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Stability of steady states and existence of travelling waves in a vector-disease model. (English) Zbl 1065.35059

The authors consider a host-vector model for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modelled in the partial integro-differential equation by a diffusion term. Throughout recorded history, non-indigenous vectors that arrive, establish and spread in new areas have fomented epidemics of human diseases such as malaria, yellow fever, typhus, plague and West Nile. Such vector-borne diseases are now major public health problems throughout the world. The spatial spread of newly introduced diseases is a subject of continuing interest to both theoreticians and empiricists.
In the paper for the general model, the stability of the steady states is studied using the contracting-convex-sets technique. When the spatial variable is one dimensional and the delay kernel assumes some special form, the existence of travelling wave solutions is established by using the linear chain trick and the geometric singular perturbation method.

MSC:

35B35 Stability in context of PDEs
35A18 Wave front sets in context of PDEs
35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
35B25 Singular perturbations in context of PDEs
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