Optimal control strategies for the reliable and competitive mathematical analysis of Covid-19 pandemic model.

To understand dynamics of the COVID-19 disease realistically, a new SEIAPHR model has been proposed in this article where the infectious individuals have been categorized as symptomatic, asymptomatic, and super-spreaders. The model has been investigated for existence of a unique solution. To measure the contagiousness of COVID-19, reproduction number R 0 is also computed using next generation matrix method. It is shown that the model is locally stable at disease-free equilibrium point when R 0 < 1 and unstable for R 0 > 1 . The model has been analyzed for global stability at both of the disease-free and endemic equilibrium points. Sensitivity analysis is also included to examine the effect of parameters of the model on reproduction number R 0 . A couple of optimal control problems have been designed to study the effect of control strategies for disease control and eradication from the society. Numerical results show that the adopted control approaches are much effective in reducing new infections.

INTRODUCTION

In December 2019, a new kind of Corona virus was experienced in Wuhan (China). This new type of virus embraced the whole city in few days and later on spread over the whole world in a very short span of time. The new virus was named as Covid‐19 by the World Health Organization (WHO), and disease due to virus was declared as pandemic in March 2020. , According to the WHO, Covid‐19 transmitted from snake and bats to human population through seafood market of Wuhan. The deadly virus not only troubled the entire world in the fields related to health but also ruined the world economy, education system, and social life of humans. As of 28 December 2021, the WHO reported that the Corona virus has infected more than 280,119,931 confirmed cases with 5,403,662 deaths. With the new variants like Delta and Omicron, the number of confirmed cases are continuously increasing. Omicron has the capability to spread quickly as compared with other variants. A lot of investigations are being carried out to determine the transmissibility and severity of Omicron. However, more than 220,951,891 individuals have also been recovered from deadly disease and its variants.

Dry cough, sneezing, trouble in breathing, headache, fatigue, loss of smell and taste, vomiting, sore throat, and body pain are usual symptoms of Covid‐19. Covid‐19 damages the human's lungs, liver, kidney, and so on. The mortality caused by Covid‐19 rates differently in different countries depending on the environment and food situation of the countries. , , , , , , , , , , , The mortality rate also relies upon the ratio of young and old people. The people over the age of 60 and having diseases like diabetes, cancer, cardiac, obesity, blood pressure, and lungs issues are at high risk of getting a severe infection that may end up with death. , , , , Covid‐19 has an incubation period of 2 to 14 days.

Dynamics of the disease and its transmission patterns are continuously being observed by both the public health experts and policy makers to suggest appropriate solutions for disease control and eradication. The foremost step in case of Covid‐19 is to run awareness and self‐protection campaigns in the community. For instance, the general public should be educated to wear face masks, to avoid large indoor gatherings, to keep social distance of at leat 6 ft, and to wash hands for at least 20 s. Other precautionary measures include smart lock down strategy, isolating exposed or infected individuals, and vaccinating the susceptible individuals.

In the field of mathematical modeling and optimal control design, researchers are continuously trying to develop different mathematical models of Covid‐19 according to physical situations or requirements and are presenting a variety of control strategies for disease control. , , , , , , , , , , , , , , , , , , In this study, we design a new Covid‐19 model termed as SEIAPHR where the infectious humans are placed in three compartments such as symptomatic infected , asymptomatic , and super‐spreaders . The division of infectious individuals in three compartments makes the model more realistic for the sake of analysis and control of disease. To restrict the spread of Covid‐19 in human population, a few of nonpharmaceutical strategies such as quarantine, health awareness, self‐protection, and social distancing are also proposed and incorporated in the Covid‐19 SEIAPHR model.

The rest of the article is managed as follows: Section 2 deals with the formulation of a nonlinear coupled mathematical model for Covid‐19. Fixed point theorem is implemented to prove existence of a unique solution in Section 3. The model is made more reliable and realistic by showing positivity and boundedness of the state variables in Section 4. The disease‐free equilibrium (DFE) point and the endemic equilibrium (EE) point are also calculated in Section 5. In Section 6, next generation approach is employed to determine the reproduction number . Both the local and global stabilities at equilibrium points are examined in Section 7. Section 8 is devoted for the sensitivity analysis of the parameters involved in reproduction number.

