key: cord-0700987-xja5uf8x
authors: Baba, Isa Abdullahi; Nasidi, Bashir Ahmad; Baleanu, Dumitru; Saadi, Sultan Hamed
title: A mathematical model to optimize the available control measures of COVID – 19
date: 2021-05-11
journal: nan
DOI: 10.1016/j.ecocom.2021.100930
sha: 62b7f7bcee7c57666ddcbf9d19ad9c153223329d
doc_id: 700987
cord_uid: xja5uf8x

In the absence of valid medicine or vaccine for treating the pandemic Coronavirus (COVID-19) infection, other control strategies like; quarantine, social distancing, self- isolation, sanitation and use of personal protective equipment are effective tool used to prevent and curtail the spread of the disease. In this paper, we present a mathematical model to study the dynamics of COVID-19. We then formulate an optimal control problem with the aim to study the most effective control strategies to prevent the proliferation of the disease. The existence of an optimal control function is established and the Pontryagin maximum principle is applied for the characterization of the controller. The equilibrium solutions (DFE & endemic) are found to be locally asymptotically stable and subsequently the basic reproduction number is obtained. Numerical simulations are carried out to support the analytic results and to explicitly show the significance of the control. It is shown that Quarantine/isolating those infected with the disease is the best control measure at the moment.

The highly contagious corona virus infection that cause the current global pandemic was first identified in the late December, 2019 in Wuhan city, China [1] . It is significantly less severe than the other two corona viruses; Severe Acute Respiratory Syndrome (SARS-COV) and Middle East Respiratory Syndrome (MERS-COV) that caused an outbreak in 2002 and 2008 respectively [2] .Although the source of the virus is not yet known, but genetic investigation revealed thatCOVID-19 virus has the same genetic characteristics with SARS-COV which were likely to be originated from bats [1] .

The COVID-19 virus spread from infected human to healthy human through eyes, nose or mouth via a droplet produced of coughing, sneezing or contact with contaminated surfaces, object and equipment of personal use. After the incubation period of 2-14 days [2] the infection developed further to cause mild symptoms that include respiratory symptoms, fever, cough, and shortness of breath and breathing difficulties. In more severe cases, infection can cause pneumonia, SARS, kidney failure and even death [3] . Scientists have not yet developed a vaccine or medicine to cure the COVID-19 infection, but the standard recommendation to prevent the spread of the infection include; quarantine, social distancing, self-isolation, use of personal protective equipment (such as face mask, hand globes, overall gown, e.t.c) regular hand washing using sanitizer, avoiding having contact with person showing the symptoms, reporting any suspected case, and compliance with orientation exercises.

However, many governments engaged in widely public orientation on distancing from public gathering that include social and religious, banning both the local and international trip except for an essential purpose, closing both public and private institutions that may attract large gathering, contact tracing and isolation of infected individuals, providing sanitizers at public domains like markets and car park, fumigating exercise, and to the large extent imposing stay at home curfew.

Since the inception of SIR epidemiological model by Kermack and Merckendric [4] in 1927, significant contributions in modeling human diseases and their controls have been offered, as such models of HIV-AIDS [5, 6] , illicit drug use [7] , multi-mutation and drug resistance [8] , e.t.c . In fact, a collection of such related articles can be found in Kumar and Singh [9] .

The impact of the ongoing global pandemic of COVID-19 epidemic in terms of morbidity, mortality and socio-economic aspects require urgent and effective control measures. However, to decline the apocalyptic proportion of such predicaments, mathematical model should be applied to determine the transmission dynamics and optimize the possible control measures [33 -37] .

Optimal control is an effective mathematical tool use to optimize the control problems arising in different field including epidemiology, aeronautic engineering, economics and finance, robotic, e.t.c. [10] .

In an effort to have a quick response about the situations, several contributions have been made as such: Mishra et al [11] who considered asymptomatic and quarantine classes for SARS-COV2.

Sunhwa et al [12] , established deterministic mathematical model (SEIHR) to suit Korean outbreak, in which he estimated the reproduction number and the effect of preventive measures.

