Séminaire de Probabilités XLIX pp 221-327 | Cite as

# Large Deviations for Infectious Diseases Models

## Abstract

We study large deviations of a Poisson driven system of stochastic differential equations modeling the propagation of an infectious disease in a large population, considered as a small random perturbation of its law of large numbers ODE limit. Since some of the rates vanish on the boundary of the domain where the solution takes its values, thus making the action functional possibly explode, our system does not obey assumptions which are usually made in the literature. We present the whole theory of large deviations for systems which include the infectious disease models, and apply our results to the study of the time taken for an endemic equilibrium to cease, due to random effects.

## Keywords

Poisson process driven SDE Lareg deviations Freidlin-Wentzell theory Epidemic models## Notes

### Acknowledgements

The authors thank an anonymous Referee, whose careful reading and detailed remarks allowed us to improve an earlier version of this paper.

This research was supported by the ANR project MANEGE, the DAAD, and the Labex Archimède.

## References

- 1.K.B. Athreya, P.E. Ney,
*Branching Processes*(Springer, New York, 1972)CrossRefGoogle Scholar - 2.P. Billingsley,
*Convergence of Probability Measures*(Wiley, New York, 1999)CrossRefGoogle Scholar - 3.A. Dembo, O. Zeitouni,
*Large Deviations Techniques and Applications*(Springer, Berlin, 2009)zbMATHGoogle Scholar - 4.P. Dupuis, R.S. Ellis,
*A Weak Convergence Approach to the Theory of Large Deviations*(Wiley, New York, 1997)CrossRefGoogle Scholar - 5.P. Dupuis, R.S. Ellis, A. Weiss, Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Probab.
**19**(3), 1280–1297 (1991)zbMATHGoogle Scholar - 6.J. Feng, T.G. Kurtz,
*Large Deviations for Stochastic Processes*(American Mathematical Society, Providence, 2006)CrossRefGoogle Scholar - 7.M.I. Freidlin, A.D. Wentzell,
*Random Perturbations of Dynamical Systems*(Springer, Berlin, 2012)CrossRefGoogle Scholar - 8.P. Kratz, E. Pardoux, B.S. Kepgnou, Numerical methods in the context of compartmental models in epidemiology. ESAIM Proc.
**48**, 169–189 (2015)MathSciNetCrossRefGoogle Scholar - 9.T.G. Kurtz, Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl.
**6**(3), 223–240 (1978)MathSciNetCrossRefGoogle Scholar - 10.K. Pakdaman, M. Thieullen, G. Wainrib, Diffusion approximation of birth-death processes: comparison in terms of large deviations and exit points. Stat. Probab. Lett.
**80**(13–14), 1121–1127 (2010)MathSciNetCrossRefGoogle Scholar - 11.D. Revuz, M. Yor,
*Continuous Martingales and Brownian Motion*(Springer, Berlin, 2005)zbMATHGoogle Scholar - 12.H. Roydon,
*Real Analysis*(Collier-Macmillan, London, 1968)Google Scholar - 13.A. Shwartz, A. Weiss,
*Large Deviations for Performance Analysis*(Chapman Hall, London, 1995)zbMATHGoogle Scholar - 14.A. Shwartz, A. Weiss, Large deviations with diminishing rates. Math. Oper. Res.
**30**(2), 281–310 (2005)MathSciNetCrossRefGoogle Scholar - 15.M. Sion, On general minimax theorems. Pac. J. Math.
**8**, 171–176 (1958)MathSciNetCrossRefGoogle Scholar