key: cord-0806118-prnq5qvu
authors: Acheampong, Edward; Okyere, Eric; Iddi, Samuel; Bonney, Joseph H.K.; Wattis, Jonathan A.D.; Gomes, Rachel L.; Asamoah, Joshua Kiddy K.
title: Mathematical modelling of earlier stages of COVID-19 transmission dynamics in Ghana
date: 2022-01-14
journal: Results Phys
DOI: 10.1016/j.rinp.2022.105193
sha: 66f3c2a180d23d0610af5f78be0f11b2a22bcbb9
doc_id: 806118
cord_uid: prnq5qvu

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, [Formula: see text]. In particular, we have shown that the disease-free equilibrium is globally asymptotically stable when [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] , with the 95% confidence interval being [Formula: see text] , and the mean value being [Formula: see text]. Of the 32 parameters in the model, we find that just six have a significant influence on [Formula: see text] , these include the rate of testing, where an increasing testing rate contributes to the reduction of [Formula: see text].

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, Nonlinear mathematical models have significantly contributed to the understanding of transmission dynamics of infectious diseases, see, e.g., [7] [8] [9] [10] [11] , and the recent COVID-19 pandemic is of no exception [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] . Qianying et al. [24] 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 [25] 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 [26] 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 [27] to examine the dynamics of the infectious disease in India. Pang et al. [28] parametrise a nonlinear SEIHR model to estimate the value and sensitivity of 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 [29] 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. [30] , who estimate parameters and performed a out global sensitivity analysis of their model parameters on R 0 .

A nonlinear SEIQR COVID-19 epidemic model is introduced by Zeb et al. [31] who present a local and global stability analysis for their model. The spread of COVID-19 in China due to undetected infections in is examined by Ivorra et al. [32] . Chen et al. [33] 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. [34] , 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 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 confirmedpositive for SARS-CoV-2 (denoted by P ). Those identified as suspected exposed individuals are denoted by Q J o u r n a l P r e -p r o o f Journal Pre-proof 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 [35] , 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, R 0 . We have used Latin-Hypercube Sampling-Partial Rank Correlation Coefficient (LHS-PRCC) technique to quantify the uncertainty in R 0 as well as to identify those parameters to which 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 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.

Compartmental models are useful means of qualitatively understanding the dynamics of disease transmissions within a population [1, 36] . 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 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

(1) tible 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 J o u r n a l P r e -p r o o f 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 δ 1 ) or enter the ordinary hospitalised class (rate δ 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 τ . We assume that individuals in all the compartments can die of COVID-19 (rates d j ) in addition to natural death (rate µ) with the exception of the susceptible compartment with only natural death. A summary of all the parameter definitions is given in Table 2 . Progression rate of exposed individuals to infectious class per day 2 Progression rate of exposed individuals to quarantined class 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 [7, 36] . This principle has been shown in many studies to be a plausible assumption [7] . If α is the average number of sufficient contacts for to susceptibles S(t) becoming infected [7] . We use ϕ to denote the effective transmission rate from an infectious individual while α 1 , α 2 , α 3 , β 1 , β 2 and β 3 denote the transmission probabilities, of exposed individuals, quarantined 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

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

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

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 (3). The epidemiologically feasible region of interest of the model

In the following sections we present a mathematical analysis of the model with respect to positivity and boundedness of the feasible region, Ω, 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 (3) with initial conditions (4) in Ω remain in Ω. 

Proof. Considering the first equation of the CoVCom9 model (3), one can clearly see that

where

Following a similar argument, it can be shown that the rest of the model variables have nonnegative solutions

Furthermore, we prove that the solutions are bounded. Adding the right-hand side of the CoVCom9 model (3) yields

Since dN/dt ≤ Λ − µN , it follows that 

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

J o u r n a l P r e -p r o o f

where N (0) is the initial value of the total population. Thus N (t) ≤ Λ/µ, as t → ∞. Therefore all feasible solutions of system (3) enter the region Ω defined by (5), which is a positively invariant set of the system (3). This implies that all solutions in Ω remain in Ω ∀t ≥ 0. It is therefore sufficient to study the dynamics of CoVCom9 model system (3) in Ω.

