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Coexistence and Noncoexistence of Markovian Viruses and their Hosts

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Jakob E. Björnberg  and
Erik I. Broman
Jakob E. Björnberg*
Affiliation:
Uppsala University
Erik I. Broman*
Affiliation:
Uppsala University
*
∗ Postal address: Department of Mathematics, Uppsala University, Box 480, Uppsala, 751 06, Sweden.
∗ Postal address: Department of Mathematics, Uppsala University, Box 480, Uppsala, 751 06, Sweden.
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Abstract

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Examining possibilities for the coexistence of two competing populations is a classic problem which dates back to the earliest ‘predator-prey’ models. In this paper we study this problem in the context of a model introduced in Björnberg et al. (2012) for the spread of a virus infection in a population of healthy cells. The infected cells may be seen as a population of ‘predators’ and the healthy cells as a population of ‘prey’. We show that, depending on the parameters defining the model, there may or may not be coexistence of the two populations, and we give precise criteria for this.

Keywords

Coexistencebranching processinteracting branching process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 191 - 208
Copyright
© Applied Probability Trust 

References

Arnold, B. C. and Groeneveld, R. A. (1979). Bounds on expectations of linear systematic statistics based on dependent samples. Ann. Statist. 7, 220–223.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Björnberg, J. E., Britton, T., Broman, E. I. and Natan, E. (2012). A stochastic model for virus growth in a cell population. Preprint. Available at http://arxiv.org/abs/1112.4694v2.Google Scholar
Garet, O. and Marchand, R. (2005). Coexistence in two-type first-passage percolation models. Ann. Appl. Prob. 15, 298–330.Google Scholar
Gumbel, E. J. (1954). The maxima of the mean largest value and of the range. Ann. Math. Statist. 25, 76–84.Google Scholar
Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Prob. 35, 683–692.CrossRefGoogle Scholar
Häggström, O. and Pemantle, R. (2000). Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. Stoch. Process. Appl. 90, 207–222.Google Scholar
Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13–20.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Hartley, H. O. and David, H. A. (1954). Universal bounds for mean range and extreme observation. Ann. Math. Statist. 25, 85–99.Google Scholar
Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Prob. 15, 739–747.Google Scholar
Kalinkin, A. V. (2002). Markov branching processes with interaction. Russian Math. Surveys 57, 241–304.CrossRefGoogle Scholar