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Special issue In memory of Fred Brauer

Epidemic highs and lows: a stochastic diffusion model for active cases

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Article: 2189001 | Received 12 Jul 2022, Accepted 04 Mar 2023, Published online: 15 Mar 2023
 

Abstract

We derive a stochastic epidemic model for the evolving density of infective individuals in a large population. Data shows main features of a typical epidemic consist of low periods interspersed with outbreaks of various intensities and duration. In our stochastic differential model, a novel reproductive term combines a factor expressing the recent notion of ‘attenuated Allee effect’ and a capacity factor is controlling the size of the process. Simulation of this model produces sample paths of the stochastic density of infectives, which behave much like long-time Covid-19 case data of recent years. Writing the process as a stochastic diffusion allows us to derive its stationary distribution, showing the relative time spent in low levels and in outbursts. Much of the behaviour of the density of infectives can be understood in terms of the interacting drift and diffusion coefficient processes, or, alternatively, in terms of the balance between noise level and the attenuation parameter of the Allee effect. Unexpected results involve the effect of increasing overall noise variance on the density of infectives, in particular on its level-crossing function.

This article is part of the following collections:
An article collection in honour of Fred Brauer

Acknowledgments

We are grateful for the reviewer's comments which have helped us to improve our manuscript. This research benefited from a workshop at the American Institute of Mathematics, attended by P.G. in 2019. D.S. is thankful to the Office of Research at Utah State University for economic support.

Disclosure statement

No potential conflict of interest was reported by the author(s).