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

• M. Akbari ^[1] ; S. Lamei ^[1]
  1. [1] University of Guilan

     University of Guilan

     Irán

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 3, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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 (Rs 0). Specifically, we demonstrate that if Rs 0 < 1, the disease will almost surely die out, whereas if Rs 0 > 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.

• Referencias bibliográficas
  • 1. Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press (2008)
  • 2. Anderson, R. M., May, R. M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press (1991)
  • 3. Allen, L. J. S.: An Introduction to Stochastic Processes with Applications to Biology. Pearson Education (2010)
  • 4. Allen, L. J. S.: Stochastic Population and Epidemic Models: Persistence and Extinction. Springer (2015)
  • 5. Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing (2007)
  • 6. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer (2003)
  • 7. Gray, A., Greenhalgh, D., Hu, L., Mao, X., Pan, J.: A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71(3), 876–902...
  • 8. Cai, Y., Kang, Y., Banerjee, M., Wang, W.: A stochastic SIRS epidemic model with infectious force under intervention strategies. J. Differential...
  • 9. Hethcote, H.W.: The Mathematics of Infectious Diseases. SIAM Rev. 42(4), 599–653 (2000)
  • 10. Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42(1–2), 43–61 (1978)
  • 11. Applebaum, D.: Levy Processes and Stochastic Calculus. Cambridge University Press (2009)
  • 12. Hu, J.N., Wen, B.Y., Zeng, T., Teng, Z.D.: Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination...
  • 13. Lv, L., Yao, X.J.: Qualitative analysis of a nonautonomous stochastic SIS epidemic model with Lévy jumps. Math. Biosci. Eng. 18, 1352–1369...
  • 14. Kiouach, D., Sabbar, Y.: The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps....
  • 15. Ding, Y.M., Fu, Y.X., Kang, Y.M.: Stochastic analysis of COVID-19 by a SEIR model with Lévy noise. Chaos 31, 043132 (2021)
  • 16. Zhang, X.B., Shi, Q.H., Ma, S.H., Huo, H.F., Li, D.G.: Dynamic behavior of a stochastic SIQS epidemic model with Lévy jumps. Nonlinear...
  • 17. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission....
  • 18. Ma, Z., Zhou, Y., Wu, J.: Modeling and Dynamic of Infectious Disease. Higher Education Press, Beijing (2009)
  • 19. Liu, Q., Chen, Q.: Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Physica A 428, 140–153...
  • 20. Khasminskii, R.: Stochastic stability of differential equations. Springer, Berlin, Heidelberg (2012)
  • 21. Roozen, H.: An asymptotic solution to a two-dimensional exit problem arising in population dynamics. SIAM J. Appl. Math. 49(6), 1793–1810...
  • 22. Tian, X., Ren, C.: Linear equations, superposition principle and complex exponential notation. College Physica 23, 23–5 (2004)
  • 23. Zhang, X., Ma, S.: Dynamics and density function analysis of a stochastic SVI epidemic model with half-saturated incidence rate. Chaos,...