A stochastic SICR model for hepatitis C virus transmission with Ornstein-Uhlenbeck process

Shuping He; Yuanlin Ma; Shuping He; Yuanlin Ma

doi:10.3934/math.2026603

AIMS Mathematics

2026, Volume 11, Issue 5: 14693-14714. doi: 10.3934/math.2026603

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Research article

A stochastic SICR model for hepatitis C virus transmission with Ornstein-Uhlenbeck process

Shuping He ,
Yuanlin Ma ^,

School of Economics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China

Received: 08 April 2026 Revised: 11 May 2026 Accepted: 19 May 2026 Published: 25 May 2026
MSC : 34F05, 37H10, 60J70, 92B05

This paper proposes a stochastic susceptible-acute-chronic-recovery (SICR) model for hepatitis C virus (HCV) transmission, in which environmental fluctuations in the transmission rate are modeled by a logarithmic Ornstein-Uhlenbeck process. This formulation preserves the positivity of parameters while capturing the mean-reverting stochasticity. A sufficient condition for disease extinction is established in terms of a stochastic threshold $ \mathcal{R}_0^e $: if $ \mathcal{R}_0^e < 1 $, then the infection becomes extinct almost surely. Conversely, when $ \mathcal{R}_0^s > 1 $, the model admits at least one ergodic stationary distribution, which implies a long-term persistence of the disease. Furthermore, under the condition $ \mathcal{R}_0 > 1 $, an explicit probability density function for the stationary distribution near the quasi-endemic equilibrium is derived, thus providing a characterization of stationary fluctuations. Numerical simulations are conducted to validate the theoretical findings and to elucidate the influence of noise parameters on the disease dynamics. The results demonstrate that environmental fluctuations can sustain HCV transmission even when the deterministic reproduction number is below unity, thus underscoring the critical role of stochasticity in HCV transmission.
- hepatitis C virus,
- stochastic epidemic model,
- Ornstein-Uhlenbeck process,
- extinction,
- stationary distribution,
- probability density function
Citation: Shuping He, Yuanlin Ma. A stochastic SICR model for hepatitis C virus transmission with Ornstein-Uhlenbeck process[J]. AIMS Mathematics, 2026, 11(5): 14693-14714. doi: 10.3934/math.2026603

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Abstract

This paper proposes a stochastic susceptible-acute-chronic-recovery (SICR) model for hepatitis C virus (HCV) transmission, in which environmental fluctuations in the transmission rate are modeled by a logarithmic Ornstein-Uhlenbeck process. This formulation preserves the positivity of parameters while capturing the mean-reverting stochasticity. A sufficient condition for disease extinction is established in terms of a stochastic threshold $ \mathcal{R}_0^e $: if $ \mathcal{R}_0^e < 1 $, then the infection becomes extinct almost surely. Conversely, when $ \mathcal{R}_0^s > 1 $, the model admits at least one ergodic stationary distribution, which implies a long-term persistence of the disease. Furthermore, under the condition $ \mathcal{R}_0 > 1 $, an explicit probability density function for the stationary distribution near the quasi-endemic equilibrium is derived, thus providing a characterization of stationary fluctuations. Numerical simulations are conducted to validate the theoretical findings and to elucidate the influence of noise parameters on the disease dynamics. The results demonstrate that environmental fluctuations can sustain HCV transmission even when the deterministic reproduction number is below unity, thus underscoring the critical role of stochasticity in HCV transmission.

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