Parameter Estimation for SIR epidemics in households

Consider the following model (investigated in Ball, Mollison and Scalia-Tomba (1997)) for the spread of an epidemic among a closed nite population partitioned into households. For n = 1; 2; : : :, let mn be the number of households of size n and N = P 1 n=1 nmn be the total number of individuals. The epidemic is initiated by a number of individuals becoming infected at time t = 0. The infectious periods of di erent infectives are assumed to be independent and identically distributed according to a random variable TI , having an arbitrary but speci ed distribution. Throughout its infectious period a given infective makes infectious contacts with each susceptible in the population at the points of a homogeneous Poisson process having rate =N , and, additionally, with each susceptible in its own household at the points of a homogeneous Poisson process having rate . All the Poisson processes describing infectious contacts (whether or not either or both of the individuals involved are the same), as well as the random variables describing infectious periods, are assumed to be mutually independent. A susceptible becomes infective as soon as it is contacted by an infective and is removed (and plays no further part in the epidemic) at the end of its infectious period. The epidemic ceases as soon as there are no infectives present in the population.