Modelling COVID-19 Transmission Dynamics in Ghana
Edward Acheampong, Eric Okyere, Samuel Iddi, Joseph H. K. Bonney, Jonathan A. D. Wattis, Rachel L. Gomes
MModelling COVID-19 Transmission Dynamics in Ghana
Edward Acheampong a,b,d, ∗ , Eric Okyere c , Samuel Iddi d , Joseph H. K.Bonney e , Jonathan A. D. Wattis a , Rachel L. Gomes b a School of Mathematical Sciences, University of Nottingham,University Park, Nottingham, NG7 2RD, UK b Food Water Waste Research Group, Faculty of Engineering,University of Nottingham, University Park, Nottingham, NG7 2RD, UK c Department of Mathematics and Statistics University of Energyand Natural Resources, P.O. Box 214, Sunyani, B/A Ghana d Department of Statistics and Actuarial Science University of Ghana,P.O. Box LG 115, Legon, Ghana e 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 identifiedin Wuhan, China and later spread to every corner of the globe. Whilstthe number of infection-induced deaths in Ghana, West Africa are minimalwhen compared with the rest of the world, the impact on the local healthservice is still significant. Compartmental models are a useful framework forinvestigating transmission of diseases in societies. To understand how theinfection will spread and how to limit the outbreak. We have developed amodified SEIR compartmental model with nine compartments (CoVCom9)to describe the dynamics of SARS-CoV-2 transmission in Ghana. We havecarried out a detailed mathematical analysis of the CoVCom9, includingthe derivation of the basic reproduction number, R . In particular, wehave shown that the disease-free equilibrium is globally asymptotically stablewhen R < ∗ Corresponding authors
Email addresses:
[email protected]/[email protected] (EdwardAcheampong),
[email protected] (Jonathan A. D. Wattis),
[email protected] (Rachel L. Gomes)
Preprint submitted to * 8th February 2021 a r X i v : . [ q - b i o . P E ] F e b reported data for confirmed-positive cases and deaths from March 13 toAugust 10, 2020, we have parametrised the CoVCom9 model. The results ofthis fit show good agreement with data. We used Latin hypercube sampling-rank correlation coefficient (LHS-PRCC) to investigate the uncertainty andsensitivity of R since the results derived are significant in controlling thespread of SARS-CoV-2. We estimate that over this five month period, thebasic reproduction number is given by R = 3 . . ≤ R ≤ . R = 2 . R , these include the rate of testing, where an increasing testingrate contributes to the reduction of R . Keywords:
Transmission model, SARS-CoV-2, Uncertainty, Sensitivity,Mathematical analysis, Monte Carlo-least squares.
1. Introduction
The recent COVID-19 pandemic has caused a devastating burden on theglobal economy. Since there are currently no widely-available vaccines tobring down or reduce the infection levels on the susceptible human popula-tion, many governmental decision-makers worldwide have resorted to intens-ive non-pharmaceutical interventions such as wearing of face-masks, socialdistancing, cleaning of suspected infected surfaces, avoiding crowded places,the use of hand sanitizers. These non-pharmaceutical interventions havesignificantly helped to reduce the risk of transmission of COVID-19.Mathematical and statistical modelling tools are important in providingkey epidemiological parameters of infectious diseases such as infection ortransmission rate, recovery rate, incubation period, isolation and hospital-ization 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 reportedcases of infection, helps estimate the basic reproduction number, R whichis a crucial epidemiological parameter that determines whether the infection2ersists 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 un-derstanding of transmission dynamics of infectious diseases, see, e.g., (Het-hcote, 2000; Chowell and Hyman, 2016; Brauer et al., 2008; Khan et al.,2015), and the recent COVID-19 pandemic is of no exception (Blyuss andKyrychko, 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 studieda data-driven SEIR type epidemic for the recent COVID-19 outbreak inWuhan which captures the effects of governmental actions and individuals’behaviour. This literature is growing rapidly; Abou-Ismail (2020) has re-viewed the fundamentals in SIR/SEIR modelling of the recent COVID-19outbreak; here we give a brief overview of literature relevant to our work.Buonomo (2020) describes a susceptible-infected-recovered-infected com-partmental model to investigate the effects of information-dependent vaccin-ation behavior on COVID-19 infections. A simple SEIR COVID-19 epidemicmodel with nonlinear incidence rates that capture governmental control hasbeen designed by Rohith and Devika (2020) to examine the dynamics ofthe infectious disease in India. Pang et al. (2020) parametrise a nonlinearSEIHR model to estimate the value and sensitivity of R using data fromWuhan from December 31st, 2019. A classic SEIR epidemic is used to studythe spreading dynamics of the 2019 coronavirus disease in Indonesia (Annaset al., 2020) using vaccination and isolation as model parameters. They con-structed a Lyapunov function to conduct global stability of the disease-freeequilibrium point. A data-driven epidemiological model that examines theeffect of delay in the diagnosis of the deadly COVID-19 disease is formulatedand studied by Rong et al. (2020), who estimate parameters and performeda out global sensitivity analysis of their model parameters on R .A nonlinear SEIQR COVID-19 epidemic model is introduced by Zebet 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 examinedby Ivorra et al. (2020). Chen et al. (2020) propose a model based on di-3iding the total population into five non-overlapping classes: susceptible,exposed, infected (symptomatic infection), asymptomatic infected, and re-covered. Sardar et al. (2020), investigate the effects of lockdown using anSEIR model. Using reported cases of this highly infectious disease in somecities and the whole of India, they performed a global sensitivity analysis onthe computed R .The exposed and infectious epidemiological classes used in formulatinginfectious diseases models mentioned above have been left as abstract con-cepts. In reality, especially regarding SARS-CoV-2, it is hard to distinguishbetween individuals exposed to or infected with SARS-CoV-2, due to thepresence of asymptomatic carriers. In this present study, we introduce twoepidemiological classes, which are: (1) an identified group of exposed in-dividuals 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 aredenoted by Q because they are quarantined as required by the COVID-19 protocols in Ghana. Likewise, confirmed-positives ( P ) are treated asinfectious 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 withinthe Ghanaian population. Using published data from March 13 to August10, 2020 (Ritchie, 2020), we have parametrised our model using a MonteCarlo-least squares technique together with information derived from liter-ature.The purpose of this research is to investigate the transmission dynamicsof SARS-CoV-2 in Ghana using these more specific epidemiological classes toestimate the basic reproduction number, R . We have used Latin-HypercubeSampling-Partial Rank Correlation Coefficient (LHS-PRCC) technique toquantify the uncertainty in R as well as to identify those parameters towhich R is most sensitive. We have organised the subsequent sections ofthe paper as follows: in Section 2 we present a detailed formulation of anepidemiological model of SARS-CoV-2 transmission in Ghana, together withcorresponding mathematical analysis of the positivity and boundedness of4olutions, a derivation of the basic reproduction number, and global stabilityanalysis of the disease-free equilibrium, which are given in Section 3. Sec-tion 4 is dedicated to parameter estimation and numerical simulation. Theuncertainty and sensitivity analysis of R and its implications are presen-ted in Section 5, together with some simulations predicting possible futuredynamics of the epidemic. Finally, we give a brief discussion and conclusionof the study in Section 6.
