SIS epidemic spreading with correlated heterogeneous infection rates

The epidemic spreading has been widely studied in homogeneous cases, where each node may get infected by an infected neighbor with the same rate. However, the infection rate between a pair of nodes, which may depend on e.g. their interaction frequency, is usually heterogeneous and even correlated with their nodal degrees in the contact network. In this paper, we aim to understand how such correlated heterogeneous infection rates influence the epidemic spreading on different network topologies. Motivated by real-world datasets, we propose a Correlated heterogeneous Susceptible–Infected–Susceptible (CSIS) model which assumes that the infection rate βij(=βji) between node i and j is correlated with the degree of the two end nodes: βij=c(didj)α, where α indicates the strength of the correlation between the infection rates and nodal degrees, and c is selected so that the average infection rate is 1 in this work. The recovery rate is the same for all the nodes. In order to understand the effect of such correlation on epidemic spreading, we consider as well the corresponding uncorrected but still heterogeneous infection rate scenario as a reference, where the original correlated infection rates in our CSIS model are shuffled and reallocated to the links of the same network topology. We compare these two scenarios in the average fraction of infected nodes in the metastable state on Erdös–Rényi (ER) and scale-free (SF) networks with a similar average degree. Through the continuous-time simulations, we find that, when the recovery rate is small, the negative correlation is more likely to help the epidemic spread and the positive correlation prohibits the spreading; as the recovery rate increases to be larger than a critical value, the positive but not negative correlation tends to help the spreading. Our findings are further analytically proved in a wheel network (one central node connects with each of the nodes in a ring) and validated on real-world networks with correlated heterogeneous interaction frequencies.