key: cord-0857163-edp2e5sb
authors: Kumar, R. Prem; Basu, Sanjoy; Santra, P. K.; Ghosh, D.; Mahapatra, G. S.
title: Optimal control design incorporating vaccination and treatment on six compartment pandemic dynamical system
date: 2022-03-31
journal: Results in Control and Optimization
DOI: 10.1016/j.rico.2022.100115
sha: f0ce1fd3669d1955f080d3c8da0c2fceb025ee49
doc_id: 857163
cord_uid: edp2e5sb

In this paper, a mathematical model of the COVID-19 pandemic with lockdown that provides a more accurate representation of the infection rate has been analyzed. In this model, the total population is divided into six compartments: the susceptible class, lockdown class, exposed class, asymptomatic infected class, symptomatic infected class, and recovered class. The basic reproduction number ( R 0 ) is calculated using the next-generation matrix method and presented graphically based on different progression rates and effective contact rates of infective individuals. The COVID-19 epidemic model exhibits the disease-free equilibrium and endemic equilibrium. The local and global stability analysis has been done at the disease-free and endemic equilibrium based on R 0 . The stability analysis of the model shows that the disease-free equilibrium is both locally and globally stable when R 0 < 1 , and the endemic equilibrium is locally and globally stable when R 0 > 1 under some conditions. A control strategy including vaccination and treatment has been studied on this pandemic model with an objective functional to minimize. Finally, numerical simulation of the COVID-19 outbreak in India is carried out using MATLAB, highlighting the usefulness of the COVID-19 pandemic model and its mathematical analysis.

A new infectious disease known as coronavirus disease (COVID-19) was reported first time on 30 January 2020 in India as an outbreak of severe acute respiratory syndrome (SARS) [1] . The virus that causes this transmittable disease is mainly transmitted through dewdrops generated when an infected person coughs, sneezes, or exhales. There is a chance of infection of COVID-19 by breathing in the virus if you are adjacent to someone who has COVID-19 infection or by touching a contaminated surface and then your mouth, eyes, or nose. That's why it is essential to wear a mask in the mouth and sanitize the hands rigorously of every person to rescue them from this infection. Most infected people pass through some mild to moderate symptoms of coronavirus and recover without special treatment. There are two types J o u r n a l P r e -p r o o f Journal Pre-proof In this section, a six-compartmental model has been studied. To derive a realistic model, we divide the total population N (t) in to six different classes, namely, susceptible class S(t), lockdown class L(t), exposed class E(t), infected but asymptomatic class I A (t), infected but symptomatic class I S (t) and recovered class R(t). The transfer diagram of the proposed pandemic model is represented in Fig. 1 . The progression rate from the susceptible class to the lockdown class

The progression rate from the lockdown class to the susceptible class β Effective contact rate of asymptomatic infective individuals d Natural death rate in each class γ 1 Rate of exposed individuals gets infected and remain asymptotic to COVID-19 γ 2 Rate of exposed individuals get recovered δ 1 Rate of asymptotic infected individuals become symptomatic to the disease δ 2 Rate of asymptomatic infected individuals gets recovered µ 1 Death rate of the asymptomatic infected individuals due to the COVID-19 disease µ 2 Death rate of symptomatic infected individuals due to the COVID-19 disease

The COVID-19 pandemic model has the following assumptions:

(a) The susceptible population (S) consists of humans who are not yet infected by COVID-19 disease.

Still, it is assumed that the humans of this class are infected when there is an effective contact with asymptomatic infected individuals (I A ) and the transmission rate of infection is given by βSI A .

J o u r n a l P r e -p r o o f Journal Pre-proof (b) The lockdown population (L) consists of humans moving from susceptible class and confine due to lockdown, and the rate of transmission from susceptible class is given by α 1 S and move out from the lockdown class (L) to the susceptible class with the transmission rate α 2 L.

