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Limit theorems for an epidemic model on the complete graph. (English) Zbl 1160.60334

Summary: We study the following random walks system on the complete graph with \(n\) vertices. At time zero, there is a number of active and inactive particles living on the vertices. Active particles move as continuous-time, rate 1, random walks on the graph, and, any time a vertex with an inactive particle on it is visited, this particle turns into active and starts an independent random walk. However, for a fixed integer \(L\geq 1\), each active particle dies at the instant it reaches a total of \(L\) jumps without activating any particle. We prove a law of large numbers and a central limit theorem for the proportion of visited vertices at the end of the process.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60J27 Continuous-time Markov processes on discrete state spaces
92D30 Epidemiology
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