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

The basic reproduction number is defined as the number of secondary infections produced by a single infectious individual during the entire infectious period [37] . 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 R 0 is the dominant eigenvalue of the next generation matrix [37, 38] . We apply the method formulation in Van den Driessche and Watmough [37] to obtain an expression of R 0 for the proposed CoVCom9 (3). Let x = E, I, Q, P, H, C, F T , then the system (3) can be written in the form

where

and

J o u r n a l P r e -p r o o f

The Jacobian of F(x) and V(x) evaluated at the disease free equilibrium E 0 are, respectively,

The basic reproduction number, R 0 is given by the dominant eigenvalue of

which can be written as

where the effective reproduction number, R 0 is made up of contributions from secondary infections from the exposed group E (R 0E ) generated by asymptomatic individuals; the infected (symptomatic) group I (R 0I ); asymptomatic quarantined individuals -class-Q (R 0Q ); confirmed positive individuals -class P (R 0P ); hospitalised cases (H, R 0H ); intensive care (C) cases, (R 0C ); and those self-isolating at home (F , R 0F ). Equation (20) 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 [37] , the disease-free steady state E 0 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1. In the next section we provide stability results for the disease-free equilibrium state. In this section, we prove global stability results for the CoVCom9 model (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 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 R 0 < 1 [15] . 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 (3) is globally asymptotically stable in

Proof. We construct a candidate Lyapunov function (22) for the CoVCom9 model (3) as

where Φ i , i = 1, 2, · · · , 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 (3), the derivative of V with respect to t can be bounded by

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

J o u r n a l P r e -p r o o f

where the parameter groupings π * are given by (17) .

After some simplifications using (17), the time derivative of the Lyapunov function can be written as

It is now clear that if R 0 < 1 then dV /dt ≤ 0. Furthermore, dV /dt = 0 if E = 0 and R 0 < 1. Thus, when R 0 < 1, the largest compact invariant set in (S, E, I, Q, P, H, C, F, R) ∈ Ω |V ≤ 0 is the single state E 0 . LaSalle's Invariance

Principle then implies that E 0 is globally asymptotically stable in Ω if R 0 < 1.

In this section, we briefly describe the parameter estimation and numerical simulation process used to investigate how well the proposed CoVCom9 model (3) 

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 nonconvergence of the optimisation of the objective function. We implement the following practical principles to choose reasonable initial parameter values:

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, µ, recruitment rate, Λ, incubation period, 1 , and recovery rate of quarantine/self-isolation at home individual, δ 1 are obtained. We assumed that the life expectancy of people in Ghana is estimated as 64.35

years [16] , then the natural death rate is estimated as µ = 1/(64.35 × 365) ≈ 4.258 × 10 −5 per day. The population of Ghana in 2020 is estimated to be N = 30, 960, 000 [39] , and the recruitment rate of humans is estimated as Λ = µN ≈ 1.318 × 10 3 people per day. The incubation period is 3-7 days, here we choose 1 = 1/5.88 per day as estimated by Pang et al. [28] which is consistent with the wider literature [40, 41] . The self-isolated positive-confirmed individuals on medication take 14 days on average to recover, thus we assume δ 1 = 1/14 per day.

2. Exploring the model using the available data (also known as 'system exploratory analysis' (SEA) [42] ). 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 confirmedpositive 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 ( θ j ) of the model parameters (θ, listed in Table 2 ) by minimising the error between the observed data (confirmed-positive cases and deaths), Y j and the simulated data from the CoVCom9 model (3), Y sim j given by the variables listed in Table 1 . Denoting the total number of data-points by n and using i (1 ≤ i ≤ M ) to enumerate the Monte Carlo simulations, we have

Finally, for the M Monte Carlo samples of θ, we obtain the mean and covariance matrix of the estimator, θ M of θ as

We also give a 95% confidence interval of the Monte Carlo samples Table 3 Estimated initial values of model variables for the system (2 using Monte Carlo least squares (MC-LS) method). R 0 [35] 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 J o u r n a l P r e -p r o o f Journal Pre-proof 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 quarantined (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 ϕ =0.02495 per day, . Table 4 Estimated values of the model parameters for the system (2) 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 κ 1 = 0.008619 and the rate at which these individuals losing immunity and becoming susceptible to the SARS-CoV-2 infection is τ = 1.538 × 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 (20) and the parameter estimates in Table 4 , the basic reproduction number, 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 R 0I = 2.417),

• hospitalised cases (class H, contributing R 0H = 0.212),

• positively tested individuals (class P giving R 0P = 0.207),

• infections due asymptomatic cases (class E, giving R 0E = 0.123),

• self-isolating individuals (class F contributing R 0F = 0.116),

• intensive care cases (class C, giving R 0C = 0.020),

• quarantined individuals (class Q contributing R 0Q = 0.015).

