\varPhi_{3}\pi_{Q}\\!\biggr{)}Q\\!+\\!\biggl{(}\\!\varPhi_{1}\theta\alpha_{4}\\!+\\!\varPhi_{5}\rho_{1}\\!+\\!\varPhi_{6}\rho_{2}\\!+\\!\varPhi_{7}\rho_{3}\\!-\\!\varPhi_{4}\pi_{P}\\!\biggr{)}P\\!$
$\displaystyle\\!+\biggl{(}\varPhi_{1}\varphi\beta_{1}+\varPhi_{6}\kappa_{2}+\varPhi_{7}\kappa_{3}-\varPhi_{5}\pi_{H}\biggr{)}H+\biggl{(}\varPhi_{1}\varphi\beta_{2}+\varPhi_{5}\eta-\varPhi_{6}\pi_{C}\biggr{)}C$
$\displaystyle\\!+\biggl{(}\varPhi_{1}\varphi\beta_{3}+\varPhi_{5}\delta_{2}-\varPhi_{7}\pi_{F}\biggr{)}F,\qquad\qquad\text{since}\qquad
S/N<1.$

Requiring the bracketed coefficients of $E$, $I$, $U$, $P$, $H$, $C$, and $Q$
to zero, we obtain expressions for the previously undetermined parameters
$\varPhi_{i}$, which are thus given by

$\begin{array}[]{c}\varPhi_{1}=1,\qquad\varPhi_{2}=\dfrac{\varphi+\varPhi_{4}\gamma_{1}}{\pi_{I}},\qquad\varPhi_{3}=\dfrac{\varphi\alpha_{2}+\varPhi_{4}\upsilon_{1}}{\pi_{Q}},\\\\[12.91663pt]
\varPhi_{4}=\dfrac{1}{\pi_{P}}\biggl{[}\varphi\alpha_{3}+\dfrac{\varphi\beta_{2}\rho_{2}}{\pi_{C}}+\dfrac{\varphi\beta_{3}\rho_{3}}{\pi_{Q}}+\biggl{(}\rho_{1}+\dfrac{\eta\rho_{2}}{\pi_{C}}+\dfrac{\delta_{2}\rho_{3}}{\pi_{F}}\biggr{)}\varPhi_{5}\biggr{]},\\\\[12.91663pt]
\varPhi_{5}=\varphi\biggl{(}\dfrac{\beta_{1}\pi_{C}\pi_{F}+\beta_{2}\kappa_{2}\pi_{F}+\beta_{3}\kappa_{3}\pi_{C}}{\pi_{H}\pi_{C}\pi_{F}-\eta\kappa_{2}\pi_{F}-\delta_{2}\kappa_{3}\pi_{C}}\biggr{)},\\\\[12.91663pt]
\varPhi_{6}=\dfrac{\varphi\beta_{2}+\varPhi_{5}\eta}{\pi_{C}},\quad\text{and
}\quad\varPhi_{7}=\dfrac{\varphi\beta_{3}+\varPhi_{5}\delta_{2}}{\pi_{F}},\end{array}$
(3.19)

where the parameter groupings $\pi_{*}$ are given by (3.12).

After some simplifications using (3.12), the time derivative of the Lyapunov
function can be written as

$\dfrac{dV}{dt}\leq\pi_{E}\bigg{(}\mathcal{R}_{0}-1\bigg{)}E.$ (3.20)

It is now clear that if $\mathcal{R}_{0}<1$ then $dV/dt\leq 0$. Furthermore,
$dV/dt=0$ if $E=0$ and $\mathcal{R}_{0}<1$. Thus, when $\mathcal{R}_{0}<1$,
the largest compact invariant set in
$\left\\{\left(S,E,I,Q,P,H,C,F,R\right)\in\Omega\;|\;\dot{V}\leq 0\right\\}$
is the single state $\mathcal{E}_{0}$. LaSalle’s Invariance Principle then
implies that $\mathcal{E}_{0}$ is globally asymptotically stable in $\Omega$
if $\mathcal{R}_{0}<1$. ∎

##  4 CoVCom9 model estimation and numerical simulations

###  4.1 Methodology

In this section, we briefly describe the parameter estimation and numerical
simulation process used to investigate how well the proposed CoVCom9 model
(2.3) agrees with the confirmed cases and deaths in Ghana. Here, we consider
the SARS-CoV-2 confirmed cases and deaths from March 13, 2020 to August 10,
2020 as reported in Ghana. The data are obtained from Our World in Data
(Ritchie, 2020).

The CoVCom9 model (2.3) has nine state variables; to obtain the disease-
induced mortality ($D$), we introduce the extra equation

$\allowdisplaybreaks\dfrac{dD}{dt}=d_{1}E+d_{2}I+d_{3}Q+d_{4}P+d_{5}H+d_{6}C+d_{7}F,$
(4.1)

which introduces no additional parameters. The CoVCom9 model has a total of 35
parameters to estimate using limited data (confirmed-positive cases and deaths
only). This results in identifiability issues causing the non-convergence of
the optimisation of the objective function. We implement the following
practical principles to choose reasonable initial parameter values:

  1. 1.

