Abstract. The expected number o f secondary cases produced by a typical infected individual during its entire period o f infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.

The basic reproduction number R 0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of R 0 where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain. The three matrices together ref lect the range of possibilities encountered in the literature for the characterization of R 0 . We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number R 0 . Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that R 0 defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r, the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that R 0 . 1 if and only if r . 0, and R 0 ¼ 1 if and only if r ¼ 0.

No abstract

No abstract

Mylius, S.D. and Diekman, O. 1995. On evolutionarily stable life histories, optimization and the need to be specific about density dependence.-Oikos 74: 218-224. 1 Concentrating on monomorphic populations in demographic steady state, we give three different conditions under which the evolutionarily stable life-history strategy can be characterized as the life-history strategy at which a relatively simple function is maximal. Depending on the way density dependence acts, this function, or fitness measure, can be the lifetime production of offspring, the population growth rate, or another quantity from a large range of possible optimization criteria. We illustrate this by examining the optimal age at maturity for a hypothetical example organism. All of this demonstrates that, when studying the evolutionary aspects of life-history characteristics, one cannot escape the task of specifying how density dependence limits population growth.