We define and study an open stochastic SIR (Susceptible–Infected–Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same time scale. Population transition intensities are assumed to be density-dependent with a constant component, the amplitude of which determines the overall scale of the population process. Standard branching approximation results for the epidemic process are first given, along with a numerical computation method for the probability of a major epidemic outbreak. This procedure is illustrated using real data on trade-related cattle movements from a densely populated livestock farming region in western France (Finistère) and epidemiological parameters corresponding to an infectious epizootic disease. Then we exhibit an exponential lower bound for the extinction time and the total size of the epidemic in the stable endemic case as a scaling parameter goes to infinity using results inspired by the Freidlin–Wentzell theory of large deviations from a dynamical system.
Multitype SIR model Epidemic and demography over the same time scale Continuous-time multitype branching processes Markovian process Major outbreak probability Basic reproduction number Real network Epidemic extinction time Epidemic total size Endemicity
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This work is part of a PhD Thesis supervised by Vincent Bansaye (CMAP, École Polytechnique) and Elisabeta Vergu (MaIAGE, INRA), whom I warmly thank for their guidance and support. It was supported by the French Research Agency within projects ANR-16-CE32-0007-01 (CADENCE) and ANR-16-CE40-0001 (ABIM), and by Chaire Modélisation Mathématique et Biodiversité Veolia-X-MNHM-FX.
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