2. Formulation of the model
Compartmental models are useful means of qualitatively understandingthe dynamics of disease transmissions within a population (Martcheva, 2015;Chowell et al., 2009). In formulating our compartmental model to gain in-sight into COVID-19 transmission dynamics, the total human population isdivided into nine distinct epidemiological classes which are summarised inTable 1. The numbers of individuals in each category is treated as a continu-ous variable, the classes being: susceptible, S ( t ), exposed, E ( t ), infectious, I ( t ), quarantined suspects Q ( t ), confirmed-positive P ( t ), hospitalised in theordinary 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 populationis 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)5 able 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 ig. 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 passfrom one class to another. The susceptible class ( S ) represents individualsnot exposed to the SARS-CoV-2 virus, and the exposed class ( E ) representsindividuals that have recently been exposed to the SARS-CoV-2 virus so arestill in the incubation period and can infect others (that is, asymptomaticindividuals). An individual in an exposed class can infect another personbut with a probability lower than an individual in the infectious class ( I ).This rate of infection is given by the nonlinear function f which dependson the parameters ϕ, α , α , α , β , β , β . Individuals in an infectious classshow clear symptoms and have high infectivity. These individuals have not7et been clinically confirmed-positive, and thus can spread the disease tothe susceptibles. Individuals in class Q are quarantined, that is, individu-als identified to have had contact with an exposed individual and so mightbe 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 arequarantined as a result of enforcement of COVID-19 protocols. These indi-viduals may either enter the susceptible class if test is confirmed negativeor 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 indi-viduals may either enter the intensive care hospitalised class, or be admittedto the ordinary hospitalised class or enter the self-isolated class after thisperiod. The rates of the these processes are governed by the parameters γ , v , ρ , ρ , ρ . The individuals in the ordinary Hospitalised class showssome level of sickness due to infection that need to be cared for at the or-dinary ward. Though there is chance of entering into recovery class, theseindividuals’ conditions may deteriorate causing them to enter the intensivecare hospitalised class. Individuals move between these categories with ratesdetermined by κ , κ , δ , η . These individuals can still infect other individu-als 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 η ).Individuals in the self-isolated class ( F ) are on medication at home andcan still infect other individuals. These individuals ( F ) may either enterthe recovered class ( R , at rate δ ) or enter the ordinary hospitalised class(rate δ ). Individuals who have recovered from SARS-CoV-2 virus enterinto 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 thesusceptible compartment with only natural death. A summary of all theparameter definitions is given in Table 2.8 able 2 Description of the CoVCom9 model parameters.
