The probability of extinction of ISAv in one and two patches

TL;DR

This paper develops stochastic models to analyze the extinction probability of Infectious Salmon Anemia virus in one or two patches, comparing CTMC and multitype branching process approaches and highlighting their agreement and limitations.

Contribution

It introduces a CTMC model for ISA virus spread in two patches and compares it with multitype branching processes, emphasizing their applicability and differences in extinction probability calculations.

Findings

CTMC models relate to deterministic models for parameter selection

MTBP approximations are valid in the supercritical regime

Partial extinction events are identified and their significance discussed

Abstract

Single type and multitype branching process 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 MTBP's are not suitable for making these calculations.