An upper bound for the epidemic threshold in exact Markovian SIR and SIS epidemics on networks

Exploiting the power of the expectation operator and indicator (or Bernoulli) random variables, we present the exact governing equations for both the SIR and SIS epidemic models on networks. Although SIR and SIS are basic epidemic models, deductions from their exact stochastic equations without making approximations (such as the common mean-field approximation) are scarce. An exact analytic solution of the governing equations is highly unlikely to be found (for any network) due to the appearing pair (and higher order) correlations. Nevertheless, the maximum average fraction y_I of infected nodes in both SIS and SIR can be written as a quadratic form of the graph´s Laplacian. Only for regular graphs, the expression for the maximum of y_I can be simplified to exhibit the explicit dependence on the spectral radius. From our new Laplacian expression, we deduce a general upper bound for the epidemic SIS threshold in any graph.