Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks.

While the foundations of modern epidemiology are based upon deterministic models with homogeneous mixing, it is being increasingly realized that both spatial structure and stochasticity play major roles in shaping epidemic dynamics. The integration of these two confounding elements is generally ascertained through numerical simulation. Here, for the first time, we develop a more rigorous analytical understanding based on pairwise approximations to incorporate localized spatial structure and diffusion approximations to capture the impact of stochasticity. Our results allow us to quantify, analytically, the impact of network structure on the variability of an epidemic. Using the susceptible–infectious–susceptible framework for the infection dynamics, the pairwise stochastic model is compared with the stochastic homogeneous-mixing (mean-field) model—although to enable a fair comparison the homogeneous-mixing parameters are scaled to give agreement with the pairwise dynamics. At equilibrium, we show that the pairwise model always displays greater variation about the mean, although the differences are generally small unless the prevalence of infection is low. By contrast, during the early epidemic growth phase when the level of infection is increasing exponentially, the pairwise model generally shows less variation.

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Mathematical models provide a rational basis to inform how, where and when to control disease. Assuming an accurate spatially explicit simulation model can be fitted to spread data, it is straightforward to use it to test the performance of a range of management strategies. However, the typical complexity of simulation models and the vast set of possible controls mean that only a small subset of all possible strategies can ever be tested. An alternative approach—optimal control theory—allows the best control to be identified unambiguously. However, the complexity of the underpinning mathematics means that disease models used to identify this optimum must be very simple. We highlight two frameworks for bridging the gap between detailed epidemic simulations and optimal control theory: open-loop and model predictive control. Both these frameworks approximate a simulation model with a simpler model more amenable to mathematical analysis. Using an illustrative example model, we show the benefits of using feedback control, in which the approximation and control are updated as the epidemic progresses. Our work illustrates a new methodology to allow the insights of optimal control theory to inform practical disease management strategies, with the potential for application to diseases of humans, animals and plants. This article is part of the theme issue ‘Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control’. This theme issue is linked with the earlier issue ‘Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes’.

Beat-to-beat variability in repolarization (BVR) has been proposed as an arrhythmic risk marker for disease and pharmacological action. The mechanisms are unclear but BVR is thought to be a cell level manifestation of ion channel stochasticity, modulated by cell-to-cell differences in ionic conductances. In this study, we describe the construction of an experimentally-calibrated set of stochastic cardiac cell models that captures both BVR and cell-to-cell differences in BVR displayed in isolated canine action potential measurements using pharmacological agents. Simulated and experimental ranges of BVR are compared in control and under pharmacological inhibition, and the key ionic currents determining BVR under physiological and pharmacological conditions are identified. Results show that the 4-aminopyridine-sensitive transient outward potassium current, Ito1, is a fundamental driver of BVR in control and upon complete inhibition of the slow delayed rectifier potassium current, IKs. In contrast, IKs and the L-type calcium current, ICaL, become the major contributors to BVR upon inhibition of the fast delayed rectifier potassium current, IKr. This highlights both IKs and Ito1 as key contributors to repolarization reserve. Partial correlation analysis identifies the distribution of Ito1 channel numbers as an important independent determinant of the magnitude of BVR and drug-induced change in BVR in control and under pharmacological inhibition of ionic currents. Distributions in the number of IKs and ICaL channels only become independent determinants of the magnitude of BVR upon complete inhibition of IKr. These findings provide quantitative insights into the ionic causes of BVR as a marker for repolarization reserve, both under control condition and pharmacological inhibition.

The Wright-Fisher model is an Itô stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0, 1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0, 1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0, 1]. Communicated

To inform mathematical modelling of the impact of chlamydia screening in England since 2000, a complete picture of chlamydia testing is needed. Monitoring and surveillance systems evolved between 2000 and 2012. Since 2012, data on publicly funded chlamydia tests and diagnoses have been collected nationally. However, gaps exist for earlier years. We collated available data on chlamydia testing and diagnosis rates among 15–44-year-olds by sex and age group for 2000–2012. Where data were unavailable, we applied data- and evidence-based assumptions to construct plausible minimum and maximum estimates and set bounds on uncertainty. There was a large range between estimates in years when datasets were less comprehensive (2000–2008); smaller ranges were seen hereafter. In 15–19-year-old women in 2000, the estimated diagnosis rate ranged between 891 and 2,489 diagnoses per 100,000 persons. Testing and diagnosis rates increased between 2000 and 2012 in women and men across all age groups using minimum or maximum estimates, with greatest increases seen among 15–24-year-olds. Our dataset can be used to parameterise and validate mathematical models and serve as a reference dataset to which trends in chlamydia-related complications can be compared. Our analysis highlights the complexities of combining monitoring and surveillance datasets.

Since the beginning of the COVID-19 pandemic, the reproduction number [Formula: see text] has become a popular epidemiological metric used to communicate the state of the epidemic. At its most basic, [Formula: see text] is defined as the average number of secondary infections caused by one primary infected individual. [Formula: see text] seems convenient, because the epidemic is expanding if [Formula: see text] and contracting if [Formula: see text]. The magnitude of [Formula: see text] indicates by how much transmission needs to be reduced to control the epidemic. Using [Formula: see text] in a naïve way can cause new problems. The reasons for this are threefold: (1) There is not just one definition of [Formula: see text] but many, and the precise definition of [Formula: see text] affects both its estimated value and how it should be interpreted. (2) Even with a particular clearly defined [Formula: see text], there may be different statistical methods used to estimate its value, and the choice of method will affect the estimate. (3) The availability and type of data used to estimate [Formula: see text] vary, and it is not always clear what data should be included in the estimation. In this review, we discuss when [Formula: see text] is useful, when it may be of use but needs to be interpreted with care, and when it may be an inappropriate indicator of the progress of the epidemic. We also argue that careful definition of [Formula: see text], and the data and methods used to estimate it, can make [Formula: see text] a more useful metric for future management of the epidemic.