<p>This study explores a novel stochastic SIRS epidemic model that incorporates Lévy noise, capturing both continuous and discontinuous random fluctuations in disease dynamics. Unlike previous research, our proposed model integrates a saturated incidence rate and stochastic perturbations driven by Lévy noise, offering a more realistic depiction of disease spread under environmental noise. We first establish the existence and uniqueness of a globally positive solution to ensure its biological validity. Using stochastic stability theory and ergodic analysis, we derive necessary and sufficient threshold conditions for disease extinction and persistence, characterized by a stochastic basic reproduction number (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_0^{s}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mn>0</mn> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation>). Specifically, we demonstrate that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_0^{s}<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the disease will almost surely die out, whereas if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_0^{s}>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>0</mn> <mi>s</mi> </msubsup> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the system admits a unique stationary distribution, indicating long-term endemic behavior. Furthermore, we explicitly derive the probability density function of the solution near the endemic equilibrium, providing a comprehensive statistical description of epidemic dynamics under Lévy noise. Numerical simulations are performed to validate the theoretical findings and illustrate their practical implications. This work enhances the understanding of stochastic epidemic models by incorporating Lévy noise and offers a robust framework for analyzing infectious disease dynamics under complex environmental influences.</p>

Analysis of a Stochastic SIRS Epidemic Model Driven by Lévy Noise: Stability, Stationary Distribution, and Long-Term Behavior

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发表日期：

2026年4月20日

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M. Akbari,
S. Lamei

摘要

This study explores a novel stochastic SIRS epidemic model that incorporates Lévy noise, capturing both continuous and discontinuous random fluctuations in disease dynamics. Unlike previous research, our proposed model integrates a saturated incidence rate and stochastic perturbations driven by Lévy noise, offering a more realistic depiction of disease spread under environmental noise. We first establish the existence and uniqueness of a globally positive solution to ensure its biological validity. Using stochastic stability theory and ergodic analysis, we derive necessary and sufficient threshold conditions for disease extinction and persistence, characterized by a stochastic basic reproduction number ( $R_0^{s}$ $R_{0}^{s}$ ). Specifically, we demonstrate that if $R_0^{s}<1$ $R_{0}^{s} < 1$ , the disease will almost surely die out, whereas if $R_0^{s}>1$ $R_{0}^{s} > 1$ , the system admits a unique stationary distribution, indicating long-term endemic behavior. Furthermore, we explicitly derive the probability density function of the solution near the endemic equilibrium, providing a comprehensive statistical description of epidemic dynamics under Lévy noise. Numerical simulations are performed to validate the theoretical findings and illustrate their practical implications. This work enhances the understanding of stochastic epidemic models by incorporating Lévy noise and offers a robust framework for analyzing infectious disease dynamics under complex environmental influences.