Expert review process which involves asking health experts and/or consulting
the relevant literature as well as individuals’ experience of the infection.
Accordingly, an estimate of the model parameters, natural birth rate, $\mu$,
recruitment rate, $\Lambda$, incubation period, $\epsilon_{1}$, and recovery
rate of quarantine/self-isolation at home individual, $\delta_{1}$ are
obtained. We assumed that the life expectancy of people in Ghana is estimated
as 64.35 years (Asamoah et al., 2020), then the natural death rate is
estimated as $\mu=1/(64.35\times 365)\approx 4.258\times 10^{-5}$ per day. The
population of Ghana in 2020 is estimated to be $N=30,960,000$ ((2020), GSS),
and the recruitment rate of humans is estimated as $\Lambda=\mu N\approx
1.318\times 10^{3}$ people per day. The incubation period is 3–7 days, here we
choose $\epsilon_{1}=1/5.88$ per day as estimated by Pang et al. (2020) which
is consistent with the wider literature (Anderson et al., 2020; Li et al.,
2020a). The self-isolated positive-confirmed individuals on medication take 14
days on average to recover, thus we assume $\delta_{1}=1/14$ per day.

  2. 2.

Exploring the model using the available data (also known as ‘system
exploratory analysis’ (SEA) (Cside18, 2018)). This process helps identify
ranges of parameter values where the trajectories of the CoVCom9 are
consistent with the data, and regions of parameter space where trajectories
deviate from the times series data of confirmed-positive cases and deaths. The
motivation for this approach is to restrict the ranges of the parameters and
so reduce risk of the Monte Carlo simulation getting trapped at a local
optima. Since we have 31 remaining model parameters to infer, applying this
SEA technique yields upper and lower bounds for the model parameters which are
presented in (Table A.1).

We use a Monte Carlo least squares method to infer model parameter since it is
reliable and efficient. This method seeks to generate the best Monte Carlo
estimate ($\widehat{\theta}_{j}$) of the model parameters ($\theta$, listed in
Table 2) by minimising the error between the observed data (confirmed-positive
cases and deaths), $\bm{Y}_{j}$ and the simulated data from the CoVCom9 model
(2.3), $\bm{Y}_{j}^{sim}$ given by the variables listed in Table 1. Denoting
the total number of data-points by $n$ and using $i$ ($1\leq i\leq M$) to
enumerate the Monte Carlo simulations, we have

$\allowdisplaybreaks\widehat{\bm{\theta}}_{j}^{(i)}=\arg\min_{\theta}\sum_{j=1}^{n}\biggl{(}\bm{Y}_{j}-\bm{Y}_{j}^{sim}\biggr{)}^{2},\quad(i=1,2,3,\cdots,M).$
(4.2)

Finally, for the $M$ Monte Carlo samples of $\widehat{\bm{\theta}}$, we obtain
the mean and covariance matrix of the estimator, $\widehat{\theta}_{M}$ of
$\bm{\theta}$ as

$\widehat{\bm{\theta}}_{M}=\frac{1}{M}\sum_{i=1}^{M}\widehat{\bm{\theta}}^{(i)},$
(4.3)

$\widehat{\bm{\Sigma}}_{M}=\frac{1}{M-1}\sum_{i=1}^{M}\left(\widehat{\bm{\theta}}^{(i)}-\widehat{\bm{\theta}}_{M}\right)\left(\widehat{\bm{\theta}}^{(i)}-\widehat{\bm{\theta}}_{M}\right)^{T}.$
(4.4)

We also give a 95% confidence interval of the Monte Carlo samples
$\\{\widehat{\bm{\theta}}^{(i)}\\}_{i=1}^{M}$ as

$\biggl{(}\widehat{\bm{\theta}}_{M}^{*(0.025)}\,,\;\widehat{\bm{\theta}}_{M}^{*(0.975)}\biggr{)},$
(4.5)

where $\widehat{\bm{\theta}}_{M}^{*(0.025)}$ and
$\widehat{\bm{\theta}}_{M}^{*(0.975)}$ are respectively the
$\widehat{\bm{\theta}}^{*(i)}$ in the 2.5% and 97.5% positions of the ordered
Monte Carlo samples $\\{\widehat{\bm{\theta}}^{*(i)}\\}_{i=1}^{M}$.

During parameter estimation, we use a logarithmically transformed parameter
vector, $\log\bm{\theta}$, since: (i) this conveniently ensures that all
parameters are positive, $\theta>0$; and (ii) this improves the numerical
search of the parameter space across a wide range of $\bm{\theta}$ (Bland and
Altman, 1996; Acheampong et al., 2019). All computations use MATLAB, 2018a .

