Host-parasite models with density-dependent (mass action) incidence and a critical Allee effect in host growth can explain both species decline and disappearance (extinction). The behaviour of the model is consistent with both the novel pathogen hypothesis and the endemic pathogen hypothesis for chytridiomycosis. Mathematically, the transition from decline to disappearance is mediated by a Hopf bifurcation and is marked by the occurrence of a heteroclinic orbit. The Hopf bifurcation is supercritical if intra-specific host competition increases with host density at a large power and subcritical if the power is small. In the supercritical case, host-parasite coexistence can be at equilibrium or periodic; in the subcritical case it is only at equilibrium.

The question whether a vertically and a horizontally transmitted parasite strain can coexist under complete cross protection is investigated in a host-parasite model with susceptibles and infectives only. It is shown that coexistence is possible even if the vertically transmitted strain would go extinct on its own provided that it is considerably less virulent than the horizontally transmitted strain. While the vertical transmitted strain is without benefit to the host as such, it protects the host against the more harmful horizontally transmitted strain. The coexistence is shown in the form of uniform strong persistence of the host and both parasite strains.

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Spatially implicit metapopulation models with discrete patch-size structure and host-macroparasite models which distinguish hosts by their parasite loads lead to infinite systems of ordinary differential equations. In several papers, a this-related theory will be developed in sufficient generality to cover these applications. In this paper the linear foundations are laid. They are of own interest as they apply to continuous-time population growth processes (Markov chains). Conditions are derived that the solutions of an infinite linear system of differential equations, known as Kolmogorov's differential equations, induce a C0-semigroup on an appropriate sequence space allowing for first moments. We derive estimates for the growth bound and the essential growth bound and study the asymptotic behavior. Our results will be illustrated for birth and death processes with immigration and catastrophes.