As a first optimal control strategy, the model SEIAPHR is reformulated in Section 9 to adjust a class of quarantined people . With the isolation strategy for disease control, the optimal control problem is designed and solved numerically in Section 9. The corresponding graphical results are also illustrated here. As a second nonpharmaceutical control strategy, the SEIAPHR model is once again updated in Section 10 to include three additional parameters (controls) named as public health information, personal, protection and medication. The corresponding optimal control problem is also formulated in this section. Optimality conditions are also derived and solved numerically for presentation of graphical results. The findings of the study are summarized in Section 11.

DESIGN FOR COVID‐19 MODEL

In the field of epidemiology, mathematical models play pivotal role in understanding the disease transmission dynamics. A carefully designed model helps the policy makers to foresee the disease patterns and to make right decisions in restricting the spread of disease. In this section, we design a new realistic SEIAPHR model where we have categorized the infectious individuals into three classes named symptomatic, asymptomatic, and super‐spreaders. In the first place, our focus is to study the disease dynamics by analyzing the model mathematically and then to suggest some control strategies to optimally control the disease.

We categorize the total population into seven classes as follows: susceptible , exposed , symptomatically infected , asymptomatic , super‐spreader , hospitalized , and recovered . Therefore, the whole population at any time is given as follows:

The first class is known as the susceptible class, denoted by . In this class, we consider those individuals who are at risk and can easily be infected after the transmittable interaction with infectious individuals. When a person of susceptible class has such an interaction, the person will move in exposed class. The exposed class, denoted by , is restrained for those who are infected but not infectious yet. In this class pathogenic microbiological agent develops and consistently strengthen. Then, the third class comes up from the exposed class containing those who are now infectious and experiencing the symptoms of corona disease identified as an symptomatic infected class, denoted by . The fourth compartment comprises those particular group of exposed individuals who are infectious now, but they are not facing the signs of corona disease, known as asymptomatic class. This class is denoted by . The fifth one is named as super‐spreader class in which we have considered those who are the rapid carrier of virus, for example, public transporter, salespersons, delivery staff, and shopkeepers. This category is represented by . The patients with severe health conditions are forming the hospitalized class. This class is indicated by . In the end, people who recovered with medicated treatment or by their strong immune system will lie in the recovered class symbolized by . It is also assumed that the recovered individuals will occupy this class for the whole life. The real‐valued state variables are also considered to be continuously differentiable functions of . Figure 1 explains the flow pattern of disease in the above said compartments.

FIGURE 1

Flow diagram of Covid‐19 disease transmission [Colour figure can be viewed at wileyonlinelibrary.com]

Mathematically, disease flow pattern is described in the form of following nonlinear coupled ordinary differential equations, called SEIAPHR model.

with nonnegative initial conditions:

The values and physical interpretation of the parameters considered in model (2) are given in Table 1.

TABLE 1

Parametric values

Parameter	Description	Values	Source
α_1	Transmission rate from S to E due to contact with I	0.866	Assumed
α_2	Transmission rate from S to E due to contact with A	0.16	7
α_3	Transmission rate from S to E due to contact with P	0.8	Assumed
α_4	Transmission rate from S to E due to contact with H	0.0131	7
α_5	Transmission rate from E to I	0.235	7
α_6	Transmission rate from E to A	0.26	7
α_7	Transmission rate from E to P	0.56	Assumed
α_8	Transmission rate from I to H	0.45	7
α_9	Transmission rate from I to R	0.6381	7
φ_1	Transmission rate from A to R	0.08	7
φ_2	Transmission rate from P to H	0.1	Assumed
φ_3	Transmission rate from P to R	0.3	Assumed
φ_4	Transmission rate from H to R	0.5431	7
δ_I	Death rate due to disease in I	0.08	Assumed
δ_Q	Death rate due to disease in Q	0.01	7
δ_P	Death rate due to disease in P	0.0412	Assumed
δ_H	Death rate due to disease in H	0.485	Assumed
Π	Birth rate	2.5	Assumed
μ	Natural death rate	0.241	Assumed

