# A stochastic model for transmission, extinction and outbreak of *Escherichia coli* O157:H7 in cattle as affected by ambient temperature and cleaning practices

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## Abstract

Many infectious agents transmitting through a contaminated environment are able to persist in the environment depending on the temperature and sanitation determined rates of their replication and clearance, respectively. There is a need to elucidate the effect of these factors on the infection transmission dynamics in terms of infection outbreaks and extinction while accounting for the random nature of the process. Also, it is important to distinguish between the true and apparent extinction, where the former means pathogen extinction in both the host and the environment while the latter means extinction only in the host population. This study proposes a stochastic-differential equation model as an approximation to a Markov jump process model, using *Escherichia coli* O157:H7 in cattle as a model system. In the model, the host population infection dynamics are described using the standard susceptible-infected-susceptible framework, and the *E. coli* O157:H7 population in the environment is represented by an additional variable. The backward Kolmogorov equations that determine the probability distribution and the expectation of the first passage time are provided in a general setting. The outbreak and apparent extinction of infection are investigated by numerically solving the Kolmogorov equations for the probability density function of the associated process and the expectation of the associated stopping time. The results provide insight into *E. coli* O157:H7 transmission and apparent extinction, and suggest ways for controlling the spread of infection in a cattle herd. Specifically, this study highlights the importance of ambient temperature and sanitation, especially during summer.

## Keywords

*Escherichia coli*O157:H7 Stochastic \({SIS}_\mathrm{E} {{ model}}\) Kolmogorov equations Extinction outbreak

## Mathematics Subject Classification (2000)

92D30 60H10 60H30## Notes

### Acknowledgments

We thank three anonymous referees and the editor for their suggestions that improved this paper. This work was supported by the National Science Foundation grant NSF-EF-0913367 to RI funded under the American Recovery and Reinvestment Act of 2009. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. This publication is based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).

## Supplementary material

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