Abstract
Two stochastic models of susceptible/infected/removed (SIR) type are introduced for the spread of infection through a spatially-distributed population. Individuals are initially distributed at random in space, and they move continuously according to independent diffusion processes. The disease may pass from an infected individual to an uninfected individual when they are sufficiently close. Infected individuals are permanently removed at some given rate α. Such processes are reminiscent of so-called frog models, but differ through the action of removal, as well as the fact that frogs jump whereas snails slither.
Two models are studied here, termed the ‘delayed diffusion’ and the ‘diffusion’ models. In the first, individuals are stationary until they are infected, at which time they begin to move; in the second, all individuals start to move at the initial time 0. Using a perturbative argument, conditions are established under which the disease infects a.s. only finitely many individuals. It is proved for the delayed diffusion model that there exists a critical value for the survival of the epidemic.
Funding Statement
ZL’s research was supported by National Science Foundation grant 1608896 and Simons Foundation grant 638143.
Acknowledgements
GRG thanks Alexander Holroyd and James Norris for useful conversations. The authors are very grateful to three referees for their detailed and valuable reports, which have led to significant corrections. The work reported here was influenced in part by the Covid-19 pandemic of 2020.
Citation
Geoffrey R. Grimmett. Zhongyang Li. "Brownian snails with removal: epidemics in diffusing populations." Electron. J. Probab. 27 1 - 31, 2022. https://doi.org/10.1214/22-EJP804
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