SIR dynamics in random networks with heterogeneous connectivity |
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Authors: | Erik Volz |
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Institution: | (1) Department of Integrative Biology, University of Texas, Austin, TX, USA |
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Abstract: | Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the
problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how
the SIR dynamics can be modeled with a system of three nonlinear ODE’s. The method makes use of the probability generating
function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities
such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds
or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining
the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution
among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays
on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions
are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree
distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the
theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale
epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.
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Keywords: | Epidemic disease SIR Networks Degree distribution |
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