We are interested in the spatiotemporal dynamics of infectious diseases in animal populations. Hence, besides epidemiological aspects the vital dynamics as well as the spatial spread become important. However, there is still not so much known about the interplay of the latter ones. By way of example, the circulation of the Feline Immunodeficieny Virus (FIV) within domestic cats (Felis catus, L.) is considered. FIV induces AIDS in cats.
We construct a two-compartmental reaction-diffusion model consisting of susceptibles and infected. We shall mainly consider the case, in which the vital dynamics are assumed to exhibit an Allee effect, i.e. there is a minimum viable population density below which the population goes extinct. Two different cases of transmission between susceptibles and infected are considered.
(i) In the case of the standard incidence, global stability results of the local dynamics are presented including the non-existence of periodic solutions. Adding diffusion, the emergence of travelling frontal waves is numerically demonstrated. Critical spatial size phenomena can be observed and the situatiuon, in which an introduced pathogen induces the host population to go extinct, if the host itself invades open space.
(ii) In the case of mass action transmission, the situation becomes much more complex. The local dynamics may show Hopf, saddle-node and homoclinic bifurcations, thus giving rise to Bogdanov-Takens and cusp points as well as to multiple equilibria including tri-stability. Adding diffusion, examples of numerically simulated target patterns and spiral waves are given.
Finally, these results are contrasted with vital dynamics of the logistic kind, demonstrating that the Allee effect adds interesting and qualitatively new dynamics to epidemiology.