Time series models are ubiquitous in science, arising in any situation where researchers seek to understand how a system's behaviour changes over time. A key problem in time series modelling is inference; determining properties of the underlying system based on observed time series. For both statistical and mechanistic models, inference involves finding parameter values, or distributions of parameters values, which produce outputs consistent with observations. A wide variety of inference techniques are available and different approaches are suitable for different classes of problems. This variety presents a challenge for researchers, who may not have the resources or expertise to implement and experiment with these methods. PINTS (Probabilistic Inference on Noisy Time Series-https://github.com/pints-team/pints) is an open-source (BSD 3-clause license) Python library that provides researchers with a broad suite of non-linear optimisation and sampling methods. It allows users to wrap a model and data in a transparent and straightforward interface, which can then be used with custom or pre-defined error measures for optimisation, or with likelihood functions for Bayesian inference or maximum-likelihood estimation. Derivative-free optimisation algorithms-which work without harder-to-obtain gradient information-are included, as well as inference algorithms such as adaptive Markov chain Monte Carlo and nested sampling, which estimate distributions over parameter values. By making these statistical techniques available in an open and easy-to-use framework, PINTS brings the power of these modern methods to a wider scientific audience.

We investigate the effect of school closure and subsequent reopening on the transmission of COVID-19, by considering Denmark, Norway, Sweden, and German states as case studies. By comparing the growth rates in daily hospitalisations or confirmed cases under different interventions, we provide evidence that the effect of school closure is visible as a reduction in the growth rate approximately 9 days after implementation. Limited school attendance, such as older students sitting exams or the partial return of younger year groups, does not appear to significantly affect community transmission. A large-scale reopening of schools while controlling or suppressing the epidemic appears feasible in countries such as Denmark or Norway, where community transmission is generally low. However, school reopening can contribute to significant increases in the growth rate in countries like Germany, where community transmission is relatively high. Our findings underscore the need for a cautious evaluation of reopening strategies that ensure low classroom occupancy and a solid infrastructure to quickly identify and isolate new infections.

Uncertainty quantification (UQ) is a vital step in using mathematical models and simulations to take decisions. The field of cardiac simulation has begun to explore and adopt UQ methods to characterize uncertainty in model inputs and how that propagates through to outputs or predictions; examples of this can be seen in the papers of this issue. In this review and perspective piece, we draw attention to an important and under-addressed source of uncertainty in our predictions—that of uncertainty in the model structure or the equations themselves. The difference between imperfect models and reality is termed model discrepancy , and we are often uncertain as to the size and consequences of this discrepancy. Here, we provide two examples of the consequences of discrepancy when calibrating models at the ion channel and action potential scales. Furthermore, we attempt to account for this discrepancy when calibrating and validating an ion channel model using different methods, based on modelling the discrepancy using Gaussian processes and autoregressive-moving-average models, then highlight the advantages and shortcomings of each approach. Finally, suggestions and lines of enquiry for future work are provided. This article is part of the theme issue ‘Uncertainty quantification in cardiac and cardiovascular modelling and simulation’.