Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments
The basic reproduction ratio and its computation formulae are established for a large class of compartmental epidemic models in periodic environments. It is proved that a disease cannot invade the disease-free state if the ratio is less than unity and can invade if it is greater than unity. It is also shown that the basic reproduction number of the time-averaged autonomous system is applicable in the case where both the matrix of new infection rate and the matrix of transition and dissipation within infectious compartments are diagonal, but it may underestimate and overestimate infection risks in other cases. The global dynamics of a periodic epidemic model with patch structure is analyzed in order to study the impact of periodic contacts or periodic migrations on the disease transmission.
KeywordsCompartmental models Reproduction ratio Periodicity Threshold dynamics
Mathematics Subject Classification (2000)34D20 37B55 92D30
Unable to display preview. Download preview PDF.
- 2.Arino, J., van den Driessche, P.: The basic reproduction number in a multi-city compartmental epidemic model, Positive Systems (Rome, 2003) pp. 135–142, Lecture Notes in Control and Information Science, vol. 294. Springer, Berlin (2003)Google Scholar
- 8.Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, Chichester (2000)Google Scholar
- 9.Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations, Lecture notes in biomath, vol. 11, pp. 1–5. Berlin-Heidelberg-New York: Springer (1976)Google Scholar
- 16.Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Company, INC, Malabar, Florida (1980)Google Scholar
- 18.Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Series 247. Longman Scientific and Technical (1991)Google Scholar
- 29.Smith, H.L., Waltman, P.: The Theory of the Chemostat. Cambridge University Press (1995)Google Scholar