Abstract
The probability is introduced to formulate the death of individuals, the recovery of the infected individuals and incidence of epidemic disease. Based on the assumption that the number of individuals in a population is a constant, discrete-time SI and SIS epidemic models with vital dynamics are established respectively corresponding to the case that the infectives can recover from the disease or not. For these two models, the results obtained in this paper show that there is the same dynamical behavior as their corresponding continuous ones, and the threshold determining its dynamical behavior is found. Below the threshold the epidemic disease dies out eventually, above the threshold the epidemic disease becomes an endemic eventually, and the number of the infectives approaches a positive constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ma Z, Zhou Y, Wang W, Jin Z. Epidemic dynamics and its mathematical modelling[M]. Beijing: Chinese Science Press, 2004 (in Chinese).
Anderson R M, may RM. Infectious diseases of humans, dynamics and control[M]. Oxford: Oxford Univ Press, 1991.
Li Jianquan, Zhang Juan, Ma Zhien. Global analysis of some epidemic models with general contact rate and constant immigration[J]. Applied Mathematics and Mechanics (English Edition), 2004, 25(4):396–404.
Wang Ladi, Li Jianquan. Qualitative analysis of an SEIS epidemic model with nonlinear incidence rate[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(5):667–672.
Allen L J S. Some discrete-time SI, SIR, and SIS epidemic models[J]. Math Biosci, 1994, 124:83–105.
Allen L J S, Burgin A M. Comparison of deterministic and stochastic SIS and SIR models in discrete time[J]. Math Biosci, 2000, 163:1–33.
Castillo-Chavez C, Yakubu A-A. Dispersal, disease and life-history evolution[J]. Math Biosci, 2001, 173:35–53.
Castillo-Chavez C, Yakubu A-A. Discrete-time S-I-S models with complex dynamics[J]. Nonlinear Analysis, 2001, 47:4753–4762.
Zhou Y, Fergola P. Dynamics of a discrete age-structured SIS models[J]. Discrete and Continuous Dynamical System, Series B, 2004, 4:841–850.
Li X, Wang W. A discrete epidemic model with stage structure[J]. Chaos, Solitons and Fractals, 2005, 26:947–958.
Zhang Weinian. Dynamical systems[M]. Beijing: Higher Education Press, 2001 (in Chinese).
Cull P. Local and global stability for population models[J]. Biol Cybern, 1986, 54:141–149.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by GUO Xing-ming
Project supported by the National Natural Science Foundation of China (Nos. 10531030, 10701053), and the Natural Science Foundation of Shanxi Province of China (No. 2005Z010)
Rights and permissions
About this article
Cite this article
Li, Jq., Lou, J. & Lou, Mz. Some discrete SI and SIS epidemic models. Appl. Math. Mech.-Engl. Ed. 29, 113–119 (2008). https://doi.org/10.1007/s10483-008-0113-y
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10483-008-0113-y