Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold

This is the authors' abstract. We don't add key points for this paper.

Exploiting the power of the expectation operator and indicator (or Bernoulli) variables, we present the exact governing equations for both the SIR and epidemic models on . Although SIR and SIS are basic epidemic, deductions from their exact stochastic equations approximations (such as the common mean-field approximation) are scarce. exact analytic solution of the governing equations is highly unlikely to be (for any network) due to the appearing pair (and higher order). Nevertheless, the maximum average fraction yI of infected in both SIS and SIR can be written as a quadratic form of the graph's. Only for regular graphs, the expression for the maximum of yI be simplied to exhibit the explicit dependence on the spectral radius. From new Laplacian expression, we deduce a general bound for the SIS threshold in any graph.