S. Zhao et al. [13] developed a Susceptible, Un-quarantined infected, Quarantined infected, Confirmed infected (SUQC) model to characterized the dynamic od COVID-19 and explicitly parameterized the intervention effects of control measures.

In [14] , C. Yang et al., proposed a new model to study the current outbreak COVID-19 in Wuhan, China. M. Tahir et al. [15] developed mathematical model (for MERS) in form of nonlinear system of differential equations, in which he considered a camel to be the source of infection that spread the virus to infective human population, then human to human transmission, then to clinic center then to care center. However, they constructed the Lyapunov candidate function to investigate the local and global stability analysis of the equilibriums solution and subsequently obtained the basic reproduction number or roughly, a key parameter describing transmission the infection.

T. M Chen et al. [16] Developed a Bats-Hosts-Reservoir-People (BHRP) transmission network model for the potential transmission from the infection source (probably bats) to the human infection,which focus on calculating .

Q. lin et al. [17] modeled (based on SEIR) the outbreak in Wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalization and quarantine) in which they estimated the preliminary magnitude of different effect of individual reaction and governmental action. [21] developed a fractional order model to study the brief interaction among the bats and unknown hosts, and then among people and infection reservoir. Their concern is on the origin of the virus; bats and sea food market.

Manyof the models in literature have a general shortcoming of not taking into consideration the time dependent control strategies. Some of these time dependent control strategies were studied [14 -19, 22 -24] .This type of control strategy can be used to suggest or design epidemic control programs [20] .

To mimic the ongoing global pandemic of COVID-19, we modified the model by M. Tahir et al. in which we incorporated the susceptible human population. We also formulated anoptimal control problem subject to the model with the aim of minimizing the transmission in infected human population , in human to human transmission population , in infected individuals to family members , in patient to clinic center transmission population , and in the patient to care center transmission population .

In the absence of valid medicine or vaccine for treating the pandemic Coronavirus (COVID- 19) infection, other control strategies like; quarantine, social distancing, self-isolation, sanitation and use of personal protective equipment are effective tool used to prevent and curtail the spread of the disease. The main contribution of this paper is incorporating these available control measures in the model as function of time and studying the end result if they were to be applied optimally.

The paper is arranged in the following order: chapter one gives an introduction, chapter two deals with the model formulation, chapter three studies the optimal control problem, chapter four discusses local stability analysis and the derivation of the reproduction number and lastly chapter five gives numerical simulation to support the analytic result, and then the discussion follows.

With bats as the origin of the novel Covid-19 virus, it is assumed that the new born of bats are born into susceptible class , at the rate , which joined the infectious class at rate .It is also assumed that the new born of humans are born into susceptible class which later became infectious as a result of contact with an infected bats at the rate . Then the virus spreads from an infected human to human , to a family member , then to clinic center and care center at the rates , , and respectively. The dynamics of the model is illustrated in figure 1 , and the meaning of parameters and variables are given in table 1 and 2 respectively. The transmission dynamics can be described by the nonlinear system of the first order differential equations;

Where, .

In this chapter we give detail of the formation of the optimal control problem together with the analysis of the control function.

The goal of the control strategies is to reduce or minimize the infected human population , the human to human transmission population , the infected individuals to family members , patient to clinic center transmission population , and the patient to care center transmission population .

Let the control functions, be the rate of quarantine of infected individuals which assume to minimize the infected human be the rate of social distance which assume to minimize human to human transmission population be the rate of self-isolation which assume to minimize transmission to family member , be the rate of sanitation which assume to minimize patient to clinic center transmission population , and be the rate of use of personal protective equipment which is assumed to minimize patient to care center transmission population .

The dynamics of control system can be described by the following system of nonlinear ODE; are constants. are defined on the closed interval [ ] The control functions are bounded and Lebesgue integrable. The coefficients denote the corresponding weight constants which balance the cost elements on the basis of their size and importance in the parts of the objective functional. Our goal is to reduce the number of infectious population and the cost for implementing strategy. The objective functional is defined as; ∫ the goal is to find the optimal control such that,

where the constraint is

The existence, uniqueness and the characteristics will be discussed in the following subsection.

