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An SIRS model with a nonlinear incidence rate

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  • Jin, Yu
  • Wang, Wendi
  • Xiao, Shiwu

Abstract

The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.

Suggested Citation

  • Jin, Yu & Wang, Wendi & Xiao, Shiwu, 2007. "An SIRS model with a nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1482-1497.
    • Handle: RePEc:eee:chsofr:v:34:y:2007:i:5:p:1482-1497
    DOI: 10.1016/j.chaos.2006.04.022
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    References listed on IDEAS

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    1. Li, Xiuying & Wang, Wendi, 2005. "A discrete epidemic model with stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 947-958.
    2. Wang, Kaifa & Wang, Wendi & Liu, Xianning, 2006. "Viral infection model with periodic lytic immune response," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 90-99.
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    Cited by:

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    2. Lahrouz, A. & El Mahjour, H. & Settati, A. & Bernoussi, A., 2018. "Dynamics and optimal control of a non-linear epidemic model with relapse and cure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 299-317.
    3. Singh, Anurag & Arquam, Md, 2022. "Epidemiological modeling for COVID-19 spread in India with the effect of testing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 592(C).
    4. Xu, Rui & Ma, Zhien, 2009. "Stability of a delayed SIRS epidemic model with a nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2319-2325.
    5. Wang, Chenxu & Wang, Gaoshuai & Luo, Xiapu & Li, Hui, 2019. "Modeling rumor propagation and mitigation across multiple social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    6. Wanduku, Divine, 2017. "Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 49-76.
    7. Fatima, Bibi & Zaman, Gul, 2020. "Co-infection of Middle Eastern respiratory syndrome coronavirus and pulmonary tuberculosis," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    8. Tipsri, S. & Chinviriyasit, W., 2015. "The effect of time delay on the dynamics of an SEIR model with nonlinear incidence," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 153-172.
    9. Xu, Rui, 2012. "Global dynamics of an SEIS epidemiological model with time delay describing a latent period," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 85(C), pages 90-102.
    10. Rahman, Ghaus ur & Shah, Kamal & Haq, Fazal & Ahmad, Naveed, 2018. "Host vector dynamics of pine wilt disease model with convex incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 113(C), pages 31-39.
    11. Li, Xue-Zhi & Zhou, Lin-Lin, 2009. "Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 874-884.
    12. Zhou, Jiaying & Zhao, Yi & Ye, Yong, 2022. "Complex dynamics and control strategies of SEIR heterogeneous network model with saturated treatment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 608(P2).
    13. Buonomo, Bruno & Giacobbe, Andrea, 2023. "Oscillations in SIR behavioural epidemic models: The interplay between behaviour and overexposure to infection," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    14. Lanconelli, Alberto & Perçin, Berk Tan, 2022. "On a new method for the stochastic perturbation of the disease transmission coefficient in SIS models," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    15. Florin Avram & Rim Adenane & Lasko Basnarkov & Matthew D. Johnston, 2023. "Algorithmic Approach for a Unique Definition of the Next-Generation Matrix," Mathematics, MDPI, vol. 12(1), pages 1-40, December.

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