(c) The exposed population (E) is composed of humans who are infected by COVID-19 disease and are not capable of spreading the disease. After testing positive for COVID-19 disease, the humans are assumed to be asymptomatic and move to the asymptomatic infected class (I A ) with the rate of transmission given by γ 1 E. Some humans of this class naturally recover from COVID-19 disease and move to the recovered class (R) with the transmission rate γ 2 E. (f) It is assumed that in every compartment, the natural death of humans occurs. In the compartments, I A and I S there exists the death of humans related to the COVID-19 disease in addition to the natural death.

A mathematical model of COVID-19 that provides a more accurate representation of the infection rate, which is useful for prevention and control, is given by the following system of nonlinear differential equations:

Here all coefficients are positive with their initial conditions :

3 Basic properties of the model 3.1 Non-negativity solutions Theorem 1. All solution of the system (1) with initial conditions (2) are non-negative for all t ≥ 0.

J o u r n a l P r e -p r o o f

Proof. The functions on the right-hand side of the system (1) are completely continuous and locally Lipschitzian on C 1 , and hence the unique solution (S(t), L(t), E(t), I A (t), I S (t), R(t)) of the system (1) with the initial conditions (2) exists on the interval [0, u) where 0 < u ≤ ∞. From the first equation of

Integrating the second equation of (1) with L(0) > 0, we get dL dt > −(α 2 + d)L, and hence L(t) > L(0)e −(α2+d)t > 0. From the remaining equations of the system (1) with

Theorem 2. All solutions of the system (1) which lies in R 6 + are uniformly bounded and are restricted to the invariant region D = {(S, L, E, I A , I S , R) ∈ R 6

Proof. Let us assume that (S, L, E, I A , I S , R) be the solution of (1). Let

Applying the theory of differential inequality [65] , we get 0 < Q(S, L, E,

Thus every solution of (1) which pledge in R 6 + are uniformly bounded and restricted to the region D = {(S, L, E, I A , I S , R) ∈ R 6 + :

Hence the region D with the initial conditions (2) is a positively invariant region under the flow induced by the system (1) in R 6 + .

Remark 3. Since every solution of (1) have non-negative components with non-negative initial values in D for t ≥ 0 and globally attracting in R 6 + based on the system (1), and further, the last equation of the system (1) does not depend on the other equations, we confine our attention to the dynamics of the system (1) without involving the last compartment in Γ = {(S, L, E, I A , I S ) ∈ R 5 + : 0 < S(t) + L(t) + E(t) + I A (t) + I S (t) ≤ Λ h }. Thus the system (1) defined on Γ is well-posed mathematically and epidemiologically. So, it is sufficient to study the dynamics of the system (1) defined on Γ.

To find the equilibria of the system (1), we set the righthand side of the system to equal zero. Then we get two equilibria in the coordinate (S, L, E,

. It is observed that DFE P 0 always exists. (ii) The endemic equilibrium (EE) P 1 (S * , L * , E * , I * A , I * S ) with positive components:

and I * S = δ1dc3(R0−1) c4c5β which are positive . Hence the EE point P 1 exists if R 0 > 1.

This section represents the basic reproduction number, denoted by R 0 , that is "the number of secondary cases which one case would produce in a completely susceptible population" [66] . Using the method of next generation matrix [67] , we determine the expression for R 0 at P 0 (S 0 , L 0 , 0, 0, 0). Let x = (E, I A , I S , S, L) T , then the system (1) can be written as

The Jacobian matrices of F(x) and V(x) at the DFE P 0 are given by

Then the matrices F and V can be written as

The spectral radius of F V −1 is ρ(F V −1 ) which is the basic reproduction number R 0 = ρ(F V −1 ) = βΛc4γ1 dc1c2c3 .

6 Local stability analysis 6 .1 Local stability of DFE Theorem 4. The DFE P 0 of (1) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

Proof. The Jacobian matrix of (1) at P 0 is given by 

The local stability of the endemic equilibrium P 1 is proved using the Routh-Hurwitz criterion [68, 69] .

