<p>This paper presents a stochastic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathsf {S_{h}V_{h}A_{h}C_{h}H_{h}R_{h}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">h</mi> </msub> <msub> <mi mathvariant="sans-serif">V</mi> <mi mathvariant="sans-serif">h</mi> </msub> <msub> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">h</mi> </msub> <msub> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">h</mi> </msub> <msub> <mi mathvariant="sans-serif">H</mi> <mi mathvariant="sans-serif">h</mi> </msub> <msub> <mi mathvariant="sans-serif">R</mi> <mi mathvariant="sans-serif">h</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> model to analyze the transmission dynamics of Hepatitis B virus (HBV) by incorporating random noise, time delays, and vaccination effects. The perturbations are modeled as linear, assuming that immunity wanes after a certain period post-vaccination. Initially, a deterministic model is developed to calculate the basic reproduction number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_{h0}^D$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <mi>h</mi> <mn>0</mn> </mrow> <mi>D</mi> </msubsup> </math></EquationSource> </InlineEquation>. This deterministic model is then extended to its stochastic counterpart. We prove the existence of a globally bounded and positive solution for the stochastic model, ensuring its mathematical and biological feasibility. Furthermore, we demonstrate that the model is ergodic and possesses a unique stationary distribution. Numerical simulations are performed to illustrate the theoretical findings, showing how stochastic perturbations can significantly influence HBV transmission dynamics. These simulations underscore the importance of effective control measures, such as vaccination and quarantine strategies, to mitigate the spread of HBV. The study provides valuable information for public health policymakers and contributes to designing more effective HBV control programs.</p>

Stochastic modeling for the transmission of hepatitis B virus with multiple time-delays and vaccination effect

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发表日期：

2025年6月25日

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Sayed Murad Ali Shah,
Yufeng Nie,
Anwarud Din,
Abdulwasea Alkhazza,
Alishba Arshad,
Bushra Younas

摘要

This paper presents a stochastic $\mathsf {S_{h}V_{h}A_{h}C_{h}H_{h}R_{h}}$ $S_{h} V_{h} A_{h} C_{h} H_{h} R_{h}$ model to analyze the transmission dynamics of Hepatitis B virus (HBV) by incorporating random noise, time delays, and vaccination effects. The perturbations are modeled as linear, assuming that immunity wanes after a certain period post-vaccination. Initially, a deterministic model is developed to calculate the basic reproduction number $R_{h0}^D$ $R_{h 0}^{D}$ . This deterministic model is then extended to its stochastic counterpart. We prove the existence of a globally bounded and positive solution for the stochastic model, ensuring its mathematical and biological feasibility. Furthermore, we demonstrate that the model is ergodic and possesses a unique stationary distribution. Numerical simulations are performed to illustrate the theoretical findings, showing how stochastic perturbations can significantly influence HBV transmission dynamics. These simulations underscore the importance of effective control measures, such as vaccination and quarantine strategies, to mitigate the spread of HBV. The study provides valuable information for public health policymakers and contributes to designing more effective HBV control programs.