Degree dependent transmission rates in epidemic processes

Abstract

The outcome of SIR epidemics with heterogeneous infective lifetimes, or heterogeneous susceptibilities, can be mapped onto a directed percolation process on the underlying contact network. In this paper we study SIR models where heterogeneity is a result of the degree dependence of disease transmission rates. We develop numerical methods to determine the epidemic threshold, the epidemic probability and epidemic size close to the threshold for configuration model contact networks with arbitrary degree distribution and an arbitrary matrix of transmission rates (dependent on transmitting and receiving node degree). For the special case of separable transmission rates we obtain analytical expressions for these quantities. We propose a categorization of spreading processes based on the ratio of the probability of an epidemic and the expected size of an epidemic, and demonstrate that this ratio has a complex dependence on the degree distribution and the degree-dependent transmission rates. For scale-free contact networks and transmission rates that are power functions of transmitting and receiving node degrees, the epidemic threshold may be finite even when the degree distribution powerlaw exponent is below $\gamma <3$. We give an expression, in terms of the degree distribution and transmission rate exponents, for the limit at which the epidemic threshold vanishes. We find that the expected epidemic size and the probability of an epidemic may grow nonlinearly above the epidemic threshold, with exponents that depend not only on the degree distribution powerlaw exponent, but on the parameters of the transmission rate degree dependence functions, in contrast to ordinary directed percolation and previously studied variations of the SIR model.

arXiv comment32 pages, 9 figures

References in corpus (15)

• Epidemic spreading in scale-free networks
• Epidemic processes in complex networks
• The spread of epidemic disease on networks
• Critical phenomena in complex networks
• Epidemic spreading in correlated complex networks
• Percolation Critical Exponents in Scale-Free Networks
• Non mean-field behavior of the contact process on scale-free networks
• Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation
• Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing
• Nonequilibrium phase transitions and finite size scaling in weighted scale-free networks
• Collective versus hub activation of epidemic phases on networks
• Nonlinear spread of rumor and inoculation strategies in the nodes with degree dependent tie strength in complex networks
• Trading interactions for topology in scale-free networks
• Assessing node risk and vulnerability in epidemics on networks
• A weighted configuration model and inhomogeneous epidemics

Cited by in corpus (1)

• Generating functions for message-passing on weighted networks: directed bond percolation and SIR epidemics