dependence among individuals, is described in a two step procedure. Thus the feedback loop that creates density dependence, i.e. Next the model specifies the impact of individuals on the environmental variables. Model specification involves, first of all, a description of changes in time of the i-state as influenced by i-state itself and the relevant environmental variables that capture the influence of the outside world. This means that we start from the notion of state at the individual level, called i-state (where i stands for individual). 2015), our model formulation is in the tradition of physiologically structured population models (PSPM Metz and Diekmann 1986 Diekmann et al. Physiologically structured population modelsĪs in our earlier paper (Leung et al. We end the introduction with an outline of the structure of the rest of the paper. Individuals are decomposed into conditionally independent components (the ‘binding sites’) and we discuss how the dynamics of these binding sites can be specified. Next, we consider three different settings based on the time scales of disease spread, partnership dynamics, and demographic turnover. In the rest of this introduction we first discuss the model formulation used and the relation between our work and existing literature. 2012, 2015) that also takes birth and death into account. In case of HIV, the disease spreads on the time scale of demographic turnover and this motivated our earlier work (Leung et al. But if the disease spreads at the time scale of formation and dissolution of partnerships, we need to take these partnership dynamics into account and next indeed rely on wishful thinking (though the answer may very well be ‘yes’). 2012 Barbour and Reinert 2013 Janson et al. If we are willing to assume that the network is constructed by the configuration procedure (Durrett 2006 van der Hofstad 2015), the answer is indeed ‘yes’ (Decreusefond et al. When considering an outbreak of a rapidly spreading disease, we can consider the network as static. And even if the true answer is still ‘no’, we may indulge in wishful thinking and answer ‘to good approximation’. But if we are willing to make assumptions about the structure (and to consider the limit of the number of individuals going to infinity), the answer might be ‘yes’. Is it possible to predict the future spread of the disease on the basis of this statistical description? The answer is ‘no’, simply because the precise network structure is important for transmission and we cannot recover the structure from the description. In this spirit, we may provide a statistical description of the network at a particular point in time by listing, for each such label, the fraction of the population carrying it. Its disease status in terms of the S, I, R classification, where, as usual, S stands for susceptible, I for infectious and R for recovered (implying immunity) We are interested in the disease status of the individual, but also in the presence of the infection in its immediate surroundings that are formed by the individual’s partners. Consider an individual in the network at a particular point in time. Suppose an infectious disease can be transmitted from an infectious individual to any of its susceptible partners and thus spread over the network. Thus we are able to characterize population-level epidemiological quantities, such as \(R_0\), r, the final size, and the endemic equilibrium, in terms of the corresponding variables.Ĭonsider an empirical network consisting of individuals that form partnerships with other individuals. In particular, individual-level probabilities are obtained from binding-site-level probabilities by combinatorics while population-level quantities are obtained by averaging over individuals in the population. The Markov chain dynamics of binding sites are described by only a few equations. A key characteristic of the network models is that individuals can be decomposed into a number of conditionally independent components: each individual has a fixed number of ‘binding sites’ for partners. These environmental variables are population level quantities. Influences from the ‘outside world’ on an individual are captured by environmental variables. In the tradition of physiologically structured population models, the formulation starts on the individual level. We distinguish three different levels: (1) binding sites, (2) individuals, and (3) the population. We formulate models for the spread of infection on networks that are amenable to analysis in the large population limit.
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