The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches

The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth–death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle’s study of a susceptible–infected–recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of Mathematical Biology Springer Journals

The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Society for Mathematical Biology
Subject
Mathematics; Mathematical and Computational Biology; Life Sciences, general; Cell Biology
ISSN
0092-8240
eISSN
1522-9602
D.O.I.
10.1007/s11538-017-0355-5
Publisher site
See Article on Publisher Site

Abstract

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth–death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle’s study of a susceptible–infected–recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations.

Journal

Bulletin of Mathematical BiologySpringer Journals

Published: Nov 2, 2017

References

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