<p>This paper considers the transmission characteristics of the virus and focuses on the dynamic properties of a stochastic HIV model with general incidence rate, accounting for virus-to-cell infection, cell-to-cell transmission, and immune impairment. The model incorporates both productively infected cells and latently infected cells, with the contact rates governed by two mean-reverting Ornstein–Uhlenbeck processes. Under certain assumptions, we first prove that the stochastic system has a unique positive global solution. We further explore the asymptotic behavior of the solution, particularly near the equilibrium points of the corresponding deterministic model. By constructing suitable Lyapunov functions, the existence of stationary distribution is confirmed when <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10160_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}_{0}^{S}>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">R</mi> <mrow> <mn>0</mn> </mrow> <mi>S</mi> </msubsup> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Biologically, stationary distribution indicates that HIV infection can persist for an extended period within the host. Furthermore, it is proved that the virus can be eliminated when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10160_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {R}}_{0}^{E}<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">R</mi> <mrow> <mn>0</mn> </mrow> <mi>E</mi> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Notably, we calculate the exact expression for the probability density function near the quasi-endemic equilibrium via solving the corresponding Fokker–Planck equation, which reflects the statistical properties of the stochastic system. Finally, numerical simulations validate our theoretical results, and we investigate the effect of stochastic perturbations on the model behavior and give the sensitivity index of each parameter to virus propagation.</p>

Understanding the Dynamics of Latent Viral Infection: A Stochastic HIV Model with General Incidence Rate, Cell-To-Cell Transmission, Immune Impairment, and Ornstein–Uhlenbeck Process

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发表日期：

2025年4月23日

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Xinhong Zhang,
Xinxin Su

摘要

This paper considers the transmission characteristics of the virus and focuses on the dynamic properties of a stochastic HIV model with general incidence rate, accounting for virus-to-cell infection, cell-to-cell transmission, and immune impairment. The model incorporates both productively infected cells and latently infected cells, with the contact rates governed by two mean-reverting Ornstein–Uhlenbeck processes. Under certain assumptions, we first prove that the stochastic system has a unique positive global solution. We further explore the asymptotic behavior of the solution, particularly near the equilibrium points of the corresponding deterministic model. By constructing suitable Lyapunov functions, the existence of stationary distribution is confirmed when ${\mathcal {R}}_{0}^{S}>1$ $R_{0}^{S} > 1$ . Biologically, stationary distribution indicates that HIV infection can persist for an extended period within the host. Furthermore, it is proved that the virus can be eliminated when ${\mathcal {R}}_{0}^{E}<1$ $R_{0}^{E} < 1$ . Notably, we calculate the exact expression for the probability density function near the quasi-endemic equilibrium via solving the corresponding Fokker–Planck equation, which reflects the statistical properties of the stochastic system. Finally, numerical simulations validate our theoretical results, and we investigate the effect of stochastic perturbations on the model behavior and give the sensitivity index of each parameter to virus propagation.