The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic

Abstract

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Letαbe its Malthusian parameter,ßits velocity of propagation,Z(A,t) the number of individuals in the setAat timet,andA√(ßt)= [√(ßt)r:r ∈ A]. The mean square convergence of the random variableW(A, t)= e–αtZ(A√(ßt),t) to a limit variableW(A) is established.