Theorem 5. If R 0 > 1, then the EE P 1 of (1) exists and is locally asymptotically stable if it satisfies the

Proof. The Jacobian matrix of (1) at P 1 is given by

The characteristic equation of the above jacobian matrix is given by (

. Clearly one of the roots of the the characteristic equation of J(P 1 ) is -c 5 < 0. The remaining roots can be determined from the following equation λ 4 + A 1 λ 3 + A 2 λ 2 + A 3 λ + A 4 = 0. Analysing the polynomial by using the Routh-Hurwitz criterion [69] , we get that the EE P 1 is locally asymptotically stable if A i > 0 for i = 1, 3, 4 and

Since c 4 k 1 − α 1 α 2 > 0 and c 1 c 2 − k 2 γ 1 = 0, we get A i > 0 for i = 1, 2, 3, 4. The equilibrium point P 1 exists iff R 0 > 1 and is locally asympotically stable if it satisfies the condition

7 Global stability analysis of the model 7.1 Global stability of DFE Theorem 6. The DFE P 0 of the system (1) is globally asymptotically stable if R 0 < 1.

Proof. Since P 0 is locally asymptotically stable when R 0 < 1, it is sufficient to show that P 0 is globally attractive. In section 3, it has been proved that every solution of (1) is non-negative and bounded. For a bounded and continuous real valued function g(t) (say) defined on R + , let g = lim sup t→∞ g(t) and g = lim inf t→∞ g(t). Hence by the Fluctuation lemma [70] (using the following notations in [70] ), there is a sequence σ n → ∞ such that S(σ n ) → S and S (σ n ) → 0 whenever n → ∞. From the first equation of (1), we get

J o u r n a l P r e -p r o o f

Letting n → ∞, we get

and using the remaining equations of (1), we get the following

Next, we shall show that E = 0. Suppose that E > 0, using (6) and (7), we get (3) and (4), we get (6) and (7), we get βS (3) and (4), (8) and (9), we get dR ≤ I S ≤ δ1 c5 I A = 0. Therefore, R = 0 and I S = 0 which implies lim t→0 R(t) = 0 and lim t→0 I S (t) = 0. Using Fluctuation lemma [70] , we get a sequence ρ n → ∞ such that

we get c 6 S = Λ + α 2 L and from (5), we get L (ρ n ) + c 4 L(ρ n ) = α 1 S(ρ n ) and then c 4 L = α 1 S. Therefore,

dc3 . Further using (4) and (5), we get c6c4L

We now analyse the global stability of the endemic equilibrium (EE) of the system (1) using Lyapunov functional method [71] .

Theorem 7. The system (1) is globally asymptotically stable around the EE point P 1 (S * , L * , E * , I * A , I * S ), if R 0 > 1 and the following conditions are satisfied

J o u r n a l P r e -p r o o f

Proof. Let us consider the Lyapunov functional L : Γ → R + as follows:

where a i ∈ R + (i = 1, 2, 3, 4, 5) and their values are assumed in the following steps. Clearly L(S, L, E, I A , I S ) >0

on Γ − (S * , L * , E * , I * A , I * S ) and L(S * , L * , E * , I * A , I * S ) = 0. Differentiating (10) with respect to time t, we get

We get the following result after some mathematical computations

Equation (13) is written asL

where 

The endemic equilibrium point P 1 would be globally asymptotically stable ifL < 0, i.e., if the real quadratic form X T ξX is negative definite. From Frobenius theorem [71] , the real symmetric matrix ξ must be negative definite for the negativity of the quadratic form X T ξX and hence must satisfy (−1) n D n > If we choose a i = 1, i = 1, 2, 3, 4, 5, then we have the following conditions,

8 Dynamics of the system with control after lockdown

In this section, an optimal control system based on the CoV SARS-2 pandemic model (1) has been set up so that this vulnerable situation can be normalised. Here, we introduce three optimal control variables v 1 (t), v 2 (t) and v 3 (t). The optimal control v 1 (t) represents the vaccination on exposed population per unit time at t, the control v 2 (t) represents the recovery rate of the asymptomatic infected individuals under treatment per unit time at t, and the control v 3 (t) represents the recovery rate of the symptomatic infected individual under treatment per unit time at t. Then, the pandemic model with vaccine and treatments after lockdown becomes:

satisfying the initial conditions

The effect of infection on exposed, asymptomatic and symptomatic infected phases are negative for recovered individuals around them. Thus it is essential to minimize them.