###  4.2 Results of CoVCom9 model parameter estimation

Table 3: Estimated initial values of model variables for the system (2.2 using Monte Carlo least squares (MC-LS) method).  Variables | Initial values | 95% Confidence Interval | Reference
---|---|---|---
$N$ |  30,955,202 |  |  (2020) (GSS)
$S$ |  30,954,982 |  |
$E$ |  214.0 |  79.49 - 261.3 |  MC-LS
$I$ |  0.346689 |  0.1959 - 3.579033 |  MC-LS
$Q$ |  2.932 |  1.732 - 11.85 |  MC-LS
$P$ |  2 |  |  Ritchie (2020)
$H$ |  0 |  |  Ritchie (2020)
$C$ |  0 |  |  Ritchie (2020)
$F$ |  0 |  |  Ritchie (2020)
$R$ |  0 |  |  Ritchie (2020)

In this section, the results obtained using the Monte Carlo least-squares
technique described in Section 4 are presented. Table 3 shows initial values
of the state variables; those for compartments $E$, $I$ and $Q$ are estimated
from the reported data. From Table 3, we infer that while on March 13, 2020
two individuals are reported to be confirmed-positive of SARS-CoV-2 infection,
the corresponding number of individuals in the exposed ($E$), infectious ($I$)
and quarantine-suspected ($Q$) compartments are approximately 213, 1, and 3
respectively.

Table 4 gives the parameter values obtained together with their confidence
intervals. We note that the infectivity of the individuals in the infected
compartment ($I$) is stronger than the other compartments: in decreasing
order, the infectivities are due to the groups $E$, $F$, $Q$, $H$, $C$, and
$P$. The overall transmission rate of the SARS-CoV-2 infection in Ghana for
the duration of the data considered in this study is $\varphi=$0.02495 per
day, .

Table 4: Estimated values of the model parameters for the system (2.2) using Monte Carlo least squares (MC-LS) method. Parameter | Value | 95% Confidence Interval | Reference
---|---|---|---
$\alpha_{1}$ |  0.8412 |  (0.2981 $,\;$ 1.0000) |  MC-LS
$\varphi$ |  0.02495 |  (0.02300 $,\;$ 0.03969) |  MC-LS
$\alpha_{2}$ |  0.3152 |  (0.2106 $,\;$ 0.7553) |  MC-LS
$\alpha_{3}$ |  0.05744 |  (0.02168 $,\;$ 0.08098) |  MC-LS
$\beta_{1}$ |  0.2606 |  (0.09697 $,\;$ 0.3576) |  MC-LS
$\beta_{2}$ |  0.1205 |  (0.06108 $,\;$ 0.2436) |  MC-LS
$\beta_{3}$ |  0.4857 |  (0.1787 $,\;$ 0.6772) |  MC-LS
$\epsilon_{1}$ |  1/5.882 |  (1/7 $,\;$ 1/3) |  Zhang et al. (2020)
$\epsilon_{2}$ |  0.001144 |  (0.000873 $,\;$ 0.003217) |  MC-LS
$\gamma_{1}$ |  0.01004 |  (0.008402 $,\;$ 0.02817) |  MC-LS
$\gamma_{2}$ |  0.000163 |  (8.663$\times 10^{-5}$ $,\;$ 3.300$\times 10^{-4}$) |  MC-LS
$\upsilon_{1}$ |  0.000524 |  (0.000379 $,\;$ 0.001293) |  MC-LS
$\upsilon_{2}$ |  1.418$\times 10^{-9}$ |  (6.864$\times 10^{-10}$ $,\;$ 2.745$\times 10^{-9}$) |  MC-LS
$\rho_{1}$ |  0.001971 |  (0.000749 $,\;$ 0.002364) |  MC-LS
$\rho_{2}$ |  5.075$\times 10^{-6}$ |  (2.494$\times 10^{-6}$ $,\;$ 9.911$\times 10^{-6}$) |  MC-LS
$\rho_{3}$ |  0.004711 |  (0.001950 $,\;$ 0.005565) |  MC-LS
$\kappa_{1}$ |  0.008619 |  (0.005728 $,\;$ 0.02076) |  MC-LS
$\kappa_{2}$ |  5.844$\times 10^{-6}$ |  (2.874$\times 10^{-6}$ $,\;$ 1.150$\times 10^{-5}$) |  MC-LS
$\kappa_{3}$ |  3.009$\times 10^{-5}$ |  (1.488$\times 10^{-5}$ $,\;$ 5.949$\times 10^{-5}$) |  MC-LS
$\delta_{1}$ |  1/14 |  (1/23 $,\;$ 1/11) |  Assumed
$\delta_{2}$ |  5.865$\times 10^{-9}$ |  (2.863$\times 10^{-9}$ $,\;$ 1.145$\times 10^{-8}$) |  MC-LS
$\eta$ |  9.771$\times 10^{-5}$ |  (4.862$\times 10^{-5}$ $,\;$ 0.000194) |  MC-LS
$\tau$ |  1.538$\times 10^{-8}$ |  (7.741$\times 10^{-9}$ $,\;$ 3.096$\times 10^{-8}$) |  MC-LS
$d_{1}$ |  7.780$\times 10^{-10}$ |  (3.893$\times 10^{-10}$ $,\;$ 1.557$\times 10^{-9}$) |  MC-LS
$d_{2}$ |  1.249$\times 10^{-13}$ |  (6.222$\times 10^{-14}$ $,\;$ 2.488$\times 10^{-13}$) |  MC-LS
$d_{3}$ |  0.002877 |  (0.001032 $,\;$ 0.003985) |  MC-LS
$d_{4}$ |  6.004$\times 10^{-10}$ |  (2.997$\times 10^{-10}$ $,\;$ 1.199$\times 10^{-9}$) |  MC-LS
$d_{5}$ |  1.392$\times 10^{-12}$ |  (6.839$\times 10^{-13}$ $,\;$ 2.735$\times 10^{-12}$) |  MC-LS
$d_{6}$ |  6.967$\times 10^{-14}$ |  (3.413$\times 10^{-14}$ $,\;$ 1.365$\times 10^{-13}$) |  MC-LS
$d_{7}$ |  2.455$\times 10^{-12}$ |  (1.201$\times 10^{-12}$ $,\;$ 4.804$\times 10^{-12}$) |  MC-LS

