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Large deviation principle for epidemic models

  • Etienne Pardoux (a1) and Brice Samegni-Kepgnou (a1)

Abstract

We consider a general class of epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes and we show the large deviation principle for such models. We generalise the approach followed by Dolgoarshinnykh (2009) in the case of the SIR epidemic model. Thanks to an additional assumption which is satisfied in many examples, we simplify the recent work of Kratz and Pardoux (2017).

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Corresponding author

* Postal address: CNRS UMR 7373 (I2M), Aix-Marseille Université, CMI, 39 rue F. Joliot Curie, F-13453 Marseille, France.
** Email address: etienne.pardoux@univ-amu.fr

References

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[1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
[2] Britton, T. and Pardoux, E. (2017). Stochastic epidemics in a homogeneous community. In preparation.
[3] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Springer, Berlin.
[4] Dolgoarshinnykh, R. (2009). Sample path large deviations for SIRS epidemic processes. Submitted.
[5] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.
[6] Dupuis, P., Ellis, R. S. and Weiss, A. (1991). Large deviations for Markov processes with discontinuous statistics. I. General upper bounds. Ann. Prob. 19, 12801297.
[7] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. John Wiley, Hoboken, NJ.
[8] Feng, J. and Kurtz, T. G. (2006). Large Deviations for Stochastic Processes (Math. Surveys Monogr. 131). American Mathematical Society, Providence, RI.
[9] Fierro, R. (1996). Large-sample analysis for a stochastic epidemic model and its parameter estimators. J. Math. Biol. 34, 843856.
[10] Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd edn. Springer, Heidelberg.
[11] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.
[12] Kratz, P. and Pardoux, E. (2017). Large Deviation for Infections Diseases. In Séminaire de Probabilités XLIX (Lecture Notes Math.), Springer (to appear).
[13] Kribs-Zaleta, C. M. and Velasco-Hernández, J. X. (2000). A simple vaccination model with multiple endemic states. Math. Biosci. 164, 183201.
[14] Kurtz, T. G. (1978). Strong approximation theorems for density dependent Markov chains. Stoch. Process. Appl. 6, 223240.
[15] Léonard, C. (1990). Some epidemic systems are long range interacting particle systems. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86), Springer, Berlin, pp. 170183.
[16] Safan, M., Heesterbeek, H. and Dietz, K. (2006). The minimum effort required to eradicate infections in models with backward bifurcation. J. Math. Biol. 53, 703718.
[17] Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis. Chapman & Hall, London.
[18] Shwartz, A. and Weiss, A. (2005). Large deviations with diminishing rates. Math. Operat. Res. 30, 281310.
[19] Sokol, A. and Hansen, N. R. (2015). Exponential martingales and change of measure for counting processes. Stoch. Anal. Appl. 33, 823843.

Keywords

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Large deviation principle for epidemic models

  • Etienne Pardoux (a1) and Brice Samegni-Kepgnou (a1)

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