Analysis of Exact and Approximated Epidemic Models over Complex Networks
Abstract
We study the spread of discretetime epidemics over arbitrary networks for wellknown propagation models, namely SIS (susceptibleinfectedsusceptible), SIR (susceptibleinfectedrecovered), SIRS (susceptibleinfectedrecoveredsusceptible) and SIV (susceptibleinfectedvaccinated). 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 "meanfield" approximation, and its linearization, are often studied instead. We provide a complete global analysis of the epidemic dynamics of the nonlinear meanfield 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 socalled endemic state where a constant fraction of the nodes remain infected). A similar result has also been shown in the continuoustime case. We tie in these results with the "true" underlying Markov chain model by showing that the linear model is the tightest upperbound on the true probabilities of infection that involves only marginals, and that, even though the nonlinear model is not an upperbound on the true probabilities in general, it does provide an upperbound on the probability of the chain not being absorbed. As a consequence, we also show that when the diseasefree fixed point is globally stable for the meanfield 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 arXiv:
 arXiv:1609.09565
 Bibcode:
 2016arXiv160909565A
 Keywords:

 Computer Science  Social and Information Networks;
 Mathematics  Dynamical Systems
 EPrint:
 Submitted to the IEEE Transactions on Network Science and Engineering