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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.

Malaria is a vector-borne disease that exacts a grave toll in the Global South. The epidemiology of Plasmodium vivax, the most geographically expansive agent of human malaria, is characterised by the accrual of a reservoir of dormant parasites known as hypnozoites. Relapses, arising from hypnozoite activation events, comprise the majority of the blood-stage infection burden, with implications for the acquisition of immunity and the distribution of superinfection. Here, we construct a hybrid transmission model for P. vivax that concurrently accounts for the accrual of the hypnozoite reservoir, (blood-stage) superinfection and the acquisition of immunity. We begin by analytically characterising within-host dynamics as a function of mosquito-to-human transmission intensity, extending our previous model (comprising an open network of infinite server queues) to capture a discretised immunity level. To model transmission-blocking and antidisease immunity, we allow for geometric decay in the respective probabilities of successful human-to-mosquito transmission and symptomatic blood-stage infection as a function of this immunity level. Under a hybrid approximation -- whereby probabilistic within-host distributions are cast as expected population-level proportions -- we couple host and vector dynamics to recover a deterministic compartmental model in line with Ross-Macdonald theory. We then perform a steady-state analysis for this compartmental model, informed by the (analytic) distributions derived at the within-host level. To characterise transient dynamics, we derive a reduced system of integrodifferential equations (IDEs), likewise informed by our within-host queueing network, allowing us to recover population-level distributions for various quantities of epidemiological interest. Our model provides insights into important, but poorly understood, epidemiological features of P. vivax.

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., Comment: This is a pre-copyedit version of an article published in Bulletin of Mathematical Biology. The final authenticated version is available online at: https://doi.org/10.1007/s11538-020-00706-1