Differential equation approximations of stochastic network processes: An operator semigroup approach
András Bátkai Istvan Z. Kiss Eszter Sikolya Péter L. Simon
Networks & Heterogeneous Media 2012, 7(1): 43-58 doi: 10.3934/nhm.2012.7.43
The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size ($N$). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as $N$ tends to infinity. Using only elementary semigroup theory we can prove the order $\mathcal{O}(1/N)$ convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.
keywords: one-parameter operator semigroup birth-and-death process mean field approximation. Dynamic network
On bounding exact models of epidemic spread on networks
Péter L. Simon Istvan Z. Kiss
Discrete & Continuous Dynamical Systems - B 2018, 23(5): 2005-2020 doi: 10.3934/dcdsb.2018192

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.

keywords: Mean-field model cooperative system differential inequality

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