Journal Pre-proof [15-17, 31-34, 46-50] .

Using the estimated parameter values given in Tables 3 and 4 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.

Uncertainty and sensitivity analysis of the basic reproduction number

The proposed CoVCom9 model (3) Moreover, this analysis also identifies which parameters need to be known precisely when estimating R 0 from data [51] .

The application of the combined LHS-PRCC methodology in infectious disease modelling are fully described elsewhere, for example, in [51, 52] . This method generally involves:

(i) generating LHS parameters in matrix form, together with a ranking of outcome measures 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 [51, 53] .

We induce the correlation between the input parameters using the rank-based method of Iman and Conover [54] .

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 1 and δ 1 and other parameters, since these two parameters are not included in the parameter estimation. Figure 4 . This histogram depicts the uncertainty in 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 R 0 , the mean, 5th, and 95th percentiles being respectively 2.623, 2.042, and 3.240. Using the best-fit values of all the parameters given in Table 4 yields an estimate of 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 R 0 . We note that there is some uncertainty in 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 R 0 , the upper and lower ends of the interval are with ±24% of the mean value; thus overall, the uncertainty in the estimate of R 0 is less than that of the individual parameters. From Figure 5 , we identify six parameters as most influential on the basic reproduction number, R 0 , these are:

• ϕ -the transmission rate of infectious individuals,

• α 2 -the probability of transmission of quarantined individuals,

• α 3 -the probability of transmission of confirmed-positive infectious individuals,

• 1 -the progression rate of exposed individuals to infectious class

• γ 1 -the progression rate of infectious individuals to confirmed-positive class, and

• υ 1 -the progression rate of quarantined individuals to the class of confirmed-positive cases.

In particular, R 0 increases with increases in ϕ, α 2 and α 3 , while R 0 decreases with increases in 1 , γ 1 and υ 1 .

It is therefore critical that intervention strategies should be aimed at decreasing the values of ϕ, α 2 and α 3 and

increasing the values of 1 , γ 1 and υ 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 (β 1 ), the progression rate of exposed individuals to quarantined individuals class ( 2 ), and recovery rate of hospitalised individual (κ 1 ).

The simulation presented in Figure 3 show One-year simulation dynamics of CoVCom9 model from March 13 2020 when there is a 68% reduction in ϕ, that is, to ϕ = 0.008. Here E, I, Q, P , H, C, F and D are respectively exposed, infectious, quarantined, confirmed-positive, hospitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis on a log-scale. 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. 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 α 1 , α 2 , α 3 , β 1 , β 2 , β 3 , in the formula (2) for the spread of the disease. We leave the topic of more detailed models of the effects of lockdown for future work. One-year simulation dynamics of CoVCom9 model from March 13 2020 when there is a 84% reduction in ϕ, that it, to ϕ = 0.004. As above, E, I, Q, P , H, C, F and D are respectively exposed, infectious, quarantined, confirmed-positive, hospitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis on a log-scale. The effect on R 0 is a change from 3.110 to 0.498.

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 R 0 < 1.

Using the reported data [35] from March 13, 2020, to August 10, 2020, for both confirmed-positive cases and 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, quarantined and infectious.

We have used Latin Hypercube Sampling-Rank Correlation Coefficient (LHS-PRCC) to investigate the uncertainty and sensitivity of the reproduction number 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 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 R 0 is most sensitive to six model parameters (ϕ, α 2 , α 3 , 1 , γ 1 , and υ 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. 

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.

J o u r n a l P r e -p r o o f Journal Pre-proof Here E, I, Q, P , H, C, F and D are respectively exposed, infectious, quarantined, confirmed-positive, hospitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis on a log-scale. The effect on R 0 is a change from 3.110 to 0.995. Here E, I, Q, P , H, C, F and D are respectively exposed, infectious, quarantined, confirmed-positive, hospitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis on a log-scale. The effect on R 0 is a change from 3.110 to 2.053. Here E, I, Q, P , H, C, F and D are respectively exposed, infectious, quarantined, confirmed-positive, hospitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis on a log-scale. The effect on R 0 is a change from 3.110 to 0.498.

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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.

In Table A 3. These graphs should be compared with Figure 3 . In Figure A. 1 we use ϕ = 0.008, which is chosen to make our estimate of 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 ϕ to 0.016which is the midpoint of the standard value ϕ = 0.02495 and that required to reduce 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 ϕ is reduced to half that needed for R 0 = 1, that is ϕ = 0.004. This suggests that the epidemic can be controlled and eliminated within a year.