COMPARTMENTAL SEIRW COVID-19 OPTIMAL CONTROL MODEL

With no specific treatment identified, COVID-19 remains a major threat to the world, as the scientific community continues to look for a better understanding of the epidemiological cycle and dynamics of the virus. Mathematical modeling of the 2019 coronavirus disease may provide a better insight into the virus complex dynamics and lay preventive measures that can be employed to contain the disease from spreading. In this research article, a new compartmental SEIRW COVID-19 model is introduced and examined, to describe the dynamics of the disease. We have established both local and global stability analysis for the model equilibria computed from the mathematical model. Additionally, using personal protection, treatment, and spraying of disinfectant as timedependent control functions, we have developed a SEIRW optimal control COVID-19 epidemic model. Applying the well-known Pontryagin’s maximum principle and the constructed Hamiltonian function, we have formulated the optimality system for the nonlinear COVID-19 epidemic model. We have numerically solved the optimality system for the COVID-19 compartmental model using the efficient fourth-order Runge-Kutta iterative scheme with the forward-backward sweep method.


INTRODUCTION
The globally threatening respiratory illness; COVID-19, was identified first in Wuhan city, China in December, 2019. The Wuhan city became the epicenter of the deadly disease outbreak in China. The disease has since then spread across the globe, resulting in fear and panic in nations, causing the cancellation of major events in the world, such as postponement of sports seasons, compulsory closure of universities and colleges, and forcing many into self-imposed quarantine and isolation [1]. The disease invaded other countries and continents through the migration of infected people in China. The spreading of the disease to the different parts of the world caused Europe; especially, Italy, Spain, and France, to be the epicenters, and then the UK and the USA also became epicenters. Finally, the African continent, which was initially considered to be unfavorably continent for the disease, due to the hotness nature of its temperature, became susceptible to the disease [2]. The aggressiveness of the disease and its ability to cause fatalities in humans caused governments to impose partial and complete lockdowns [3].
According to (WHO 2020) [4], coronaviruses consist of viruses identified with causing illness, comprising of sicknesses such as the common cold and other complicated diseases. The virus is transmitted through contact with the droplets, sneezes, or exhales from the infected person or becoming contact with contaminated surfaces. Infected individuals may be symptomatic with headache, aches, cough, fever, dyspnea, sore throat, diarrhea, and loss of taste or smell. The COVID-19 infection can be prevented by adhering to the underlying measures: wearing of a face mask, application of hand sanitizers, regular washing of hand with soap under running water, avoidance of unnecessary mass gathering, social and personal distancing, self-isolation. As of September 3rd, 2020, COVID-19 global confirmed cases reported by WHO are 25, 605, 665, with 852, 758 deaths, reported by WHO globally. The alarming number of intensified deaths and confirmed cases reported globally each day prompted the World Health Organization to declare the disease pandemic and consider it a global health emergency [5].
The study of infectious diseases using reliable mathematical and statistical modeling tools are very useful for understanding the spread of diseases in susceptible populations and can help health policy makers to suggest possible control measures [6][7][8][9][10][11][12][13][14]. Even though it does not provide all-round insight to epidemic outbreaks, it has been of utmost importance to humans, as it employs scientific investigation to uncover the causes and effects of infection. The complex transmission dynamics of the 2019 coronavirus disease pandemic in the world has attracted a lot of attention in computational epidemiological modeling see, e.g., [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. In [32], the authors generated a conceptual COVID-19 outbreak model in Wuhan considering the behavior reaction of the people and governmental actions. The estimates from the two main components of zoonotic introduction and emigration in 1918 influenza pandemic in London, United Kingdom were used to compute the disease's future trends, and the reporting ratio. In [33], Peng et al. proposed and discussed a model on the dynamics of COVID-19 in China. In [34], the authors examined the possibility of scrutinizing the available information on the ongoing outbreak of and evaluating the burden on healthcare systems by using an SIR model to project the actual numbers of infected cases, and identify Intensive care units and isolation wards burdens.
