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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allen LJS (2003) An introduction to stochastic process with applications to biology. Pearson/Prentice Hall, Upper Saddle River
Allen LJS, Lahodny GE (2012) Extinction thresholds in deterministic and stochastic epidemic models. J Biol Dyn 6(2):590–611
Allen LJS, Lahodny GE (2013) Probability of a disease outbreak in stochastic multipatch epidemic models. Bull Math Biol 75(7):1157–1180
Allen LJS, van den Driessche P (2013) Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models. Math Biosci 243(1):99–108
Ball FG (1983) The threshold behaviour of epidemic models. J Appl Prob 20(7):227–241
Ball FG, Donnelly D (1995) Strong approximations for epidemic models. Stoch Proc Appl 55(1):1–21
Beretta E, Kuang Y (1998) Modeling and analysis of a marine bacteriophage infection. Math Biosci 149:57–76
Britton T (2010) Stochastic epidemic models: a survey. Math Biosci 225:24–35
Butler G, Freedman HI, Waltman P (1986) Uniformly persistent systems. Proc Am Math Soc 96:425–430
Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and computation of the basic reproduction ratio R0 in models of infectious disease in heterogeneous populations. J Math Biol 28:365–382
Dorman KS, Sinsheimer JS, Lange K (2004) In the garden of branching processes. SIAM Rev 46(2):202–229
Falk K, Namork E, Rimstad E, Mjaaland S, Dannevig BH (1997) Characterization of infectious salmon anemia virus, an Orthomyxo-like virus isolated from Atlantic salmon (Salmo salar L.). J Virol 71(12):9016–23
Fonda A (1988) Uniformly persistent semidynamical systems. Proc Am Math Soc 104:111–116
Freedman HI, Ruan S, Tang M (1994) Uniform persistence and flows near a closed positively invariant set. J Dyn Differ Equ 6(4):583–600
Garay B (1989) Uniform persistence and chain recurrence. J Math Anal Appl 139:372–381
Griffiths M, Greenhalgh D (2011) The probability of extinction in a bovine respiratory syncytial virus epidemic model. Math Biosci 231(2):144–158
Harris TE (1963) The theory of branching processes. Springer, Berlin
Hofbauer J, So JW-H (1989) Uniform persistence and repellors for maps. Proc Am Math Soc 107(4):1137–1142
Kimmel M, Axelrod DE (2002) Branching processes in biology. Springer, New York
Mardones FO, Perez AM, Carpenter TE (2009) Epidemiologic investigation of the re-emergence of infectious salmon anemia virus in Chile. Dis Aquat Org 84(2):105–14
Milliken E, Pilyugin SS (2016) A model of infectious salmon anemia virus with viral diffusion between wild and farmed patches. DCDS-B, Accepted
Mode CJ (1971) Multitype branching processes theory and applications. Elsevier, New York
Nowak MA, May RM (2000) Virus dynamics. Oxford University Press, New York
Perelson AS, Nelson PW (1999) Mathematical analysis of HIV-I: dynamics in vivo. SIAM Rev 41(1):3–44
Seneta E (1998) IJ Bieneymé [1796-1878]: criticality, inequality and internationalization. Int Stat Rev 66(3):291–301
Thieme HR (1993) Persistence under relaxed point dissipativity (with applications to an endemic model). SIAM J Math Anal 24(2):407–435
van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48
Vike S, Nylund S, Nylund A (2009) ISA virus in Chile: evidence of vertical transmission. Arch Virol 154(1):1–8
Watson HW, Galton F (1875) On the probability of the extinction of families. J Anthropol Inst Gt Britain Irel 4:138–144
Whittle P (1955) The outcome of a stochastic epidemic- a note on Bailey’s paper. Biometrika 42:116–122
Acknowledgements
This work was conducted with the support from NSF Grants DMS-1411853, DMS-1515661 and the Center for Applied Mathematics at University of Florida. The author would like to thank the referees for their helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Milliken, E. The Probability of Extinction of Infectious Salmon Anemia Virus in One and Two Patches. Bull Math Biol 79, 2887–2904 (2017). https://doi.org/10.1007/s11538-017-0355-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-017-0355-5