##
**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.

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.

Reviewer: Michael I. Gil’ (Beer-Sheva)

### 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 |