The objective functional [72] [73] [74] [75] [76] is defined as (17) where W i (i = 1, 2, 3, 4, 5, 6) are positive weight factors that balance the size of the terms in the integrand. 

, for i = 1, 2, 3} and we have to seek the optimal control (v * 1 , v * 2 , v * 3 ) such that the objective functional is to be minimized, i.e.,

Here, we shall show that there exists an optimal control (v * 1 , v * 2 , v * 3 ) for the control system (15) with initial condition (16) . Let E(t), I A (t) and I S (t) be the state variables with controls v 1 (t), v 2 (t) and v 3 (t) respectively.

Theorem 8. For the control system (15) with initial condition (16) , there exists an optimal control

Proof. The optimal control system (16) can be expressed as the following form:

where

where the constants p 1 > 0 and p 2 > 0 are independent of the variables S(t) and I A (t).

Hence,

where p = p 1 + p 2 + ||C|| < ∞. Therefore, G(φ) is said to be uniformly Lipschitz continuous function.

From the definition of V and restriction on S(t), E(t), I A (t), I S (t), R(t) ≥ 0, we can say that a solution of the system (18) exist [65] . In this case, all the state variables and control variables are non-negative.

The convexity [76] (17)) is satisfied in the minimizing optimal control problem (15) . As the set of control variable (v 1 , v 2 , v 3 ) ∈ V is closed and convex, the system of optimal control is bounded [77] which determines the compactness required for the existence of the optimal control

. Again, we observed that the integrand of (17) i.e.,

is convex on the control set V . Since the state variables are bounded, ∃ n > 1 and positive real numbers k 1 and k 2 such than J To describe the necessary conditions for the optimal control variables, the Pontryagin's maximum principle [78] is applied and it follows the Hamiltonian (H) as the form:

where τ i (t), i = 1, 2, 3, 4, 5, are the adjoint functions to be determined duly.

Theorem 9. Let S * (t), E * (t), I * A (t), I * S (t) and R * (t) are optimal solutions for the optimal control problem (15) with initial conditions (16) associated with the optimal control variables v * 1 (t), v * 2 (t) and v * 3 (t). Then there exist five ad-joint variables τ 1 , τ 2 , τ 3 , τ 4 and τ 5 which satisfy

with the transversality conditions τ i (t e ) = 0 for all i = 1 to 5.

Furthermore, the solutions of optimal control variables are given as follows:

Proof. To determine the all five ad-joint functions and the transversality conditions, Hamiltonian (19) has been used. After setting S(t) = S * (t), E(t) = E * (t), I A = I * A (t), I S = I * S (t) and R(t) = R * (t) and differentiating the Hamiltonian (19) with respect to S, E, I A , I S and R, we obtain equations (20) . From the Pontryagin's Maximum Principle [78] , we obtained the following optimality condition:

which represents the ultimate result of (22-24).

The solution of the optimal control variables (v * 1 , v * 2 , v * 3 ) is given by equations (22) (23) (24) . The optimal control and the state variables are obtained after solving the optimality system consisting of the state system (15), the adjoint system (20) , initial condition (16), the transversality condition (21) and the characterization of the optimal control (22) (23) (24) . Further it is noticed that the second derivative of the integrand of J from (17) with respect to the control variables v 1 , v 2 and v 2 is positive, which guarantees that the optimal problem is minimum at the controls v * 1 , v * 2 , v * 2 . Substituting the optimal control values v * 1 , v * 2 , v * 2 in the control system (15), we find the system as follows

