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Dangerous connections: on binding site models of infectious disease dynamics. (English) Zbl 1364.92048

This paper represents an elaborate attempt to model the spread of a disease on a network of individuals. The modeling is done at three different levels: binding sites, individuals and population. While the latter two levels are the more common building blocks for disease propagation models, the former is introduced through the assumptions that each individual has a fixed number of binding sites and that two free binding sites corresponding to different individuals can be joined together to form a partnership between these individuals. It is assumed that the binding sites are coupled through the disease status of the individual they belong to and are otherwise independent. At both the binding site and the individual level, one finds a Markov chain dynamics described by a limited number of equations. At the population level, the model becomes deterministic. The authors study \(R_0\), the basic reproduction number, \(r\) the Malthusian parameter, and the final size of the epidemic. First, the authors consider the case of a static network, for which an one-dimensional renewal equation is derived. As a result, \(R_0\), \(r\) and the final size of the epidemic can be derived via straightforward arguments from the renewal equation. In this case, the number of binding sites is allowed to vary from individual to individual. Then, the case of a dynamic network without demographic turnovers, but with partnership changes, is considered. In this case, the dynamics of the susceptible binding sites is given by a six-dimensional closed system of ODEs or as a more involved renewal equation for the fraction of binding sites which are infected. In this case, the final size of the epidemic could only be characterized in an implicit manner, rather than explicit. Finally, the case of a dynamic network with not only partnership formation and separation, but also with demography, is considered under the assumption that the age of individuals is exponentially distributed. A system of three renewal equations is derived to describe the dynamics of binding sites. Alternatively, this could be replaced with a much higher-dimensional system of ODEs. It is observed, however, that the system of renewal equations can be used to obtain \(R_0\) and \(r\) via linearization near the disease-free equilibrium. The paper concludes with a comprehensive list of open problems and conjectures which are indicated as possible directions of further research. These are given along with remarks regarding generalizations concerning the network structure or concerning the propagation of the disease and comments regarding an attempt to prove the existence and uniqueness of the steady state.

MSC:

92D30 Epidemiology
92D25 Population dynamics (general)
34D20 Stability of solutions to ordinary differential equations
60J28 Applications of continuous-time Markov processes on discrete state spaces
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