Optimal Control Model for the Transmission of Novel COVID-19

: As the corona virus (COVID-19) pandemic ravages socio-economic activities in addition to devastating infectious and fatal consequences, optimal control strategy is an effective measure that neutralizes the scourge to its lowest ebb. In this paper, we present a mathematical model for the dynamics of COVID-19, and then we added an optimal control function to the model in order to effectively control the outbreak. We incorporate three main control efforts (isolation, quarantine and hospitalization) into the model aimed at controlling the spread of the pandemic. These efforts are further subdivided into five functions; u 1 ( t ) (isolation of the susceptible communities), u 2 ( t ) (con-tact track measure by which susceptible individuals with contact history are quarantined), u 3 ( t ) (contact track measure by which infected individualsare quarantined), u 4 ( t ) (control effort of hospitalizing the infected I 1 ) and u 5 ( t ) (control effort of hospitalizing the infected I 2 ). We establish the existence of the optimal control and also its characterization by applying Pontryaging maximum principle. The disease free equilibrium solution (DFE) is found to be locally asymptotically stable and subsequently we used it to obtain the key parameter; basic reproduction number. We constructed Lyapunov function to which global stability of the solutions is established. Numerical simulations show how adopting the available control measures optimally, will drastically reduce the infectious populations.


Introduction
The novel coronavirus pneumonia which was officially named as Corona Virus Disease 2019  by World Health Organization (WHO) was reported first in late December 2019, in Wuhan, China [1]. The source of the virus is not yet known, but genetic investigation revealed that COVID-19 virus has the same genetic characteristics with SARS-CoV2 (which was likely to be originated from bats) [2]. It is also found to be significantly less severe than the other two The scourge does not only cause apocalyptic proportion in terms of infection, morbidity and fatality, but also socio-economic consequences. To control the above mentioned problems, there is need to have better understanding on the transmission dynamics of the disease. This could be achieved by developing mathematical model that optimizes the possible control measures.
Optimal control is considered as an effective mathematical tool used to optimize the control problems arising in different field including epidemiology, aeronautic engineering, economics, finance, robotics, etc [5]. Mathematical model offers an insight in to the transmission and control of infectious disease [6][7][8][9][10][11][12][13]. Zhao et al. [14] developed a Susceptible, Un-quarantined infected, Quarantined infected, Confirmed infected (SUQC) model to characterize the dynamics of COVID-19 and explicitly parameterize the intervention effects of control measures. Song et al. [15] established deterministic mathematical model (SEIHR) to suit Korean outbreak, in which he estimated the reproduction number and the effect of preventive measures.
Tahir et al. [16] developed a mathematical model (for MERS) in form of nonlinear system of differential equations, in which he considered a camel to be the source of the infection. The virus is then spread to human population, then human to human transmission, then human to clinic center and then human to care center. They used Lyapunov function to investigate the global stability analyses of the equilibrium solutions and subsequently obtained the basic reproduction number or roughly, a key parameter describing transmission of the infection.
Yang et al. [17] proposed a mathematical model to investigate the current outbreak of the coronavirus disease  in Wuhan, China. The model described the multiple transmission pathways in the infection dynamics, and emphasized the role of environmental reservoir in the transmission and spread of the disease. However, the model employed non-constant transmission rates which change with the epidemiological status and environmental conditions and which reflect the impact of the ongoing disease control measures.
Chen et al. [18] 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. Sunhwa et al. [19] developed a Bats-Hosts-Reservoir-People (BHRP) transmission network model for the potential transmission from the infection source (probably bats) to the human, which focuses on calculating R 0 . Elhia et al. [20] developed a mathematical model based on the epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and close contacts.
Most of these models have a general shortcoming of not taking into consideration time dependent control strategies. For the model to be more realistic, it has to be time dependent [21][22][23][24][25][26]. Here, we modified the work of Elhia et al. [20], by incorporating control functions with the aim of deriving optimal control that drastically minimizes the spread of the infection.
The paper is arranged in the following order: Chapter 1 gives the introduction, Chapter 2 gives preliminary definitions and theorems, Chapter 3 is the model formulation, Chapter 4 discusses the formulation and analysis of optimal control, Chapter 5 presents local and global stability analyses of the solutions of the model and the derivation of the reproduction number and lastly Chapter 6 gives numerical simulation results and then the discussion follows.

Preliminary Definitions and Theorem
Definition 1 (Optimal Control) [27]: A fairly general continuous time optimal control problem can be defined as follows: Problem i: To find the control vector trajectory u : [t 0 , t f ] ∈ R → R n that minimizes the performance index: Subject to: . . , f n ) T and ϕ : R n × R → R is a terminal cost function.
Problem ii: Find t f and u (t) to minimize: Subject to: This special type of optimal control problem is called the minimum time problem.
Definition 2 (Hamiltonian): A time varying Lagrange's multiplier function λ : [t 0 , t f ] → R, also known as the co-state define Hamiltonian function H as: such that Theorem 1 (Pontryagin Maximum Principle): If u * (t) , x * (t) t ∈ t 0 , t f is a solution of the optimal control problem Eqs. (1) and (2) then there exists a non-zero absolutely continuous function λ(t) such that λ (t) , x * (t) , u * (t) satisfy the system such that, for almost all t ∈ t 0 , t f the function in Eq. (3) attains its maximum: , and such that at terminal time t f the conditions If the functions λ(t), x(t), u(t) satisfy the relation Eqs. (5) and (6) (i.e., x (t) , u (t) are Portryagin extremals), then the condition Remark 1: Becerra states that for a minimum, it is necessary for the stationary (optimality) condition to give:

Model Formulation
We know that most people are susceptible to COVID-19 and the patients in the incubation period can infect healthy people. We denote the population of susceptible people with S, the patients in the incubation period and the patients that are yet to be diagnosed by I, patients in the hospital by H, removed people by R, respectively. Here the infectivity of the patients in the incubation period and the patients that are yet to be diagnosed are assumed to be the same.
After the outbreak of COVID-19, susceptible people are advised to lock themselves down at home, and all close contacts of infected individuals tracked are quarantined. Therefore, we divide the population of susceptible people into; susceptible people (S 1 ), the quarantined susceptible people (by close contacts tracked measure) (S 2 ) and general isolated susceptible people (due to community lockdown) (S 3 ). Infected people population is divided into general infected people, including the patients in the incubation period and the infected people that are yet to be diagnosed (I 1 ) and infected people that are quarantined (I 2 ). Here we assume that all susceptible people isolated at home cannot be infected and all infected people isolated at home cannot infect healthy people. Thus, we establish the transmission dynamics of the disease as in Fig. 1.
The transmission dynamics can be described by the nonlinear system of first order differential equations as follows: where, When patients go to hospital and are diagnosed (I 1 + I 2 ), by the close contacts tracked measure, susceptible people (q 1 ) and infected people (q 2 ) are quarantined by the proposition b. Here the number of quarantined susceptible people (q 1 ) and quarantined infected people (q 2 ) are less than the number of susceptible people (S 1 ) and infected people (I 1 ) respectively. Then After the isolation of 14 days 1 α 1 , the quarantined susceptible individuals become susceptible. When quarantined infected people have symptoms, they are hospitalized and diagnosed (I 2 ). After the time of treatment 1 δ , they are removed from the hospital. The communities are isolated and healthy people are also advised to isolate themselves at home unless they have something urgent to deal with. Then μS 1 and α 2 S 3 denote the weak movements of population from susceptible to isolated susceptible and from isolated susceptible to susceptible, respectively.

Optimal Control
Here the detail formulation and analysis of the optimal control problem with respect to the model Eqs. (8)-(14) is given.

Formation of an Optimal Control
The aim of the control strategy is to prevent the susceptible population from becoming infected and reduce the infected population by increasing hospitalization which eventually reduces the number of new cases.
Let the control functions be the rate at which susceptible communities are isolated.
be the contact track measure by which susceptible individuals with contact history are quarantined.
be the control effort of hospitalizing the infected I 1 .
The dynamics of control system can be described by the following system of nonlinear ODE; For a fixed terminal time t f , the problem to minimize the objective functional associated to system Eq. (15) through Eq. (21) is where, . ., 10 denote the weights parameters that balanced the size of the terms.
We seek for optimal control u * such that where U is the set of admissible controls defined by

Existence of Optimal Control
The system of nonlinear ODE Eqs. (15)- (21) can be written as, where, Theorem 3: The optimal control system Eq. (23) is Lipschitz continuous. Proof where,

Characterization of Optimal Control
To formulate the optimal control strategy, we define the Hamiltonian as: H (t) , R (t)) with associated optimal control variables u 1 , u 2 , u 3 , u 4 , u 5 , then there exists a co-state variable satisfying: Proof: Applying the co-state (adjoint) condition of Eq. (5) yielḋ subject to the following transversality conditions; Applying the optimality conditions, we get Solving the optimality system requires initial and transversality conditions together with characterization obtained in Eqs. (38)-(42), in addition, from the Largrangian equation we can see that the second derivative with respect to u 1 , u 2 , u 3 , u 4 , and u 5 is positive. This shows that the optimal control problem is minimum at controls u 1 , u 2 , u 3 , u 4 , and u 5 respectively. Now by substituting Eqs. (38)-(42) into the system Eqs. (15)-(21) we have; dH dt = γ +min 1,max 0, (λ 4 −λ 6 )I 1

Local Stability of the Equilibria
We construct the Jacobian matrix from Eqs. (15)- (20) as: where, Theorem 5: The disease free equilibrium E 0 is locally asymptotically stable. Proof: The eigenvalue is obtained from;

Basic Reproduction Number
For the DFE to be locally asymptotically stable, the eigenvalue K 4 must be negative. That is: Now, define the basic reproduction ratio (R 0 ) to be:

Global Stability Analysis
Here the global stability analyses of the two equilibrium points are carried out.
Theorem 6: The disease free equilibrium is globally asymptotically stable. Proof: Let the Lyapunov candidate function be, Clearly the above function V (S 1 , S 2 , S 3 , I 1 , I 2 , H) > 0.

Numerical Simulations
In this chapter numerical simulations are carried out to support the analytic results and to show the significance of the controller. Most of the data used in the simulation for the parameters and the variables is from china as in [7]. The values can be found in Tabs. 1 and 2 below.  It can be seen from Fig. 2, that when no any control measure is observed and people were allowed to behave as usual the number of infected individuals will escalate. On the other hand if the control measures were observed optimally, that is; susceptible communities are isolated, susceptible individuals that have contact with infected individuals are quarantined, asymptomatic individuals are quarantined, and infected individuals are traced and hospitalized, then the number of infected individuals will drastically be reduced as shown in Fig. 3. Although these control measures aren't easy to be observed but their significance can easily be seen from the above graphs. It is clearly shown that when individuals and governments at various levels put hands together the spread of the disease will be curbed. From the above two graphs it can be seen that the number of people that will be removed from the population (either by death or by natural recovery) will be reduced from about 2.5 × 10 8 when there is no control to less than 9000 people when control is observed optimally.