and the Hamiltonian (19) can be rewritten as follows

To determine the optimal control and state variables, it is required to solve the system (25) and (26) numerically. assumed that some percentage of the exposed population are recovered from the infection with in 10 days due to low virus load, which is considered as the observation period of some individuals in the exposed population and hence it is assumed that γ 2 = 1 10 = 0.1. For best fitting, we assume that γ 2 = 0.101. The incubation period for the coronavirus is between two and fourteen days after an effective contact with the asymptomatic infected individuals of this COVID-19 disease. A report published earlier in the pandemic period states that more than 97% of people who contract SARS-CoV-2 show symptoms within 12 days after having effective contact with the asymptomatic infected individuals. It appears that transmission can occur between one to three days before any symptoms appear. So, we assume some individuals move from exposed class (E) to asymptomatic infected (I A ) class within eight days. Therefore γ 1 = 1 8 = 0.125. Those with a mild case of COVID-19 infection usually recover between one to two weeks. Recovery can take six weeks or more for severe cases where the vital organs like the heart, kidneys, lungs and brain are damaged. So, we assume that some individuals of the asymptomatic infected (I A ) class move to the recovered class within twelve days and hence it is assumed that δ 2 = 1 12 = 0.08. In our model, the symptomatic case means the confirmed infected cases tested and declared by the Government. The COVID-19 testing process takes between two to three days, and not all infected people are tested due to a lack of infrastructure and hence considering all these factors, we assume that some percentage of individuals move from asymptomatic infected (I A ) class to symptomatic infected (I S ) class within six days. Therefore δ 1 = 1 6 = 0.17. We assume that the recovery time for symptomatic infected (I S ) class is 13 days and hence = 1 13 = 0.077. Analytical works can never be completed without numerical simulation results. Here, firstly we consider the cases when R 0 value is less than unity using the parameter values Λ = 6 × 10 4 , α 1 = 5 × 10 −3 , α 2 = 10 × 10 −4 , β = 7 × 10 −10 , γ 1 = 8 × 10 −2 , γ 2 = 5 × 10 −2 , δ 1 = 7.5 × 10 −2 , δ 2 = 5 × 10 −2 , µ 1 = 10 × 10 −5 , µ 2 = 10×10 −4 , d = 4×10 −5 , = 7×10 −2 . Using these values for various initial conditions, the model's dynamics are analysed and presented in figures 2(A)-2(E). These figures clearly shows that when R 0 = 0.89 < 1, the susceptible population(S) and lockdown population(L) persists but the exposed population(E), asymptomatic infected population(I A ) and symptomatic infected population(I S ) tends to zero as t → ∞, i.e., the system approaches the disease free equilibrium P 0 (2.58278 × 10 8 , 1.24172 × 10 9 , 0, 0, 0) in long run.

These numerical results supports the results of the theorem 4.

Next, we consider the case when R 0 = 1.75 > 1, using the parameter values Λ = 6 × 10 4 , α 1 = 5 × 10 −3 , α 2 = 10 × 10 −4 , β = 7 × 10 −10 , γ 1 = 8 × 10 −2 , γ 2 = 5 × 10 −2 , δ 1 = 7.5 × 10 −2 , δ 2 = 5 × From figure 4(A) , it is clear that when the progression rate from the susceptible class to the lockdown class increases, the basic reproduction number (R 0 ) decreases and goes below one. So, the system approaches the DFE P 0 , which is globally stable. Hence, the more the population is in lockdown, the more likely it is that the disease will become extinct.

From figure 4(B) , it is clear that as the progression rate from lockdown class to susceptible class increases, the basic reproduction number (R 0 ) increases steadily and goes over unity and, as a result, endemic equilibrium is stable. Hence, if lockdown is not strictly enforced, the disease persists in society for a long time.

From figure 4(C) , it is obvious that as the effective contact rate of asymptomatic infective individuals increases the basic reproduction number (R 0 ) increases steadily and goes over-unity hence the endemic equilibrium is stable and the disease persists in society for a long period.

