Statistical and agent-based modelling of the transmissibility of different SARS-CoV-2 variants in England and impact of different interventions.

The English SARS-CoV-2 epidemic has been affected by the emergence of new viral variants such as B.1.177, Alpha and Delta, and changing restrictions. We used statistical models and the agent-based model Covasim, in June 2021, to estimate B.1.177 to be 20% more transmissible than the wild type, Alpha to be 50-80% more transmissible than B.1.177 and Delta to be 65-90% more transmissible than Alpha. Using these estimates in Covasim (calibrated 1 September 2020 to 20 June 2021), in June 2021, we found that due to the high transmissibility of Delta, resurgence in infections driven by the Delta variant would not be prevented, but would be strongly reduced by delaying the relaxation of restrictions by one month and with continued vaccination. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Multi-scale models of local control of calcium induced calcium release.

Calcium-induced-calcium-release in cardiac myocytes is the release of Ca(2+) from the sarcoplasmic reticulum (SR) triggered by Ca(2+) entering the cell through L-type Ca(2+) channels. The Ca(2+) is released through ryanodine receptors which 'sense' local [Ca(2+)] in the small region (the diadic space) positioned between the t-tubules and the SR. The length-scale of a single diad is of the order of 10nm and the diffusion time-scale is of order of 1 micros with each cell containing approximately 10,000 diadic spaces which act independently. However, typically one is interested in Ca(2+) currents at the whole cell level and higher. This is a multi-scale problem and cannot be solved by direct computation. In this paper we develop a general framework for deriving approximate solutions of these models.

The effects of static magnetic field on action potential propagation and excitation recovery in nerve.

Calculations using the Hodgkin-Huxley and one-dimensional cable equations have been performed to determine the expected sensitivity of conduction and refractoriness to changes in the time constant of sodium channel deactivation at negative potentials, as reported experimentally by Rosen (Bioelectromagnetics 24 (2003) 517) when voltage-gated sodium channels are exposed to a 125 mT static magnetic field. The predicted changes in speed of conduction and refractory period are very small.

Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release.

Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localised calcium stores. The aim of this paper is to analyse how the stochastic nature of individual receptors within these stores combines to create stochastic behaviour on long time-scales that may ultimately lead to waves of activity in a spatially extended cell model. Techniques from asymptotic analysis and stochastic phase-plane analysis are used to show that a large cluster of receptor channels leads to a release probability with a sigmoidal dependence on calcium density. This release probability is incorporated into a computationally inexpensive model of calcium release based upon a stochastic generalisation of the fire-diffuse-fire (FDF) threshold model. Numerical simulations of the model in one and two dimensions (with stores arranged on both regular and disordered lattices) illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. Illustrations of spreading circular waves, spirals and more irregular waves are presented. Furthermore, receptor noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously.

Stability of cardiac waves.

Cardiac waves can fail to propagate when the membrane potential of the cells in the wavefront rises too slowly. The sodium channel inactivation gates play an important role in this process of propagation block. Simple models including inactivation gates can have travelling waves of constant form with two possible velocities. A stability analysis demonstrates that the slower velocity is always unstable, and in limited parameter regimes the faster velocity can also be unstable. Waves with the lower velocity propagate a finite distance before they dissipate due to this instability and this distance is calculated. The distance can be large suggesting that they might be seen in certain pathological conditions. The analytical results are compared with numerical simulations of the simplified model and a detailed cardiac ionic model.

A mathematical analysis of the generation and termination of calcium sparks.

Calcium sparks are local regenerative releases of Ca(2+) from a cluster of ryanodine receptors on the sarcoplasmic reticulum. During excitation-contraction coupling in cardiac cells, Ca(2+) sparks are triggered by Ca(2+) entering the cell via the T-tubules (Ca(2+)-induced Ca(2+) release). However under conditions of calcium overload, Ca(2+) sparks can be triggered spontaneously. The exact process by which Ca(2+) sparks terminate is still an open question, although both deterministic and stochastic processes are likely to be important. In this article, asymptotic methods are used to analyze a single Ca(2+) spark model, which includes both deterministic and stochastic biophysical mechanisms. The analysis calculates both spark frequencies and spark duration distributions, and shows under what circumstances stochastic transitions are important. Additionally, a model of the coupling of the release channels via the FK-binding protein is analyzed.

A simplified local control model of calcium-induced calcium release in cardiac ventricular myocytes.

Calcium (Ca2+)-induced Ca2+ release (CICR) in cardiac myocytes exhibits high gain and is graded. These properties result from local control of Ca2+ release. Existing local control models of Ca2+ release in which interactions between L-Type Ca2+ channels (LCCs) and ryanodine-sensitive Ca2+ release channels (RyRs) are simulated stochastically are able to reconstruct these properties, but only at high computational cost. Here we present a general analytical approach for deriving simplified models of local control of CICR, consisting of low-dimensional systems of coupled ordinary differential equations, from these more complex local control models in which LCC-RyR interactions are simulated stochastically. The resulting model, referred to as the coupled LCC-RyR gating model, successfully reproduces a range of experimental data, including L-Type Ca2+ current in response to voltage-clamp stimuli, inactivation of LCC current with and without Ca2+ release from the sarcoplasmic reticulum, voltage-dependence of excitation-contraction coupling gain, graded release, and the force-frequency relationship. The model does so with low computational cost.

Interplay between SERCA and sarcolemmal Ca2+ efflux pathways controls spontaneous release of Ca2+ from the sarcoplasmic reticulum in rat ventricular myocytes.

Waves of calcium-induced calcium release occur in a variety of cell types and have been implicated in the origin of cardiac arrhythmias. We have investigated the effects of inhibiting the SR Ca(2+)-ATPase (SERCA) with the reversible inhibitor 2',5'-di(tert-butyl)-1,4-benzohydroquinone (TBQ) on the properties of these waves. Cardiac myocytes were voltage clamped at a constant potential between -65 and -40 mV and spontaneous waves evoked by increasing external Ca(2+) concentration to 4 mm. Application of 100 microm TBQ decreased the frequency of waves. This was associated with increases of resting [Ca(2+)](i), the time constant of decay of [Ca(2+)](i) and the integral of the accompanying Na(+)-Ca(2+) exchange current. There was also a decrease in propagation velocity of the waves. There was an increase of the calculated Ca(2+) efflux per wave. The SR Ca(2+) content when a wave was about to propagate decreased to 91.7 +/- 3.2%. The period between waves increased in direct proportion to the Ca(2+) efflux per wave meaning that TBQ had no effect on the Ca(2+) efflux per unit time. We conclude that (i) decreased wave frequency is not a direct consequence of decreased Ca(2+) pumping by SERCA between waves but, rather, to more Ca(2+) loss on each wave; (ii) inhibiting SERCA increases the chance of spontaneous Ca(2+) release propagating at a given SR content.