Bifurcation analysis of an SIS model with a modified nonlinear incidence rate

Jianzhi Cao; Xuhong Ji; Pengmiao Hao; Peiguang Wang; Jianzhi Cao; Xuhong Ji; Pengmiao Hao; Peiguang Wang

doi:10.3934/era.2025173

Electronic Research Archive

2025, Volume 33, Issue 6: 3901-3930. doi: 10.3934/era.2025173

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Research article

Bifurcation analysis of an SIS model with a modified nonlinear incidence rate

Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China

Received: 01 April 2025 Revised: 22 May 2025 Accepted: 03 June 2025 Published: 23 June 2025

A modified nonlinear incidence rate in an SIS epidemic model was investigated. When a new disease emerged or an old one resurged, the infectivity was initially high. Subsequently, the psychological effect attenuated the infectivity. Eventually, due to the crowding effect, the infectivity reached a saturation state. The nonlinearity of the functional form of the infection incidence was modified to enhance its biological plausibility. The stability of the associated equilibria was examined, and the basic reproduction number and the critical value that governed the dynamics of the model were deduced. Bifurcation analyses were presented, encompassing backward bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and Hopf bifurcation. Numerical simulations were conducted to validate our findings.
- SIS model,
- stability,
- saddle-node bifurcation,
- Bogdanov-Takens bifurcation,
- Hopf bifurcation
Citation: Jianzhi Cao, Xuhong Ji, Pengmiao Hao, Peiguang Wang. Bifurcation analysis of an SIS model with a modified nonlinear incidence rate[J]. Electronic Research Archive, 2025, 33(6): 3901-3930. doi: 10.3934/era.2025173

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Abstract

A modified nonlinear incidence rate in an SIS epidemic model was investigated. When a new disease emerged or an old one resurged, the infectivity was initially high. Subsequently, the psychological effect attenuated the infectivity. Eventually, due to the crowding effect, the infectivity reached a saturation state. The nonlinearity of the functional form of the infection incidence was modified to enhance its biological plausibility. The stability of the associated equilibria was examined, and the basic reproduction number and the critical value that governed the dynamics of the model were deduced. Bifurcation analyses were presented, encompassing backward bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and Hopf bifurcation. Numerical simulations were conducted to validate our findings.

References

[1]	M. De la Sen, R. Nistal, S. Alonso-Quesada, A. Ibeas, Some formal results on positivity, stability, and endemic steady-state attainability based on linear algebraic tools for a class of epidemic models with eventual incommensurate delays, Discrete Dyn. Nat. Soc., 2019 (2019), 8959681. https://doi.org/10.1155/2019/8959681 doi: 10.1155/2019/8959681
[2]	M. De la Sen, A. Ibeas, S. Alonso-Quesada, R. Nistal, On a new epidemic model with asymptomatic and dead-infective subpopulations with feedback controls useful for ebola disease, Discrete Dyn. Nat. Soc., 2017 (2017), 4232971. https://doi.org/10.1155/2017/4232971 doi: 10.1155/2017/4232971
[3]	W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ. — The problem of endemicity, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 138 (1932), 55–83. https://doi.org/10.1098/rspa.1932.0171 doi: 10.1098/rspa.1932.0171
[4]	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
[5]	D. M. Xiao, Y. G. Zhou, Qualitative analysis of an epidemic model, Can. Appl. Math. Q., 14 (2006), 469–492.
[6]	Y. G. Zhou, D. M. Xiao, Y. L. Li, Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, Chaos, Solitons Fractals, 32 (2007), 1903–1915. https://doi.org/10.1016/j.chaos.2006.01.002 doi: 10.1016/j.chaos.2006.01.002
[7]	M. Lu, J. C. Huang, S. G. Ruan, P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differ. Equations, 267 (2019), 1859–1898. https://doi.org/10.1016/j.jde.2019.03.005 doi: 10.1016/j.jde.2019.03.005
[8]	P. van den Driessche, J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000), 525–540. https://doi.org/10.1007/s002850000032 doi: 10.1007/s002850000032
[9]	M. H. Alharbi, Global investigation for an "SIS" model for COVID-19 epidemic with asymptomatic infection, Math. Biosci. Eng., 20 (2023), 5298–5315. https://doi.org/10.3934/mbe.2023245 doi: 10.3934/mbe.2023245
[10]	Z. Shuai, P van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Appl. Math., 73 (2013), 1513–1532. https://doi.org/10.1137/120876642 doi: 10.1137/120876642
[11]	P van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[12]	Y. A. Kuznetsov, Numerical Analysis of Bifurcations, Cambridge University Press, 2004.
[13]	Z. Zhang, Qualitative Theory of Differential Equations, Science Press, 1992.
[14]	S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, 2003.
[15]	J. C. Huang, Y. J. Gong, S. G. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101–2121. https://doi.org/10.3934/dcdsb.2013.18.2101 doi: 10.3934/dcdsb.2013.18.2101
[16]	Y. L. Tang, D. Q. Huang, S. G. Ruan, W. N. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621–639. https://doi.org/10.1137/070700966 doi: 10.1137/070700966
[17]	B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf bifurcation, Cambridge University Press, 1981.
[18]	J. E. Marsden, M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, 2012.
[19]	R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funct. Anal. Appl., 9 (1975), 144–145. https://doi.org/10.1007/BF01075453 doi: 10.1007/BF01075453
[20]	F. Takens, Forced oscillations and bifurcations, in Global Analysis of Dynamical Systems, CRC Press, (2001), 11–71.
[21]	L. Perko, Differential Equations and Dynamical Systems, Springer Science & Business Media, 2013.
[22]	S. G. Ruan, W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135–163. https://doi.org/10.1016/S0022-0396(02)00089-X doi: 10.1016/S0022-0396(02)00089-X
[23]	C. Castillo-Chavez, B. J. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/mbe.2004.1.361 doi: 10.3934/mbe.2004.1.361

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