The year 2020 has seen the emergence of a global pandemic as a result of the disease COVID-19. This report reviews knowledge of the transmission of COVID-19 indoors, examines the evidence for mitigating measures, and considers the implications for wintertime with a focus on ventilation.

Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data are typically akin to a boundary value-type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this paper, we reformulate the classical susceptible–infectious–recovered system in terms of the number of detected positive infected cases at different times to yield what we term the observational model. We then prove the existence and uniqueness of a solution to the boundary value problem associated with the observational model and present a numerical algorithm to approximate the solution. This article is part of the theme issue ‘Technical challenges of modelling real-life epidemics and examples of overcoming these’.

Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data is typically akin of a boundary value type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this paper, we reformulate the classical Susceptible-Infectious-Recovered system in terms of the number of detected positive infected cases at different times, we then prove the existence and uniqueness of a solution to the derived boundary value problem and then present a numerical algorithm to approximate the solution.

A mathematical model is proposed for the flow of nutrients in an inflatable hydroponics module being developed by Phytoponics. Simple experiments were performed via the injection of dye into the system enabling a basic understanding of the time and length scales of nutrient flow and mixing. Four different flow regimes are identified. At the scale of a single root, a Stokes-flow approximation may be used. Brinkman flow operates at the individual plant scale which homogenises into a 1D model for macro-scale flow of nutrients. A shear flow model is used to predict the flow in regions dominated by plant roots. Finally, simplified two-phase flow equations are derived for the more turbulent bubble flow during aeration. These are solved within the software COMSOL. The overall conclusion is that both the periodic flow of nutrients and the aeration are required to enable even nutrient spread.