Threshold of disease transmission in a patch environment. (English) Zbl 1021.92039
Summary: An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that the reproduction number is a threshold of uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen.
It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.
It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.
MSC:
| 92D30 | Epidemiology |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 92D40 | Ecology |
References:
| [1] | Anderson, R. M.; May, R. M., Infectious Diseases of Humans, Dynamics and Control (1991), Oxford Univ. Press: Oxford Univ. Press Oxford |
| [2] | Beretta, E.; Kuang, Y., Modeling and analysis of a marine bacteriophage infection, Math. Biosci., 149, 57-76 (1998) · Zbl 0946.92012 |
| [3] | Brauer, F.; van den Driessche, P., Models for transmission of disease with immigration of infectives, Math. Biosci., 171, 143-154 (2001) · Zbl 0995.92041 |
| [4] | Castillo-Chavez, C.; Yakubu, A. A., Dispersal, disease and life-history evolution, Math. Biosci., 173, 35-53 (2001) · Zbl 1005.92029 |
| [5] | Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018 |
| [6] | Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033 |
| [7] | Hirsch, W. M.; Smith, H. L.; Zhao, X.-Q., Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 1, 107-131 (2001) · Zbl 1129.37306 |
| [8] | Thieme, H. R., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24, 407-435 (1993) · Zbl 0774.34030 |
| [9] | Thieme, H. R., Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166, 173-201 (2000) · Zbl 0970.37061 |
| [10] | van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48 (2002) · Zbl 1015.92036 |
| [11] | W. Wang, X.-Q. Zhao, An epidemic model in a patchy environment, to appear; W. Wang, X.-Q. Zhao, An epidemic model in a patchy environment, to appear · Zbl 1048.92030 |
| [12] | Zhao, X.-Q.; Zou, X., Threshold dynamics in a delayed SIS epidemic model, J. Math. Anal. Appl., 257, 282-291 (2001) · Zbl 0988.92027 |
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