Signature of the initial state in absorbing Markov chains: Theory and an application to Batrachochytrium dendrobatidis infection

Background/Question/Methods

Demographic Markov chain models treat population dynamics as a series of jumps between discrete numbers of individuals. When some number of individuals is a dead end the Markov chain is said to have an absorbing state. For example, a closed population with zero individuals is extinct and cannot recover. For an open population of pathogens infecting a particular host, the number of pathogens that induces host mortality is also an absorbing state. In either case, the probability of reaching the absorbing state by a given time is determined by a very large system of differential equations. We show that, in the long run, the probability of absorption is approximated by just two quantities, both of which can be determined without explicit solution of the large system of equations. One of these quantities is the mean time to absorption for an “established” population. The second quantity takes account of initial conditions. Our approach is applicable to both the absorbing upper threshold of an infection process and extinction in a classic birth-death process.

Results/Conclusions

For a demonstration of the general theory, we analyzed a simple model of Batrachochytrium dendrobatidis infection. Chytridiomycosis, the disease caused by the globally dispersed B. dendrobatidis fungus, emerged in the past decade to become a major threat to amphibian populations. Outbreaks of chytridiomycosis have been observed in two Rana species native to the California Sierra Nevada (R. muscosa and R. sierrae), where individual frogs are observed to carry widely varying loads of encysted B. dendrobatidis zoospores. We modeled the spore load on individual frogs as a Markov chain, with the lethal level of infection comprising an absorbing state. Initial spore load is determined at metamorphosis, since the tadpole stage hosts zoospores without showing disease symptoms, and is moreover the target of proposed mitigation by in situ treatment of metamorphic frogs with fungicides. Following the general theory, we calculate the long run probability of lethal infection including the effect of initial spore load. The results allow us to quantify reduction in the long run risk of frog mortality achieved by the proposed mitigation strategy.