Stochastic SICA Epidemic Model with Jump L\'{e}vy Processes

TL;DR

This paper introduces a stochastic SICA epidemic model for HIV/AIDS incorporating both Brownian motion and jump Lévy noise, analyzing its mathematical properties and implications for disease extinction and persistence.

Contribution

It extends existing SICA models by including jump Lévy noise, providing a more realistic stochastic framework for HIV/AIDS epidemic modeling.

Findings

Proves existence and uniqueness of solutions

Derives conditions for HIV/AIDS extinction

Provides numerical simulations illustrating results

Abstract

We propose and study a shifted SICA epidemic model, extending the one of Silva and Torres (2017) to the stochastic setting driven by both Brownian motion processes and jump L\'evy noise. L\'evy noise perturbations are usually ignored by existing works of mathematical modelling in epidemiology, but its incorporation into the SICA epidemic model is worth to consider because of the presence of strong fluctuations in HIV/AIDS dynamics, often leading to the emergence of a number of discontinuities in the processes under investigation. Our work is organised as follows: (i) we begin by presenting our model, by clearly justifying its used form, namely the component related to the L\'evy noise; (ii) we prove existence and uniqueness of a global positive solution by constructing a suitable stopping time; (iii) under some assumptions, we show extinction of HIV/AIDS; (iv) we obtain sufficient conditions assuring persistence of HIV/AIDS; (v) we illustrate our mathematical results through numerical simulations.