Analysis of a stochastic SEI u I r R epidemic model incorporating the Ornstein-Uhlenbeck process

Mhammed Mediani; Abdeldjalil Slama; Ahmed Boudaoui; Thabet Abdeljawad

doi:10.1016/j.heliyon.2024.e35749

Analysis of a stochastic SEI _u I _r R epidemic model incorporating the Ornstein-Uhlenbeck process

Heliyon. 2024 Aug 6;10(16):e35749. doi: 10.1016/j.heliyon.2024.e35749. eCollection 2024 Aug 30.

Authors

Mhammed Mediani¹, Abdeldjalil Slama¹, Ahmed Boudaoui¹, Thabet Abdeljawad^{2

3

4

5}

Affiliations

¹ Laboratory of Mathematics, Modeling and Applications (LaMMA), University of Adrar, Adrar, Algeria.
² Department of Mathematics and Sciences, Prince Sultan University Riyadh, Saudi Arabia.
³ Department of Medical Research, China Medical University, Taichung 40402, Taiwan.
⁴ Center for Applied Mathematics and Bioinformatics (CAMB), Gulf University for Science and Technology, Hawally, 32093, Kuwait.
⁵ Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa.

Abstract

This article aims to analyze a stochastic epidemic model $S E I_{u} I_{r} R$ (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein-Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.

Keywords: Disease extinction; Ornstein–Uhlenbeck process; Stationary distributions; Stochastic epidemic model.