EXISTENCE AND UNIQUENESS OF SOLUTION

In this section, we state some theorems to prove existence and uniqueness of solution of the Covid‐19 model (2). Some basic definitions and theorems from functional analysis are also presented here to support the proof of our stated theorems.

Let us put the Covid‐19 model (2) in the form

where is a real‐valued function defined by with and where are right‐hand sides of the equations of model (2). To establish existence and uniqueness of the solution of model (2), we state some basic theorems and definitions.

Let be a continuously differentiable mapping from to . Then satisfies a Lipschitz condition on each convex compact subset of with Lipchitz constant . Where is the supremum of the derivative of on , that is,

Suppose , and let be continuous on and satisfies Lipschitz condition there, then the initial value problem has a solution.

Picard mapping Given a point and a differential equation where and is a vector field over , identify Picard mapping towards mapping that takes a function to the function , such as with

The mapping is a solution to with initial condition if and only if Where with

The right‐hand side function in (3a) is Lipschitz continuous in .

Let be a convex compact subset of Let , then by mean value theorem (for several variables) such that where Hence, Since , hence, over convex compact set constant such that hence, Hence, is Lipschitz in the second argument. □

Thus, Theorem 2 implies that system (3) has a solution and Theorem 3 implies that the solution will be the fixed point of Picard mapping.

Let be a complete metric space and be a contraction on . Then has one and only one fixed point.

The solution of model (3) is unique.

To prove the uniqueness of solution, we suppose that and be two solutions of (3). Then, both will be the fixed points of the Picard mapping, that is, and Thus, As satisfies the Lipschitz condition in the second argument, so If we choose , the mapping is contraction. Here, represents the final time and is the Lipchitz constant.

Hence, Theorem 5 implies uniqueness of the solution of SEIAPHR model (2). □

BOUNDEDNESS AND POSITIVITY OF SOLUTIONS

In this section, we prove boundedness and positivity of the state variables of model (2) and also define the feasible region for the state variables.

The solution y(t) of Covid‐19 model (2) is bounded.

Differentiating Equation (1) with respect to time and then using equations of model (2), we obtain with Suppose for any initial condition, we have

Equation (4) can be put in the form:

By using Gröwnwall's inequality, (6) is solved to reach at the solution which implies that Hence, it is proved that Thus, the solution is bounded for every . □

The solution of system of Equation (2) having nonnegative initial conditions (2h) is positive for all 0.

First, suppose that Equation (2.1.2a) could be written in the form

Since the solutions of model (2) are bounded, Equation (7) can be put in the form where Equation (8) is a linear in whose integrating factor is computed to have . So Equation (8) after multiplication with the integrating factor becomes Integrating over the interval , where , we get Simplification yields us

Since , so (9) implies that for all . In the same way, we can prove that all the other state variables are positive. □

Thus, the feasible region for the model is defined as follows:

EQUILIBRIUM POINTS

Equilibrium points are computed by solving steady‐state equations of model (2). For Corona‐free or DFE point, we consider the absence of virus, whereas for Corona present or EE point, the presence of virus is assumed in the community.

Therefore, Corona‐free or the DFE point is computed to give the following: and Corona present or the EE point is given as follows: where with

REPRODUCTION NUMBER

The reproduction number is a mathematical quantity which determines the disease's dispersion. Disease is pandemic only in the case if . The number is computed using the next generation matrix method. , , It is actually a spectral radius of , where is the Jacobian of the rate of recruitment of new infections and is the Jacobian of the rate of the other transmission terms in equations involving infections. That is, where , respectively, represent state variables , and The absolute maximum eigenvalue of the matrix is computed to give the reproduction number

STABILITY ANALYSIS

This section deals with the local and global stabilities of the Covid‐19 model (2) at the DEF and EE points. Global stabilities are investigated using the Lyapunov theory with LaSalle invariant principle and Castillo‐Chavez approach.