Parameters DescriptionΛ Recruitment rate µ Natural death rate ϕ Transmission rate of infectious individuals ( I ) α Probability of transmission of exposed individuals ( E ) α Probability of transmission of quarantine suspected infectious individuals( Q ) α Probability of transmission of confirmed-positive infectious individuals ( P ) β Probability of transmission of hospitalised at ordinary ward individuals ( H ) β Probability of transmission of hospitalised at intensive care individuals ( C ) β Probability of transmission of self-isolation at home individuals ( F ) (cid:15) Progression rate of exposed individuals to infectious class per day (cid:15) Progression rate of exposed individuals to quarantine suspected per day γ Progression rate of infectious individuals to confirmed-positive per day γ Recovery rate of infectious individuals per day υ Progression rate of quarantined ( Q ) to confirmed cases ( P ) per day υ Progression rate of quarantined cases to susceptible cases per day ρ Progression rate of confirmed-positive infectives to hospital class per day ρ Progression rate of confirmed-positive to intensive care class per day ρ Progression rate of confirmed-positive ( P ) to self-isolationat home ( F ) class per day κ Recovery rate of hospitalised ( H ) individual per day κ Progression rate of hospitalised (ordinary, H ) to intensive care ( C ) per day κ Progression rate of hospitalised (ordinary ward, H ) toself-isolation at home ( F ) class per day δ Recovery rate of self-isolation at home individual per day δ Progression rate of self-isolation at hometo hospitalised at ordinary ward ( H ) class per day η Progression rate of intensive care to ordinary ward class per day τ Progression rate of recovery individuals to susceptible class per day d Disease-induced death rate of exposed individuals per day d Disease-induced death rate of infectious individuals per day d Disease-induced death rate of quarantine suspected infectives per day d Disease-induced death rate of confirmed-positive individuals per day σ Disease-induced death rate of intensive care individuals per day δ Disease-induced death rate of self-isolated ( F ) cases per day d Disease-induced death rate of hospitalised individuals per day
The standard form of incidence which is formulated from the basic prin-ciples that effective transmission rates are independent of the population9ize N for human diseases is used in this study (Martcheva, 2015; Hethcote,2000). This principle has been shown in many studies to be a plausibleassumption (Hethcote, 2000). If α is the average number of sufficient con-tacts for transmission of an individual per unit time, then αI/N is theaverage number of contacts with infectives per unit time of one suscept-ible, and ( αI/N ) S is the incidence. That is, the number of new cases perunit time at time t due to susceptibles S ( t ) becoming infected (Hethcote,2000). We use ϕ to denote the effective transmission rate from an infectiousindividual while α , α , α , β , β and β denote the transmission prob-abilities, of exposed individuals, quarantine suspected exposed individuals,confirmed-positive individuals, ordinary hospitalised individuals, intensivecare hospitalised individuals, and self-isolated individual, respectively. Allthese probabilities lie between zero and one. The incidence is therefore givenby f ( S, E, I, Q, P, H, C, F ) = ϕ (cid:18) α E + I + α Q + α P + β H + β C + β FN (cid:19) S. (2.2) Our COVID-19 model (CoVCom9) is obtained by ‘translating’ the com-partmental model summarised in Figure 1 into nine coupled ordinary differ-ential equations dSdt = Λ + υ U + τ R − f ( S, E, I, Q, P, H, C, F ) − µS, (2.3a) dEdt = f ( S, E, I, Q, P, H, C, F ) − ( (cid:15) + (cid:15) + µ + d ) E, (2.3b) dIdt = (cid:15) E − ( γ + γ + µ + d ) I, (2.3c) dQdt = (cid:15) E − ( υ + υ + µ + d ) Q, (2.3d) dPdt = γ I + υ Q − ( ρ + ρ + ρ + µ + d ) P, (2.3e) dHdt = ρ P + ηC + δ Q − ( κ + κ + κ + µ + d ) H, (2.3f) dCdt = ρ P + κ H − ( η + µ + d ) C, (2.3g)10 Fdt = ρ P + κ H − ( δ + δ + µ + d ) F, (2.3h) dRdt = γ I + κ H + δ F − ( τ + µ ) R, (2.3i)with t >
0. These are solved subject to the initial conditions S (0) = S ≥ , E (0) = E ≥ , I (0) = I ≥ ,Q (0) = Q ≥ , P (0) = P ≥ , H (0) = H ≥ ,C (0) = C ≥ , F (0) = F ≥ , R (0) = R ≥ . (2.4)In this paper, we will use the acronym CoVCom9 to indicate the nine com-partments of the model of SARS-CoV-2 transmission pattern in Ghana givenby Equation (2.3). The epidemiologically feasible region of interest of themodel (2.3) is the domain defined byΩ = (cid:26) ( S ( t ) , E ( t ) , I ( t ) , Q ( t ) , P ( t ) , H ( t ) , C ( t ) , F ( t ) , R ( t )) ∈ R : S + E + I + Q + P + H + C + F + R ≤ Λ µ (cid:27) . (2.5)In the following sections we present a mathematical analysis of the modelwith respect to positivity and boundedness of the feasible region, Ω, as wellas various stability results and the epidemiological threshold of interest. Inthe subsequent sections, we discuss a theorem demonstrating that solutionsof Equation (2.3) with initial condidtions (2.4) in Ω remain in Ω.
3. Mathematical analysis of CoVCom9 model
The CoVCom9 model (2.3) depicts COVID-19 transmission dynamics inthe human population, so it is vital to show that variables in (2.3) remainnonnegative and bounded for all time t ≥ emma 1 (Positivity and Boundedness) . For any given nonnegative initialconditions in Eq. (2.4), the CoVCom9 model (2.3) has a nonnegative solu-tion { 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 ≥ whenever the parameters are non-negative. Moreover lim t →∞ sup N ( t ) ≤ Λ µ . (3.1) Proof.
Considering the first equation of the CoVCom9 model (2.3), one canclearly see that dSdt ≥ − ( λ + µ ) S, (3.2)where λ = ϕ (cid:18) α E + I + α Q + α P + β H + β C + β FN (cid:19) Next, integrating Eq. (3.2), we find S ( t ) ≥ S exp (cid:20) − (cid:90) to ( λ ( ζ ) + µ ) dζ (cid:21) (3.3)Therefore S ( t ) ≥ t ≥ t ≥
0. That is, E ( t ) ≥ I ( t ) ≥ Q ( t ) ≥ P ( t ) ≥ H ( t ) ≥ C ( t ) ≥ F ( t ) ≥ R ( t ) ≥ ∀ t ≥ dNdt = Λ − µN − dE − d I − dU − d P − d H − d C − d F ≤ Λ − µN, (3.4)Since dN/dt ≤ Λ − µN , it follows thatlim t →∞ sup N ( t ) ≤ Λ µ . (3.5)12 emma 2 (Positively Invariant Region) . The region defined by the closedset, Ω in Eq. (2.5) is positively invariant for the model (2.3) with nonneg-ative initial conditions (2.4) whenever the parameters are nonnegative.Proof. As in Lemma 1, it follows from the summation of all the equationsof the CoVCom9 model (2.3) that dNdt ≤ Λ − µN. (3.6)Using the initial condition N (0) > ≤ N ( t ) ≤ Λ µ + N (0) exp( − µt ) , (3.7)where N (0) is the initial value of the total population. Thus N ( t ) ≤ Λ /µ ,as t → ∞ . Therefore all feasible solutions of system (2.3) enter the region Ωdefined by (2.5), which is a positively invariant set of the system (2.3). Thisimplies that all solutions in Ω remain in Ω ∀ t ≥
