Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold
Abstract
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 \emph{networks}. Although SIR and SIS are basic epidemic models, deductions from their exact stochastic equations \textbf{without} making approximations (such as the common meanfield 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 simplied to exhibit the explicit dependence on the spectral radius. From our new Laplacian expression, we deduce a general \textbf{upper} bound for the epidemic SIS threshold in any graph.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.1731
 Bibcode:
 2014arXiv1402.1731V
 Keywords:

 Mathematics  Dynamical Systems;
 Quantitative Biology  Populations and Evolution