Stability of an SIRS Epidemic Model with Saturated Incidence Rate and Saturated Treatment Function

Yıl 2021, Cilt: 4 Sayı: 3, 133 - 138, 27.12.2021

İrem Çay

https://doi.org/10.33187/jmsm.1009561

Öz

In this paper the global dynamics of susceptible-infected-recovered-susceptible (SIRS) epidemic model with saturated incidence rate and saturated treatment function is studied. Firstly, the basic reproduction number $R_0$ is calculated and the existence of the disease-free and positive equilibria is showed. In addition, local stability of the equilibria is investigated. Then, sufficient conditions are achieved for global stability of disease-free and endemic equilibria. Finally, the numerical examples are presented to validate the theoretical results.

Anahtar Kelimeler

Global stability, Lyapunov function, SIRS epidemic model

Kaynakça

• [1] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Berlin, Springer, 2011.
• [2] V. Capasso, G. Serio, A generalization of the Kermack–Mckendrick deterministic epidemic model, Math. Biosci. 42 (1978), 43-61.
• [3] X. Zhang, X. N. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348(1) (2008), 433–443.
• [4] E. J. Avila-Vales, A. G. Cervantes-P´erez, Global Stability for SIRS Epidemic Models with General Incidence Rate and Transfer from Infectious to Susceptible, SeMA J. Bolet´ın de la Sociedad Matem´atica Mexicana. 25 (2019), 637–658.
• [5] JP. LaSalle, The Stability of Dynamical Systems, Philadelphia, PA, USA: Soc. Ind. Appl. Math. 1976.
• [6] M. Lu, J. Huang, S. Ruan, P. Yu, Bifurcation Analysis of a SIRS Epidemic Model with a Generalized Nonmonotone and Saturated Incidence Rate, . J. Differ. Equ. 267 (2019), 1859-1898.
• [7] S. Liao, J. Wang, Global Stability Analysis of Epidemiological Models Based on Volterra Lyapunov Stable Matrices, Chaos Solitons Fractals. 45 (2012), 966-977.