Degree dependent transmission rates in epidemic processes

TL;DR

This paper investigates how degree-dependent transmission rates influence epidemic thresholds and sizes in SIR models on complex networks, providing numerical and analytical tools for diverse degree distributions and transmission dependencies.

Contribution

It introduces numerical methods and analytical expressions for epidemic thresholds and sizes considering degree-dependent transmission rates in heterogeneous networks.

Findings

Epidemic thresholds can be finite in scale-free networks with certain transmission rate dependencies.

The ratio of epidemic probability to size exhibits complex dependence on network degree distribution.

Epidemic growth above the threshold can be nonlinear, influenced by degree distribution and transmission parameters.

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.