Mean-field-like approximations for stochastic processes on weighted and dynamic networks

The explicit use of networks in modelling stochastic processes such as epidemic dynamics has revolutionised research into understanding the impact of contact pattern properties, such as degree heterogeneity, preferential mixing, clustering, weighted and dynamic linkages, on how epidemics invade, spread and how to best control them. In this thesis, I worked on mean-field approximations of stochastic processes on networks with particular focus on weighted and dynamic networks. I mostly used low dimensional ordinary differential equation (ODE) models and explicit network-based stochastic simulations to model and analyse how epidemics become established and spread in weighted and dynamic networks. I begin with a paper presenting the susceptible-infected-susceptible/recovered (SIS, SIR) epidemic models on static weighted networks with different link weight distributions. This work extends the pairwise model paradigm to weighted networks and gives excellent agreement with simulations. The basic reproductive ratio, R0, is formulated for SIR dynamics. The effects of link weight distribution on R0 and on the spread of the disease are investigated in detail. This work is followed by a second paper, which considers weighted networks in which the nodal degree and weights are not independent. Moreover, two approximate models are explored: (i) the pairwise model and (ii) the edge-based compartmental model. These are used to derive important epidemic descriptors, including early growth rate, final epidemic size, basic reproductive ratio and epidemic dynamics. Whilst the first two papers concentrate on static networks, the third paper focuses on dynamic networks, where links can be activated and/or deleted and this process can evolve together with the epidemic dynamics. We consider an adaptive network with a link rewiring process constrained by spatial proximity. This model couples SIS dynamics with that of the network and it investigates the impact of rewiring on the network structure and disease die-out induced by the rewiring process. The fourth paper shows that the generalised master equations approach works well for networks with low degree heterogeneity but it fails to capture networks with modest or high degree heterogeneity. In particular, we show that a recently proposed generalisation performs poorly, except for networks with low heterogeneity and high average degree

A class of pairwise models for epidemic dynamics on weighted networks

In this paper, we study the $SIS$ (susceptible-infected-susceptible) and $SIR$ (susceptible-infected-removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both models start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with network simulation to evaluate the impact of different weight distributions on epidemic threshold and dynamics in general. For the $SIR$ dynamics, the basic reproductive ratio $R_0$ is computed, and we show that (i) for both network models $R_{0}$ is maximised if all weights are equal, and (ii) when the two models are equally matched, the networks with a random weight distribution give rise to a higher $R_0$ value. The models are also used to explore the agreement between the pairwise and simulation models for different parameter combinations