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A stochastic SIS infection model incorporating indirect transmission

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Damian Clancy
Damian Clancy*
Affiliation:
The University of Liverpool
*
∗Postal address: Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, UK. Email address: d.clancy@liv.ac.uk
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Abstract

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We describe a stochastic susceptible–infective–susceptible (SIS) model for transmission of infectious disease through a population, incorporating both direct host–host transmission and indirect transmission via free-living infectious stages (e.g. environmental bacteria). Existence of a quasi-stationary distribution conditional upon nonextinction of infection is established. A bivariate Ornstein–Uhlenbeck approximation is used to investigate the long-term behaviour of the process conditional upon nonextinction of infection. We show that indirect transmission leads to lower variability in the number of infected hosts present in quasi-stationarity and, consequently, to a greater tendency of infection to persist, compared with a model with direct transmission only and the same average individual infectivity. Some numerical work illustrating these results is presented.

Keywords

Quasi-stationary distributionnormal approximationfree-living infectious stagepiecewise-deterministic Markov process

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 726 - 737
Copyright
© Applied Probability Trust 2005 

References

Anderson, R. M. and May, R. M. (1981). The population dynamics of microparasites and their invertebrate hosts. Phil. Trans. R. Soc. London B 291, 451–524.Google Scholar
Anderson, W. J. (1991). Continuous-Time Markov Chains. An Applications-Oriented Approach. Springer, Berlin.Google Scholar
Andersson, H. and Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. J. Appl. Prob. 35, 662–670.CrossRefGoogle Scholar
Clancy, D., O'Neill, P. D. and Pollett, P. K. (2001). Approximations for the long-term behaviour of an open-population epidemic model. Methodology Comput. Appl. Prob. 3, 75–95.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.Google Scholar
Ferrari, P., Kesten, H., Martínez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamic approach. Ann. Prob. 23, 501–521.Google Scholar
Kijima, M., Nair, M. G., Pollett, P. K. and van Doorn, E. A. (1997). Limiting conditional distributions for birth–death processes. Adv. Appl. Prob. 29, 185–204.Google Scholar
Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure Jump Markov processes. J. Appl. Prob. 7, 49–58.Google Scholar
Kurtz, T. G. (1971). Limit theorems for sequences of Jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.Google Scholar
Nåsell, I. (1996). The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895–932.Google Scholar
Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40.Google Scholar
Nåsell, I. (1999). On the time to extinction in recurrent epidemics. J. R. Statist. Soc. B 61, 309–330.Google Scholar
Pakes, A. G. (1995). Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state. Adv. Appl. Prob. 27, 120–145.Google Scholar
Pollett, P. K. (1990). On a model for interference between searching insect parasites. J. Austral. Math. Soc. Ser. B 31, 133–150.Google Scholar
Turner, J., Begon, M., Bowers, R. G. and French, N. P. (2003). A model appropriate to the transmission of a human food-borne pathogen in a multigroup managed herd. Prev. Vet. Med. 57, 175–198.Google Scholar
Weiss, G. H. and Dishon, M. (1971). On the asymptotic behaviour of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265.CrossRefGoogle Scholar
Xiao, Y., Bowers, R. G., Clancy, D. and French, N. P. (2005). Dynamics of infection with multiple transmission mechanisms in unmanaged/managed animal populations. Submitted.Google Scholar