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Applied Mathematics and Mechanics

, Volume 29, Issue 1, pp 113–119 | Cite as

Some discrete SI and SIS epidemic models

  • Li Jian-quan  (李建全)Email author
  • Lou Jie  (娄洁)
  • Lou Mei-zhi  (娄梅枝)
Article

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.

Key words

discrete epidemic model dynamical behavior fixed point stability 

Chinese Library Classification

O175.7 

2000 Mathematics Subject Classification

39A11 92D30 

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References

  1. [1]
    Ma Z, Zhou Y, Wang W, Jin Z. Epidemic dynamics and its mathematical modelling[M]. Beijing: Chinese Science Press, 2004 (in Chinese).Google Scholar
  2. [2]
    Anderson R M, may RM. Infectious diseases of humans, dynamics and control[M]. Oxford: Oxford Univ Press, 1991.Google Scholar
  3. [3]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  5. [5]
    Allen L J S. Some discrete-time SI, SIR, and SIS epidemic models[J]. Math Biosci, 1994, 124:83–105.zbMATHCrossRefGoogle Scholar
  6. [6]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Castillo-Chavez C, Yakubu A-A. Dispersal, disease and life-history evolution[J]. Math Biosci, 2001, 173:35–53.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Castillo-Chavez C, Yakubu A-A. Discrete-time S-I-S models with complex dynamics[J]. Nonlinear Analysis, 2001, 47:4753–4762.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Zhou Y, Fergola P. Dynamics of a discrete age-structured SIS models[J]. Discrete and Continuous Dynamical System, Series B, 2004, 4:841–850.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Li X, Wang W. A discrete epidemic model with stage structure[J]. Chaos, Solitons and Fractals, 2005, 26:947–958.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Zhang Weinian. Dynamical systems[M]. Beijing: Higher Education Press, 2001 (in Chinese).Google Scholar
  12. [12]
    Cull P. Local and global stability for population models[J]. Biol Cybern, 1986, 54:141–149.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. and Springer-Verlag 2008

Authors and Affiliations

  • Li Jian-quan  (李建全)
    • 1
    • 2
    Email author
  • Lou Jie  (娄洁)
    • 3
  • Lou Mei-zhi  (娄梅枝)
    • 4
  1. 1.Department of Applied Mathematics and PhysicsAir Force Engineering UniversityXi’anP. R. China
  2. 2.Department of MathematicsYuncheng CollegeYunchengP. R. China
  3. 3.Department of MathematicsShanghai UniversityShanghaiP. R. China
  4. 4.Police College of HenanZhengzhouP. R. China

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