0. It is therefore sufficientto study the dynamics of CoVCom9 model system (2.3) in Ω.
The CoVCom9 model has a disease-free equilibrium point given by E = ( S , , , , , , , , ∈ Ω , S = Λ µ . (3.8)The basic reproduction number is defined as the number of secondary infec-tions produced by a single infectious individual during the entire infectiousperiod (Van den Driessche and Watmough, 2002). In this study, the repro-duction number defined as the number of secondary SARS-CoV-2 infectionsgenerated by a single active SARS-CoV-2 individual during the entire in-fectious period. Mathematically, the basic reproduction number R is thedominant eigenvalue of the next generation matrix (Diekmann et al., 2010;Van den Driessche and Watmough, 2002). We apply the method formula-tion in (Van den Driessche and Watmough, 2002) to obtain an expression of13 for the proposed CoVCom9 (2.3). Let x = (cid:0) E, I, Q, P, H, C, F (cid:1) T , thenthe system (2.3) can be written in the form d x dt = F ( x ) − V ( x ) , (3.9)where F ( x ) = ϕ ( α E + I + α Q + α P + β H + β C + β F ) SN , (3.10) V ( x ) = π E E − (cid:15) E + π I I − (cid:15) E + π Q Q − γ I − ν Q + π P P − ρ P − ηC − δ F + π H H − ρ P − κ H + π C C − ρ P − κ H + π F F . (3.11)and π E = (cid:15) + (cid:15) + µ + d ; π I = γ + γ + µ + d ; π Q = υ + υ + µ + d ; π P = ρ + ρ + ρ + µ + d ; π H = κ + κ + κ + µ + d ; π C = η + µ + d ; π F = δ + δ + µ + d . (3.12)14he Jacobians of F ( x ) and V ( x ) evaluated at the disease free equilibrium E are, respectively, J F = ϕα ϕ ϕα ϕα ϕβ ϕβ ϕβ , (3.13) J V = π E − (cid:15) π I − (cid:15) π Q − γ − ν π P − ρ π H − η − δ − ρ − κ π C
00 0 0 − ρ − κ π F . (3.14)The basic reproduction number, R is given by the dominant eigenvalue of J F J − V R = ϕ (cid:26) α π E + (cid:15) π E π I + α (cid:15) π E π Q + α π P (cid:18) (cid:15) γ π E π I + (cid:15) υ π E π Q (cid:19) + β π P (cid:18) ρ π C π F + δ ρ π C + ηρ π F π H π C π F − δ κ π C − ηκ π F (cid:19)(cid:18) (cid:15) π E γ π I + (cid:15) π E υ π Q (cid:19) + β π P (cid:18) ρ π H π F + κ ρ π F + δ ( κ ρ − κ ρ ) π H π C π F − δ κ π C − ηκ π F (cid:19)(cid:18) (cid:15) γ π E π I + (cid:15) υ π E π Q (cid:19) + β π P (cid:18) ρ π H π C + κ ρ π C + η ( κ ρ − κ ρ ) π H π C π F − δ κ π C − ηκ π F (cid:19)(cid:18) (cid:15) γ π E π I + (cid:15) υ π E π Q (cid:19)(cid:27) , (3.15)which can be written as R = R E + R I + R Q + R P + R H + R C + R F , (3.16)15here the effective reproduction number, R is made up of contributionsfrom secondary infections from the exposed group E ( R E ) generated byasymptomatic individuals; the infected (symptomatic) group I ( R I ); asymp-tomatic quarantine suspected individuals - class- Q ( R Q ); confirmed positiveindividuals - class P ( R P ); hospitalised cases ( H , R H ); intensive care ( C )cases, ( R C ); and those self-isolating at home ( F , R F ). Equation (3.15)implies that intervention strategies of SARS-CoV-2 infections should targetthose 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 is locally asymptotically stable if R < R >
1. In the next section we provide stability results forthe disease-free equilibrium state.
In this section, we prove global stability results for the CoVCom9 model(2.3). The epidemiological implication of the local stability is that a smallnumber of the infected individuals will not generate large outbreaks so in thelong run, resulting in SARS-CoV-2 dying out provided R <
1. The globalstability result helps demonstrate that the disappearance of SARS-CoV-2disease is independent of the size of the initial subpopulations in the model,provided R < E is established using a candidate Lyapunov function. Theorem 1.