The corresponding best fits of the model to the reported data and the two-year
simulations based on the estimated parameter estimates are shown in Figure 2.

Figure 2: Dynamics of CoVCom9 model showing model fit (blue line) and reported
data (red and black dots) for (Left panel) daily numbers of confirmed cases
simulated from the CoVCom9 model and the numbers from the report data (Left
panel) daily numbers of confirmed deaths simulated from the CoVCom9 model and
the numbers from the report data from March 13, 2020 to August 10, 2020.

The rate at which individuals transfer from Classes $E$ to $Q$ is
$\epsilon_{2}=0.001144$, indicating that each day only 0.11% of the
individuals exposed to the SARS-CoV-2 infection are identified with the
suspicion of carrying the infection, and can be contact traced and quarantine
in order to clinically confirm their status as either positive or negative of
the infection. The rate at which these suspected exposed individuals are
confirmed positive is estimated to be $\upsilon_{1}=0.000524$. For individuals
confirmed-positive with the SARS-CoV-2 infection, we can infer from Table 4
that the rate at which individuals progress to Intensive Care (Class $C$) is
low compared to the rate at which they progress to either $H$ or $F$ Classes
(standard hospital ward or self-isolating at home), with the rate of
progression from $P$ to $F$ Classes the highest (that is from positive test to
home isolation). The recovery rate of individuals in Class $H$ is estimated as
$\kappa_{1}=0.008619$ and the rate at which these individuals losing immunity
and becoming susceptible to the SARS-CoV-2 infection is $\tau=1.538\times
10^{-8}$; indicating that the rate of SARS-CoV-2 re-infection in Ghana is
extremely low (full details of parameters and ranges is given in Table 4).

From equation (3.15) and the parameter estimates in Table 4, the basic
reproduction number, $\mathcal{R}_{0}$, is estimated to be 3.110. The
breakdown of this estimate is given, in decreasing order, by

  * •

primarily, symptomatic individuals (class $I$, giving
$\mathcal{R}_{0I}=2.417$),

  * •

hospitalised cases (class $H$, contributing $\mathcal{R}_{0H}=0.212$),

  * •

positively tested individuals (class $P$ giving $\mathcal{R}_{0P}=0.207$),

  * •

infections due asymptomatic cases (class $E$, giving
$\mathcal{R}_{0E}=0.123$),

  * •

self-isolating individuals (class $F$ contributing $\mathcal{R}_{0F}=0.116$),

  * •

intensive care cases (class $C$, giving $\mathcal{R}_{0C}=0.020$),

  * •

those quarantined at home (class $Q$ contributing $\mathcal{R}_{0Q}=0.015$).

The basic reproduction number of COVID-19 based on the proposed CoVCom9 model
for Ghana is higher than that of many other countries, which indicates a
greater epidemic risk in Ghana. A recent study by Asamoah et al. (2020)
provides a similar estimate of $\mathcal{R}_{0}$ in Ghana of $2.64$, differs
by only 15% from our estimate. However, the number of deaths reported in Ghana
is low compared to that of other countries in the world. For published values
for other countries, please see Ali et al. (2020); Zeb et al. (2020); Brandi
et al. (2020); Mwalili et al. (2020); Kumar et al. (2020); Mushayabasa et al.
(2020); Götz and Heidrich (2020); Chen et al. (2020); Sardar et al. (2020);
Ivorra et al. (2020); Asamoah et al. (2020); Rahman et al. (2020).

Using the estimated parameter values given in Tables 3 and 4, the one-year
simulation transmission dynamics of the CoVCom9 model offers insight into the
SARS-CoV-2 among Ghanaian with respect to the COVID-19 protocols which are in
place in the country. Figure 3 depicts the one-year simulation dynamics for
the classes $E$, $I$, $Q$, $P$, $H$, $C$, $F$, and deaths ($D$).

