2013/08/26 by Svante Janson, Janson, Svante, Malwina Luczak +3
Mathematics · Medicine · Physics and Astronomy · #05C80 #60F99 #60J28 #92D30 #COVID-19 epidemiological studies #Complex Network Analysis Techniques #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1308.5493
openalex publication_date 2013/08/26 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We study the susceptible-infective-recovered (SIR) epidemic on a random graph\nchosen uniformly subject to having given vertex degrees. In this model\ninfective vertices infect each of their susceptible neighbours, and recover, at\na constant rate.\n Suppose that initially there are only a few infective vertices. We prove\nthere is a threshold for a parameter involving the rates and vertex degrees\nbelow which only a small number of infections occur. Above the threshold a\nlarge outbreak occurs with probability bounded away from zero. Our main result\nis that, conditional on a large outbreak, the evolutions of certain quantities\nof interest, such as the fraction of infective vertices, converge to\ndeterministic functions of time.\n We also consider more general initial conditions for the epidemic, and derive\ncriteria for a simple vaccination strategy to be successful.\n In contrast to earlier results for this model, our approach only requires\nbasic regularity conditions and a uniformly bounded second moment of the degree\nof a random vertex.\n En route, we prove analogous results for the epidemic on the configuration\nmodel multigraph under much weaker conditions. Essentially, our main result\nrequires only that the initial values for our processes converge, i.e. it is\nthe best possible.\n