Branching Approximation for the Collective Epidemic Model

A new approach is developed that allows us to establish and analyze a branching-type approximation for the collective epidemic model. Firstly, a necessary and sufficient condition is obtained for the vague convergence of the final size of the epidemic to the total progeny in an appropriate branching model, as the initial number of susceptibles tends to infinity. Then, an upper bound for the L ₁ distance between the statistics under study is derived, showing inter alia that the approximation may hold even when the initial number of infectives is arbitrarily large. The results are illustrated with several particular models of special interest.