Figure 3: One-year simulation dynamics of CoVCom9 model from March 13 2020
where $E$, $I$, $Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed,
infectious, quarantine suspected-expose, confirmed-positive, hospitalised at
ordinary ward, hospitalised at intensive care unit, and deaths with the
vertical axis on a log-scale.

As shown in Figure 3, all state variables in the CoVCom9 model show an
increasing trend, indicating that Ghana continuing the same protocols may not
be enough to eradicate the SARS-CoV-2 infection. This has been further
complicated by theopening of the the borders, meaning that new control
measures are needed to mitigate the spread (both in and out). Our projections
show that with Ghana exercising current COVID-19 protocols the actual cases
substantially exceed those reported (whether hospitalised or only positively
tested). We thus expect the exponential growth to continue.

In the next section we discuss the derivation of the basic reproduction number
from the CoVCom9 model, and identify influential parameters that intervention
strategies should focus on in order to control the spread of the virus.

##  5 Uncertainty and sensitivity analysis of the basic reproduction number

###  5.1 Methodology

The proposed CoVCom9 model (2.3) has many unknown parameters. Due to the
limited data available, there is substantial uncertainty in calibrating the
values of the 31 CoVCom9 model (2.3) parameters (Marino et al., 2008).
However, in all cases the ratio of the upper bounds of the 95 % confidence
interval is less than five times the lower bound, and more often four or
below, thus so the order of magnitude of all parameters is well established.
Since the intervals are derived using the logarithm of parameter values, and
our best estimates lie in the centre of this band, each upper bound is
approximately twice the estimate and the lower bound half of it. This
uncertainty in model parameters results in some variability in the prediction
of the basic reproduction number $\mathcal{R}_{0}$. Latin Hypercube Sampling-
Partial Rank Correlation Coefficient (LHS-PRCC) sensitivity analysis was used
to evaluate variabilities in the basic reproduction number $\mathcal{R}_{0}$.
The LHS-PRCC approach provides an opportunity to examine the entire parameter
space of the CoVCom9 model (2.3) with computer simulations.

We analyse the impacts of the LHS parameters on the basic reproduction number
$\mathcal{R}_{0}$ of the CoVCom9 model (2.3) via standard Monte Carlo
procedure. The key parameters to which $\mathcal{R}_{0}$, given by (3.15), is
most sensitive are determined using the PRCCs values, suggesting the most
effective way of controlling SARS-CoV-2 infection. Moreover, this analysis
also identifies which parameters need to be known precisely when estimating
$\mathcal{R}_{0}$ from data (Marino et al., 2008).

The application of the combined LHS-PRCC methodology in infectious disease
modelling are fully described elsewhere, for example, in (Wu et al., 2013;
Marino et al., 2008). This method generally involves:

(i)

generating LHS parameters in matrix form, together with a ranking of outcome
measures $\mathcal{R}_{0}$;

(ii)

construction of two linear regression models in response to each parameter and
outcome measure, and

(iii)

computation of a Pearson rank correlation coefficient for the residuals from
the two regression models to obtain the PRCC values for that particular
parameter (Marino et al., 2008; Orwa et al., 2019).

We induce the correlation between the input parameters using the rank-based
method of Iman and Conover (Iman and Conover, 1982). The correlation matrix
for the 28 model parameters (listed in Table 2) is obtained from the parameter
estimation in Section 4, where no correlation is assumed between the
parameters $\epsilon_{1}$ and $\delta_{1}$ and other parameters, since these
two parameters are not included in the parameter estimation.

###  5.2 Results of Analysing the LHS-PRCC for the CoVCom9 model

The result of the uncertainty analysis of the basic reproduction number,
$\mathcal{R}_{0}$ (3.15) of the CoVCom9 obtained by generating 1000 LHS
samples using the Monte Carlo technique is presented in Figure 4. This
histogram depicts the uncertainty in $\mathcal{R}_{0}$, where the degree of
uncertainty quantified via the 95% confidence intervals is indicated by the
dashed lines. Figure 5 shows the distribution of obtained values for
$\mathcal{R}_{0}$, the mean, 5th, and 95th percentiles being respectively
2.623, 2.042, and 3.240.

Figure 4: Uncertainty analysis of the basic reproduction number
$\mathcal{R}_{0}$ depicted by the histogram with plot showing 95% confidence
interval (dashed lines), mean (solid line) and an estimate (red dotted-dashed
line) of $\mathcal{R}_{0}$ (3.15).

Using the best-fit values of all the parameters given in Table 4 yields an
estimate of $\mathcal{R}_{0}$ towards the upper end of the distribution,
namely a value of 3.110 (see the red dotted-dashed line). In general, the
higher the uncertainty, the wider the spread of the distribution of
$\mathcal{R}_{0}$. We note that there is some uncertainty in $\mathcal{R}_{0}$
due to the model parameter estimates in Table 4; however, this is less than
for most parameters. In Table 4, for almost all parameters, the upper and
lower 95% confidence intervals differ from the best fit value by a factor of
two. However, for $\mathcal{R}_{0}$, the upper and lower ends of the interval
are with $\pm$24% of the mean value; thus overall, the uncertainty in the
estimate of $\mathcal{R}_{0}$ is less than that of the individual parameters.

