Bulletin of Mathematical Biology

, Volume 69, Issue 8, pp 2537–2559

On a Nonautonomous SEIRS Model in Epidemiology

Original Article

Abstract

In this paper, we derive some threshold conditions for permanence and extinction of diseases that can be described by a nonautonomous SEIRS epidemic model. Under the quite weak assumptions, we establish some sufficient conditions to prove the permanence and extinction of disease. Some new threshold values are determined.

Keywords

Nonautonomous SEIRS epidemic model Disease Permanence Extinction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, R.M., May, R.M., 1978. Regulation and stability of host-parasite population interactions II: destabilizing process. J. Anim. Ecol. 47, 219–267. CrossRefGoogle Scholar
  2. Anderson, R.M., May, R.M., 1979. Population biology of infectious diseases: Part I. Nature 280, 361–367. CrossRefGoogle Scholar
  3. Anderson, R.M., May, R.M., 1992. Infectious Disease of Humans, Dynamical and Control. Oxford University Press, Oxford. Google Scholar
  4. Brauer, F., Castillo-Chavez, C., 2001. Mathematical Models in Population Biology and Epidemiology. Tests in Applied Mathematics. Springer, Berlin. MATHGoogle Scholar
  5. Capasso, V., 1993. Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, vol. 97. Springer, Berlin. MATHGoogle Scholar
  6. Cull, P., 1981. Global stability for population models. Bull. Math. Biol. 43, 47–58. MATHMathSciNetGoogle Scholar
  7. Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, Chichester. Google Scholar
  8. Dowell, S.F., 2001. Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerg. Infect. Dis. 7, 369–374. Google Scholar
  9. Earn, D.J.D., Dushoff, J., Levin, S.A., 2002. Ecology and evolution of the flu. Trends Ecol. Evol. 17, 334–340. CrossRefGoogle Scholar
  10. Herzog, G., Redheffer, R., 2004. Nonautonomous SEIRS and Thron models for epidemiology and cell biology. Nonlinear Anal. RWA 5, 33–44. MATHCrossRefMathSciNetGoogle Scholar
  11. Hethcote, H.W., 2000. The mathematics of infectious diseases. SIAM Rev. 42, 599–653. MATHCrossRefMathSciNetGoogle Scholar
  12. Kermark, M.D., Mckendrick, A.G., 1927. Contributions to the mathematical theory of epidemics: Part I. Proc. Roy. Soc. 115, 700–721. CrossRefGoogle Scholar
  13. Li, M.Y., Graef, J.R., Wang, L., Karsai, J., 1999. Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160, 191–213. MATHCrossRefMathSciNetGoogle Scholar
  14. Liu, W., Hethcote, H.W., Levin, S.A., 1987. Dynamical behavior of epidemiological models in epidemiology. J. Math. Biol. 25, 359–380. MATHCrossRefMathSciNetGoogle Scholar
  15. London, W., Yorke, J.A., 1973. Recurrent outbreaks of measles, chickenpox and mumps: I. seasonal variation in contact rates. Am. J. Epidemiol. 98, 453–468. Google Scholar
  16. Ma, Z., Zhou, Y., Wang, W., Jin, Z., 2004. Mathematical Modelling and Research of Epidemic Dynamical Systems. Science, Beijing. Google Scholar
  17. Mena-Lorca, J., Hethcote, H.W., 1992. Dynamic models of infectious diseases as regulators of population sizes. J. Math. Biol. 30, 693–716. MATHCrossRefMathSciNetGoogle Scholar
  18. Takeuchi, Y., Cui, J., Rinko, M., Saito, Y., 2006a. Permanence of delayed population model with dispersal loss. Math. Biosci. 201, 143–156. MATHCrossRefMathSciNetGoogle Scholar
  19. Takeuchi, Y., Cui, J., Rinko, M., Saito, Y., 2006b. Permanence of dispersal population model with time delays. J. Comp. Appl. Math. 192, 417–430. MATHCrossRefGoogle Scholar
  20. Teng, Z., Chen, L., 2003. Permanence and extinction of periodic predator-prey systems in a patchy environment with delay. Nonlinear Anal. RWA 4, 335–364. MATHCrossRefMathSciNetGoogle Scholar
  21. Teng, Z., Li, Z., 2000. Permanence and asymptotic behavior of the N-species nonautonomous Lotka–Volterra competitive systems. Comput. Math. Appl. 39, 107–116. MATHCrossRefMathSciNetGoogle Scholar
  22. Teng, Z., Yu, Y., 1999. The extinction in nonautonomous prey-predator Lotka–Volterra systems. Acta Math. Appl. Sin. 15, 401–408. MATHCrossRefMathSciNetGoogle Scholar
  23. Thieme, H.R., 1999. Uniform weak implies uniform strong persistence also for non-autonomous semiflows. Proc. Am. Math. Soc. 127, 2395–2403. MATHCrossRefMathSciNetGoogle Scholar
  24. Thieme, H.R., 2000. Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166, 173–201. MATHCrossRefMathSciNetGoogle Scholar
  25. Thieme, H.R., 2003. Mathematics in Population Biology. Princeton University Press, Princeton. MATHGoogle Scholar
  26. Zhang, J., Lou, J., Ma, Z., Wu, J., 2005. A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China. Appl. Math. Comput. 162, 909–924. MATHCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  1. 1.College of Mathematics and System SciencesXinjiang UniversityUrumqiPeople’s Republic of China

Personalised recommendations