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In the infectious disease literature, significant effort has been devoted to studying dynamics at a single scale. For example, compartmental models describing population-level dynamics are often formulated using differential equations. In cases where small numbers or noise play a crucial role, these differential equations are replaced with memoryless Markovian models, where discrete individuals can be members of a compartment and transition stochastically. Classic stochastic simulation algorithms, such as Gillespie's algorithm and the next reaction method, can be employed to solve these Markovian models exactly. The intricate coupling between models at different scales underscores the importance of multiscale modelling in infectious diseases. However, several computational challenges arise when the multiscale model becomes non-Markovian. In this paper, we address these challenges by developing a novel exact stochastic simulation algorithm. We apply it to a showcase multiscale system where all individuals share the same deterministic within-host model while the population-level dynamics are governed by a stochastic formulation. We demonstrate that as long as the within-host information is harvested at a reasonable resolution, the novel algorithm we develop will always be accurate. Moreover, the novel algorithm we develop is general and can be easily applied to other multiscale models in (or outside) the realm of infectious diseases., Comment: 23 pages, 7 figures

Plasmodium vivax is the most geographically widespread malaria parasite due to its ability to remain dormant (as a hypnozoite) in the human liver and subsequently reactivate. Given the majority of P. vivax infections are due to hypnozoite reactivation, targeting the hypnozoite reservoir with a radical cure is crucial for achieving P. vivax elimination. Stochastic effects can strongly influence dynamics when disease prevalence is low or when the population size is small. Hence, it is important to account for this when modelling malaria elimination.cWe use a stochastic multiscale model of P. vivax transmission to study the impacts of multiple rounds of mass drug administration (MDA) with a radical cure, accounting for superinfection and hypnozoite dynamics. Our results indicate multiple rounds of MDA with a high-efficacy drug are needed to achieve a substantial probability of elimination. This work has the potential to help guide P. vivax elimination strategies by quantifying elimination probabilities for an MDA approach.

Multi-scale systems often exhibit stochastic and deterministic dynamics. Capturing these aspects in a compartmental model is challenging. Notably, low occupancy compartments exhibit stochastic dynamics and high occupancy compartments exhibit deterministic dynamics. Failing to account for stochasticity in small populations can produce 'atto-foxes', e.g. in the Lotka-Volterra ordinary differential equation (ODE) model. This limitation becomes problematic when studying extinction of species or the clearance of infection, but it can be resolved by using discrete stochastic models e.g. continuous time Markov chains (CTMCs). Unfortunately, simulating CTMCs is infeasible for most realistic populations. We develop a novel mathematical framework, to couple continuous ODEs and discrete CTMCs: 'Jump-Switch-Flow' (JSF). In this framework populations can reach extinct states ("absorbing states"), thereby resolving atto-fox-type problems. JSF has the desired behaviours of exact CTMC simulation, but is substantially faster than existing alternatives. JSF's utility for simulation-based inference, particularly multi-scale problems, is demonstrated by several case-studies. In a simulation study, we demonstrate how JSF can enable a more nuanced analysis of the efficacy of public health interventions. We also carry out a novel analysis of longitudinal within-host data from SARS-CoV-2 infections to quantify the timing of viral clearance. JSF offers a novel approach to compartmental model development and simulation.

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.

In genetic studies, haplotype data provide more refined information than data about separate genetic markers. However, large-scale studies that genotype hundreds to thousands of individuals may only provide results of pooled data, where only the total allele counts of each marker in each pool are reported. Methods for inferring haplotype frequencies from pooled genetic data that scale well with pool size rely on a normal approximation, which we observe to produce unreliable inference when applied to real data. We illustrate cases where the approximation breaks down, due to the normal covariance matrix being near-singular. As an alternative to approximate methods, in this paper we propose exact methods to infer haplotype frequencies from pooled genetic data based on a latent multinomial model, where the observed allele counts are considered integer combinations of latent, unobserved haplotype counts. One of our methods, latent count sampling via Markov bases, achieves approximately linear runtime with respect to pool size. Our exact methods produce more accurate inference over existing approximate methods for synthetic data and for data based on haplotype information from the 1000 Genomes Project. We also demonstrate how our methods can be applied to time-series of pooled genetic data, as a proof of concept of how our methods are relevant to more complex hierarchical settings, such as spatiotemporal models., Comment: 35 pages, 16 figures, 3 algorithms, submitted to Biometrics journal