Figure 5: Sensitivity of the basic reproduction number $\mathcal{R}_{0}$ to
changes in the CoVCom9 parameters using PRCC index.

Figure 5 shows the sensitivity of the reproduction number $\mathcal{R}_{0}$ to
each of the parameters in the underlying model (2.3). PRCC assigns each
parameter a value between $-1$ and $+1$. The magnitude of PRCC shows the
parameter importance while the sign of PRCC gives the direction of the
relationship between the input parameter and the model output of interest.
Negative PRCC values mean that as the parameter value increases, the value of
the model output of interest decreases and vice versa. The results of the
PRCCs depicted help identify which parameters are primarily responsible for
the uncertainty in $\mathcal{R}_{0}$, which suggests those interventions which
should be most efficacious in controlling the spread of the virus by reducing
$\mathcal{R}_{0}$. A PRCC value of zero gives an indication of no association
between the input parameter and model output of interest. The most significant
model parameters are those associated with small $p-$values ($p<0.05$) and
large magnitude PRCC values ($0.5\leq|PRCC|\leq 1$).

From Figure 5, we identify six parameters as most influential on the basic
reproduction number, $\mathcal{R}_{0}$, these are:

  * •

$\varphi$ \- the transmission rate of infectious individuals,

  * •

$\alpha_{2}$ \- the probability of transmission of quarantine suspected
infectious individuals,

  * •

$\alpha_{3}$ \- the probability of transmission of confirmed-positive
infectious individuals,

  * •

$\epsilon_{1}$ \- the progression rate of exposed individuals to infectious
class

  * •

$\gamma_{1}$ \- the progression rate of infectious individuals to confirmed-
positive class, and

  * •

$\upsilon_{1}$ \- the progression rate of quarantine suspected infectives to
the class of confirmed-positive cases.

In particular, $\mathcal{R}_{0}$ increases with increases in $\varphi$,
$\alpha_{2}$ and $\alpha_{3}$, while $\mathcal{R}_{0}$ decreases with
increases in $\epsilon_{1}$, $\gamma_{1}$ and $\upsilon_{1}$. It is therefore
critical that intervention strategies should be aimed at decreasing the values
of $\varphi$, $\alpha_{2}$ and $\alpha_{3}$ and increasing the values of
$\epsilon_{1}$, $\gamma_{1}$ and $\upsilon_{1}$.

These recommendations should not be interpreted as discounting the value of
considering efforts to alter other significant model parameters such as
probability of transmission of hospitalised individuals at ordinary ward
($\beta_{1}$), the progression rate of exposed individuals to quarantine
suspected exposed class ($\epsilon_{2}$), and recovery rate of hospitalised
individual ($\kappa_{1}$).

###  5.3 Predicting the effects of lockdown

The simulation presented in Figure 3 show a worrying trend of exponential
growth with no sign of plateau or reduction in the effects of the pandemic.
Many countries have implemented a ‘lockdown’, that is regulations to restrict
social interactions and so reduce the spread of the disease. Here, we model
the effects of lockdown by a simple reduction in the parameter $\varphi$, and
simulate the spread by solving the model using the standard value of $\varphi$
for the first 350 days, and a lower value of $\varphi$ for the time period
$350\leq t\leq 700$ days. The results are presented in Figures 6, 7, 8, for
the values $\varphi=0.008$, 0.016, 0.004, the first value of $\varphi$ being
chosen so as to reduce the expected value of $\mathcal{R}_{0}$ from 3.110 to
0.995, the threshold required for containment of the epidemic. The second and
third values are chosen to be double and half of this critical value. Note
that the vertical scales in Figures 6, 7, and 8 are not identical.

Figure 6: One-year simulation dynamics of CoVCom9 model from March 13 2020
when there is a 68% reduction in $\varphi$, that is, to $\varphi=0.008$. Here
$E$, $I$, $Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed,
infectious, quarantine suspected-expose, confirmed-positive, hospitalised at
ordinary ward, hospitalised at intensive care unit, and deaths with the
vertical axis on a log-scale. $\mathcal{R}_{0}$ changes from 3.110 to 0.995.

Figure 6 shows a clear almost instant reduction in the number of exposed
people ($E$), followed by a plateau, whilst the sizes of most other sub-
populations plateau. However, the numbers of hospitalised cases ($H$ and $C$)
both continue to rise slowly. We see that this strength of lockdown stops the
exponential growth. The less severe lockdown simulated in Figure 7 causes a
brief reduction in the number of exposed cases; however, the exponential
growth is quickly resumed, in the size of all sub-populations, albeit with a
slightly smaller growth rate. The more severe lockdown simulated in Figure 8
shows a sudden and sharp reduction in the number of exposed ($E$), followed by
a steady exponential decrease. The numbers of infected, quarantined and
positive cases is also seen to fall exponentially, whilst the cases of
hospitalised, intensive care, and self-isolated all plateau, as the total
number of deaths slowly increases.

