Mathematical Analysis, Forecasting and Optimal Control of HIV/AIDS Spatiotemporal Transmission with a Reaction Diffusion SICA Model

TL;DR

This paper develops a mathematical spatiotemporal SICA model for HIV/AIDS transmission, incorporating diffusion and control strategies, with theoretical analysis, numerical simulations, and optimal control to minimize infection and treatment costs.

Contribution

It introduces a novel reaction-diffusion SICA model with proven existence and uniqueness of solutions, and establishes optimal control strategies for epidemic management.

Findings

Existence and uniqueness of global positive solutions proven.

Numerical simulations illustrate model behavior.

Optimal control reduces infection density and treatment costs.

Abstract

We propose a mathematical spatiotemporal epidemic SICA model with a control strategy. The spatial behavior is modeled by adding a diffusion term with the Laplace operator, which is justified and interpreted both mathematically and physically. By applying semigroup theory on the ordinary differential equations, we prove existence and uniqueness of the global positive spatiotemporal solution for our proposed system and some of its important characteristics. Some illustrative numerical simulations are carried out that motivate us to consider optimal control theory. A suitable optimal control problem is then posed and investigated. Using an effective method based on some properties within the weak topology, we prove existence of an optimal control and develop an appropriate set of necessary optimality conditions to find the optimal control pair that minimizes the density of infected individuals and the cost of the treatment program.