Local stability at DFE

Jacobian matrix method is computed for model (2) to evaluate its local stability at DFE. The Jacobian matrix evaluated at is given as follows:

We present the following theorem for local stability of the model (2) at DFE point .

Model (2) is locally asymptotically stabile (LAS) at if and unstable for .

With the assistance of Maple software, we attain the following eigenvalues of the Jacobian matrix (14).

where It is evident from (15) that all the eigenvalues are negative when but not in the case when . Thus, it is proved that model (2) is LAS for and unstable for . □

Global stability

In order to show that DFE point is globally stable, we use the approach given by Castillo‐Chavez et al. and rewrite the model (2) in the form where represents uninfected humans who are susceptible and denotes the number of people who are exposed, symptomatic, asymptomatic, super‐spreader, and hospitalized, with and . We omit the model's last equation since it is independent to the others. is the DFE point. To prove the global asymptotical stability (GAS) of DFE point, the conditions given below must be fulfilled.

Here, is an M‐matrix and is the model's feasible region. Thus, due to Castillo‐Chavez et al., we state the following theorem.

The DFE point of model (2) is GAS if and the conditions and are met.

Let represent uninfected persons; represent exposed, symptomatic infected, asymptomatic, super‐spreader, and hospitalized individuals; and is the DFE point. So

If , then , that is,

From Equation (20) as . Therefore, is GAS. Now, where and .

It is evident that is an M‐matrix. Since at DFE point each of , thus, column matrix . So DFE point is GAS. □

Global stability at EE

We present the next theorem that shows the global stability of model (2) at EE point .

The EE point of model (2) is stable provided and unstable when .

We consider a Volterra‐type Lyapunov function defined as follows: where is an EE point.

Taking derivative with respect to time and simplifying, we get

Replacing the time derivatives of state variables with the right‐hand sides of the ODEs of model (2), we reach at where and

Since each of the parameters in model (2) is nonnegative, hence, we have when and when . The case implies that , , and .

So according to LaSalle's invariance principle, the EE point is globally asymptotically stable. □

SENSITIVITY ANALYSIS

Sensitivity analysis plays a vital role to make the best strategies to control a pandemic. Researchers used the tool of sensitivity analysis to mark the parameters with high sensitivity. There are many techniques defined and used for sensitivity analysis. We use the technique named as normalized sensitivity index or elasticity index defined by Using this approach, sensitivity analysis of the parameters of model (2) is given in Table 2. From the table data, we observe that has the highest sensitivity impact on the reproduction number . , and are other parameters which also have higher influence on the number as compared with the rest of the parameters.

TABLE 2

Sensitivity index for

Transmission rates	Sensitivity index	Transmission rates	Sensitivity index
Π	0	α_8	−0.0938
μ	−0.4080	α_9	−0.1291
α_1	0.2746	φ_1	−0.0213
α_2	0.0130	φ_2	−0.0996
α_3	0.6732	φ_3	−0.3297
α_4	0.0392	φ_4	−0.0147
α_5	−0.0577	δ_I	−0.0064
α_6	−0.1827	δ_P	−0.0133
α_7	0.3612	δ_H	−0.0114

OPTIMAL CONTROL STRATEGY I

The most suitable mathematical theory to resolve the problems related to deploying the best choice to get a certain target is optimal control theory. The theory by Pontryagin and Boltyansikii for optimal control problems has been implemented on various integer and fractional order epidemic models to obtain utmost benefits in taking upcoming decision. , , , , ,

In this section, we design an optimal control problem with a strategy to isolate the infected people from the rest of the population. For isolating or self quarantining the infected individuals, a quarantine compartment is added in the existing SEIAPHR model (2). We suppose that the infectious people are isolated at the rates , and from symptomatic, asymptomatic, and super‐spreader compartments, respectively. In addition to this, we also assume that the isolated people with sever disease symptoms are hospitalized at the rate . Isolated people also die at the rate . With these considerations for the newly introduced compartment , model (2) is updated to get the following system of equations. with initial conditions where initial values are nonnegative.

