key: cord-0579163-q1vjp4bk authors: Okyere, Samuel; Ackora-Prah, Joseph; Bonyah, Ebenezer; Darkwah, Kwaku Forkuoh title: An Optimal Control Model of transmission dynamics of COVID-19 in Ghana date: 2022-01-26 journal: nan DOI: nan sha: 139ec3793d70d94b35a6a116ff27936bc834acb2 doc_id: 579163 cord_uid: q1vjp4bk Almost every country in the world is battling to limit the spread of COVID-19. As the world strives to get an effective medication to control the disease, appropriate intervention measures, for now, remains one of the effective methods to reduce the spread of the disease. Optimal control strategies have proven to be an effective method in curtailing the spread of infectious diseases. In this paper, a model has been formulated to study transmission dynamics of the disease. Basic properties of the model such as the basic reproduction number, equilibrium points and stability of the equilibrium points have been determined. Sensitivity analysis was carried on to determine the impact of the model parameters on the basic reproduction number. We also introduced a compartment for the deceased and examined the behaviour of COVID-19 related deaths. The numerical simulation prediction is consistent with real data from Ghana for the period March 2020 to March 2021. The simulation revealed the disease had less impact on the population during the first seven months of the outbreak. To help contain the spread of the disease, time dependent optimal controls were incorporated into the model and Pontryagin maximum principle was used to characterize vital conditions of the optimal control model. Numerical simulations of the optimal control model showed that, combination of optimal preventive strategies such as nose mask and vaccination are effective to significantly decrease the number of COVID-19 cases in different compartments of the model. Vaccination decreases the susceptibility to the disease whereas mask usage preserved the susceptible population from extinction. Ghanaians by the government. Notwithstanding all these protocols, the government has introduced vaccination, and the Health Authorities are vaccinating all individuals aged 18 years and above [28] . The transmission dynamics of infectious diseases have been studied and analyzed by researchers using mathematical models. Models offer a simplified representation of reality and may be used to predict future outcomes of diseases and possible interventions. COVID -19 has been studied [2, 6, 12, 13, 29 -33] , but since many aspects related to the COVID-19 virus are unknown, researchers continue to propose models that will best describe the dynamics of the disease. Ndaïrou et al. 2020 [29] , proposed a compartmental mathematical model for the spread of the disease with a special focus on the transmissibility of super-spreaders individuals in Wuhan China. Mugisha et al. 2021 [30] , proposed a mathematical model that incorporates the currently known disease characteristics and tracks various intervention measures in Uganda. They modeled the trace-and-isolate protocol in which some of the latently infected individuals tested positive while in strict isolation. Garba et al. (2020) [31] proposed a compartmental model to analyze the transmission dynamics of the disease in South Africa. A notable feature of their method was the incorporation of the role of environmental contamination by COVID-infected individuals. Fu et al. (2020) [32] , applied Boltzmann-function-based regression analyses to estimate the number of SARS-CoV-2 confirmed cases in China. Nana-Kyere et al. 2022 [33] , proposed the SEQIAHR compartmental model of the disease to provide insight into the dynamics of the disease by underlining tailored strategies designed to minimize the pandemic. Dwomoh et al. (2021) [34] proposed the SEIQHRS (susceptible-exposed-infectious-quarantine-hospitalized-recovered-susceptible) model that predicts the trajectory of the epidemic to help plan an effective control strategy for COVID-19 in Ghana using the generalized growth model. Optimal control methods applied in various models have contributed greatly in engineering intervention strategies in curtailing the spread of infectious diseases [37, 40] . Optimal control methods have been used to model diseases such as tuberculosis [40 -42] , HIV/AIDS [43] , diabetes [44] and recently COVID-19 [45 -49] . In [49] , a mathematical model to study transmission mechanism in Senegal incorporating vital dynamics of the disease and two key therapeutic measures: vaccination of susceptible individuals and recovery/treatment of infected individuals was formulated. Nana -Kyere et al. (2020) [45] , used an SEQIAHR compartmental model to study the disease dynamics and modified the model incorporating optimal controls such as personal protection and vaccination of the susceptible individuals. In [47] , a model was used to study the disease transmission in Ethiopia. The model was modified incorporating optimal control strategies such as public health education, personal protective measures and treatment of hospitalized cases. Motivated by the available COVID-19 works in all these literature, we seek to formulate an optimal control model which accurately predicts and suggest intervention strategies to curtail the spread of the disease in Ghana. The model considers two preventive optimal strategies: the use of nose masks and vaccination. Both strategies among several measures have been adopted by several countries to limit the spread of the disease. As the world strives to get an effective medication to control the