Research article

Dynamical analysis of a network-based SIR model with saturated incidence rate and nonlinear recovery rate: an edge-compartmental approach

  • Received: 28 December 2023 Revised: 12 February 2024 Accepted: 27 February 2024 Published: 14 March 2024
  • A new network-based SIR epidemic model with saturated incidence rate and nonlinear recovery rate is proposed. We adopt an edge-compartmental approach to rewrite the system as a degree-edge-mixed model. The explicit formula of the basic reproduction number R0 is obtained by renewal equation and Laplace transformation. We find that R0<1 is not enough to ensure global asymptotic stability of the disease-free equilibrium, and when R0>1, the system can exist multiple endemic equilibria. When the number of hospital beds is small enough, the system will undergo backward bifurcation at R0=1. Moreover, it is proved that the stability of feasible endemic equilibrium is determined by signs of tangent slopes of the epidemic curve. Finally, the theoretical results are verified by numerical simulations. This study suggests that maintaining sufficient hospital beds is crucial for the control of infectious diseases.

    Citation: Fang Wang, Juping Zhang, Maoxing Liu. Dynamical analysis of a network-based SIR model with saturated incidence rate and nonlinear recovery rate: an edge-compartmental approach[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5430-5445. doi: 10.3934/mbe.2024239

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  • A new network-based SIR epidemic model with saturated incidence rate and nonlinear recovery rate is proposed. We adopt an edge-compartmental approach to rewrite the system as a degree-edge-mixed model. The explicit formula of the basic reproduction number R0 is obtained by renewal equation and Laplace transformation. We find that R0<1 is not enough to ensure global asymptotic stability of the disease-free equilibrium, and when R0>1, the system can exist multiple endemic equilibria. When the number of hospital beds is small enough, the system will undergo backward bifurcation at R0=1. Moreover, it is proved that the stability of feasible endemic equilibrium is determined by signs of tangent slopes of the epidemic curve. Finally, the theoretical results are verified by numerical simulations. This study suggests that maintaining sufficient hospital beds is crucial for the control of infectious diseases.





    [1] V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61. https://doi.org/10.1016/0025-5564(78)90006-8 doi: 10.1016/0025-5564(78)90006-8
    [2] R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991. https://doi.org/10.1093/oso/9780198545996.001.0001
    [3] W. Wang, S. Ruan, Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2015), 775–793. https://doi.org/10.1016/j.jmaa.2003.11.043 doi: 10.1016/j.jmaa.2003.11.043
    [4] J. Eckalbar, W. Eckalbar, Dynamics of an epidemic model with quadratic treatment, Nonlinear Anal. Real., 12 (2011), 320–332. https://doi.org/10.1016/j.nonrwa.2010.06.018 doi: 10.1016/j.nonrwa.2010.06.018
    [5] C. Shan, H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differ. Equations, 257 (2014), 1662–1688. https://doi.org/10.1016/j.jde.2014.05.030 doi: 10.1016/j.jde.2014.05.030
    [6] Q. Cui, Z. Qiu, W. Liu, Z. Hu, Complex dynamics of an SIR epidemic model with nonlinear saturate incidence and recovery rate, Entropy, 19 (2017), 305–320. https://doi.org/10.3390/e19070305 doi: 10.3390/e19070305
    [7] C. Li, A. M. Yousef, Bifurcation analysis of a network-based SIR epidemic model with saturated treatment function, Chaos, 29 (2019), 033129. https://doi.org/10.1063/1.5079631 doi: 10.1063/1.5079631
    [8] Y. Huang, C. Li, Backward bifurcation and stability analysis of a network-based SIS epidemic model with saturated treatment function, Phys. A, 527 (2019), 121407. https://doi.org/10.1016/j.physa.2019.121407 doi: 10.1016/j.physa.2019.121407
    [9] X. Wang, J. Yang, Dynamical analysis of a mean-field vector-borne diseases model on complex networks: An edge based compartmental approach, Chaos, 30 (2020), 013103. https://doi.org/10.1063/1.5116209 doi: 10.1063/1.5116209
    [10] X. Wang, J. Yang, A bistable phenomena induced by a mean-field SIS epidemic model on complex networks: A geometric approach, Front. Phys., 9 (2021), 681268. https://doi.org/10.3389/fphy.2021.681268 doi: 10.3389/fphy.2021.681268
    [11] J. Yang, X. Duan, X. Li, Complex patterns of an SIR model with a saturation treatment on complex networks: An edge-compartmental approach, Appl. Math. Lett., 123 (2022), 107573. https://doi.org/10.1016/j.aml.2021.107573 doi: 10.1016/j.aml.2021.107573
    [12] J. Jiang, T. Zhou, The influence of time delay on epidemic spreading under limited resources, Phys. A, 508 (2018), 414–423. https://doi.org/10.1016/j.physa.2018.05.114 doi: 10.1016/j.physa.2018.05.114
    [13] R. Rao, Z. Lin, X. Ai, J. Wu, Synchronization of epidemic systems with neumann boundary value under delayed impulse, Mathematics, 10 (2022), 2064. https://doi.org/10.3390/math10122064 doi: 10.3390/math10122064
    [14] Y. Zhao, L. Wang, Practical exponential stability of impulsive stochastic food chain system with time-varying delays, Mathematics, 11 (2023), 147. https://doi.org/10.3390/math11010147 doi: 10.3390/math11010147
    [15] B. Wang, Q. Zhu, S. Li, Stability analysis of discrete-time semi-Markov jump linear systems with time delay, IEEE. Trans. Autom. Control, 68 (2023), 6758–6765. https://doi.org/10.1109/TAC.2023.3240926 doi: 10.1109/TAC.2023.3240926
    [16] H. Xu, Q. Zhu, W. Zheng, Exponential stability of stochastic nonlinear delay systems subject to multiple periodic impulses, IEEE Trans. Autom. Control, (2023), 1–8. https://doi.org/10.1109/TAC.2023.3335005 doi: 10.1109/TAC.2023.3335005
    [17] L. Fan, Q. Zhu, W. Zheng, Stability analysis of switched stochastic nonlinear systems with state-dependent delay, IEEE Trans. Autom. Control, (2023), 1–8. https://doi.org/10.1109/TAC.2023.3315672 doi: 10.1109/TAC.2023.3315672
    [18] Q. Yang, X. Wang, X. Cheng, B. Du, Y. Zhao, Positive periodic solution for neutral-type integral differential equation arising in epidemic model, Mathematics, 11 (2023), 2701. https://doi.org/10.3390/math11122701 doi: 10.3390/math11122701
    [19] J. Yang, F. Xu, The computational approach for the basic reproduction number of epidemic models on complex networks, IEEE Access, 7 (2019), 26474–26479. https://doi.org/10.1109/ACCESS.2019.2898639 doi: 10.1109/ACCESS.2019.2898639
    [20] A. Abdelrazec, J. Belair, C. Shan, H. Zhu, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016), 136–145. https://doi.org/10.1016/j.mbs.2015.11.004 doi: 10.1016/j.mbs.2015.11.004
    [21] G. Li, Y. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, PLoS ONE, 12 (2017), e0175789. https://doi.org/10.1371/journal.pone.0175789 doi: 10.1371/journal.pone.0175789
    [22] G. Lan, S. Yuan, B. Song, The impact of hospital resources and environmental perturbations to the dynamics of SIRS model, J. Franklin Inst., 358 (2021), 2405–2433. https://doi.org/10.1016/j.jfranklin.2021.01.015 doi: 10.1016/j.jfranklin.2021.01.015
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