Figure 7: One-year simulation dynamics of CoVCom9 model from March 13 2020
when there is a 34% reduction in $\varphi$, that is, to $\varphi=0.016$.
Again, $E$, $I$, $Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed,
infectious, quarantine suspected-expose, confirmed-positive, hospitalised at
ordinary ward, hospitalised at intensive care unit, and deaths with the
vertical axis on a log-scale. The effect on $\mathcal{R}_{0}$ is a change from
3.110 to 2.053.

It should be noted that these simulations are only a crude model of the
effects of lockdown, in reality a lockdown could cause changes to other
parameters, particularly $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$,
$\beta_{1}$, $\beta_{2}$, $\beta_{3}$, in the formula (2.2) for the spread of
the disease. We leave the topic of more detailed models of the effects of
lockdown for future work.

Figure 8: One-year simulation dynamics of CoVCom9 model from March 13 2020
when there is a 84% reduction in $\varphi$, that it, to $\varphi=0.004$. As
above, $E$, $I$, $Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed,
infectious, quarantine suspected-expose, confirmed-positive, hospitalised at
ordinary ward, hospitalised at intensive care unit, and deaths with the
vertical axis on a log-scale. The effect on $\mathcal{R}_{0}$ is a change from
3.110 to 0.498.

##  6 Discussion and conclusions

We have developed a mathematical model (CoVCom9) in the form of a system of
coupled ordinary differential equations to describe SARS-CoV-2 transmission
dynamics in Ghana. This categorises every member of the population into one of
9 classes, including various classes well-defined and measurable classes, such
as those who have tested positive for SARS-Cov-2 and are hospitalised
(ordinary wards/intensive care), quarantined, etc, as well as unmeasurable but
clinically important classes, such as those who have been exposed to the
virus, those who are infectious but not yet tested positive. We investigated
the epidemiological well-posedness of the CoVCom9 model, shown that solutions
remain positive, and analysed the stability of the equilibrium solution. Using
a candidate Lyapunov function, we have shown that the disease-free equilibrium
is globally asymptotically stable when the basic reproduction number is
$\mathcal{R}_{0}<1$.

Using the reported data (Ritchie, 2020) from March 13, 2020, to August 10,
2020, for both confirmed-positive cases and deaths of SARS-CoV-2 disease, we
have parameterised the CoVCom9 model, with other parameters being estimated
based from the literature. During the parameter estimation exercise, we used
system exploratory analysis (SEA) to find practical parameter spaces. The
estimated parameter values provided best fits that are in good agreement with
both reported confirmed-positive cases and deaths. Also, the results point
that on March 13, 2020, while two individuals are confirmed-positive,
approximately 213, 3, and 1 persons were respectively exposed, quarantine
suspected and infectious.

We have used Latin Hypercube Sampling-Rank Correlation Coefficient (LHS-PRCC)
to investigate the uncertainty and sensitivity of the reproduction number
$\mathcal{R}_{0}$. The results derived are of significant epidemiological
value in SARS-CoV-2 control. We estimate that over the period, March-August
2020, the average basic reproduction number for Ghana was
$\mathcal{R}_{0}=3.110$, which has the 95% confidence percentile interval
(2.042 - 3.240, in approximate centre of this interval is the mean value of
2.623). From Figure 5, we note that $\mathcal{R}_{0}$ is most sensitive to six
model parameters ($\varphi$, $\alpha_{2}$, $\alpha_{3}$, $\epsilon_{1}$,
$\gamma_{1}$, and $\upsilon_{1}$ whose effects are detailed in Table 2).

The proposed CoVCom9 model is a result of our effort to gain insight into the
vital features of SARS-CoV-2 transmission dynamics in Ghana. Future work will
be focused on extending the model to account for inflow into other classes due
to opening of Ghana’s borders. Further, we will consider time-dependent
optimal control intervention strategies to gain insight into the best strategy
for Ghana. Other extensions include the time-dependent force of infection and
the maximum capacity of intensive care units.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the study
reported in this paper.

## Acknowledgements

The authors (EA, JADW and RLG) are thankful for funding provided by the
Leverhulme Trust Doctoral Scholarship (DA214-024), Modelling and Analytics for
a Sustainable Society (MASS), and to the University of Nottingham.

##  Appendix A Appendix

In Table A.1 we list the parameter values used in the simulations presented in
Section 4.

