The basic reproduction number in some discrete-time epidemic models. (English) Zbl 1147.92032

Allen, Linda J. S.; van den Driessche, P.

J. Difference Equ. Appl. 14, No. 10-11, 1127-1147 (2008).

Summary: The next generation matrix approach for calculating the basic reproduction number \(\mathcal R_0\) is summarized for discrete-time epidemic models. This approach is applied to six disease models developed for the study of two emerging wildlife diseases: hantavirus in rodents and chytridiomycosis in amphibians. Two of the models include discrete spatial patches. For each model, \(\mathcal R_0\) is calculated in terms of the model parameters. For \(\mathcal R_0 <1\) , if a small number of infectives is introduced, then the wildlife disease dies out. Global stability of the disease-free equilibrium is verified for a special case of the SI hantavirus model when \(\mathcal R_0<1\) . In addition, a numerical example indicates that there is a transcritical bifurcation at \(\mathcal R_0 = 1\) , with the disease dying out if \(\mathcal R_0 < 1\) but tending to an endemic level if \(\mathcal R_0 > 1\) .

MSC:

92D30	Epidemiology
39A11	Stability of difference equations (MSC2000)

Keywords:

epidemic model on patches

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