Model (22) will serve as restrictions for our optimal control problem. The flow diagram for model (22) is shown in Figure 2.

FIGURE 2

Flow diagram of Covid‐19 disease transmission with quarantine class [Colour figure can be viewed at wileyonlinelibrary.com]

Cost functional

For formulation of optimal control problem, we consider the cost functional of the following type.

Here, is final time; are time‐dependent control; and are the corresponding costs of controls. The objective of the functional is to reduce the infected individuals with some adjustable costs. The values of constants , will be set to either zero or one to have different cost functionals for the sake of analysis.

Our aim is to find optimal control in the way that cost functional (23) is reduced to its minimum, i.e.,

Here is the set of controls specified as follows:

Necessary conditions

In this section, we formulate a Hamiltonian function in order to derive the necessary optimality conditions for optimal control problem (28). The Hamiltonian is described as follows: where and symbolize the state variables; are the associated adjoint variables; and are the right‐hand sides of the equations of system (22). The Hamiltonian for the control problem (24) in expanded form is written as follows:

The first optimality condition of the Pontryagin maximum principle provides us the equations for control variables With bounds on controls, we have

The second optimality condition of the Pontryagin maximum principle offers the system of coupled linear adjoint equations: supported with the conditions at the final time, that is,

Finally, variation of with respect to adjoint variables yield us the ODEs for the state variables as given by (26).

Solution algorithm

To solve the optimality conditions for the control problem (28), we follow the steps of the following algorithm. symbolizes the state variable , adjoint variable , and the control , whereas is the tolerance set as per requirement.

Optimal solutions

This section consists of the presentation and discussion on optimal solutions of the optimal control problem (24). The solutions are obtained by implementing Algorithm 1 along with Matlab code. State and adjoint variables are approximated using RK4.

To implement RK4 method, we discretize continuous time domain into equal subintervals each of width such that the corresponding discrete points are given as . Solutions are approximated at these discrete points. The integral in the cost functional (23) is evaluated by Simpson's one third formula. For sake of analysis, we consider the following two cases of the optimal control problem with two different cost functionals.

Case 1: As a first case, we consider the cost functional (23) with . For this case, the optimizer of the optimal control problem (24) is shown in Figure 3. These variables, respectively, offer us the optimum quarantined and hospitalized rates to attain the minimum cost. Graph for the cost functional is given in Figure 4. From the figure, it is evident that the functional has reached to its minimum under the optimal controls in the 14th iteration.

FIGURE 3

Optimal controls (Case 1) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4

Cost functional (Case 1) [Colour figure can be viewed at wileyonlinelibrary.com]

Figure 5 demonstrates graphical curves of state variables before and after optimization. We notice a remarkable decrease in the curves representing exposed, symptomatic, asymptomatic, super‐spreader, and hospitalized compartments after optimization. The curves from these compartments have moved to disease‐free state. From the figure, we also observe that there is a need to quarantine and hospitalize more people in the beginning. However, there is an increase in the number of susceptible individuals. Thus, decline in infected individuals is an accomplishment of this optimal control strategy.

FIGURE 5

State variables before and after optimization (Case 1) [Colour figure can be viewed at wileyonlinelibrary.com]

Case 2: Now we consider the cost functional (23) with for and for . This means that we are not considering the quarantine state and the hospitalized class in cost functional. We also omit the control from the cost functional. Figure 6 shows the optimal curves for optimizers (control variables ) of the control problem (24) under assumptions of case 2. We notice that all the controls vary with time but are restricted with in the bounds. The corresponding cost functional (23) with , shown in Figure 7, approaches to its least value in the 29th iteration of the optimization algorithm.

