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On the outcome of epidemics with detections

Part of: Markov processes

Published online by Cambridge University Press:  15 September 2017

Claude Lefèvre  and
Philippe Picard
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles and Université de Lyon
Philippe Picard*
Affiliation:
Université de Lyon
*
* Postal address: Département de Mathématique, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium. Email address: [email protected]
** Postal address: Ecole ISFA, 50 Avenue Tony Garnier, F-69007 Lyon, France. Email address: [email protected]
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Abstract

The classical SIR epidemic model is generalized to incorporate a detection process of infectives in the course of time. Our purpose is to determine the exact distribution of the population state at the first detection instant and the following ones. An extension is also discussed that allows the parameters to change with the number of detected cases. The followed approach relies on simple martingale arguments and uses a special family of Abel–Gontcharoff polynomials.

Keywords

General epidemicdetection of infectiveschange of parametersstopping timeAbel–Gontcharoff polynomial

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 890 - 904
Copyright
Copyright © Applied Probability Trust 2017 

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References

Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and their Statistical Analysis (Lecture Notes Statist. 151). Springer, New York. CrossRefGoogle Scholar
Ball, F. and O'Neill, P. (1999). The distribution of general final state random variables for stochastic epidemic models. J. Appl. Prob. 36, 473–491. CrossRefGoogle Scholar
Bootsma, M. C. J., Wassenberg, M. W. M., Trapman, P. and Bonten, M. J. M. (2011). The nosocomial transmission rate of animal-associated ST398 meticillin-resistant Staphylococcus aureus . J. R. Soc. Interface 8, 578–584. Google Scholar
Daley, D. J. and Gani, J. (1999). Epidemic Modelling: An Introduction. Cambridge University Press. Google Scholar
Hernández-Suárez, C. M., Castillo-Chavez, C., Montesinos-López, O. and Hernández-Cuevas, K. (2010). An application of queuing theory to SIS and SEIS epidemic models. Math. Biosci. Eng. 7, 809–823. Google Scholar
Lambert, A. and Trapman, P. (2013). Splitting trees stopped when the first clock rings and Vervaat's transformation. J. Appl. Prob. 50, 208–227. Google Scholar
Lefèvre, C. and Picard, P. (2015). Risk models in insurance and epidemics: a bridge through randomized polynomials. Prob. Eng. Inf. Sci. 29, 399–420. CrossRefGoogle Scholar
Picard, P. and Lefèvre, C. (1990). A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Adv. Appl. Prob. 22, 269–294. Google Scholar
Trapman, P. and Bootsma, M. C. J. (2009). A useful relationship between epidemiology and queueing theory: the distribution of the number of infectives at the moment of the first detection. Math. Biosci. 219, 15–22. Google Scholar