Sandar et al. [35], examined the COVID-19 transmission dynamics of a mathematical model which considered the difference in transmission between symptomatic and asymptomatic population and the lockdown effect on the disease. The effect of 21 days lockdown was assessed to lower the number of deaths and confirmed cases, using cases from three states in India.
Hou et al. [36]; investigated the possibility of applying a three-stage SEIR meta-population model, which is in continuous-time, with consideration of the characteristics and the disease dynamics, to explain the spatial dynamics of the epidemic. In [37], a transmission network model of Bat-Host-Reservoir-People was formulated to determine the potentiality of the disease's transmission from infection source; bat to the host; human. Nishiura et al. [38] employed information from 28 infector-infectee pairs to derive the serial interval estimation of COVID-19. The estimation was done by collecting dates of infectors illnesses from the onset and that of infectee from research article and investigation reports, and analyses were performed on both.
Zhang et al. [39] illustrated the possibility of MERS-CoV's transmissibility estimation and indicated the effective way of containing the infection from spreading by employing quantitative methodologies. Wu et al. [40], forecasted the national and global spread of COVID-19, by effectively accounting for Wuhan's metropolitan-wide quarantine surrounding cities, from Jan 23 − 24, 2020. In [41], the authors examined the appropriate way of preventing COVID-19 at the various departmental levels of China by proposing a time series model that predicts the disease's trend. Based on the simulated results, various predictions were made about the future trend of the disease. In [42], a COVID-19 model was developed, which used stochastic transmission to evaluate the potency of contact tracing and isolation to curtail the disease.
In [43], the authors modified their existing transmission model of COVID-19 [44] by including contact and diagnosis rates that are time-dependent, to refit the available new data. With this, they predicted when the reproduction number would be less than 1. In [45], an application of spatial statistics, specifically, moran's I was employed to examine the spatial dynamics of the COVID-19 epidemic in mainland China. In [46], the authors examined the growth model of the three inflammatory lungs disease of MERS, SARS, and COVID-19, by considering the inhibition constant and growth rate of the diseases. The application of nonlinear fitting was used to determine the parameters of the models for the diseases. The results from the parametric analysis indicated that the COVID-19 growth rate is twice that of SARS and MERS. Sari et.al [47], introduced a simple SEIR model, which includes vaccination and treatment with the sole aim of minimizing the expose and infected. In [48], wang et al. estimated the trend of the COVID-19 epidemic in Wuhan with two assumptions of the basic reproduction number, by proposing a simple SEIR model for the disease dynamics.
The application of nonlinear differential equations with optimal control dynamics in epidemic modeling has widely been studied . Djidjou-Demasse et al. [78] explored some useful control strategies that needs to be implemented to lower treatment costs and deaths of the infected individuals. Grigorieva et al. [79] described the COVID-19 spread in humans, using optimal control to derive an effective quarantine strategy for the SEIR model type. A COVID-19 optimal control mathematical that captures four time-dependent control functions is studied in [80]. In the work of the authors in [81], they developed and studied an epidemic deterministic model for the transmission dynamics and control of COVID-19 disease. They constructed Lyapunov functions and investigated global stability for both disease-free and endemic equilibrium points. They also considered global sensitivity analysis and cost-effectiveness analysis in their study. Further recent works on optimal control modeling of COVID-19 infectious disease can be found in [82][83][84][85][86][87].
In this research article, a new compartmental SEIRW COVID-19 model is considered to describe the dynamics of the disease. Our present study is motivated by deterministic compartmental modeling concepts, nonlinear differential equations, optimal control theory, and the forward-backward sweep iterative method. This research is also inspired by the aforementioned literature on mathematical modeling of COVID-19 and other infectious diseases.
The rest of this research work is organized as follows: Section 2 deals with formulating a deterministic non-optimal control COVID-19 mathematical model. In section 3, local and global stability analysis are investigated. In section 4, a new SEIRW optimal control mathematical model is constructed and analyzed using personal protection, treatment, and spraying of disinfectant as time-dependent control functions. Finally, in section 5, optimal control strategies for the new Optimal control COVID-19 model are numerically solved and simulations results are discussed.

