Analytical solution of susceptible-infected-recovered models on homogeneous networks

The ability to actually implement epidemic models is a crucial stake for public institutions, as they may be overtaken by the increasing complexity of current models and sometimes tend to revert to less elaborate models such as the susceptible-infected-recovered (SIR) model. In our work, we study a simple epidemic propagation model, called SIR-k, which is based on a homogeneous network of degree k, where each individual has the same number k of neighbors. This model represents a refined version of the basic SIR which assumes a completely homogeneous population. We show that nevertheless, analytical expressions, simpler and richer than the ones existing for the SIR model, can be derived for this SIR-k model. In particular, we obtain an exact implicit analytical solution for any k, from which quantities such as the epidemic threshold or the total number of agents infected during the epidemic can be obtained. We furthermore obtain simple exact explicit solutions for small ks, and in the large k limit we find a new formulation of the analytical solution of the basic SIR model, which comes with new insights.