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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

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

Enzyme-catalysed reactions involve two distinct timescales. There is a short timescale on which enzymes bind to substrate molecules to produce bound complexes, and a comparatively long timescale on which the complex is transformed into a product. The rate at which the substrate is converted into product is characteristically non-linear and is traditionally derived by applying singular perturbation theory to the system's governing equations. Central to this analysis is the assumption that complex formation is effectively instantaneous on the timescale over which significant substrate degradation occurs. This prevents accurate modelling of enzyme kinetics by many particle-based simulations of reaction-diffusion systems as they rely on proximity-based reaction conditions that do not correctly model the fast reactions associated with the complex on the long timescale. In this paper we derive a new proximity-based reaction condition that correctly incorporates the reactions that occur on the short timescale for a specific enzymatic system. We present proof of concept particle-based simulations and demonstrate that non-linear reaction rates typical of enzyme kinetics can be reproduced without needing to explicitly simulate reactions on the short timescale., Comment: 18 pages, 6 figures

\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

Turing patterns in reaction-diffusion (RD) systems have classically been studied only in RD systems which do not explicitly depend on independent variables such as space. In practise, many systems for which Turing patterning is important are not homogeneous with ideal boundary conditions. In heterogeneous systems with stable steady states, the steady states are also necessarily heterogeneous which is problematic for applying the classical analysis. Whilst there has been some work done to extend Turing analysis to some heterogeneous systems, for many systems it is still difficult to determine if a stable patterned state is driven purely by system heterogeneity or if a Turing instability is playing a role. In this work, we try to define a framework which uses numerical continuation to map heterogeneous RD systems onto a sensible nearby homogeneous system. This framework may be used for discussing the role of Turing instabilities in establishing patterns in heterogeneous RD systems. We study the Schnakenberg and Gierer-Meinhardt models with spatially heterogeneous production as test problems. It is shown that for sufficiently large system heterogeneity (large amplitude spatial variations in morphogen production) it is possible that Turing-patterned and base states become coincident and therefore impossible to distinguish. Other exotic behaviour is also shown to be possible. We also study a novel scenario in which morphogen is produced locally at levels that could support Turing patterning but on intervals/patches which are on the scale of classical critical domain lengths. Without classical domain boundaries, Turing patterns are allowed to bleed through; an effect noted by other authors. In this case, this phenomena effectively changes the critical domain length. Indeed, we even note that this phenomena may also effectively couple local patches together and drive instability in this way., Comment: 10 figures