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; _{1}(_{2}(_{3}(_{4}(_{1}) and _{5}(_{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.

The novel coronavirus pneumonia which was officially named as Corona Virus Disease 2019 (COVID-19) by World Health Organization (WHO) was reported first in late December 2019, in Wuhan, China [

As scientists all over the world are busy trying to develop a cure and vaccine, all hands must be put together to support and comply with the standard recommendations that can lower the transmissions of the disease. This is why, the following measures must be taken; social distancing, self-isolation, use of personal protective equipment (such as face mask, hand globes, overall gown, etc.), regular hand washing using soap or sanitizer, avoid having contact with person showing the symptoms and report any suspected case. Moreover, relevant authorities must engage in widely public orientation exercise for sensitization and enlightenment, banning of social (or religious) gathering and local (or international) trip, contact tracing and isolation of infected individuals, providing sanitizers at public domains like markets and car parks, fumigating exercise, and to the large extent imposing lockdown.

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 [

Tahir et al. [

Yang et al. [

Chen et al. [_{0}. Elhia et al. [

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 [

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.

Definition 1 (Optimal Control) [

Subject to:

where

_{f}

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

such that

Theorem 1 (Pontryagin Maximum Principle): If

such that, for almost all

and such that at terminal time _{f}

If the functions

Remark 1: Becerra states that for a minimum, it is necessary for the stationary (optimality) condition to give:

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 (_{1}), the quarantined susceptible people (by close contacts tracked measure) (_{2}) and general isolated susceptible people (due to community lockdown) (_{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 (_{1}) and infected people that are quarantined (_{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

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 (_{1} +_{2}), by the close contacts tracked measure, susceptible people (_{1}) and infected people (_{2}) are quarantined by the proposition b. Here the number of quarantined susceptible people (_{1}) and quarantined infected people (_{2}) are less than the number of susceptible people (_{1}) and infected people (_{1}) respectively. Then

After the isolation of 14 days _{2}). After the time of treatment

Here the detail formulation and analysis of the optimal control problem with respect to the model

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

_{1}.

_{2}.

The dynamics of control system can be described by the following system of nonlinear ODE;

For a fixed terminal time _{f}

where,

We seek for optimal control ^{*} such that

The system of nonlinear ODE

where,

Theorem 3: The optimal control system

Proof

To formulate the optimal control strategy, we define the Hamiltonian as:

Theorem 4: Let _{1}, _{2}, _{3}, _{4}, _{5}, then there exists a co-state variable satisfying:

Proof:

Applying the co-state (adjoint) condition of

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

Now by substituting

In this chapter, two equilibrium points; Disease Free and Endemic Equilibria are found. Basic reproduction ratio is obtained. Global stability analyses of the equilibrium solutions are carried out.

Since there does not appear the state variable

Disease free equilibrium _{0} is obtained by substituting _{1} = _{2} =

The endemic equilibrium

Since the endemic equilibrium is positive, then

Also

We construct the Jacobian matrix from

Theorem 5: The disease free equilibrium _{0} is locally asymptotically stable.

Proof:

The eigenvalue is obtained from;

For the DFE to be locally asymptotically stable, the eigenvalue _{4} must be negative. That is:

Now, define the basic reproduction ratio (_{0}) to be:

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

Also

Clearly,

Theorem 7: The endemic equilibrium is globally asymptotically stable.

Proof:

Let the Lyapunov candidate function be,

Clearly,

Also

Clearly,

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 [

Model variables | Descriptions | Mean value |
---|---|---|

Total population | ||

Initial susceptible population | ||

Initial quarantine susceptible population | 3762 | |

Initial isolated susceptible population | ||

Initial infected population | 4101 | |

Initial quarantine infected population | 700 | |

Initial hospitalized population | 3886 | |

Initial removed individuals | 64 |

Model parameters | Description | Value |
---|---|---|

time of isolation at home for susceptible population | 1/14 | |

Transfer rate from isolated susceptible population | ||

Transmission rate of COVID-19 | 0.3567 | |

Transfer rate from susceptible to isolated susceptible population | ||

Hospitalization rate of infected population | 0.1429 | |

Discharge rate from hospital | 0.0949 | |

Isolation coefficient | 12 |

It can be seen from

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

We thank the reviewers for their valuable contributions.