The disease-free equilibrium state, E of the CoVCom9 model(2.3) is globally asymptotically stable in Ω if R < and unstable if R > .Proof. We construct a candidate Lyapunov function (3.17) for the CoVCom9model (2.3) as V ( E, I, Q, P, H, C, F ) = Φ E + Φ I + Φ Q + Φ P + Φ H + Φ C + Φ F, (3.17)where Φ i , i = 1 , , · · · , V . Assumingthat the variables are solutions of the model (2.3), the derivative of V with16espect to t can be bounded by dVdt = Φ (cid:18) ϕ (cid:0) α E + I + α Q + α P + β H + β C + β F (cid:1)(cid:0) SN (cid:1) − π E E (cid:19) + Φ (cid:18) (cid:15) E − π I I (cid:19) + Φ (cid:18) (cid:15) E − π Q Q (cid:19) + Φ (cid:18) γ I + υ Q − π P P (cid:19) + Φ (cid:18) ρ P + ηC + δ F − π H H (cid:19) + Φ (cid:18) ρ P + κ H − π C C (cid:19) + Φ (cid:18) ρ P + κ H − π F F (cid:19) ≤ (cid:18) Φ ϕα + Φ (cid:15) + Φ (cid:15) − Φ π E (cid:19) E + (cid:18) Φ ϕ + Φ γ − Φ π I (cid:19) I + (cid:18) Φ ϕα + Φ υ − Φ π Q (cid:19) Q + (cid:18) Φ θα + Φ ρ + Φ ρ + Φ ρ − Φ π P (cid:19) P + (cid:18) Φ ϕβ + Φ κ + Φ κ − Φ π H (cid:19) H + (cid:18) Φ ϕβ + Φ η − Φ π C (cid:19) C + (cid:18) Φ ϕβ + Φ δ − Φ π F (cid:19) F, since S/N < . (3.18)Requiring the bracketed coefficients of E , I , U , P , H , C , and Q to zero, weobtain expressions for the previously undetermined parameters Φ i , whichare thus given by Φ = 1 , Φ = ϕ + Φ γ π I , Φ = ϕα + Φ υ π Q ,Φ = 1 π P (cid:20) ϕα + ϕβ ρ π C + ϕβ ρ π Q + (cid:18) ρ + ηρ π C + δ ρ π F (cid:19) Φ (cid:21) ,Φ = ϕ (cid:18) β π C π F + β κ π F + β κ π C π H π C π F − ηκ π F − δ κ π C (cid:19) ,Φ = ϕβ + Φ ηπ C , and Φ = ϕβ + Φ δ π F , (3.19)where the parameter groupings π ∗ are given by (3.12).After some simplifications using (3.12), the time derivative of the Lya-17unov function can be written as dVdt ≤ π E (cid:18) R − (cid:19) E. (3.20)It is now clear that if R < dV /dt ≤
0. Furthermore, dV /dt = 0 if E = 0 and R <
1. Thus, when R <
1, the largest compact invariant setin (cid:110) ( S, E, I, Q, P, H, C, F, R ) ∈ Ω | ˙ V ≤ (cid:111) is the single state E . LaSalle’sInvariance Principle then implies that E is globally asymptotically stablein Ω if R <
4. CoVCom9 model estimation and numerical simulations
In this section, we briefly describe the parameter estimation and numer-ical simulation process used to investigate how well the proposed CoVCom9model (2.3) agrees with the confirmed cases and deaths in Ghana. Here, weconsider the SARS-CoV-2 confirmed cases and deaths from March 13, 2020to August 10, 2020 as reported in Ghana. The data are obtained from OurWorld 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 dDdt = d E + d I + d Q + d P + d H + d C + d F, (4.1)which introduces no additional parameters. The CoVCom9 model has atotal of 35 parameters to estimate using limited data (confirmed-positivecases and deaths only). This results in identifiability issues causing the non-convergence of the optimisation of the objective function. We implement thefollowing practical principles to choose reasonable initial parameter values:1. Expert review process which involves asking health experts and/orconsulting the relevant literature as well as individuals’ experienceof the infection. Accordingly, an estimate of the model parameters,natural birth rate, µ , recruitment rate, Λ, incubation period, (cid:15) , and18ecovery rate of quarantine/self-isolation at home individual, δ areobtained. We assumed that the life expectancy of people in Ghanais estimated as 64.35 years (Asamoah et al., 2020), then the naturaldeath rate is estimated as µ = 1 / (64 . × ≈ . × − per day.The population of Ghana in 2020 is estimated to be N = 30 , ,
000 (,GSS), and the recruitment rate of humans is estimated as Λ = µN ≈ . × people per day. The incubation period is 3–7 days, here wechoose (cid:15) = 1 / .
88 per day as estimated by Pang et al. (2020) whichis consistent with the wider literature (Anderson et al., 2020; Li et al.,2020a). The self-isolated positive-confirmed individuals on medicationtake 14 days on average to recover, thus we assume δ = 1 /
14 per day.2. Exploring the model using the available data (also known as ‘sys-tem exploratory analysis’ (SEA) (Cside18, 2018)). This process helpsidentify ranges of parameter values where the trajectories of the CoV-Com9 are consistent with the data, and regions of parameter spacewhere trajectories deviate from the times series data of confirmed-positive cases and deaths. The motivation for this approach is torestrict the ranges of the parameters and so reduce risk of the MonteCarlo simulation getting trapped at a local optima. Since we have31 remaining model parameters to infer, applying this SEA techniqueyields upper and lower bounds for the model parameters which arepresented in (Table A.1).We use a Monte Carlo least squares method to infer model parametersince it is reliable and efficient. This method seeks to generate the bestMonte Carlo estimate ( (cid:98) θ j ) of the model parameters ( θ , listed in Table 2) byminimising the error between the observed data (confirmed-positive casesand deaths), Y j and the simulated data from the CoVCom9 model (2.3), Y simj given by the variables listed in Table 1. Denoting the total number ofdata-points by n and using i (1 ≤ i ≤ M ) to enumerate the Monte Carlo19imulations, we have (cid:98) θ ( i ) j = arg min θ n (cid:88) j =1 (cid:18) Y j − Y simj (cid:19) , ( i = 1 , , , · · · , M ) . (4.2)Finally, for the M Monte Carlo samples of (cid:98) θ , we obtain the mean andcovariance matrix of the estimator, (cid:98) θ M of θ as (cid:98) θ M = 1 M M (cid:88) i =1 (cid:98) θ ( i ) , (4.3) (cid:98) Σ M = 1 M − M (cid:88) i =1 (cid:16) (cid:98) θ ( i ) − (cid:98) θ M (cid:17) (cid:16) (cid:98) θ ( i ) − (cid:98) θ M (cid:17) T . (4.4)We also give a 95% confidence interval of the Monte Carlo samples { (cid:98) θ ( i ) } Mi =1 as (cid:18) (cid:98) θ ∗ (0 . M , (cid:98) θ ∗ (0 . M (cid:19) , (4.5)where (cid:98) θ ∗ (0 . M and (cid:98) θ ∗ (0 . M are respectively the (cid:98) θ ∗ ( i ) in the 2.5% and 97.5%positions of the ordered Monte Carlo samples { (cid:98) θ ∗ ( i ) } Mi =1 .During parameter estimation, we use a logarithmically transformed para-meter vector, log θ , since: (i) this conveniently ensures that all parametersare positive, θ >
0; and (ii) this improves the numerical search of the para-meter space across a wide range of θ (Bland and Altman, 1996; Acheamponget al., 2019). All computations use MATLAB, 2018a .20 .2. Results of CoVCom9 model parameter estimation Table 3
Estimated initial values of model variables for the system (2.2 using Monte Carloleast squares (MC-LS) method).
