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Analysis of a stochastic epidemic model driven by bilinear incidence rate with two different transmission mechanisms

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Abstract

Study on double epidemics is attracting the attention of researchers in recent times to make models that are close to reality. In this paper, an epidemic model with two transmission mechanisms, Susceptible-Infected-Recovered and Susceptible-Infected-Recovered-Susceptible, is considered. The model is stochastically perturbed at transmission rates and the resulting stochastic model is examined for the existence of a positive global solution. Further, extinction, persistence and co-persistence results are studied with suitable sufficient conditions. Finally, some numerical examples are presented to support the theoretical results established.

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References

  1. Martcheva, M. 2015. An Introduction to Mathematical Epidemiology, vol. 61. Berlin: Springer.

    Book  Google Scholar 

  2. Cui, J., Y. Sun, and H. Zhu. 2008. The impact of media on the control of infectious diseases. Journal of Dynamics and Differential Equations 20: 31–53.

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  3. Hattaf, K., N. Yousfi, and A. Tridane. 2012. Mathematical virus dynamics model with general incidence rate and cure rate. Nonlinear Analysis Real World Applications 13: 1866–1872. https://doi.org/10.1016/j.nonrwa.2011.12.015.

    Article  MathSciNet  Google Scholar 

  4. Lahrouz, A., L. Omari, D. Kiouach, and A. Belmaâti. 2012. Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination. Applied Mathematics and Computation 218(11): 6519–6525.

    Article  MathSciNet  Google Scholar 

  5. Li, G., and J. Zhen. 2005. Global stability of an SEI epidemic model with general contact rate. Chaos, Solitons and Fractals 23: 997–1004.

    ADS  MathSciNet  Google Scholar 

  6. M’Kendrick, A. 1925. Applications of mathematics to medical problems. Proceedings of the Edinburgh Mathematical Society 44: 98–130.

    Article  Google Scholar 

  7. Jacquez, J.A. 1987. A note on chain-binomial models of epidemic spread: What is wrong with the reed-frost formulation? Mathematical Biosciences 87(1): 73–82.

    Article  MathSciNet  Google Scholar 

  8. Bartlett, M. 1949. Some evolutionary stochastic processes. Journal of the Royal Statistical Society. Series B (Methodological) 11(2): 211–229.

    Article  MathSciNet  Google Scholar 

  9. Meng, X., Z. Li, and X. Wang. 2010. Dynamics of a novel nonlinear SIR model with double epidemic hypothesis and impulsive effects. Nonlinear Dynamics 59(3): 503–513.

    Article  MathSciNet  PubMed  Google Scholar 

  10. Abbas, S., and M. Banerjee. 2010. Stochastically perturbed allelopathic phytoplankton model. Electronic Journal of Differential Equations (EJDE) 98: 1–15.

    Google Scholar 

  11. Miao, A., X. Wang, T. Zhang, W. Wang, and B. Sampath Aruna Pradeep. 2017. Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. Advances in Difference Equations 1: 1–27.

    MathSciNet  Google Scholar 

  12. Qi, H., L. Liu, and X. Meng. 2017. Dynamics of a nonautonomous stochastic SIS epidemic model with double epidemic hypothesis. Complexity 2017: 4861391.

    Article  MathSciNet  Google Scholar 

  13. Zhang, X., D. Jiang, T. Hayat, and B. Ahmad. 2017. Dynamics of a stochastic SIS model with double epidemic diseases driven by Lévy jumps. Physica A: Statistical Mechanics and its Applications 471: 767–777.

    Article  ADS  MathSciNet  Google Scholar 

  14. Abbas, S., M. Niezabitowski, and S.R. Grace. 2021. Global existence and stability of Nicholson blowflies model with harvesting and random effect. Nonlinear Dynamics 103: 2109–2123.

    Article  Google Scholar 

  15. Ng, T.W., G. Turinici, and A. Danchin. 2003. A double epidemic model for the SARS propagation. BMC Infectious Diseases 3(1): 1–16.

    Article  Google Scholar 

  16. Chang, Z., X. Meng, and X. Lu. 2017. Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates. Physica A: Statistical Mechanics and its Applications 472: 103–116.

    Article  ADS  MathSciNet  Google Scholar 

  17. Meng, X., S. Zhao, T. Feng, and T. Zhang. 2016. Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. Journal of Mathematical Analysis and Applications 433(1): 227–242.

    Article  MathSciNet  Google Scholar 

  18. Boukanjime, B., M. El Fatini, A. Laaribi, and R. Taki. 2019. Analysis of a deterministic and a stochastic epidemic model with two distinct epidemics hypothesis. Physica A: Statistical Mechanics and its Applications 534: 122321.

    Article  MathSciNet  Google Scholar 

  19. Selvan, T.T., and M. Kumar. 2023. Dynamics of a deterministic and a stochastic epidemic model combined with two distinct transmission mechanisms and saturated incidence rate. Physica A: Statistical Mechanics and its Applications 619: 128741.

    Article  MathSciNet  Google Scholar 

  20. Mao, X. 2007. Stochastic Differential Equations and Applications. Amsterdam: Elsevier.

    Google Scholar 

  21. Liptser, R.S. 1980. A strong law of large numbers for local martingales. Stochastics 3(1–4): 217–228.

    Article  MathSciNet  Google Scholar 

  22. Ji, C., and D. Jiang. 2014. Threshold behaviour of a stochastic SIR model. Applied Mathematical Modelling 38(21–22): 5067–5079.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, NIT Calicut, Kozhikode, Kerala, 673601, India

    T. Tamil Selvan & M. Kumar

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  1. T. Tamil Selvan
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  2. M. Kumar
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All the authors contributed equally to the preparation of the present paper.

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Correspondence to M. Kumar.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by S. Ponnusamy.

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Tamil Selvan, T., Kumar, M. Analysis of a stochastic epidemic model driven by bilinear incidence rate with two different transmission mechanisms. J Anal 32, 509–527 (2024). https://doi.org/10.1007/s41478-023-00665-5

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  • DOI: https://doi.org/10.1007/s41478-023-00665-5

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