Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

Abstract

We present the generalised mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalised model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma-and uniformly-distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalised pairwise model lies in approximating the time evolution of the epidemic.

An SIR rumor pairwise model with spreading age

“…The present paper extend the Röst SIR model [13] and the other literature [9,15,21] to establish a rumor pairwise model and analyze the effect of spreading age on the speed of the rumor transmission in complex network. The rest of this article is arranged as following.…”

A monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling

“…For many epidemics, the infectious period has great importance and it is measured empirically. Recently, pairwise approximations of the SIR dynamics with non-Markovian recovery have been derived, see [12,26,21,22]. In the special case of fixed recovery time σ , the meanfield model is given by…”

“…Considering a general distribution for the recovery period, the pairwise model can be formulated as a system of integro-differential equations [22,26], which is given bẏ…”

Pairwise models for non-Markovian epidemics on networks

“…We develop the models for the fixed and the general case, simulate explicitly the stochastic process for comparison with the solutions of the resulting systems and study the most important features, such as reproduction number and implicitly given final size relation. These results are summarised in three scientific papers of István Kiss, Gergely Röst and the author of this thesis, see [49], [70] and [71]. We consider SIR dynamics, but the introduced framework may be applied to more exotic dynamics as well.…”

“…We introduced these definitions in [49] and use these concepts and results in our further papers in this topic ( [70], [71]). The pairwise model is written at the level of links and describes the dynamics of susceptible (S − S) and infected (S − I) links.…”