Dynamical behaviors of a stochastic HIV epidemic model with different infection stages driven by Ornstein-Uhlenbeck process

Wenhe Li; Kang Wang; Huili Wei; Wenhe Li; Kang Wang; Huili Wei

doi:10.3934/era.2026256

Electronic Research Archive

2026, Volume 34, Issue 8: 5758-5794. doi: 10.3934/era.2026256

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Research article

Dynamical behaviors of a stochastic HIV epidemic model with different infection stages driven by Ornstein-Uhlenbeck process

School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, China

Received: 02 March 2026 Revised: 10 May 2026 Accepted: 10 June 2026 Published: 06 July 2026

This paper presents a stochastic human immunodeficiency vrius transmission model driven by a log-normal Ornstein–Uhlenbeck process, designed to capture the stochastic fluctuations in the transmission rates across multiple stages of infection while incorporating co-infection characteristics. First, by constructing suitable Lyapunov functions, the existence and uniqueness of the global positive solution to the model are proved. Subsequently, it is shown that the system admits a stationary distribution when $ R_0^S > 1 $; under this condition, an analytical expression for the probability density function around the quasi-equilibrium state is derived. Furthermore, a sufficient condition for disease extinction is established, namely $ R_0^E < 1 $. Finally, the theoretical results are validated through numerical simulations, and the impacts of the stochastic process's parameters on disease transmission are investigated. The results indicate that a smaller reversion speed and a larger fluctuation intensity of the Ornstein–Uhlenbeck process tend to destabilize the epidemic system.
- stochastic HIV epidemic model,
- log-normal Ornstein–Uhlenbeck process,
- stationary distribution,
- probability density function,
- extinction
Citation: Wenhe Li, Kang Wang, Huili Wei. Dynamical behaviors of a stochastic HIV epidemic model with different infection stages driven by Ornstein-Uhlenbeck process[J]. Electronic Research Archive, 2026, 34(8): 5758-5794. doi: 10.3934/era.2026256

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Abstract

This paper presents a stochastic human immunodeficiency vrius transmission model driven by a log-normal Ornstein–Uhlenbeck process, designed to capture the stochastic fluctuations in the transmission rates across multiple stages of infection while incorporating co-infection characteristics. First, by constructing suitable Lyapunov functions, the existence and uniqueness of the global positive solution to the model are proved. Subsequently, it is shown that the system admits a stationary distribution when $ R_0^S > 1 $; under this condition, an analytical expression for the probability density function around the quasi-equilibrium state is derived. Furthermore, a sufficient condition for disease extinction is established, namely $ R_0^E < 1 $. Finally, the theoretical results are validated through numerical simulations, and the impacts of the stochastic process's parameters on disease transmission are investigated. The results indicate that a smaller reversion speed and a larger fluctuation intensity of the Ornstein–Uhlenbeck process tend to destabilize the epidemic system.

References

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