J.A.P. Heesterbeek

On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations

The expected number of secondary cases produced by a typical infected individual during its entire period of 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 construction of next-generation matrices for compartmental epidemic models.

The basic reproduction number ℛ0 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 ℛ0 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 reflect the range of possibilities encountered in the literature for the characterization of ℛ0. 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 ℛ0. 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 ℛ0 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 ℛ0 > 1 if and only if r > 0, and ℛ0 = 1 if and only if r = 0.

Reconciling complexity with stability in naturally assembling food webs

Understanding how complex food webs assemble through time is fundamental both for ecological theory and for the development of sustainable strategies of ecosystem conservation and restoration. The build-up of complexity in communities is theoretically difficult, because in random-pattern models complexity leads to instability. There is growing evidence, however, that nonrandom patterns in the strengths of the interactions between predators and prey strongly enhance system stability. Here we show how such patterns explain stability in naturally assembling communities. We present two series of below-ground food webs along natural productivity gradients in vegetation successions. The complexity of the food webs increased along the gradients. The stability of the food webs was captured by measuring the weight of feedback loops of three interacting ‘species’ locked in omnivory. Low predator–prey biomass ratios in these omnivorous loops were shown to have a crucial role in preserving stability as productivity and complexity increased during succession. Our results show the build-up of food-web complexity in natural productivity gradients and pin down the feedback loops that govern the stability of whole webs. They show that it is the heaviest three-link feedback loop in a network of predator–prey effects that limits its stability. Because the weight of these feedback loops is kept relatively low by the biomass build-up in the successional process, complexity does not lead to instability.

A new method for estimating the effort required to control an infectious disease

We propose a new threshold quantity for the analysis of the epidemiology of infectious diseases. The quantity is similar in concept to the familiar basic reproduction ratio, R0, but it singles out particular host types instead of providing a criterion that is uniform for all host types. Using this methodology we are able to identify the long–term effects of disease–control strategies for particular subgroups of the population, to estimate the level of control necessary when targeting control effort at a subset of host types, and to identify host types that constitute a reservoir of infection. These insights cannot be obtained by using R0 alone.

The saturating contact rate in marriage- and epidemic models

In this note we show how to derive, by a mechanistic argument, an expression for the saturating contact rate of individual contacts in a population that mixes randomly. The main assumption is that the individual interaction times are typically short as compared to the time-scale of changes in, for example, individual-type, but that the interactions yet make up a considerable fraction of the time-budget of an individual. In special cases an explicit formula for the contact rate is obtained. The result is applied to mathematical epidemiology and marriage models.

The Basic Reproduction Number for Complex Disease Systems : Defining R0 for Tick-Borne Infections

Characterizing the basic reproduction number, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape

Prioritizing emerging zoonoses in The Netherlands.

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Vector-borne diseases and the basic reproduction number: a case study of African horse sickness

\end{document} , for many wildlife disease systems can seem a complex problem because several species are involved, because there are different epidemiological reactions to the infectious agent at different life‐history stages, or because there are multiple transmission routes. Tick‐borne diseases are an important example where all these complexities are brought together as a result of the peculiarities of the tick life cycle and the multiple transmission routes that occur. We show here that one can overcome these complexities by separating the host population into epidemiologically different types of individuals and constructing a matrix of reproduction numbers, the so‐called next‐generation matrix. Each matrix element is an expected number of infectious individuals of one type produced by a single infectious individual of a second type. The largest eigenvalue of the matrix characterizes the initial exponential growth or decline in numbers of infected individuals. Values below 1 therefore imply that the infection cannot establish. The biological interpretation closely matches that of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape