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Population dynamics of mosquito-borne disease: Persistence in a completely heterogeneous environment. (English) Zbl 0647.92015
The m/n model of mosquito-borne disease is studied. m/n model means that infected hosts and mosquitos are divided into groups (patches) and a mosquito belonging to any one of n vector patches can take blood from any of m host patches. The condition for extinction or persistence of the infection involves the so-called basic reproductive rate for the m/n model R(m/n).
If R(m/n)\(\leq 1\), the infection becomes extinct, whereas the number of infectives and susceptibles approaches constant positive levels in all patches if \(R(m/n)>1\). The basic reproductive rate for the m/n model can be estimated against the basic reproductive rate for the corresponding m/1, 1/1 and 1/m models.
On the other hand each of the four R(m/n), R(m/1), R(1/1), R(1/m) is the product of two factors, one is the basic reproductive rate for the 1/1 model, the second is called heterogeneity factor and denoted by f(m/1), f(1/1), f(1/m), respectively. Lower and upper estimates for the heterogeneity factor are given.
An accurate analysis of the mathematical model is made. The study of the stability of the 0 solution and the existence and the form of the eigenvalues of a suitable matrix leads to the main results about R(m/n).
Reviewer: S.Totaro

MSC:
92D25 Population dynamics (general)
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