Theorem 1: There exist optimal controls such that; satisfy the control system (9) -(16) with the initial condition (17) .

Proof: This result will be yielded by the general discussion in [17] and the following Lemma.

The control system (9) -(16) with the initial condition (17) satisfy the following conditions i. Г is nonempty and the interval is closed and convex. ii.

Let , . Then the righthand function of the control system (9) 

The control goal is to search the optimal control function such that, reaches its minimum. The necessary condition for the optimal solution is given by Pontryagin's maximum principle, which converts the optimal control (9) -(16) into a problem of minimizing the Hamiltonian function. Let us consider the Hamiltonian function given by; + where is the adjoint variable and is the relaxing variable. By Pontryagin's maximum principle in [10] and the existence result for the optimal control, the following theorem is formulated. 

Through the range of we can obtain the properties of Furthermore, the second derivative of the optimal control is positive, which means that the optimal problem arrives minimum at controls. Then combined with the adjoint equations, the state equations, the initial and transversality conditions, the optimal system can be formulated as;

In this chapter we study boundedness of the solutions, obtain disease free and endemic equilibria, calculate the basic reproduction ratio and carry out local and global stability analysis of the equilibria. 

Disease free equilibrium is obtained by equating to zero. Hence we get;

( )

The endemic equilibrium is obtained by taking all the variables to be different from zero, and solving the system simultaneously;

Solving

From equation (9) 

For the DFE to be locally asymptotically stable, we need the eigenvalue Simplifying, we get

We let basic reproduction ratio to be 

Here we study global stability analysis of the equilibria.

The disease free equilibrium is globally asymptotically stableif .

Let the Lyapunov candidate function be Clearly the above function , Also, at

Clearly,

The endemic equilibrium is globally asymptotically stableif .

Let the Lyapunov candidate function be Clearly the above function ,

The remaining result is,

Substituting equation (9) -(16) in above we get

Where,

Hence the endemic equilibrium is globally asymptotically stable.

In this chapter, numerical simulations are carried out. Variable and parameter values in table 3 and 4 were adopted from [15] . The following can be observed from the graphs; Figure 2 , when no any control measure is taken, it can clearly be seen that there will be epidemics. Figure 3 , shows the dynamics of the optimal control functions. Figure 4 , shows how effective these control measures can be when applied optimally. That is when infected individuals enter quarantine, people applied social distancing, family of infected individuals undergo self-isolation, sanitation is adopted for medical center individuals and use of personal protective equipment for care center individuals. Although all these control measures can't be applied at a time in some settings, Figure 5 shows that if all the measures will be taken except quarantining the infected individuals, there will still be epidemics. Lastly, from Figure 6 , we can understand how important quarantine is, that if it can be applied optimally with time the epidemics will be eliminated, even if all the remaining control measures were not applied.

In conclusion, this paper studies a model that consists of a system of eight nonlinear ordinary differential equations. The model studies the dynamics of COVID-19. An optimal control problem was constructed with the aim to study the most effective control strategies to prevent the proliferation of the disease. The existence of an optimal control function was established and the Pontryagin maximum principle was applied for the characterization of the controller. Two equilibrium solutions; disease free equilibrium (DFE) and endemic equilibrium (EE) were found. Local stability analysis was carried out, and it was established that both DFE and EE depends on the magnitude of a threshold quantity, basic reproduction ratio ( ). DFE is locally asymptotically stable when , whereas EE is locally asymptotically stable when .

Numerical simulations were carried out to support the analytic results and to explicitly show the significance of the control measures. It was obtained that when no control measures taken. But when infected individuals enter quarantine, people applied social distancing, family of infected individuals undergo self-isolation, sanitation adopted for medical center individuals and personal protective equipment for care center individuals were used, then the pandemic will automatically be controlled. The significance of quarantine over the remaining control measures was also shown.

Funding: There is no funding to this research Authors Contribution: All authors of this research paper have directly participated in the planning, execution, or analysis of this study

European Center for Disease Prevention and Control (ECDC), "disease background of COVID-19

World Health Organization (WHO): "coronavirus" available from: www.who.int/healthtopics/corona-virus

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