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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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##### References:
  Aron, J.L.; May, R.M., The population dynamics of malaria, (), 139-179  Bailey, N.T.J., The mathematical theory of infectious diseases and its applications, (1975), Griffin London · Zbl 0115.37202  Bailey, N.T.J., The biomathematics of malaria, (1982), Griffin London · Zbl 0494.92018  Dye, C.; Hasibeder, G., Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others, Trans. R. soc. trop. med. hyg, 80, 69-77, (1986)  Holling, C.S., The functional response to prey density and its role in mimicry and population regulation, Mem. entomol. soc. canad, 45, 1-60, (1965)  Lajmanovich, A.; Yorke, J.A., A deterministic model for gonorrhea in a non–homogeneous population, Math. biosci, 28, 221-236, (1976) · Zbl 0344.92016  Macdonald, G., The epidemiology and control of malaria, (1957), Oxford Univ. Press London  Nold, A., Heterogeneity in disease-transmission modeling, Math. biosci, 52, 227-240, (1980) · Zbl 0454.92020  Ross, R., The prevention of malaria, (1911), Murray London  Seneta, E., Non-negative matrices and Markov chains, (1981), Springer New York · Zbl 0471.60001
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