FIGURE 6

Optimal controls (Case 2) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7

Cost functional (Case 2) [Colour figure can be viewed at wileyonlinelibrary.com]

Figure 8 demonstrates the solution curves of state variables before and after optimization. The solution curves representing exposed, symptomatic, asymptomatic, super‐spreader, and hospitalized states move to the Corona‐free state. In addition to this, we also experience a significant drop of the hospitalized individuals with this case. However, we need to quarantine more individuals in the beginning in order to get optimal reduction in infected individuals.

FIGURE 8

State variables before and after optimization (Case 2) [Colour figure can be viewed at wileyonlinelibrary.com]

OPTIMAL CONTROL STRATEGY II

The second strategy to control Covid‐19 is to tell people about Covid‐19 effects, self‐protection, and medication of hospitalized individuals. By educating susceptible about health risks, effects, and safety planes of Covid‐19, we can restrict the spread of Covid‐19. The habit of wearing mask and the practice of social distancing are the key factors of self‐protection. The best available medicine for the cure of patients with severe condition is also offered to save their lives. To execute this approach, we transform our nonlinear epidemic model (2) to incorporate nonpharmaceutical parameters.

Modified model

Covid‐19 model (2) is restructured to include nonpharmaceutical parameters such as , and . We assume that susceptible people are recovered at rate due to health awareness campaign and the susceptible, exposed, symptomatic, asymptomatic, super‐spreader, and hospitalized get well again at the rate by self‐protection. The hospitalized individuals move in recovered class at rate with possible medication in hospital. With these assumptions, model (2) is modified to give us the following system of ODEs. with the set of initial conditions:

Model (28) will play the role of a set of restrictions for optimal control problem (30) defined in Section 10.2. The flow diagram for model (28) is shown in the Figure 9.

FIGURE 9

Flow diagram of Covid‐19 disease transmission with nonpharmaceutical control parameters [Colour figure can be viewed at wileyonlinelibrary.com]

To achieve the target of controlling disease with nonpharmaceutical parameters, an optimal control problem is designed by devising a cost functional in the ongoing section. In addition, for the optimal solution of the problem, necessary conditions are also computed.

With the new strategy, the cost functional is defined as follows: where is final time, represent time‐dependent controls, represent state variables, and costs associated with controls are .

As our goal is to find optimal control such that functional (29) is minimized, that is,

is the space of controls specified as follows:

To build up the necessary conditions for optimal control problem (30), we implement Pontryagin's maximum principle. The optimality conditions are derived from Hamiltonian defined as follows: where , symbolize the state variables, related adjoint variables are , and are the RHS of system (28). Thus, the Hamiltonian for the optimal control (30) is given below.

The first optimality condition of the Pontryagin maximum principle provides us the equations for control variables and with bound restriction, updated controls are expressed as follows:

The second optimality condition of the Pontryagin maximum principle offers us the system of linear adjoint equations with conditions at final time

Derivative of Hamiltonian with respect to the adjoint variables , lead us to system of state Equation (32).

Optimizer of the optimal control problem (30) is determined by implementing the following algorithm. symbolizes the state variables , adjoint variables , and the control variable . The parameter is the accepted tolerance set as per requirement.

In this section, we present and discuss optimal solutions of the optimal control problem (30). The solutions are obtained by implementing Algorithm 2 through Matlab code. Time space is discretized into equal subintervals each of length with discrete points . Solutions of state and adjoint equations are obtained at discrete points by exploiting RK4 method. Cost functional is approximated by Simpson's one third rule. We categorize here two optimal control problems by considering two different cost functionals each with different controls. For this, we present and discuss the following two cases.

Case 1: For the first case, we consider all the three controls , and in the cost functional (29). Through Algorithm 2, we get graphs of time‐dependent optimal control variables as shown in Figure 10. These are the optimum rates for health information ( ), self‐protection ( ), and medication ( ) that not only minimize the cost functional but also help in reducing the spread of disease.

