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 approximationto phase diagrams typical of one-dimensional systems.In addition,we have performed computer simulations of the same model and shown thatwhile the CPA with a constant correlation parametercannot describe the global behaviour of the model,a reasonable description of the endemic equilibriaas well as of the phase diagram may be obtainedby allowing the parameter to depend on the demographic rate.