Cost and social distancing dynamics in a mathematical model of COVID-19 with application to Ontario, Canada.

A mathematical model of COVID-19 is presented where the decision to increase or decrease social distancing is modelled dynamically as a function of the measured active and total cases as well as the perceived cost of isolating. Along with the cost of isolation, we define an overburden healthcare cost and a total cost. We explore these costs by adjusting parameters that could change with policy decisions. We observe that two disease prevention practices, namely increasing isolation activity and increasing incentive to isolate do not always lead to optimal health outcomes. We demonstrate that this is due to the fatigue and cost of isolation. We further demonstrate that an increase in the number of lock-downs, each of shorter duration can lead to minimal costs. Our results are compared with case data in Ontario, Canada from March to August 2020 and details of expanding the results to other regions are presented.

Quasi-steady uptake and bacterial community assembly in a mathematical model of soil-phosphorus mobility.

We mathematically model the uptake of phosphorus by a soil community consisting of a plant and two bacterial groups: copiotrophs and oligotrophs. Four equilibrium states emerge, one for each of the species monopolising the resource and dominating the community and one with coexistence of all species. We show that the dynamics are controlled by the ratio of chemical adsorption to bacterial death permitting either oscillatory states or quasi-steady uptake. We show how a steady state can emerge which has soil and plant nutrient content unresponsive to increased fertilization. However, the additional fertilization supports the copiotrophs leading to community reassembly. Our results demonstrate the importance of time-series measurements in nutrient uptake experiments.

Regularization of the Ostwald supersaturation model for Liesegang bands.

In a previous paper, we analysed the Keller-Rubinow formulation of Ostwald's supersaturation theory for the formation of Liesegang rings or Liesegang bands, and found that the model is ill-posed, in the sense that after the termination of the first crystal front growth, secondary bands form, as in the experiment, but these are numerically found to be a single grid space wide, and thus an artefact of the numerical method. This ill-posedness is due to the discontinuity in the crystal growth rate, which itself reflects the supersaturation threshold inherent in the theory. Here we show that the ill-posedness can be resolved by the inclusion of a relaxation mechanism describing an impurity coverage fraction, which physically enables the transition in heterogeneous nucleation from precipitate-free impurity to precipitate-covered impurity.

On the Keller-Rubinow model for Liesegang ring formation.

We study the model of Keller & Rubinow (Keller & Rubinow 1981 J. Chem. Phys74, 5000-5007. (doi:10.1063/1.441752)) describing the formation of Liesegang rings due to Ostwald's supersaturation mechanism. Keller and Rubinow provided an approximate solution both for the growth and equilibration of the first band, and also for the formation of secondary bands, based on a presumed asymptotic limit. However, they did not provide a parametric basis for the assumptions in their solution, nor did they provide any numerical corroboration, particularly of the secondary band formation. Here, we provide a different asymptotic solution, based on a specific parametric limit, and we show that the growth and subsequent cessation of the first band can be explained. We also show that the model is unable to explain the formation of finite width secondary bands, and we confirm this result by numerical computation. We conclude that the model is not fully posed, lacking a transition variable which can describe the hysteretic switch across the nucleation threshold.

Numerical simulations of drumlin formation.

We summarize the present form of the instability theory for drumlin formation, which describes the coupled subglacial flow of ice, water and sediment. This model has evolved over the last 20 years, and is now at the point where it can predict instabilities corresponding to ribbed moraine, drumlins and mega-scale glacial lineations, but efforts to provide numerical solutions of the model have been limited. The present summary adds some slight nuances to previously published versions of the theory, notably concerning the constitutive description of the subglacial water film and its flow. A new numerical method is devised to solve the model, and we show that it can be solved for realistic values of most of the parameters, with the exception of that corresponding to the water film thickness. We show that evolved bedforms can be three-dimensional and of the correct sizes, and we explore to some extent the variation of the solutions with the model's parameters.