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For a heterogeneous host population, the basic reproduction number of an infectious disease, $\cR_0$, is defined as the spectral radius of the next generation operator (NGO). The threshold properties of the basic reproduction number are typically established by imposing conditions that make $\cR_0$ an eigenvalue of the NGO associated with a positive eigenvector and a positive eigenfunctional (eigenvector of the dual of the NGO). More general results can be obtained by imposing conditions that associate $\cR_0$ just with a positive eigenfunctional. The next generation operator is conveniently expressed by a measure kernel or a Feller kernel which enables the use of analytic rather than functional analytic methods., Comment: 47 pages

The suggestion by K.L. Cooke (1967) that infected individuals become infective if they are exposed often enough for a natural disease resistance to be overcome is built into a Kermack-McKendrick type epidemic model with infectivity age. Both the case that the resistance may be the same for all hosts and the case that it is distributed among the host population are considered. In addition to the familiar threshold behavior of the final size of the epidemic with respect to a basic reproductive number, an Allee effect is generated with respect to the final cumulative force of infection exerted by the initial infectives. This offers a deterministic explanation of why geographic areas that appear to be epidemiologically similar have epidemic outbreaks of quite different severity.

We study the long-time behavior of solutions to a measure-valued selection-mutation model that we formulated in \cite{CLEVACK}. We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy (ASS). We also show that when the strategy space is discrete, the selection-mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy.

It is well known that an ordered normed vector space $X$ with normal cone $X_+$ has an order-preserving norm that is equivalent to the original norm. Such an equivalent order-preserving norm is given by \begin{equation} \sharp x \sharp = \max \{ d(x, X_+), d(x, - X_+)\}, \qquad x \in X. \end{equation} This paper explores the properties of this norm and of the half-norm $\psi(x) = d(x,-X_+)$ independently of whether or not the cone is normal. We use $\psi$ to derive comparison principles for the solutions of abstract integral equations, derive conditions for point-dissipativity of nonlinear positive maps, compare Collatz-Wielandt numbers, bounds, and order spectral radii for bounded homogeneous maps and give conditions for a local upper Collatz-Wielandt radius to have a lower positive eigenvector., Comment: An earlier version had a mistake in the definition of the lower local Collatz-Wielandt radius which affected various proofs in the paper

This paper considers homogeneous order preserving continuous maps on the normal cone of an ordered normed vector space. It is shown that certain operators of that kind which are not necessarily compact themselves but have a compact power have a positive eigenvector that is associated with the cone spectral radius. We also derive conditions for the existence of homogeneous order preserving eigenfunctionals. Our results are illustrated in a model for spatially distributed two-sex populations., Comment: 41 pages