In figure 5 and figure 6 , the analysis is made on the change of R 0 with respect to α 1 and α 2 , α 1 and β, α 2 and β respectively, fixing other all parameter values as in table 2. It is seen in figures 5(A) and

6(A) that as α 2 increases, R 0 increases sharply, exceeding unity, thus stabilizing endemic equilibrium. As a result, the disease persists in society for a long time. With figures 5(B) and 6(B), it is apparent that as β, the effective contact rate of infected individuals, rises, R 0 value rises in proportion exceeding unity, thereby maintaining the stability of endemic equilibrium, which ensures that the disease persists in society.

From figure 5(C) and figure 6(C), it is obvious that as α 2 which is the progression rate from lockdown class to susceptible class, increases, there is a high chance of individuals in susceptible compartment getting in contact with asymptomatic infective individuals, which is represented by the effective contact rate β also increases and as a result, the R 0 value exceeds unity. Hence there is a wide spread of the disease in the society. Table (2) and Table ( 3) respectively, for the period 23 rd March to 31 st December, 2020. In figure (7) , it is depicted that the real data of the total infected almost coincide with our proposed model curve from 23 rd March to 31 st December, 2020. It is seen that, the proposed epidemic model is best fitted to the current situation of India. Figure ( The optimal control graph for the controls v 1 (t), v 2 (t) and v 3 (t) are presented in figures 11(A)-11(C).

It is obvious from these figures that more effort must be given to the controls, namely, vaccination control v 1 (t) on the exposed class, treatment control on asymptomatic infected class v 2 (t) and treatment control on symptomatic infected class v 3 (t) at the beginning of the disease outbreak. Therefore, it is so important that these controls are applied to the respective compartments at the start of the COVID-19 pandemic in India so that the rapid spread of the disease is controlled. (2) and (3) respectively

From figure (14), it is noticed that the single strain COVID-19 waves are formed in our epidemic model when α 1 = 0. The study showed that if the lockdown was completely relaxed, a single strain COVID-19

wave was observed.

In this paper, we have considered a COVID-19 epidemic model consisting of six population classes, namely, susceptible population (S), lockdown population (L), exposed population (E), asymptomatic infected population (I A ), symptomatic infected population (I S ), recovered population (R) and analyzed the dynamic behavior of the system. The system has two equilibrium points, namely disease-free equilibrium P 0 and endemic equilibrium P 1 . The basic reproduction number R 0 , which is an important threshold parameter used to study the dynamical behavior of the system, has been calculated and is given by

It is found that the DFE P 0 is globally asymptotically stable when R 0 < 1 and the EE P 1 is globally asymptotically stable under some conditions when R 0 > 1. From the sensitivity analysis of R 0 with respect to the parameters α 1 , it is noticed that increase in the progression rate from susceptible class to lockdown class makes R 0 decrease and hence the spread of infection in society is drastically reduced. If α 2 increases, i.e., relaxation in lockdown is announced by the Government, then the value of R 0 start increasing steadily and hence there is a rapid spread of the disease in society. Furthermore, the rise of the effective contact rate of infective individuals also increases the value of R 0 which in turn increases the number of infected individuals in the society.

The main aim of this paper is to establish an optimal control problem related to the COVID-19 epidemic model such as to minimize the spread of infection and the cost of treatment. We have used three controls, namely, vaccination control v 1 (t), treatment control v 2 (t) on asymptomatic infected compartment and treatment control v 3 (t) on symptomatic infected compartment. (2) is presented and the optimal control are obtained theoretically and finally presented graphically. Controlling the spread of the epidemic is a very important task, and it is a vital issue to make detailed studies on control strategies. Predicting and identifying cost-effective control strategies to control the epidemic and minimize the cost of implementing control strategies are important tasks of health administrators and researchers. Many research articles analysed the dynamics of the COVID-19 models without control strategies with real data belonging to various other countries and the results from our proposed COVID-19 pandemic model considered the data sets from Indian population during the pandemic and it suggested that the COVID-19 epidemic is well controlled by implementing the lockdown, and after analyzing the optimal control problem without lockdown relative to our basic model, we see that 

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We are grateful to the Editor and anonymous referees for their valuable comments and helpful suggestions which have helped us to improve the presentation of this work significantly.