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A deterministic epidemic model taking account of repeated contacts between the same individuals. (English) Zbl 0940.92023

From the introduction: The transmission of an infectious agent is often predicated upon some form of contact between two hosts. A submodel for the host contact process is, therefore, an important constituent of a model for the spread of an infectious disease. The law of mass action asserts that contact rates are proportional to the densities of the types of individuals involved, where it is allowed that the population is stratified according to spatial or social position, the latter capturing a large variety of context dependent distinctions. With finite rates, the expected number of contacts of an individual in a given finite interval of time is finite. In the deterministic limit the subpopulation sizes are, by assumption, infinite and hence the probability that two particular individuals have contact equals zero. As a consequence, the probability that two individuals have contact twice is zero as well.
Such properties of the contact model contradict our daily experience of (human and animal) social relationships (and therefore motivated ad hoc modifications). So it seems relevant to study the effect of repeated contacts with a (possibly large, either fixed or changeable) group of ‘acquaintances’ on the spread of an infectious disease. The acquaintance relation defines a network of connections among the members of the population and the modelling now entails a specification of this network. In spatial lattice models one assumes a very regular contact network. In random graph models one only prescribes certain statistical properties of the network. In pair formation models one assumes that contacts are restricted to partners that form an isolated pair for an extended period of time. Yet another class of models stratifies the population according to households.
Here we consider a class of models that combines features of spatial lattice and (the limit of) random graph models. We picture an individual as surrounded by a circle of \(k\) acquaintances. But these acquaintances do not contact each other. In fact we assume that the probability that two acquaintances have other acquaintances in common equals, in the infinite population, zero. Admittedly this again contradicts our daily experience and we think that, ultimately, models should be constructed in which the correlation between acquaintances of acquaintances is neither as rigid as it is in spatial lattices nor as absent as we postulate it here.

MSC:

92D30 Epidemiology
05C80 Random graphs (graph-theoretic aspects)
60J85 Applications of branching processes
05C90 Applications of graph theory
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