Dynamic properties and $ \alpha $-path of an uncertain SIRS epidemic model

Jianye Zhang; Zhiming Li; Jianye Zhang; Zhiming Li

doi:10.3934/math.2026057

AIMS Mathematics

2026, Volume 11, Issue 1: 1332-1353. doi: 10.3934/math.2026057

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Research article

Dynamic properties and $ \alpha $-path of an uncertain SIRS epidemic model

Jianye Zhang ,
Zhiming Li ^,

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830017, China

Received: 06 October 2025 Revised: 25 December 2025 Accepted: 08 January 2026 Published: 16 January 2026
MSC : 60G51, 60G57, 92B05

The dynamic analysis of epidemic models plays a crucial role in understanding disease transmission mechanisms and prevention strategies. Building upon the research of Tan et al. ^[1], this paper investigates novel dynamic properties of solutions and $ \alpha $-paths of an uncertain SIRS model. First, we prove the existence, uniqueness, and stability of a solution for the SIRS model. Using the Yao-Chen formula, we derive $ \alpha $-paths of the model and prove that both the disease-free and endemic equilibria are globally asymptotically stable, thereby refining existing research. When the threshold $\mathscr {R}_0^{u} \leq 1$, the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium is globally asymptotically stable if $\mathscr {R}_0^{u} > 1$. Finally, numerical simulations demonstrate that increasing the recovery rate and decreasing the disease-induced mortality rate can effectively reduce the disease spread. The results show that the disease transmission rate and the intensity of the Liu process are essential to prevent the disease spread.
- uncertain SIRS model,
- $ \alpha $-path,
- Lyapunov function,
- globally asymptotically stability
Citation: Jianye Zhang, Zhiming Li. Dynamic properties and $ \alpha $-path of an uncertain SIRS epidemic model[J]. AIMS Mathematics, 2026, 11(1): 1332-1353. doi: 10.3934/math.2026057

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Abstract

The dynamic analysis of epidemic models plays a crucial role in understanding disease transmission mechanisms and prevention strategies. Building upon the research of Tan et al. ^[1], this paper investigates novel dynamic properties of solutions and $ \alpha $-paths of an uncertain SIRS model. First, we prove the existence, uniqueness, and stability of a solution for the SIRS model. Using the Yao-Chen formula, we derive $ \alpha $-paths of the model and prove that both the disease-free and endemic equilibria are globally asymptotically stable, thereby refining existing research. When the threshold $\mathscr {R}_0^{u} \leq 1$, the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium is globally asymptotically stable if $\mathscr {R}_0^{u} > 1$. Finally, numerical simulations demonstrate that increasing the recovery rate and decreasing the disease-induced mortality rate can effectively reduce the disease spread. The results show that the disease transmission rate and the intensity of the Liu process are essential to prevent the disease spread.

