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The epidemiological behavior of Plasmodium vivax malaria occurs across spatial scales including within-host, population, and metapopulation levels. On the within-host scale, P. vivax sporozoites inoculated in a host may form latent hypnozoites, the activation of which drives secondary infections and accounts for a large proportion of P. vivax illness; on the metapopulation level, the coupled human-vector dynamics characteristic of the population level are further complicated by the migration of human populations across patches with different malaria forces of (re-)infection. To explore the interplay of all three scales in a single two-patch model of Plasmodium vivax dynamics, we construct and study a system of eight integro-differential equations with periodic forcing (arising from the single-frequency sinusoidal movement of a human sub-population). Under the numerically-informed ansatz that the limiting solutions to the system are closely bounded by sinusoidal ones for certain regions of parameter space, we derive a single nonlinear equation from which all approximate limiting solutions may be drawn, and devise necessary and sufficient conditions for the equation to have only a disease-free solution. Our results illustrate the impact of movement on P. vivax transmission and suggest a need to focus vector control efforts on forest mosquito populations. The three-scale model introduced here provides a more comprehensive framework for studying the clinical, behavioral, and geographical factors underlying P. vivax malaria endemicity., Comment: 42 pages and 9 figures

Plasmodium vivax is one of the most geographically widespread malaria parasites in the world due to its ability to remain dormant in the human liver as hypnozoites and subsequently reactivate after the initial infection (i.e. relapse infections). More than 80% of P. vivax infections are due to hypnozoite reactivation. Mathematical modelling approaches have been widely applied to understand P. vivax dynamics and predict the impact of intervention outcomes. In this article, we provide a scoping review of mathematical models that capture P. vivax transmission dynamics published between January 1988 and May 2023 to provide a comprehensive summary of the mathematical models and techniques used to model P. vivax dynamics. We aim to assist researchers working on P. vivax transmission and other aspects of P. vivax malaria by highlighting best practices in currently published models and highlighting where future model development is required. We provide an overview of the different strategies used to incorporate the parasite's biology, use of multiple scales (within-host and population-level), superinfection, immunity, and treatment interventions. In most of the published literature, the rationale for different modelling approaches was driven by the research question at hand. Some models focus on the parasites' complicated biology, while others incorporate simplified assumptions to avoid model complexity. Overall, the existing literature on mathematical models for P. vivax encompasses various aspects of the parasite's dynamics. We recommend that future research should focus on refining how key aspects of P. vivax dynamics are modelled, including spatial heterogeneity in exposure risk, the accumulation of hypnozoite variation, the interaction between P. falciparum and P. vivax, acquisition of immunity, and recovery under superinfection.

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.

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network, and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

\textit{Plasmodium vivax} is the most geographically widespread malaria-causing parasite resulting in significant associated global morbidity and mortality. One of the factors driving this widespread phenomenon is the ability of the parasites to remain dormant in the liver. Known as hypnozoites, they reside in the liver following an initial exposure, before activating later to cause further infections, referred to as relapses. As around 79-96$\%$ of infections are attributed to relapses, we expect it will be highly impactful to apply treatment to target the hypnozoite reservoir to eliminate \textit{P. vivax}. Treatment with a radical cure to target the hypnozoite reservoir is a potential tool to control or eliminate \textit{P. vivax}. We have developed a multiscale mathematical model as a system of integro-differential equations that captures the complex dynamics of \textit{P. vivax} hypnozoites and the effect of hypnozoite relapse on disease transmission. Here, we use our model to study the anticipated effect of radical cure treatment administered via a mass drug administration (MDA) program. We implement multiple rounds of MDA with a fixed interval between rounds, starting from different steady-state disease prevalences. We then construct an optimisation model to obtain the optimal MDA interval. We also incorporate mosquito seasonality in our model to study its effect on the optimal treatment regime. We find that the effect of MDA interventions is temporary and depends on the pre-intervention disease prevalence (and choice of model parameters) as well as the number of MDA rounds under consideration. We find radical cure alone may not be enough to lead to \textit{P. vivax} elimination under our mathematical model (and choice of model parameters) since the prevalence of infection eventually returns to pre-MDA levels., Comment: 39 pages, 14 figures

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 a mosquito-borne disease with a devastating global impact. Plasmodium vivax is a major cause of human malaria beyond sub-Saharan Africa. Relapsing infections, driven by a reservoir of liver-stage parasites known as hypnozoites, present unique challenges for the control of P. vivax malaria. Following indeterminate dormancy periods, hypnozoites may activate to trigger relapses. Clearance of the hypnozoite reservoir through drug treatment (radical cure) has been proposed as a potential tool for the elimination of P. vivax malaria. Here, we introduce a stochastic, within-host model to jointly characterise hypnozoite and infection dynamics for an individual in a general transmission setting, allowing for radical cure. We begin by extending an existing activation-clearance model for a single hypnozoite, adapted to both short- and long-latency strains, to include drug treatment. We then embed this activation-clearance model in an epidemiological framework accounting for repeated mosquito inoculation and the administration of radical cure. By constructing an open network of infinite server queues, we derive analytic expressions for several quantities of epidemiological significance, including the size of the hypnozoite reservoir; the relative contribution of relapses to the infection burden; the distribution of multiple infections; the cumulative number of recurrences over time, and the time to first recurrence following drug treatment. By deriving, rather than assuming parameteric forms, we characterise the transient dynamics of the hypnozoite reservoir following radical cure more accurately than previous approaches. To yield population-level insights, our analytic within-host distributions can be embedded in multiscale models. Our work thus contributes to the epidemiological understanding of the effects of radical cure on P. vivax malaria.

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