Effect of Disease-Induced Death Rate and Latent Period on Global Stability for SIRS Epidemic Models with General Incidence Rate
DOI:
https://doi.org/10.5890/JAND.2023.09.006Abstract
In this paper, We study a class of delayed SIRS epidemic dynamical models with a general nonlinear incidence rate representing the transfer from susceptible class to infected class. Besides we incorporate a surviving probability from susceptibles to infectious. Throughout the paper, Lyapunov and Euler's stability tools are used to establish the global and local stability for both disease-free and endemic equilibriums depending on reproduction number value $R_0$ and disease-induced death rate. Finally, a sensitivity analysis over basic reproduction number with respect to controllable model's parameters and numerical simulations are presented to illustrate and explain our theoretical results.References
[1] Anderson, R.M. and May, R.M. (1978), Regulation and stability of host-parasite population interactions: I. Regulatory processes, The Journal of Animal Ecology, 47(1), 219-267.
[2] Capasso, V. and Serio, G. (1978), A generalization of Kermack-Mckendrick deterministic epidemic model, Mathematical Biosciences, 42(1-2), 41-61.
[3] Chen, L.S. and Chen, J. (1993), Nonlinear Biologicl Dynamics System, Scientific Press, China.
[4] Jiang, Z. and Wei, J. (2008), Stability and bifurcation analysis in a delayed SIR model, Chaos, Solitons & Fractals, 35(3), 609-619.
[5] Kaddar, A. (2010), Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Analysis: Modelling and Control, 15(3), 299-306.
[6] Wei, C. and Chen, L. (2008), A delayed epidemic model with pulse vaccination, Discrete Dynamics in Nature and Society, Article ID 746-951.
[7] Xu, R. and Ma, Z. (2009), Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 41(5), 2319-2325.
[8] Zhang, J.Z., Jin, Z., Liu, Q.X., and Zhang, Z.Y. (2008), Analysis of a delayed SIR model with nonlinear incidence rate, Discrete Dynamics in Nature and Society, Article ID 66153.
[9] Avila-Vales, E.J. and Cervantes-Pérez, Á.G. (2019), Global stability for SIRS epidemic models with general incidence rate and transfer from infectious to susceptible, Boletín de la Sociedad Matemática Mexicana, 25, 637-658.
[10] Gabriela, M., Gomes, M., White, L.J., and Medley, G.F. (2005), The reinfection threshold, Journal of Theoretical Biology, 236, 111-113.
[11] Tang, Q., Teng, Z., and Abdurahman, X. (2017), New Lyapunov function for SIRS epidemic models, In: Bulletin of the Malaysian Mathematical Sciences Society, 40(1), 237-258.
[12] Wang, L., Zhang, X., Liu, Z. (2018), An SEIR epidemic model with relapse and general nonlinear incidence rate with application to media impact, Qualitative Theory of Dynamical Systems, 17, 309-329.
[13] Beretta, E., Hara, T., Ma, W., and Takeuchi, Y. (2001), Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis, Theory, Methods & Applications, 47(6), 4107-4115.
[14] Cooke, K.L. (1979), Stability analysis for a vector disease model, The Rocky Mountain Journal of Mathematics, 9(1), 31-42.
[15] Takeuchi, Y., Ma, W., and Beretta, E. (2000), Global asymptotic properties of a delayed SIR epidemic model with finite incubation time, Nonlinear Analysis, Theory, Methods and Applications, 42, 931-947.
[16] Bernoussi, A., Kaddar, A., and Asserda, S. (2017), On the dynamics of an siri epidemic model with a generalized incidence function, Modélisation et optimisation avancées, 19(1), 87-96.
[17] Guo, P., Yang, X., and Yang, Z. (2014), Dynamical behaviors of an SIRI epidemic model with nonlinear incidence and latent period, Advances in Difference Equations, 164, 2-18.
[18] Beretta, E. and Breda, D. (2011), An SEIR epidemic model with constant latency time and infectious period, Mathematical Biosciences and Engineering, 8(4), 931-952.
[19] Enatsu, Y. and Nakata, Y. (2014), Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate, Mathematical Biosciences and Engineering, 11(4), 785-805.
[20] Hattaf, K., Lashari, A.A., Louartassi, Y., and Yousfi, N. (2013), A delayed SIR epidemic model with a generalized incidence rate, Electronic Journal of Qualitative Theory of Differential Equations, 3, 1-9.
[21] Wang, W. and Ruan, S. (2004), Bifurcation in epidemic model with constant removal rate infectives, Journal of Mathematical Analysis and Applications, 291, 775-793.
[22] Zhang, F., Li, Z.Z., and Zhang, F. (2008), Global stability of an SIR epidemic model with constant infectious period, Applied Mathematics and Computation, 199, 285-291.
[23] Zhou, Y. and Liu, H. (2003), Stability of periodic solutions for an SIS model with pulse vaccination, Mathematical and Computer Modelling, 38, 299-308.
[24] de Jong, M.C.M., Diekmann, O., and Heesterbeek, H. (1995), How does transmission of infection depend on population size? In: Epidemic models: their structure and relation to data, Mollison D. (Ed.), Cambridge University Press, Cambridge, 84-94.
[25] Hethcote, H.W. (2000), The Mathematics of Infectious Disease, SIAM Review, 42, 599-653.
[26] Bernoussi, A. and Hattaf, K. (2021), Global dynamics of an SIRSI epidemic model with discrete delay and general incidence rate, Journals of Applied Nonlinear Dynamics, 10(3), 545-560.
[27] Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego.
[28] LaSalle, J.P. (1976), The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
[29] Bernoussi, A. (2021), Bifurcation of periodic solution of a delayed SEIR epidemic model with nonlinear incidence rate, Journals of Applied Nonlinear Dynamics, 10(3), 351-367.
[30] Chitnis, N., Hyman, J.M., and Cushing, J.M. (2008), Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70(5), 1272-1296.
[31] Hussain, T., Ozair, M., Ali, F., Rehman, S., Assiri, T.A., and Mahmoud, E.E. (2021), Sensitivity analysis and optimal control of COVID-19 dynamics based on SEIQR model, Results in Physics, 22, 103956.
Article Metrics
Usage tracking begins September 1, 2026.