References

[1]	S. Tan, L. Zhang, Y. Sheng, Threshold dynamics of an uncertain SIRS epidemic model with a bilinear incidence, J. Intell. Fuzzy Syst., 45 (2023), 9083–9093. https://doi.org/10.3233/JIFS-223439 doi: 10.3233/JIFS-223439
[2]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. A, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[3]	Z. Ma, Y. Zhou, W. Wang, Z. Jin, Mathematical modelling and research of epidemic dynamical systems, Beijing: Science Press, 2004.
[4]	M. Lakhal, T. E. Guendouz, R. Taki, M. E. Fatini, The threshold of a stochastic SIRS epidemic model with a general incidence, Bull. Malays. Math. Sci. Soc., 47 (2024), 100. https://doi.org/10.1007/s40840-024-01696-2 doi: 10.1007/s40840-024-01696-2
[5]	S. Zhi, H.-T. Niu, Y. Su, Global dynamics of a diffusive SIRS epidemic model in a spatially heterogeneous environment, Appl. Anal., 104 (2025), 390–418. https://doi.org/10.1080/00036811.2024.2367667 doi: 10.1080/00036811.2024.2367667
[6]	G. Lan, Z. Lin, C. Wei, S. Zhang, A stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching, J. Franklin I., 356 (2019), 9844–9866. https://doi.org/10.1016/j.jfranklin.2019.09.009 doi: 10.1016/j.jfranklin.2019.09.009
[7]	S. Zhi, Y. Su, H. Niu, L. Qiang, The threshold dynamics of a waterborne pathogen model with seasonality in a polluted environment, Acta Math. Sci., 44 (2024), 2165–2189. https://doi.org/10.1007/s10473-024-0607-z doi: 10.1007/s10473-024-0607-z
[8]	J. P. Chávez, A. Gökçe, T. Götz, B. Gürbüz, An SIRS model considering waning efficacy and periodic re-vaccination, Chaos Soliton. Fract., 196 (2025), 116436. https://doi.org/10.1016/j.chaos.2025.116436 doi: 10.1016/j.chaos.2025.116436
[9]	C. Zhou, Z. Li, A stochastic SIRS epidemic model with a saturation incidence under semi-Markovian switching, Int. J. Biomath., in press. https://doi.org/10.1142/S1793524524501407
[10]	S. Zhi, D. Bai, Y. Su, Turing patterns induced by cross-diffusion in a predator-prey model with Smith-type prey growth and additive predation effects, Chaos Soliton. Fract., 202 (2026), 117522. https://doi.org/10.1016/j.chaos.2025.117522 doi: 10.1016/j.chaos.2025.117522
[11]	B. Liu, Uncertainty theory: a branch of mathematics for modeling human uncertainty, Berlin: Springer Press, 2010. https://doi.org/10.1007/978-3-642-13959-8
[12]	B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3–16.
[13]	X. Chen, B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Making, 9 (2010), 69–81. https://doi.org/10.1007/s10700-010-9073-2 doi: 10.1007/s10700-010-9073-2
[14]	K. Yao, J. Gao, Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Making, 12 (2013), 3–13. https://doi.org/10.1007/s10700-012-9139-4 doi: 10.1007/s10700-012-9139-4
[15]	T. Su, H. Wu, J. Zhou, Stability of multi-dimensional uncertain differential equation, Soft Comput, 20 (2016), 4991–4998. https://doi.org/10.1007/s00500-015-1788-0 doi: 10.1007/s00500-015-1788-0
[16]	Y. Liu, An analytic method for solving uncertain differential equations, Journal of Uncertain Systems, 6 (2012), 244–249.
[17]	K. Yao, X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825–832. https://doi.org/10.3233/IFS-120688 doi: 10.3233/IFS-120688
[18]	L. W. T. Ng, K. E. Willcox, Multifidelity approaches for optimization under uncertainty, Int. J. Numer. Meth. Eng., 100 (2014), 746–772. https://doi.org/10.1002/nme.4761 doi: 10.1002/nme.4761
[19]	I. A. Korostil, G. W. Peters, J. Cornebise, D. G. Regan, Adaptive Markov chain Monte Carlo forward projection for statistical analysis in epidemic modelling of human papillomavirus, Stat. Med., 32 (2013), 1917–1953. https://doi.org/10.1002/sim.5590 doi: 10.1002/sim.5590
[20]	W. Du, J. Su, Uncertainty quantification for numerical solutions of the nonlinear partial differential equations by using the multi-fidelity Monte Carlo method, Appl. Sci., 12 (2022), 7045. https://doi.org/10.3390/app12147045 doi: 10.3390/app12147045
[21]	Z. Li, Z. Teng, Analysis of uncertain SIS epidemic model with nonlinear incidence and demography, Fuzzy Optim. Decis. Making, 18 (2019), 475–491. https://doi.org/10.1007/s10700-019-09303-x doi: 10.1007/s10700-019-09303-x
[22]	X. Chen, J. Li, C. Xiao, P. Yang, Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19, Fuzzy Optim. Decis. Making, 20 (2021), 189–208. https://doi.org/10.1007/s10700-020-09342-9 doi: 10.1007/s10700-020-09342-9
[23]	O. O. Aybar, Biochemical models of SIR and SIRS: effects of bilinear incidence rate on infection-free and endemic states, Chaos, 33 (2023), 093120. https://doi.org/10.1063/5.0166337 doi: 10.1063/5.0166337
[24]	S. Aznague, A. Settati, A. Lahrouz, B. Harchaoui, K. E. Bakkioui, A. N. Brahim, Threshold dynamics of stochastic SIS epidemic models with logistic recruitment rate, New Math. Nat. Comput., in press. https://doi.org/10.1142/S1793005727500074
[25]	W. Chen, G. Lin, S. Pan, Propagation dynamics in an SIRS model with general incidence functions, Mathematical Biosciences and Engineering, 20 (2023), 6751–6775. http://doi.org/10.3934/mbe.2023291 doi: 10.3934/mbe.2023291
[26]	K. Yao, Uncertain differential equations, Berlin: Springer Press, 2016. https://doi.org/10.1007/978-3-662-52729-0
[27]	B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3–10.
[28]	J. M. Heffernan, R. J. Smith, L. M. Wahl, Perspectives on the basic reproductive ratio, J. Roy. Soc. Interface, 2 (2005), 281–293. https://doi.org/10.1098/rsif.2005.0042 doi: 10.1098/rsif.2005.0042

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