Law of large numbers for the number of active particles in the epidemic model

In the last decade, a large number of papers was devoted to studying probabilistic properties of epidemic models on graphs. In this paper we consider a generalization of the model proposed by Machado, Mashurian, and Matzinger. The Machado–Mashurian–Matzinger model serves as an interpretation of spread of viruses in a computer network. We assume that at each moment of time more than one node of the network can be infected. In this context we propose a more advanced model permitting jumps of several particles each time, while the number of such particles is random. We prove the optimal version of the law of large numbers for the number of infected particles in the epidemic model at hand.