disease, appropriate control measures, for now, remains one of the effective measures to reduce the spread of the disease [54] . We examine the transmission dynamics of the COVID-19 in Ghana by modifying the model of [34] by introducing a compartment for vaccination and extending the period of simulation from March 2020 to March 2021. In the new model, we assumed the infectious class is either asymptomatic or symptomatic. We first, consider a non -optimal control model, examine the basic properties of the model and then compute the basic reproduction number and later incorporate the optimal controls in the formulated model. We partition the population into six (6) compartments, namely: Susceptible individuals (S), Exposed (E), Asymptomatic (A), Symptomatic (Q), Vaccinated (V) and Recovered (R). We assumed that the population is homogeneously mixed with no restriction on age and other social factors. Once infected, you become exposed to the disease before becoming infectious. For the model, only the asymptomatic individuals transmit the virus when they come in contact with the susceptible. Individuals are recruited into the susceptible class at the rate Ω and they die at the rate μ. The transmission rate is  and the disease-induced death rate is  . The parameters  and  are the recovery rate of asymptomatic and symptomatic individuals respectively. The flowchart of the model is shown in Fig. 1 . The following ordinary differential equations describe the model: We present the following results which guarantee that system (1) is epidemiologically and mathematically well-posed in a feasible region  [51] , given as (2) is contained and bounded [51] Proof: Solving the differential inequalities yields Taking the limits as , gives . That is, all solutions are confined in the feasible region  . We now show that the solutions of , which is a contradiction, hence ( ) 0 St  Moreover, from the second equation in system (1), it is easy to see 0 ) for all 00 t  which is a contradiction, . Furthermore from the third equation in , then from the fourth equation in system (1) we have Hence, this completes the proof. The diseasefree equilibrium ( 0 ) is the steady state solution where there is no infection in the population. Thus We now calculate the basic reproduction number ( 0 ) of the model (1) using the next generation operator method [19] . Denoting F and V, respectively, as matrices for the new infections generated and the transition terms, thus The basic reproduction number is the largest positive eigenvalue of To calculate 0 at the early stage of the infection, we exclude the vaccination compartment and using the next generation operator method [19] , 0 is given as The necessary condition for the local stability of the disease -free steady state is established in the following theorem. [51] . The Jacobian matrix of system (2) is given as The Jacobian matrix evaluated at the disease-free equilibrium point is We show that all eigenvalues of system (7) are negative. The first, fourth, fifth, and sixth (5) columns give the first four (4) eigenvalues which are and  − (repeated roots). The rest are obtained from the ) 2 2 (  sub-matrix formed by excluding the first, fourth, fifth and sixth rows and columns of system (7). Hence we have The characteristic equation of system (8) is given as and this makes the disease-free equilibrium point unstable. We denote the endemic equilibrium point by ) , , , , , ( , equating the right handside of the system (1) to zero and solving yields . , We state and prove the following theorem: , then the endemic equilibrium point * E , is locally asymptotically stable [51] . The Jacobian matrix (6) evaluated at the endemic equilibrium point is given as where  − ). ( , * 2 2 32 32 23 33 22 44 44 33 44 22 23 32 33 22 11 3 44 33 44 22 44 11 23 32 33 22 33 11 22 11 2 44 33 22 11 are satisfied, then the characteristic equation above has negative real parts and hence a stable equilibrium. We state the following theorem with proof. The reduced form of the system which is globally asymptotically stable equilibrium point for the reduced system The third equation of system (14) gives Solving the second equation of system (14) gives  Solving the first equation of system (14) gives Hence, the convergence of system (2) , satisfies the following conditions given in [20] , i.e. satisfies the conditions above. After formulating a model, one important thing is to validate the model to see if it will stand the test of time. Model validation is the process of determining the degree to which a mathematical model is an accurate representation of the available data. In this section, we validate the model by using cumulative cases of Ghana from Ghana Health Service for the period March 2020 to March, 2021 [24] . We also estimate the parameters of the model and test the effect of the model parameters on the basic reproductive number. The cumulative data of confirmed COVID-19 cases for the period March 2020 to March, 2021 is depicted in figure 2 and figure 3 shows the residuals of the best fitted curve. A starting point of our simulation is March 2020 where the authorities of Ghana confirmed the first two cases of the COVID-19 [25] . According to 2020 population and housing census, the population of Ghana is 30.8 million [53] . We choose these initial conditions [21] . Persons fully vaccinated as of 31 st March 2021 was 500,000 [36] . The parameter values are given in Table 1 . We now determine which of the parameters are most influential to control the basic reproductive number. We perform a sensitivity