COVID-19 MATHEMATICAL MODEL
We construct a new compartmental SEIRW model to describe the dynamics of the disease.
The population under consideration at time t is separated into susceptible S(t), exposed E(t), infected I(t) and recovered R(t). W (t) present the virus in the environment. Let Λ N denotes the recruitment rate of susceptible individuals. The model assumed that the susceptible get the disease through contact with an exposed individual at a rate β 1 . The susceptible can again have the disease through contact with a contaminated environment at rate β 2 . This assumption is contained in the research of [88], which confirmed people becoming infected when they come in contact with surfaces contaminated with the virus. This happens as a result of the exposed and infected individuals shedding the environment, especially through droplets, sneezes, or exhales.
Exposed individuals enter the infected class at a rate γ. Recovered individuals move to the recovery class at a rate λ . The exposed and infected individuals shed the environment at rates b and α, respectively. Let δ be the disease induced death in humans and η, decay rate of the virus in the environment. All individual classes leave the population through natural death µ.
The dynamics for the model are presented in the schematic of Fig. 1. Based on these assumptions and the presentation in the flowchart, the differential equations describing the model are given as: All involving parameters are nonnegative and defined in Table 1 below. model has a disease-free equilibrium (DFE), determined by equating the right-hand side of the equations in the model (1) to zero, given by: The basic reproduction number of the model at ploying the results of the next-generation matrix of Driessche et al. [91]. We deduced the following matrices based on [91], Hence, the basic reproduction number for the COVID-19 model (1) is obtain by calculating the spectral radius of the matrix ρ(FV − 1) as: (1); there exist a unique endemic equilibrium, denoted by; (1) is asymptotically stable.

Existence of endemic equilibrium. Consider the model system
Proof. By evaluating the Jacobian matrix at the disease-free equilibrium, we obtained the simplified Jacobian matrix given by The characteristic equation of the remaining matrix is given by; where From the Routh-Hurwitz criterion [92], the characteristic equation will have negative real parts provided m 1 m 2 > m 3 > 0. Hence the disease-free equilibrium is locally asymptotically stable if and only if these conditions hold, otherwise unstable.
Then the time derivative of L is given by; It follows that dL dt ≤ 0 for R 0 ≤ 1. Also it can clearly be seen that, dL dt = 0 provided I = 0 and W = 0. Hence from [93], the disease-free equilibrium, E 0 is globally asymptotically stable when R 0 < 1.
Proof. The Jacobian matrix J EQ evaluated at the endemic equilibrium (S * , E * , I * , R * ,W * ) is given by The characteristic equation of J EQ is given by Routh Hurtwiz criterion [92], provides the conditions for determining the stability of (4), as; If these condition are satisfied, the endemic equilibrium will be asymptotically stable, otherwise unstable.
Differentiating L with respect to time.
which gives; Grouping the above expression provides the equation; Clearly, the inequality Q 1 < Q 2 can be confirmed, indicating dL dt ≤ 0 when Q 1 < Q 2 . Hence, it can be verified that dL dt = 0, when S = S * , E = E * , I = I * , R = R * and W = W * . This implies that the largest compact invariant set {(S, E, I, R,W ) ∈ τ 1 : where EQ is the endemic equilibrium. Hence, from [93], EQ is globally asymptotically stable in τ 1 .

