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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.

Increasing evidence that the strengths of interactions among populations in biological communities form patterns that are crucial for system stability requires clarification of the precise form of these patterns, how they come about, and why they influence stability. We show that in real food webs, interaction strengths are organized in trophic loops in such a way that long loops contain relatively many weak links. We show and explain mathematically that this patterning enhances stability, because it reduces maximum “loop weight” and thus reduces the amount of intraspecific interaction needed for matrix stability. The patterns are brought about by biomass pyramids, a feature common to most ecosystems. Incorporation of biomass pyramids in 104 food-web descriptions reveals that the low weight of the long loops stabilizes complex food webs. Loop-weight analysis could be a useful tool for exploring the structure and organization of complex communities.

Despite some notable successes in the control of infectious diseases, transmissible pathogens still pose an enormous threat to human and animal health. The ecological and evolutionary dynamics of infections play out on a wide range of interconnected temporal, organizational and spatial scales, which even within a single pathogen often span hours to months, cellular to ecosystem levels, and local to pandemic spread. Some pathogens are directly transmitted between individuals of a single species, while others circulate among multiple hosts, need arthropod vectors, or can survive in environmental reservoirs. Many factors, including increasing antimicrobial resistance, increased human connectivity, and dynamic human behavior, raise prevention and control from formerly national to international issues. In the face of this complexity, mathematical models offer essential tools for synthesizing information to understand epidemiological patterns, and for developing the quantitative evidence base for decision-making in global health.