analysis on the parameters of system (2) to determine which parameter will increase or decrease the basic reproductive number. We use the normalized forward sensitivity index given as  is the parameter under consideration. Positive sensitivity index means an increase in that parameter will lead to an increase in the basic reproductive number and a negative sensitivity index means an increase in the parameter will decrease the basic reproductive number. We use the parameter values given in Table 1 to determine the sensitivity indices in Table 2 below. From Table 2 , it can be seen that the most positive sensitivity indices is the parameter  . On the other hand, the most negative sensitivity index is the parameter  which is the recovery rate for the asymptomatic individuals. Using In Fig. 4 , a decline in the population of the susceptible is noticed after the first 300 days of the outbreak. The number of exposed, asymptomatic and symptomatic individual increases after the first 200 days of the outbreak as shown in Figs. 5 -7 respectively. In Fig. 8 , the vaccinated individuals increase with time. There is a rapid increase in number of individuals who recovers from the disease and those deceased after the first 200 days as depicted in Fig. 9 and 10 respectively. In this section, we look at two preventive control mechanisms i.e. control ( 1 u ) which represents the use of a mask and control ) ( 2 u vaccination of individuals. The directive to use mask started right from the beginning of the outbreak whereas vaccination of individuals started at the beginning of March 2021. To include the nose mask control in the model, we replaced the parameter with (1 − 1 ) where, 0 ≤ 1 ≤ 1 . If there are no usages of the mask, then 1 = 0 and if the entire population uses the mask then 1 = 1.We incorporate the time-dependent controls into the system (1) and we have We analyse the behaviour of system (16) . The objective function for fixed time is and In other to derive the necessary condition for the optimal control, Pontryagin maximum principle given in [23] was used. This principle converts system (16) -(18) into a problem of minimizing a Hamiltonian H, defined by and R  represents the co-state variables. The system of equations is derived by taking into account the correct partial derivatives of system (19) with respect to the associated state variables. where , with the transversality conditions Proof: The differential equations characterized by the adjoint variables are obtained by considering the right-hand side derivatives of system (23) determined by the optimal control. The adjoint equations derived are given as This gives In this section, we analysed numerically the behaviour of the optimal control model (16) using the method of forward-backward sweep method as in [45] . We develop a numerical scheme that uses matlab fourth order Runge-Kutta method [37 -39, 45 ] to solve the model's optimality system. The control 1 u is optimized for the period whiles the control 2 u is set to zero. The results of the simulation are displayed in Figs. 11 -16. In figure 11 , the case with control is higher than a case without control for the susceptible population. Though they both decline, a situation without control decline faster than when there is a control. In figures 12 -16 there is a decline in the exposed, asymptomatic, symptomatic, deceased and recovered population respectively when there is a control within 400 days. This shows that, fewer infections, recoveries and less people deceased when the nose masks usage are implemented. We now focus our attention on the vaccination control 2 . The goal is to reduce the number of In figures 17 -22 there is a decline in the susceptible, exposed, asymptomatic, symptomatic, deceased and recovered population respectively when vaccination control is implemented within 400 days. This shows that, fewer infections, recoveries and less number of individuals deceased when the vaccination is implemented fully. An optimal control model has been formulated to study and control the spread of COVID-19 in Ghana. The basic reproduction, equilibrium points and stability of the equilibrium points has been determined. The model was locally and globally stable for the diseasefree equilibrium. The model was validated using COVID-19 data for the period March 2020 to March 2021. The results of the numerical simulation were consistent with the real data from Ghana. The simulation revealed the disease had less impact on the population during the first seven months of the outbreak. Optimal controls were incorporated into the model to determine the effectiveness of two preventive control measures such as the use of a nose mask and vaccination. Both measures were very effective in curtailing the spread of the disease as there is a decline in the number of exposed, asymptomatic, symptomatic and the deceased when the controls were optimized fully. The use of masks though led to a decline in the susceptible population, this was slow as compared to a situation without its usage. The vaccination on the other hand reduces the number of the susceptible individuals faster than a non-vaccination population. This is so because the vaccinated individuals tend to develop immunity to the disease. 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The data/information supporting the formulation of the mathematical model in this paper are/is from Ghana health service website: https://www.ghs.gov.gh/covid19/ which has been cited in the manuscript. No conflict of interest regarding the content of this article