SEIR Epidemic Dynamics in Random Networks

Predicting disease transmission on complex networks has attracted considerable recent attention in the epidemiology community. In this paper, we develop a low-dimensional system of nonlinear ordinary differential equations to model the susceptible-exposed-infectious-recovered (SEIR) epidemics on random network with arbitrary degree distributions. Both the final size of epidemics and the time-dependent behaviors are derived within our simple framework. The underlying network is represented by the configuration model, which appropriately accounts for the heterogeneity and finiteness of the degree observed in a variety of real contact networks. Moreover, a generalized model where the infectious state of individual can be skipped is treated in brief. 1. Introduction Infectious diseases spread over networks of contacts between susceptible and infectious individuals. Typical mathematical representation of an epidemic assumes that the host populations are fully mixed (mass-action approximation) [1, 2], that is, every individual has an equal opportunity to infect others and the underlying network topology is modelled as a fully connected graph. However, in the real world, the contact patterns are characterized by high levels of heterogeneity and each individual only has contact with a small fraction of the population [3–5]. In recent years, a number of researches have addressed the contact patterns among individuals as random networks [6–14], which allow for more realistic and accurate capture of heterogeneities in the number of contacts compared with classical fully mixed models. Network epidemic models make use of network topology of potential contacts instead of assuming that contact is possible with the total population. Some quantities of interest such as epidemic probability and mean final size of epidemics have been precisely solved in random networks with specified degree distributions (configuration models) using ideas drawn from percolation theory [9, 10, 15]. The heterogeneity introduced in the network framework, nevertheless, makes it rather difficult to analytically describe the time-dependent properties and the dynamical course of an epidemic. Some researchers made it by using high-dimensional pair-approximation methods (or moment closure methods) [4, 16, 17], which typically neglect the correlations between the states of nodes some steps away from each other, while others adopted approximate approaches that assume all nodes of the same degree having the same infection probability at any given time [3, 18, 19]. In addition, a good deal of effort