The asymmetric exclusion process is a well-established model in statistical physics that exhibits non-equilibrium phase transitions. It has received considerable attention of late as it is widely applicable to problems in molecular biology involving the transit of component parts along specified tracks or pathways. In this paper we use a self-consistent mean-field approach, backed up by Monte Carlo simulations, to examine the case where the exit from such a track is ‘gated’ by the presence of some external component that is capable of binding and unbinding from an additional site to the track; exit from the path is only possible by the bound presence of this component. We not only compute the relevant phase diagrams for this instance both in terms of the exit and entrance rates but also the binding and unbinding rates of the ‘gate’ and comment on this model's applicability to problems in biology.

We construct and exactly solve a model of an extended Brownian ratchet. The model comprises an arbitrary number of heterogeneous, growing and shrinking filaments which together move a rigid membrane by a ratchet mechanism. The model draws parallels with the dynamics of actin filament networks at the leading edge of the cell. In the model, the filaments grow and contract stochastically. The model also includes forces which derive from a potential dependent on the separation between the filaments and the membrane. These forces serve to attract the filaments to the membrane or generate a surface tension that prevents the filaments from dispersing. We derive an N -dimensional diffusion equation for the N filament-membrane separations, which allows the steady-state probability distribution function to be calculated exactly under certain conditions. These conditions are fulfilled by the physically relevant cases of linear and quadratic interaction potentials. The exact solution of the diffusion equation furnishes expressions for the average velocity of the membrane and critical system parameters for which the system stalls and has zero net velocity. In the case of a restoring force, the membrane velocity grows as the square root of the force constant, whereas it decreases once a surface tension is introduced.

For a disease such as COVID-19, it is important to identify individuals in a population at heightened risk of infection, as well as broader patterns of infection spread. This is both to estimate burden on healthcare systems (given substantial variation in disease severity from person to person), and to better control the spread of infection. In Scotland, the circulation of SARS-CoV-2 continues to place sustained pressure on healthcare systems, even after a comprehensive vaccination programme and earlier strict non-pharmaceutical interventions. To better understand individuals at heightened risk, we analyse the spatio-temporal distribution of over 450,000 cases of COVID-19 registered in Scotland in the waves of the B.1.1.529 Omicron lineage from November 2021, and an earlier wave of the B.1.617.2 Delta lineage from May 2021. These cases are taken from a uniquely fine scale national data set specifying individual tests. We use random forest regression on local case numbers, informing the model with measures relating to local geography, demographics, deprivation, COVID-19 testing and vaccination coverage. We can then identify broader risk factors indicative of higher case numbers. Despite the Delta and Omicron waves occurring around six months apart, with different control measures and immunity from vaccination and prior infection, the overall risk factors remained broadly similar for both. We find that finer details and clusters in the case distribution are only adequately explained when incorporating a combination of all these factors, implying that variation in COVID-19 cases results from a complex interplay of individual-level behaviour, existing immunity, and willingness to test for COVID-19 at all. On comparing testing patterns to subsequent COVID-19 hospitalisations, we conjecture that the distribution of cases may not be representative of the wider pattern of infection, particularly with respect to local deprivation.

The Rényi entropy is a generalisation of the Shannon entropy that is sensitive to the fine details of a probability distribution. We present results for the Rényi entropy of the totally asymmetric exclusion process (TASEP). We calculate explicitly an entropy whereby the squares of configuration probabilities are summed, using the matrix product formalism to map the problem to one involving a six direction lattice walk in the upper quarter plane. We derive the generating function across the whole phase diagram, using an obstinate kernel method. This gives the leading behaviour of the Rényi entropy and corrections in all phases of the TASEP. The leading behaviour is given by the result for a Bernoulli measure and we conjecture that this holds for all Rényi entropies. Within the maximal current phase the correction to the leading behaviour is logarithmic in the system size. Finally, we remark upon a special property of equilibrium systems whereby discontinuities in the Rényi entropy arise away from phase transitions, which we refer to as secondary transitions. We find no such secondary transition for this nonequilibrium system, supporting the notion that these are specific to equilibrium cases.