Analysis of Exact and Approximated Epidemic Models over Complex Networks

We study the spread of discrete-time epidemics over arbitrary networks for well-known propagation models, namely SIS (susceptible-infected-susceptible), SIR (susceptible-infected-recovered), SIRS (susceptible-infected-recovered-susceptible) and SIV (susceptible-infected-vaccinated). Such epidemics are described by $2^n$- or $3^n$-state Markov chains. Ostensibly, because analyzing such Markov chains is too complicated, their $O(n)$-dimensional nonlinear "mean-field" approximation, and its linearization, are often studied instead. We provide a complete global analysis of the epidemic dynamics of the nonlinear mean-field approximation. In particular, we show that depending on the largest eigenvalue of the underlying graph adjacency matrix and the rates of infection, recovery, and vaccination, the global dynamics takes on one of two forms: either the epidemic dies out, or it converges to another unique fixed point (the so-called endemic state where a constant fraction of the nodes remain infected). A similar result has also been shown in the continuous-time case. We tie in these results with the "true" underlying Markov chain model by showing that the linear model is the tightest upper-bound on the true probabilities of infection that involves only marginals, and that, even though the nonlinear model is not an upper-bound on the true probabilities in general, it does provide an upper-bound on the probability of the chain not being absorbed. As a consequence, we also show that when the disease-free fixed point is globally stable for the mean-field model, the Markov chain has an $O(\log n)$ mixing time, which means the epidemic dies out quickly. We compare and summarize the results on different propagation models.