FIGURE 10

Optimal controls (Case 1) [Colour figure can be viewed at wileyonlinelibrary.com]

The graph of cost functional (29) associated with optimization iterations is shown in Figure 11. The graph demonstrates that cost functional has achieved its minimum value with optimal controls shown in Figure 10.

FIGURE 11

Cost functional (Case 1) [Colour figure can be viewed at wileyonlinelibrary.com]

Figure 12 illustrates graphs of state variables before and after optimization. Before optimization, the state variables are computed with some constant controls. From Figure 12, we observe a considerable reduction in the infected individuals in each of the exposed, symptomatic, asymptomatic, and super‐spreader compartments. In this case, we almost have disease‐free situation after optimization. However, the figure also shows an increase in the recovered people.

FIGURE 12

State variables before and after optimization (Case 1) [Colour figure can be viewed at wileyonlinelibrary.com]

Case 2: In this case, we optimize the optimal control problem (30) by considering only two controls (health information) and (self‐protection). We want to study the effectiveness of the two measures: health information to public and self‐protection (face masking and social distancing), on the control of disease. In this case, we omit the parameter from the model.

The curves for the optimizers of the control problem (30) are shown in Figure 13. Under these controls, the cost functional reduced to its minimum value at the 19th iteration as shown in Figure 14.

FIGURE 13

Optimal controls (Case 2) [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 14

Cost functional (Case 2) [Colour figure can be viewed at wileyonlinelibrary.com]

Behavior of state variables before and after optimization is shown in Figure 15. State variables before optimization are approximated by using some constant controls. From Figure 15, we observe a reasonable reduction in the infected individuals with the strategy of considering two nonpharmaceutical controls in our model. The rise in susceptible and recovered individuals in this case is more as compared with Case 1.

FIGURE 15

State variables before and after optimization (Case 2) [Colour figure can be viewed at wileyonlinelibrary.com]

CONCLUSIONS

In this research work, we designed a nonlinear Covid‐19 model not only to study the dynamics of the disease but also to provide a platform for suggesting nonpharmaceutical control strategies for the control of disease. This model is a transformation of the usual SEIR model with a realistic addition of symptomatic, asymptomatic, super‐spreader, and hospitalization compartments. To study dynamics of the model, we first established that the model has a unique solution and the solutions are nonnegative and bounded. We determined reproduction number as well as the equilibrium points and investigated the local and global stabilities at these points. From sensitivity analysis, we observed that the transmission parameter , rate of transmission of Corona from super‐spreader to susceptible, has the highest sensitivity index to . To reduce the expansion of the infection, two optimal control strategies were also designed in this study. Primarily, a quarantine compartment was adjusted in the already designed Covid‐19 model with an intention to disconnect the symptomatic, asymptomatic, and super‐spreaders people from the rest. With this strategy, we considered two cases and studied there the optimum quarantine and hospitalization rates (controls) that restrain the cost functional to minimum. Subsequently, we have experienced a notable decline in the infected curves with this strategy. For the next approach, the typical Covid‐19 model was rationalized to acquire three nonpharmaceutical control parameters with an intention to control the disease. By taking into consideration of a new cost functional, we defined an optimal control problem such that the newly added nonpharmaceutical parameters served as control variables. With this strategy, we again discussed two cases by considering different controls. The efficiency of the strategy to control virus is witnessed by simulation results. Conclusively, the second strategy is effortlessly applicable, resourceful, and trouble‐free to put into practice.

In both of the strategies presented here, we were able to control the spread of disease by reducing the number of infected individuals in exposed, symptomatically infected, asymptomatically infected, and super‐spreader classes. However, the first strategy looks difficult to implement practically as it requires a reasonable infrastructure and a lot of resources to isolate the huge number of people. The second strategy looks practically possible, as media can play its role in educating people about public health issues as well as about benefits of social distancing and wearing masks.

CONFLICT OF INTEREST

This work does not have any conflicts of interest with respect to research, authorship, and/or publication of this article.