# American Institute of Mathematical Sciences

July  2018, 23(5): 2005-2020. doi: 10.3934/dcdsb.2018192

## On bounding exact models of epidemic spread on networks

 1 Institute of Mathematics, Eötvös Loránd University Budapest, Hungary 2 Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary 3 School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK

* Corresponding author: Péter L. Simon

Received  March 2017 Revised  October 2017 Published  May 2018

Fund Project: The first author is supported by Hungarian Scientific Research Fund, OTKA, (grant no. 115926).

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that closures or approximations at individual or node level lead to mean-field models that bound the exact stochastic process from above. This will be done in the context of modelling epidemic spread on networks and the proof of the result relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively). The results in the paper hold for Markovian epidemics and arbitrary weighted and directed networks. Furthermore, we cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models. While deterministic initial conditions are key to obtain the negative correlation result we show that this condition can be relaxed as long as extra conditions on the edge weights are imposed.

Citation: Péter L. Simon, Istvan Z. Kiss. On bounding exact models of epidemic spread on networks. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2005-2020. doi: 10.3934/dcdsb.2018192
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The relation of the joint and marginal probabilities.
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 $\left\langle {I_i} \right\rangle$ $\left\langle {S_i} \right\rangle$ $\left\langle {I_j} \right\rangle$ a b q $\left\langle {S_j} \right\rangle$ c d 1-q p 1-p
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