Pair approximation models for disease spread

We consider a Susceptible-Infective-Recovered (<Emphasis Type="SmallCaps">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 (<Emphasis Type="SmallCaps">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 <Emphasis Type="SmallCaps">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 <Emphasis Type="SmallCaps">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. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006