Variables Initial values 95% Confidence Interval Reference N S E I Q P H C F R In this section, the results obtained using the Monte Carlo least-squarestechnique described in Section 4 are presented. Table 3 shows initial valuesof the state variables; those for compartments E , I and Q are estimated fromthe reported data. From Table 3, we infer that while on March 13, 2020 twoindividuals 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, and3 respectively.Table 4 gives the parameter values obtained together with their confid-ence intervals. We note that the infectivity of the individuals in the infectedcompartment ( I ) is stronger than the other compartments: in decreasingorder, the infectivities are due to the groups E , F , Q , H , C , and P . Theoverall transmission rate of the SARS-CoV-2 infection in Ghana for theduration of the data considered in this study is ϕ =0.02495 per day, .21 able 4 Estimated values of the model parameters for the system (2.2) using Monte Carlo leastsquares (MC-LS) method.
Parameter Value 95% Confidence Interval Reference α , ϕ , α , α , β , β , β , (cid:15) , (cid:15) , γ , γ × − , × − ) MC-LS υ , υ × − (6.864 × − , × − ) MC-LS ρ , ρ × − (2.494 × − , × − ) MC-LS ρ , κ , κ × − (2.874 × − , × − ) MC-LS κ × − (1.488 × − , × − ) MC-LS δ , δ × − (2.863 × − , × − ) MC-LS η × − (4.862 × − , τ × − (7.741 × − , × − ) MC-LS d × − (3.893 × − , × − ) MC-LS d × − (6.222 × − , × − ) MC-LS d , d × − (2.997 × − , × − ) MC-LS d × − (6.839 × − , × − ) MC-LS d × − (3.413 × − , × − ) MC-LS d × − (1.201 × − , × − ) MC-LS The corresponding best fits of the model to the reported data and thetwo-year simulations based on the estimated parameter estimates are shownin Figure 2. 22 ig. 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 fromthe CoVCom9 model and the numbers from the report data (
Left panel ) daily numbersof confirmed deaths simulated from the CoVCom9 model and the numbers from the reportdata from March 13, 2020 to August 10, 2020 . The rate at which individuals transfer from Classes E to Q is (cid:15) =0 . υ = 0 . C ) is low compared to the rateat which they progress to either H or F Classes (standard hospital ward orself-isolating at home), with the rate of progression from P to F Classes thehighest (that is from positive test to home isolation). The recovery rate ofindividuals in Class H is estimated as κ = 0 . τ = 1 . × − ; indicating that the rate of SARS-CoV-2re-infection in Ghana is extremely low (full details of parameters and rangesis given in Table 4). 23rom equation (3.15) and the parameter estimates in Table 4, the basicreproduction number, R , is estimated to be 3.110. The breakdown of thisestimate is given, in decreasing order, by • primarily, symptomatic individuals (class I , giving R I = 2 . • hospitalised cases (class H , contributing R H = 0 . • positively tested individuals (class P giving R P = 0 . • infections due asymptomatic cases (class E , giving R E = 0 . • self-isolating individuals (class F contributing R F = 0 . • intensive care cases (class C , giving R C = 0 . • those quarantined at home (class Q contributing R Q = 0 . R in Ghana of 2 .
64, differs by only15% from our estimate. However, the number of deaths reported in Ghanais low compared to that of other countries in the world. For published val-ues for other countries, please see Ali et al. (2020); Zeb et al. (2020); Brandiet al. (2020); Mwalili et al. (2020); Kumar et al. (2020); Mushayabasa et al.(2020); G¨otz 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 insightinto the SARS-CoV-2 among Ghanaian with respect to the COVID-19 pro-tocols which are in place in the country. Figure 3 depicts the one-yearsimulation dynamics for the classes E , I , Q , P , H , C , F , and deaths ( D ).24 ig. 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, anddeaths with the vertical axis on a log-scale . As shown in Figure 3, all state variables in the CoVCom9 model showan increasing trend, indicating that Ghana continuing the same protocolsmay not be enough to eradicate the SARS-CoV-2 infection. This has beenfurther complicated by theopening of the the borders, meaning that newcontrol measures are needed to mitigate the spread (both in and out). Our25rojections show that with Ghana exercising current COVID-19 protocolsthe actual cases substantially exceed those reported (whether hospitalised oronly positively tested). We thus expect the exponential growth to continue.In the next section we discuss the derivation of the basic reproductionnumber from the CoVCom9 model, and identify influential parameters thatintervention strategies should focus on in order to control the spread of thevirus.