Figures A.1, A.2, A.3 show our predictions for how the subpopulation sizes in
the model would have evolved over time if a lockdown had been imposed as soon
as the first cases entered Ghana. These predictions are obtained by keeping
all parameters at the same values as in the main model, and reducing $\varphi$
to the values used in Section 5.3. These graphs should be compared with Figure
3. In Figure A.1 we use $\varphi=0.008$, which is chosen to make our estimate
of $\mathcal{R}_{0}=1$. We see that this has the effect of bringing the
pandemic under some sort of control, but only over an extremely long
timescale. In Figure A.2 we simulate a partial lockdown, that is, reducing
$\varphi$ to 0.016 - which is the midpoint of the standard value
$\varphi=0.02495$ and that required to reduce $\mathcal{R}_{0}$ to 1. We see
that epidemic still grows, but at a slower rate than with no lockdown.
Finally, in Figure A.3, we consider the effect of a much more severe lockdown,
where $\varphi$ is reduced to half that needed for $\mathcal{R}_{0}=1$, that
is $\varphi=0.004$. This suggests that the epidemic can be controlled and
eliminated within a year.

Table A.1: Estimated initial values model variables and parameters for the
system (2.2.)

Parameters | Min | Max
---|---|---
$\alpha_{1}$ |  0.245545 |  1
$\varphi$ |  0.012654 |  0.050619
$\alpha_{2}$ |  0.210646 |  0.842586
$\alpha_{3}$ |  0.020245 |  0.080978
$\beta_{1}$ |  0.089411 |  0.357644
$\beta_{2}$ |  0.060902 |  0.243607
$\beta_{3}$ |  0.169305 |  0.677222
$\epsilon_{1}$ |  1/7 |  1/3 (a)
$\epsilon_{2}$ |  0.000873 |  0.003492
$\gamma_{1}$ |  0.008402 |  0.033610
$\gamma_{2}$ |  8.662695$\times 10^{-5}$ |  0.000347
$\upsilon_{1}$ |  0.000379 |  0.001517
$\upsilon_{2}$ |  6.863871$\times 10^{-10}$ |  2.745548$\times 10^{-9}$
$\rho_{1}$ |  0.000547 |  0.002188
$\rho_{2}$ |  2.477852$\times 10^{-6}$ |  9.911408$\times 10^{-6}$
$\rho_{3}$ |  0.001391 |  0.005565
$\kappa_{1}$ |  0.005728 |  0.022913
$\kappa_{2}$ |  2.874618$\times 10^{-6}$ |  1.149847$\times 10^{-5}$
$\kappa_{3}$ |  1.488217$\times 10^{-5}$ |  5.952870$\times 10^{-5}$
$\delta_{1}$ |  1/23 |  1/11 (b)
$\delta_{2}$ |  2.862673$\times 10^{-9}$ |  1.145069$\times 10^{-8}$
$\eta$ |  4.861253$\times 10^{-5}$ |  0.000194
$\tau$ |  7.740808$\times 10^{-9}$ |  3.096323$\times 10^{-8}$
$d_{1}$ |  3.892588$\times 10^{-10}$ |  1.557035$\times 10^{-9}$
$d_{2}$ |  6.221547$\times 10^{-14}$ |  2.488619$\times 10^{-13}$
$d_{3}$ |  0.000996 |  0.003985
$d_{4}$ |  2.996889$\times 10^{-10}$ |  1.198756$\times 10^{-9}$
$d_{5}$ |  6.838531$\times 10^{-13}$ |  2.735412$\times 10^{-12}$
$d_{6}$ |  3.412952$\times 10^{-14}$ |  1.365181$\times 10^{-13}$
$d_{7}$ |  1.200917$\times 10^{-12}$ |  4.803667$\times 10^{-12}$
$E$ |  40 |  300
$I$ |  0 |  10
$Q$ |  1 |  70

(a) denotes literature values and (b) denotes assumed value

Figure A.1: One-year simulation dynamics of CoVCom9 model from March 13 2020
when there is a 68% reduction in $\varphi$ to $\varphi=0.008$. Here $E$, $I$,
$Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed, infectious,
quarantine suspected-expose, confirmed-positive, hospitalised at ordinary
ward, hospitalised at intensive care unit, and deaths with the vertical axis
on a log-scale. The effect on $\mathcal{R}_{0}$ is a change from 3.110 to
0.995.

Figure A.2: One-year simulation dynamics of CoVCom9 model from March 13 2020
when there is a 34% reduction in $\varphi$ to $\varphi=0.016$. Here $E$, $I$,
$Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed, infectious,
quarantine suspected-expose, confirmed-positive, hospitalised at ordinary
ward, hospitalised at intensive care unit, and deaths with the vertical axis
on a log-scale. The effect on $\mathcal{R}_{0}$ is a change from 3.110 to
2.053.

Figure A.3: One-year simulation dynamics of CoVCom9 model from March 13 2020
when there is a 84% reduction in $\varphi$, to $\varphi=0.004$. Here $E$, $I$,
$Q$, $P$, $H$, $C$, $F$ and $D$ are respectively exposed, infectious,
quarantine suspected-expose, confirmed-positive, hospitalised at ordinary
ward, hospitalised at intensive care unit, and deaths with the vertical axis
on a log-scale. The effect on $\mathcal{R}_{0}$ is a change from 3.110 to
0.498.

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