Fractional Dynamics of Infectious Disease Transmission with Optimal Control

Abstract

This article investigates and studies the dynamics of infectious disease transmission using a fractional mathematical model based on Caputo fractional derivatives‎. ‎Consequently‎, ‎the population studied has been divided into four categories‎: ‎susceptible‎, ‎exposed‎, ‎infected‎, ‎and recovered. The basic reproduction rate‎, ‎existence‎, ‎and uniqueness of disease-free as well as infected steady-state‎ equilibrium points of the mathematical model have been investigated in this study‎. ‎The local and global stability of both equilibrium points has‎ been investigated and proven by Lyapunov functions‎. ‎Vaccination and drug therapy are two controllers that may be used to control the spread of diseases in society‎, ‎and the conditions for the optimal use of these two controllers have been prescribed by the principle of Pontryagin's maximum. The stated theoretical results have been investigated using numerical simulation‎. ‎The‎ numerical simulation of the fractional optimal control problem indicates that vaccination of the susceptible subjects in the community reduces‎‎horizontal transmission while applying drug control to the infected subjects reduces vertical transmission‎. ‎Furthermore‎, ‎the simultaneous use of‎ both controllers is much more effective and leads to a rapid increase in the cured population and it prevents the disease from spreading and‎ turning into an epidemic in the community‎.

Keywords

• fractional calculus‎
• ‎infectious disease‎
• ‎optimal control‎