5. Uncertainty and sensitivity analysis of the basic reproductionnumber
The proposed CoVCom9 model (2.3) has many unknown parameters.Due to the limited data available, there is substantial uncertainty in calib-rating 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 % confid-ence interval is less than five times the lower bound, and more often four orbelow, 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 isapproximately twice the estimate and the lower bound half of it. This un-certainty in model parameters results in some variability in the predictionof the basic reproduction number R . Latin Hypercube Sampling-PartialRank Correlation Coefficient (LHS-PRCC) sensitivity analysis was used toevaluate variabilities in the basic reproduction number R . The LHS-PRCCapproach provides an opportunity to examine the entire parameter space ofthe CoVCom9 model (2.3) with computer simulations.We analyse the impacts of the LHS parameters on the basic reproductionnumber R of the CoVCom9 model (2.3) via standard Monte Carlo proced-ure. The key parameters to which R , given by (3.15), is most sensitiveare determined using the PRCCs values, suggesting the most effective wayof controlling SARS-CoV-2 infection. Moreover, this analysis also identifies26hich parameters need to be known precisely when estimating R from data(Marino et al., 2008).The application of the combined LHS-PRCC methodology in infectiousdisease 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 ofoutcome measures R ; (ii) construction of two linear regression models in response to each para-meter and outcome measure, and (iii) computation of a Pearson rank correlation coefficient for the residualsfrom the two regression models to obtain the PRCC values for thatparticular 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 cor-relation matrix for the 28 model parameters (listed in Table 2) is obtainedfrom the parameter estimation in Section 4, where no correlation is assumedbetween the parameters (cid:15) and δ and other parameters, since these twoparameters are not included in the parameter estimation. The result of the uncertainty analysis of the basic reproduction number, R (3.15) of the CoVCom9 obtained by generating 1000 LHS samples usingthe Monte Carlo technique is presented in Figure 4. This histogram depictsthe uncertainty in R , where the degree of uncertainty quantified via the95% confidence intervals is indicated by the dashed lines. Figure 5 shows thedistribution of obtained values for R , the mean, 5th, and 95th percentilesbeing respectively 2.623, 2.042, and 3.240.27 ig. 4. Uncertainty analysis of the basic reproduction number R depicted by the histo-gram with plot showing 95% confidence interval (dashed lines), mean (solid line) and anestimate (red dotted-dashed line) of R (3.15). Using the best-fit values of all the parameters given in Table 4 yields anestimate of R towards the upper end of the distribution, namely a value of3.110 (see the red dotted-dashed line). In general, the higher the uncertainty,the wider the spread of the distribution of R . We note that there is someuncertainty in R 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 bya factor of two. However, for R , the upper and lower ends of the interval arewith ±
24% of the mean value; thus overall, the uncertainty in the estimateof R is less than that of the individual parameters.28 ig. 5. Sensitivity of the basic reproduction number R to changes in the CoVCom9parameters using PRCC index. Figure 5 shows the sensitivity of the reproduction number R to eachof the parameters in the underlying model (2.3). PRCC assigns each para-meter a value between − R , which suggests those interventions which shouldbe most efficacious in controlling the spread of the virus by reducing R .A PRCC value of zero gives an indication of no association between theinput parameter and model output of interest. The most significant modelparameters are those associated with small p − values ( p < .
05) and largemagnitude PRCC values (0 . ≤ | P RCC | ≤ R , these are: • ϕ - the transmission rate of infectious individuals, • α - the probability of transmission of quarantine suspected infectiousindividuals, 29 α - the probability of transmission of confirmed-positive infectiousindividuals, • (cid:15) - the progression rate of exposed individuals to infectious class • γ - the progression rate of infectious individuals to confirmed-positiveclass, and • υ - the progression rate of quarantine suspected infectives to the classof confirmed-positive cases.In particular, R increases with increases in ϕ , α and α , while R decreaseswith increases in (cid:15) , γ and υ . It is therefore critical that interventionstrategies should be aimed at decreasing the values of ϕ , α and α andincreasing the values of (cid:15) , γ and υ .These recommendations should not be interpreted as discounting thevalue of considering efforts to alter other significant model parameters suchas probability of transmission of hospitalised individuals at ordinary ward( β ), the progression rate of exposed individuals to quarantine suspectedexposed class ( (cid:15) ), and recovery rate of hospitalised individual ( κ ). The simulation presented in Figure 3 show a worrying trend of expo-nential growth with no sign of plateau or reduction in the effects of thepandemic. Many countries have implemented a ‘lockdown’, that is regula-tions 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 para-meter ϕ , and simulate the spread by solving the model using the standardvalue of ϕ for the first 350 days, and a lower value of ϕ for the time period350 ≤ t ≤
700 days. The results are presented in Figures 6, 7, 8, for thevalues ϕ = 0 . ϕ being chosen so as toreduce the expected value of R from 3.110 to 0.995, the threshold requiredfor containment of the epidemic. The second and third values are chosento be double and half of this critical value. Note that the vertical scales inFigures 6, 7, and 8 are not identical.30 ig. 6. One-year simulation dynamics of CoVCom9 model from March 13 2020 whenthere is a 68% reduction in ϕ , that is, to ϕ = 0 . . Here E , I , Q , P , H , C , F and D are respectively exposed, infectious, quarantine suspected-expose, confirmed-positive, hos-pitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the verticalaxis on a log-scale. R changes from 3.110 to 0.995 . Figure 6 shows a clear almost instant reduction in the number of ex-posed people ( E ), followed by a plateau, whilst the sizes of most othersub-populations plateau. However, the numbers of hospitalised cases ( H and C ) both continue to rise slowly. We see that this strength of lockdownstops the exponential growth. The less severe lockdown simulated in Figure31 causes a brief reduction in the number of exposed cases; however, the ex-ponential growth is quickly resumed, in the size of all sub-populations, albeitwith a slightly smaller growth rate. The more severe lockdown simulatedin 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 thecases of hospitalised, intensive care, and self-isolated all plateau, as the totalnumber of deaths slowly increases. 32 ig. 7. One-year simulation dynamics of CoVCom9 model from March 13 2020 whenthere is a 34% reduction in ϕ , that is, to ϕ = 0 . . Again, E , I , Q , P , H , C , F and D are respectively exposed, infectious, quarantine suspected-expose, confirmed-positive, hos-pitalised at ordinary ward, hospitalised at intensive care unit, and deaths with the verticalaxis on a log-scale. The effect on R is a change from 3.110 to 2.053 . It should be noted that these simulations are only a crude model ofthe effects of lockdown, in reality a lockdown could cause changes to otherparameters, particularly α , α , α , β , β , β , in the formula (2.2) for thespread of the disease. We leave the topic of more detailed models of theeffects of lockdown for future work. 33 ig. 8. One-year simulation dynamics of CoVCom9 model from March 13 2020 whenthere is a 84% reduction in ϕ , that it, to ϕ = 0 . . 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 ver-tical axis on a log-scale. The effect on R is a change from 3.110 to 0.498 .
