Abstract
In this paper, we attempt to determine the optimal duration of an anti-epidemic control strategy which targets susceptible people, under the isoperimetric condition that we could not control all individuals of this category due to restricted health resources. We state and prove the local and global stability conditions of free and endemic equilibria of a simple epidemic compartmental model devised in the form of four ordinary differential equations which describe the dynamics of susceptible-controlled-infected-removed populations and where it is taken into account that the controlled people cannot acquire long-lived immunity to move towards the removed compartment due to the temporary effect of the control parameter. Thereafter, we characterize the sought optimal control and we show the effectiveness of this limited control policy along with the research of the optimal duration that is needed to reduce the size of the infected population. The isoperimetric constraint is defined over a fixed horizon, while the objective function is defined over a free horizon present under a quadratic form in the payoff term. The complexity of this optimal control problem requires the execution of three numerical methods all combined together at the same time, namely, the forward–backward sweep method to generate the optimal state and control functions, the secant method adapted to the isoperimetric restriction, and, finally, the fixed point method to obtain the optimal final time.
Highlights
BackgroundMany epidemiological models have been interested in the study of the dynamics of susceptible, infected and removed individuals who belong to a sample of a population threatened by an infection
We study the local and global stability of our epidemic model, which is devised in the form of four ordinary differential equations, and wherein a control is introduced as a constant parameter; and, second, we seek the optimal duration needed for reaching the goal of our strategy while determining the optimal value of this control when it is changed to a function of time under the hypothesis that this anti-epidemic preventive measure can reach only a specific fraction of susceptible people due to the limited health resources
We have determined the optimal duration needed for controlling an epidemic based on a free horizon optimal control approach with an isoperimetric constraint and which has been applied to a four-compartmental epidemic model where it is supposed that the controlled population does not reach the removed class due to the temporary effect of the control
Summary
Many epidemiological models have been interested in the study of the dynamics of susceptible, infected and removed individuals who belong to a sample of a population threatened by an infection. Many theoretical models in epidemiology have been devised to show the effect of different anti-epidemic control strategies when they are followed to prevent transmission of a particular type of infection to the susceptible population. As examples of these control approaches, we can cite Refs. Other examples of control models have used vaccination of the susceptible individuals as a control policy, while considering the number of vaccinated people as additional compartment in their systems (see [3,4,5,6,7]). The present paper tries to find the optimal value of this mentioned variable through a free horizon optimal control approach applied to an epidemic model with four compartments, namely classes of susceptible, controlled, infected and removed people
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