Optimal control strategies of a fractional order model for Zika virus infection involving various transmissions

Abstract

According to the world health organization (WHO), more than 17% of infectious diseases represented in vector-borne diseases (VBDs). So, more than 700,000 deaths annually caused by the spread of these diseases. Basically, VBDs are a usually used term that characterizes a disease transmitted to a host population, by vectors like mosquitoes, insects and ticks through feeding activity. Being a vector means that it carries a disease from an infective to a healthy host. In this paper, we address the fractional mathematical model which describes the transmission dynamics for one of the VBDs, that is Zika virus. We incorporated various combinations between four controls to reduce the transmission of this disease, the measures taken to reduce the contact among vectors and humans such as asleep under a bed net, wearing full clothes, stay in places with a screen window to keep the vector outside; the efforts to reduce the transmission rate from vectors to humans by increasing the autoimmunity; the efforts made to change sexual habits among a susceptible and infected human by using a condom at the period of Zika virus outbreak; and the measures taken to increase the death rate of vectors by using the insecticide spraying. The stability of all equilibrium points (EPs) of the fractional order model (FOM) is investigated. The control reproduction number R c of the FOM is computed and the effect of the suggested controls on the behavior of R c are presented graphically. Moreover, we characterized a fractional optimal control problem (FOCP) as well as used Pontryagin’s maximum principle to derive a fractional order necessary optimality conditions (NOCs). We solved these NOCs numerically by developing the forward-backward sweep method (FBSM) using the predictor-corrector method (PCM), where the state and adjoint equations in the form of the left Caputo fractional derivative (CFD). Furthermore, the outcomes are discussed with some figures and the approximate values for the cost functional are given based on the proposed strategies. Comparing all strategies with the uncontrolled case in order to choose the best strategy for reducing the spread of Zika virus infection.