6. Discussion and conclusions
We have developed a mathematical model (CoVCom9) in the form of asystem of coupled ordinary differential equations to describe SARS-CoV-2transmission dynamics in Ghana. This categorises every member of the pop-34lation into one of 9 classes, including various classes well-defined and meas-urable classes , such as those who have tested positive for SARS-Cov-2 andare hospitalised (ordinary wards/intensive care), quarantined, etc, as well as unmeasurable but clinically important classes , such as those who have beenexposed 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 equi-librium solution. Using a candidate Lyapunov function, we have shown thatthe disease-free equilibrium is globally asymptotically stable when the basicreproduction number is R < R . The results derived are of significant epidemiological valuein SARS-CoV-2 control. We estimate that over the period, March-August2020, the average basic reproduction number for Ghana was R = 3 . R is most sensitive to six model parameters ( ϕ , α , α , (cid:15) , γ ,and υ whose effects are detailed in Table 2).The proposed CoVCom9 model is a result of our effort to gain insight intothe vital features of SARS-CoV-2 transmission dynamics in Ghana. Futurework will be focused on extending the model to account for inflow into otherclasses due to opening of Ghana’s borders. Further, we will consider time-35ependent optimal control intervention strategies to gain insight into thebest strategy for Ghana. Other extensions include the time-dependent forceof infection and the maximum capacity of intensive care units. Declaration of competing interest
The authors declare that they have no known competing financial in-terests or personal relationships that could have appeared to influence thestudy reported in this paper.
Acknowledgements
The authors (EA, JADW and RLG) are thankful for funding providedby the Leverhulme Trust Doctoral Scholarship (DA214-024), Modelling andAnalytics for a Sustainable Society (MASS), and to the University of Not-tingham.
Appendix A. Appendix
In Table A.1 we list the parameter values used in the simulations presen-ted in Section 4.Figures A.1, A.2, A.3 show our predictions for how the subpopulationsizes in the model would have evolved over time if a lockdown had beenimposed as soon as the first cases entered Ghana. These predictions areobtained by keeping all parameters at the same values as in the main model,and reducing ϕ to the values used in Section 5.3. These graphs should becompared with Figure 3. In Figure A.1 we use ϕ = 0 . R = 1. We see that this has the effect of bringingthe pandemic under some sort of control, but only over an extremely longtimescale. In Figure A.2 we simulate a partial lockdown, that is, reducing ϕ to 0.016 - which is the midpoint of the standard value ϕ = 0 . R to 1. We see that epidemic still grows, but ata slower rate than with no lockdown. Finally, in Figure A.3, we considerthe effect of a much more severe lockdown, where ϕ is reduced to half that36eeded for R = 1, that is ϕ = 0 . Table A.1
Estimated initial values model variables and parameters for the system (2.2.)
Parameters Min Max α ϕ α α β β β (cid:15) (cid:15) γ γ × − υ υ × − × − ρ ρ × − × − ρ κ κ × − × − κ × − × − δ δ × − × − η × − τ × − × − d × − × − d × − × − d d × − × − d × − × − d × − × − d × − × − E
40 300 I Q ig. A.1. One-year simulation dynamics of CoVCom9 model from March 13 2020 whenthere is a 68% reduction in ϕ to ϕ = 0 . . Here E , I , Q , P , H , C , F and D are re-spectively exposed, infectious, quarantine suspected-expose, confirmed-positive, hospitalisedat ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis ona log-scale. The effect on R is a change from 3.110 to 0.995 . ig. A.2. One-year simulation dynamics of CoVCom9 model from March 13 2020 whenthere is a 34% reduction in ϕ to ϕ = 0 . . Here E , I , Q , P , H , C , F and D are re-spectively exposed, infectious, quarantine suspected-expose, confirmed-positive, hospitalisedat ordinary ward, hospitalised at intensive care unit, and deaths with the vertical axis ona log-scale. The effect on R is a change from 3.110 to 2.053 . ig. A.3. One-year simulation dynamics of CoVCom9 model from March 13 2020 whenthere is a 84% reduction in ϕ , to ϕ = 0 . . 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