  Malaria is a parasitic disease, transmitted by mosquito vectors. Plasmodium
vivax presents particular challenges for disease control, in light of an
undetectable reservoir of latent parasites (hypnozoites) within the host liver.
Superinfection, which is driven by temporally proximate mosquito inoculation
and/or hypnozoite activation events, is an important feature of P. vivax. Here,
we present a model of hypnozoite accrual and superinfection for P. vivax. To
couple host and vector dynamics, we construct a density-dependent Markov
population process with countably many types, for which disease extinction is
shown to occur almost surely. We also establish a functional law of large
numbers, taking the form of an infinite-dimensional system of ordinary
differential equations that can also be recovered under the hybrid
approximation or a standard compartment modelling approach. Recognising that
the subset of these equations that models the infection status of human hosts
has precisely the same form as the Kolmogorov forward equations for a Markovian
network of infinite server queues with an inhomogeneous batch arrival process,
we use physical insight into the evolution of the latter to write down a
time-dependent multivariate generating function for the solution. We use this
characterisation to collapse the infinite-compartment model into a single
integrodifferential equation (IDE) governing the intensity of mosquito-to-human
transmission. Through a steady state analysis, we recover a threshold
phenomenon for this IDE in terms of a bifurcation parameter $R_0$, with the
disease-free equilibrium shown to be uniformly asymptotically stable if $R_0<1$
and an endemic equilibrium solution emerging if $R_0>1$. Our work provides a
theoretical basis to explore the epidemiology of P. vivax, and introduces a
general strategy for constructing tractable population-level models of malarial
superinfection.
