Pair Approximation Models for Disease Spread

We consider a Susceptible-Infective-Recovered (SIR) model, where the mechanism for the renewal of susceptibles is demographic, on a ring with next nearest neighbour interactions, and a family of correlated pair approximations (CPA), parametrized by a measure of the relative contributions of loops and open triplets of the sites involved in the infection process. We have found that the phase diagram of the CPA, at fixed coordination number, changes qualitatively as the relative weight of the loops increases, from the phase diagram of the uncorrelated pair approximation to phase diagrams typical of one-dimensional systems. In addition, we have performed computer simulations of the same model and shown that while the CPA with a constant correlation parameter cannot describe the global behaviour of the model, a reasonable description of the endemic equilibria as well as of the phase diagram may be obtained by allowing the parameter to depend on the demographic rate.Comment: 6 pages, 3 figures, LaTeX2e+SVJour+AmSLaTeX, NEXTSigmaPhi 2005; metadata title corrected wrt paper titl

Heterogeneous pair approximation for voter models on networks

For models whose evolution takes place on a network it is often necessary to augment the mean-field approach by considering explicitly the degree dependence of average quantities (heterogeneous mean-field). Here we introduce the degree dependence in the pair approximation (heterogeneous pair approximation) for analyzing voter models on uncorrelated networks. This approach gives an essentially exact description of the dynamics, correcting some inaccurate results of previous approaches. The heterogeneous pair approximation introduced here can be applied in full generality to many other processes on complex networks.Comment: 6 pages, 6 figures, published versio

Population dynamics on random networks: simulations and analytical models

We study the phase diagram of the standard pair approximation equations for two different models in population dynamics, the susceptible-infective-recovered-susceptible model of infection spread and a predator-prey interaction model, on a network of homogeneous degree $k$. These models have similar phase diagrams and represent two classes of systems for which noisy oscillations, still largely unexplained, are observed in nature. We show that for a certain range of the parameter $k$ both models exhibit an oscillatory phase in a region of parameter space that corresponds to weak driving. This oscillatory phase, however, disappears when $k$ is large. For $k=3, 4$, we compare the phase diagram of the standard pair approximation equations of both models with the results of simulations on regular random graphs of the same degree. We show that for parameter values in the oscillatory phase, and even for large system sizes, the simulations either die out or exhibit damped oscillations, depending on the initial conditions. We discuss this failure of the standard pair approximation model to capture even the qualitative behavior of the simulations on large regular random graphs and the relevance of the oscillatory phase in the pair approximation diagrams to explain the cycling behavior found in real populations.Comment: 8 pages, 5 figures; we have expanded and rewritten the introduction, slightly modified the abstract and the text in other sections; also, several new references have been added in the revised manuscript (Refs. [17-25,30,35])

Cluster approximations for infection dynamics on random networks

In this paper, we consider a simple stochastic epidemic model on large regular random graphs and the stochastic process that corresponds to this dynamics in the standard pair approximation. Using the fact that the nodes of a pair are unlikely to share neighbors, we derive the master equation for this process and obtain from the system size expansion the power spectrum of the fluctuations in the quasi-stationary state. We show that whenever the pair approximation deterministic equations give an accurate description of the behavior of the system in the thermodynamic limit, the power spectrum of the fluctuations measured in long simulations is well approximated by the analytical power spectrum. If this assumption breaks down, then the cluster approximation must be carried out beyond the level of pairs. We construct an uncorrelated triplet approximation that captures the behavior of the system in a region of parameter space where the pair approximation fails to give a good quantitative or even qualitative agreement. For these parameter values, the power spectrum of the fluctuations in finite systems can be computed analytically from the master equation of the corresponding stochastic process.Comment: the notation has been changed; Ref. [26] and a new paragraph in Section IV have been adde