Diseases transmission in a z-ary tree

Abstract

Consider a Galton–Watson tree [Formula: see text] of height [Formula: see text], each leaf is either infected by one of [Formula: see text] diseases or not infected at all. In other words, [Formula: see text] at generation [Formula: see text] is infected by the [Formula: see text]th infection with probability [Formula: see text] and sane with probability [Formula: see text]. Moreover, the infections are independently distributed for each leaf. Infections spread along the tree based on deterministic specific rules. We study the limit distribution of the disease of the root of [Formula: see text] as [Formula: see text] goes to infinity. We also study the specific case of a [Formula: see text]-ary tree, and we prove convergence of the distribution of the root node for [Formula: see text].