A periodic epidemic model in a patchy environment. (English) Zbl 1101.92046

Summary: An epidemic model in a patchy environment with periodic coefficients is investigated. By employing persistence theory, we establish a threshold between extinction and uniform persistence of the disease. Further, we obtain conditions under which the positive periodic solution is globally asymptotically stable. At last, we present two examples and numerical simulations.


92D30 Epidemiology
34C25 Periodic solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
92D40 Ecology
34D05 Asymptotic properties of solutions to ordinary differential equations
37N25 Dynamical systems in biology
Full Text: DOI


[1] Aronsson, G.; Kellogg, R.B., On a differential equation arising from compartmental analysis, Math. biosci., 38, 113-122, (1973) · Zbl 0375.34028
[2] Brauer, F.; van den Driessche, P., Models for transmission of disease with immigration of infectives, Math. biosci., 171, 143-154, (2001) · Zbl 0995.92041
[3] Cooke, K.; van den Driessche, P.; Zou, X., Interaction of maturation delay and nonlinear birth in population and epidemic models, J. math. biol., 39, 332-352, (1999) · Zbl 0945.92016
[4] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases, model building, analysis and interpretation, (2000), Wiley · Zbl 0997.92505
[5] Hethcote, H.W., Qualitative analysis of communicable disease models, Math. biosci., 28, 335-356, (1976) · Zbl 0326.92017
[6] Hethcote, H.W.; Levin, S.A., Periodicity in epidemiological models, ()
[7] Hirsch, M.W., Systems of differential equations that are competitive or cooperative II: convergence almost everywhere, SIAM J. math. anal., 16, 423-439, (1985) · Zbl 0658.34023
[8] Jin, Y.; Wang, W., The effect of population dispersal on the spread of a disease, J. math. anal. appl., 308, 343-364, (2005) · Zbl 1065.92044
[9] Smith, H.L.; Waltman, P., The theory of the chemostat, (1995), Cambridge Univ. Press · Zbl 0860.92031
[10] Wang, W.; Zhao, X.-Q., An epidemic model in a patchy environment, Math. biosci., 190, 97-112, (2004) · Zbl 1048.92030
[11] Zhao, X.-Q., Dynamical systems in population biology, (2003), Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.