  Malaria is an infectious disease with an immense global health burden.
Plasmodium vivax is the most geographically widespread species of malaria.
Relapsing infections, caused by the activation of liver-stage parasites known
as hypnozoites, are a critical feature of the epidemiology of Plasmodium vivax.
Hypnozoites remain dormant in the liver for weeks or months after inoculation,
but cause relapsing infections upon activation. Here, we introduce a dynamic
probability model of the activation-clearance process governing both potential
relapses and the size of the hypnozoite reservoir. We begin by modelling
activation-clearance dynamics for a single hypnozoite using a continuous-time
Markov chain. We then extend our analysis to consider activation-clearance
dynamics for a single mosquito bite, which can simultaneously establish
multiple hypnozoites, under the assumption of independent hypnozoite behaviour.
We derive analytic expressions for the time to first relapse and the time to
hypnozoite clearance for mosquito bites establishing variable numbers of
hypnozoites, both of which are quantities of epidemiological significance. Our
results extend those in the literature, which were limited due to an assumption
of non-independence. Our within-host model can be embedded readily in
multi-scale models and epidemiological frameworks, with analytic solutions
increasing the tractability of statistical inference and analysis. Our work
therefore provides a foundation for further work on immune development and
epidemiological-scale analysis, both of which are important for achieving the
goal of malaria elimination.