OPTIMAL CONTROL COVID-19 SEIRW MODEL
In this section, we develop and analyze a new optimal control COVID-19 SEIRW dynamic model. We consider three multiple time-dependent control functions for our optimal control model formulation consisting of ν 1 , denoting personal protection, ν 2 , denoting treatment of the infected individuals and ν 3 , denoting spraying of the environment with disinfectant. For the personal protection time-dependent control, we can think of wearing of face mask, application of hand sanitizer, regular washing of hand with soap under running water, social distancing, self-isolation, and other important preventive measures. The main aim for constructing this new SEIRW optimal problem is to minimise the number of exposed and infected individuals in the human population. For this purpose, we seek to minimise the objective functional given by; with control set defined as; The weight constants h 1 and h 2 are associated with exposed and infectious individuals respectively. Also the positive weights n 1 , n 2 and n 3 are associated with time-dependent control functions ν 1 , ν 2 and ν 3 respectively.
From equations (5) and (6), the Hamiltonian function related to the optimal control problem is given by We can determine an optimal solution for a given optimal control problem using the Pontryagin maximum principle developed and studied in [94]. Now, suppose that (ζ , ξ ) represent an optimal control solution for a given dynamical optimal control problem, then there exist adjoint or co-state variables, ς = (ς 1 , ς 2 , · · · , ς n ) which satisfies the equation below By applying equation (9) and the Hamiltonian function (8), the adjoint state model or system and the optimal control characterisation for the optimal control problem are given in the theorem below.
Finally, to obtain the control characterization (11), we need to solve for u * 1 , u * 2 and u * 3 from the equation below.

NUMERICAL SIMULATION AND DISCUSSION
This section deals with numerical simulations and results discussions for our optimality system.
Since analytical solutions of nonlinear optimal control problems or systems is a very difficult task, we have therefore for this purpose used the fourth-order Runge-Kutta iterative scheme in MATLAB to generate our numerical solutions. We consider the following initial conditions: 5.1. Minimizing through controls ν 1 and ν 2 . To contain the disease, the controls ν 1 and ν 2 are utilized to improve the objective functional to optimality. The exposed and infected plots showed the effectiveness of the strategy considered. Thus, with the controls ν 1 and ν 2 , the exposed and infected plots of figure 2 showed a differentiable reduction in the graphs of control compared to the scenarios without control. This implies that the strategy of ν 1 and ν 2 is efficient in containing the disease from spreading.  (ν 1 = 0, ν 2 = 0) and without optimal control functions 5.2. Minimizing through controls ν 1 and ν 3 . The controls ν 1 and ν 3 are utilized to improve the objective functional to optimality. The exposed and infected plots showed the effectiveness of the strategy considered by displaying an observable difference in the plots of figure3. With the controls ν 1 and ν 3 , the exposed and infected plots of figure 3 showed a drastic reduction in the control graphs compared to the graphs without control. This shows that the strategy of ν 1 and ν 3 is efficient in containing the disease from spreading.  To improve the objective functional to optimality, the controls ν 2 and ν 3 are utilized. The plots of the exposed and infected individuals as indicated in figure 4 showed an observable distinction in the control and without control graphs. The controls ν 2 and ν 3 , provided the desired results of explicitly lowering the number of exposed and infected individuals, suggesting the effectiveness of this strategy.  to improve the objective functional to optimality. The exposed and infected plots of figure 5 showed the effectiveness of the strategy considered. The controls ν 1 , ν 2 and ν 3 , provided the desired results of explicitly minimizing the number of exposed and infected individuals. This is confirmed in the plots of figure 5 with the control graphs compared to the scenarios without control. This is an indication that the controls; ν 1 , ν 2 , and ν 3 are effective in containing the disease from spreading. 5.6. Conclusion. In this article, a new compartmental SEIRW COVID-19 model is introduced and analyzed. We derived the model equilibria (disease-free and endemic) and the basic reproduction number for the non-optimal control SEIRW epidemic model. Stability analysis for the nonlinear SEIRW COVID-19 model is locally and globally investigated. In this study, we have used mathematical modeling concepts in optimal control theory to explore some control strategies by formulating a new optimal control problem that describes the dynamics of the deadly COVID-19 disease. The control model was analyzed by utilizing the Pontryagin's maximum principle, to derive the optimality system for the SEIRW COVID-19 model. We have generated some interesting numerical results using fourth-order Runge-Kutta scheme with the forwardbackward sweep method, which is one of the most reliable and efficient iterative schemes for nonlinear dynamical optimal control problems. The simulated results proved the incomparable impact of the control strategies considered by substantially reducing the exposed and infected individuals in the population.

DATA AVAILABILITY
Our mathematical modeling does not include data. All parameter values that were used for our simulations have been cited accordingly.