Analysis of SEIR model with a single control for COVID-19

A SEIR mathematical model for COVID 19 with a single control vaccination was formulated to analysis the flow of state variables. Properties of Pontryagin’s maximum principle is verified and found the optimal levels of control variable. Optimal values of S, E, I, R were derived by equilibrium analysis. Numerical simulations were carried out to exhibit the Susceptible, Exposed, Infectious and Recovery class with and without vaccination which directs that when the susceptible is vaccinated the flow of disease will drop eventually.


Introduction
Normally organisms such as bacteria and virus are not harmful but some of them cause disease. It is important to follow some control measures to control the infectious diseases. Here we consider vaccination as our control measure. Vaccination develops an immune in our body against the disease which leads us to control the infection of the disease. In India, we have vaccine for measles and for COVID -19 developed recently. Nowadays, mathematical model plays a key role in an analysis of spread and control of the infectious diseases. It describes the transmission process of the diseases. Mathematicians use the optimal control theory for infectious diseases to minimize the cost, infectious individuals and maximize the recovery individuals.
Many Mathematicians used the Optimal control theory for infectious diseases. Specifically, Holly Gaff [2] applied the optimal control for the SIR and SEIR models. Tunde Tajudeen Yusuf and Francis Benyah [9] formulated a SIR model with optimal control of vaccination and treatment. A.Huppert [3] analysed the SIR model with a single control strategy as vaccination.Jakia sultana and Chandra N.Podder [4] analysed Nipah virus infections by the optimal control theory. Gul Zaman [1] studied the behaviour of control strategies for various infectious diseases. Nita H.Shah [8] formulated a mathematical model for the Human -Bat population with control strategies. Moftah Bakush [6] studied the impact of vaccination on infectious diseases by the optimal control technique. Moussa Barro [7] has introduced an optimal control for a SIR model with time delay. A. Mhlannga [5] formulated and analyzed the two patch model for Ebola virus disease. The first COVID -19 case was reported on 7 March 2020 in Tamilnadu, India. Chennai,the captial of the state being worst affected. The COVID -19 outbreak was reported on 1 April 2020 in Vellore district of Tamilnadu. In this paper, we have used the data of COVID -19, Vellore district has been collected from the press release of the Vellore Collectorate and formulated a SEIR model for COVID -19 with a single control, vaccination. We have analysed the model by optimal control theory and attained that the vaccination decrease the flow of the disease in the state variables.

Formulation of Mathematical model and Control set
The SEIR model for COVID-19 with a single control is given by the following system of ordinary differential equations.
where S(t), E(t), I(t), R(t) are the Susceptible, Exposed, Infectious and Recovery state respectively, N (t) -Total population, ω -Average birth rate, µ -Average death rate, β -Transistion infectious rate, γ -Recovery rate , d -disease induced death rate, α-exposed rate. We assume the model with single control vvaccination. The transmission of SEIR model with a single control for COVID-19 is given as Figure 1: SEIR model with a single control for COVID-19 For this system of equations, we define an objective functional similar as in [6] which is a free problem system subject to infected.
Where AI(t) is the number of infected individuals, B is the weight parameter.
The control set is defined as

Pontryagin's Maximum Principle
Pontryagin's maximum principle is a powerful method to compute the optimal value of the control variable. To use this, we have to prove the properties P 1 , P 2 , P 3 and P 4 [2]. P 1 : The set of controls and the corresponding state variables is non empty. P 2 : The control set U is convex and closed. P 3 : The R.H.S. of the state system is bounded by a linear function in the state and control variables. P 4 : The integrand of the objective functional is convex on U and is bounded below To prove this we need the following theorem and lemma.  is Lipschitz continuous with respect to y on a rectangle R = {(x, y)/x 0 − c < x < x 0 + c; y 0 − b < y < y 0 + b; c < a} then there is a unique solution y(t) in R.
Lemma 3.2 If f (x, y) has continuous partial derivative ∂f i ∂y j on a bounded closed domain R, is a set of real numbers. Then it satisfies a Lipschitz condition in R.
To prove P 1 : From (1), From (4), ∂R | = µ < ∞ As the partial derivatives are continuous and bounded, it satisfies the Lipschitz condition. Hence there exist a unique solution of (1). Thus the set of controls and the corresponding state variables is non empty, P 1 is satisfied. To prove P 2 : By the definition of U , it is closed.
Hence P 2 is satisfied. To prove P 3 : From (4)- (7), The state system can be rewrite in the matrix form, which can be written as the linear combination of control v, AsS is bounded, K 2 is upper bound. Hence R.H.S. is bounded by the sum of state and control variables, P 3 is satisfied. To prove P 4 : The control and state variables are non negative and the control variable is also convex and closed. Then the integrand of the objective functional (2) is convex on the set U. Now, Here K 2 > 0 as it depends on I, K 1 > 0 and η = 2 > 1 Hence P 4 is satisfied. Therefore, the control system satisfies the properties of Pontryagin's maximum principle.

Existence of Optimal Control
As the properties for Pontryagin's maximum principle are satisfied, which converts (1) +λ R (γI − µR + vS) (8) and the adjoint system is given by Hence the optimal control is v * = min 1, max 0, S(λ S − λ R ) B

Numerical Analysis
As we found the optimal value of control variable and state variables, we analyse the flow of state variables to study the model with and without vaccination. In Vellore district, the birth rate is ω = 0.0477 and death rate is µ = 0.0203. The transmission rate of COVID-19 is β = 0.011, the recovery rate is γ = 0.96 and the  We study the flow of variable of susceptible. Figure (4) shows that the Figure 3: Flow of Variables with respect to Time without vaccination susceptible with vaccination is less than without vaccination, which shows that vaccination can decrease the spread of the disease. On the study of variable of exposed figure (5) shows that the Exposed with vaccination is reduced from day 1 and without vaccination is higher. We study the flow of variable of infectious. It is clear from figure (6) that the Infectious with vaccination is is reduced from day 1 while without vaccination is higher. On the study of variable of recovered, figure (7) shows that the recovery with vaccination is higher than without vaccination. Hence the control variable vaccination plays an important role in control the spread of COVID -19.

Conclusion
A SEIR model with single control measure vaccination is formulated. By Pontryagin's maximum principle and by equlibrium analysis, the optimal levels of controls and variables were analysed. Numerical simulations were given for transition of COVID-19 in Vellore district ,the flow of variables with respect to time without vaccination and the Susceptible, Exposed, Infectious and recovery class with and without vaccination which justifies that the flow of the disease decreases when the susceptible is vaccinated.