Geostatistical analysis of health data is increasingly used to model spatial variation in malaria prevalence, burden, and other metrics. Traditional inference methods for geostatistical modelling are notoriously computationally intensive, motivating the development of newer, approximate methods. The appeal of faster methods is particularly great as the size of the region and number of spatial locations being modelled increases. Methods We present an applied comparison of four proposed `fast' geostatistical modelling methods and the software provided to implement them -- Integrated Nested Laplace Approximation (INLA), tree boosting with Gaussian processes and mixed effect models (GPBoost), Fixed Rank Kriging (FRK) and Spatial Random Forests (SpRF). We illustrate the four methods by estimating malaria prevalence on two different spatial scales -- country and continent. We compare the performance of the four methods on these data in terms of accuracy, computation time, and ease of implementation. Results Two of these methods -- SpRF and GPBoost -- do not scale well as the data size increases, and so are likely to be infeasible for larger-scale analysis problems. The two remaining methods -- INLA and FRK -- do scale well computationally, however the resulting model fits are very sensitive to the user's modelling assumptions and parameter choices. Conclusions INLA and FRK both enable scalable geostatistical modelling of malaria prevalence data. However care must be taken when using both methods to assess the fit of the model to data and plausibility of predictions, in order to select appropriate model assumptions and approximation parameters.

\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.

\emph{In vitro} experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these \emph{in vitro} experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination., Comment: 26 pages, 7 figures

Plasmodium (P.) falciparum and P. vivax are the two most common causes of malaria. While the majority of deaths and severe morbidity are due to P. falciparum, P. vivax poses a greater challenge to eliminating malaria outside of Africa due to its ability to form latent liver stage parasites (hypnozoites), which can cause relapsing episodes within an individual patient. In areas where P. falciparum and P. vivax are co-endemic, individuals can carry parasites of both species simultaneously. These mixed infections complicate dynamics in several ways; treatment of mixed infections will simultaneously affect both species, P. falciparum can mask the detection of P. vivax, and it has been hypothesised that clearing P. falciparum may trigger a relapse of dormant P. vivax. When mixed infections are treated for only blood-stage parasites, patients are at risk of relapse infections due to P. vivax hypnozoites. We present a stochastic mathematical model that captures interactions between P. falciparum and P. vivax, and incorporates both standard schizontocidal treatment (which targets blood-stage parasites) and radical treatment (which additionally targets liver-stage parasites). We apply this model to assess the implications of different treatment coverage of radical cure for mixed and P. vivax infections and a so-called "unified radical cure" treatment strategy for P. falciparum, P. vivax and mixed infections. We find that a unified radical cure strategy, with G6PD screening, leads to a substantially lower incidence of malaria cases and deaths overall. We perform a one-way sensitivity analysis to highlight important model parameters.

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

In this paper, we present and validate a simple adaptive surface remeshing technique to transfer history dependent variables from an old distorting mesh to a new mesh during finite element simulations of elastic surface deformation. This technique allows us to reduce the error arising from excessive mesh distortion whilst preserving information about the initial configuration of the mesh and the history dependent variables. The transfer technique presented here constructs the initial configuration of the new mesh by considering the distortion incurred by the elements of the old mesh and projecting backward in time. Using this new initial configuration, the stress and strain over the new mesh can be easily calculated. After presenting the necessary steps to reconstruct the initial configuration, we show that this relatively simple transfer technique adds stability to finite element simulations and reduces the spatial error and the strain error across the domain. The novel transfer technique presented in this paper is easy to implement and provides a simple strategy to add stability to simulations undergoing large deformations.

We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity--fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, \cred{a viscous fluid model}, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the normal velocity of the outermost layer of cells, and this time--dependence is a challenge in designing and implementing a stable and fast numerical scheme. We recast the model into a form where the hyperbolic equation is defined on a fixed extended domain and retrieve the tumour boundary as the interface at which the cell volume fraction decreases below a threshold value. This procedure eliminates the need to track the tumour boundary explicitly and the computationally expensive re--meshing of the time--dependent domains. A numerical scheme based on finite volume methods for the hyperbolic conservation law, Lagrange $\mathbb{P}_2 - \mathbb{P}_1$ Taylor--Hood finite element method for the \cred{viscous} system, and mass--lumped finite element method for the parabolic equations is implemented in two spatial dimensions, and several cases are studied. We demonstrate the versatility of the numerical scheme in catering for irregular and asymmetric initial tumour geometries. We present numerical simulations for both cases and the results are consistent with theoretical and heuristic expectations such as early linear growth rate and preservation of radial symmetry when the boundary conditions are symmetric., Comment: 24 pages, 51 figures, numerical solution without nre-meshing the time-depdendent domain