We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

We extrapolate from the exact master equations of epidemic dynamics on fully connectedgraphs to non-fully connected by keeping the size of the state space N + 1, whereN is thenumber of nodes in the graph. This gives rise to a system of approximate ODEs (ordinarydifferential equations) where the challenge is to compute/approximate analytically thetransmission rates. We show that this is possible for graphs with arbitrary degreedistributions built according to the configuration model. Numerical tests confirm that:(a) the agreement of the approximate ODEs system with simulation is excellent and (b) thatthe approach remains valid for clustered graphs with the analytical calculations of thetransmission rates still pending. The marked reduction in state space gives good results,and where the transmission rates can be analytically approximated, the model provides astrong alternative approximate model that agrees well with simulation. Given that thetransmission rates encompass information both about the dynamics and graph properties, thespecific shape of the curve, defined by the transmission rate versus the number ofinfected nodes, can provide a new and different measure of network structure, and themodel could serve as a link between inferring network structure from prevalence orincidence data.

We consider the spread of an infectious disease on a heterogeneous metapopulation definedby any (correlated or uncorrelated) network. The infection evolves under transmission,recovery and migration mechanisms. We study some spectral properties of a connectivitymatrix arising from the continuous-time equations of the model. In particular we show thatthe classical sufficient condition of instability for the disease-free equilibrium, wellknown for the particular case of uncorrelated networks, works also for the general case.We give also an alternative condition that yields a more accurate estimation of theepidemic threshold for correlated (either assortative or dissortative) networks.

In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation isstudied on a special class of 3-regular graphs, called modified cycle graphs. The modifiedcycle graph is constructed from a cycle graph with N nodes by connecting nodei to thenode i +d in a way that every node has exactly three links.Monte-Carlo simulations show that the propagation process depends on the value ofd in anon-monotone way. A new theoretical model is developed to explain this phenomenon. Thisreveals a new relation between the spreading process and the average path length in thegraph.

We consider a stochastic SIS model for the spread of an epidemic amongst a population of n individuals that are equally spaced upon the circumference of a circle. Whilst infectious, an individual, i say, makes both local and global infectious contacts at the points of homogeneous Poisson point processes. Global contacts are made uniformly at random with members of the entire population, whilst local contacts are made according to a contact distribution centred upon the infective. Individuals at the end of their infectious period return to the susceptible state and can be reinfected. The emphasis of the paper is on asymptotic results as the population size n → ∞. Therefore, a contact process with global infection is introduced representing the limiting behaviour as n → ∞ of the circle epidemics. A branching process approximation for the early stages of the epidemic is derived and the endemic equilibrium of a major outbreak is obtained. Furthermore, assuming exponential infectious periods, the probability of a major epidemic outbreak and the proportion of the population infectious in the endemic equilibrium are shown to satisfy the same equation which characterises the epidemic process.

We analyse SIS epidemics among populations partitioned into households. The analysis considers both the stochastic and deterministic models and, unlike in previous analyses, we consider general infectious period distributions. For the deterministic model, we prove the existence of an endemic equilibrium for the epidemic if and only if the threshold parameter, R*, is greater than 1. Furthermore, by utilising Markov chains we show that the total number of infectives converges to the endemic equilibrium as t → ∞. For the stochastic model, we prove a law of large numbers result for the convergence, to the deterministic limit, of the mean number of infectives per household. This is followed by the derivation of a Gaussian limit process for the fluctuations of the stochastic model.