Modelling of spatial infection spread through heterogeneous population: from lattice to partial differential equation models.

We present a simple model for the spread of an infection that incorporates spatial variability in population density. Starting from first-principle considerations, we explore how a novel partial differential equation with state-dependent diffusion can be obtained. This model exhibits higher infection rates in the areas of higher population density-a feature that we argue to be consistent with epidemiological observations. The model also exhibits an infection wave, the speed of which varies with population density. In addition, we demonstrate the possibility that an infection can 'jump' (i.e. tunnel) across areas of low population density towards areas of high population density. We briefly touch upon the data reported for coronavirus spread in the Canadian province of Nova Scotia as a case example with a number of qualitatively similar features as our model. Lastly, we propose a number of generalizations of the model towards future studies.

Narrow escape problem with a mixed trap and the effect of orientation.

We consider the mean first passage time (MFPT) of a two-dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections. High-order asymptotic formulas for the MFPT and the fundamental eigenvalue of the Laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture. We obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a Green's function and a certain alignment vector. We find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle. In the scenario where the trap undergoes prescribed motion in the domain, the MFPT is seen to be particularly sensitive to the orientation of the trap.

A tale of two distributions: from few to many vortices in quasi-two-dimensional Bose-Einstein condensates.

Motivated by the recent successes of particle models in capturing the precession and interactions of vortex structures in quasi-two-dimensional Bose-Einstein condensates, we revisit the relevant systems of ordinary differential equations. We consider the number of vortices N as a parameter and explore the prototypical configurations ('ground states') that arise in the case of few or many vortices. In the case of few vortices, we modify the classical result illustrating that vortex polygons in the form of a ring are unstable for N≥7. Additionally, we reconcile this modification with the recent identification of symmetry-breaking bifurcations for the cases of N=2,…,5. We also briefly discuss the case of a ring of vortices surrounding a central vortex (so-called N+1 configuration). We finally examine the opposite limit of large N and illustrate how a coarse-graining, continuum approach enables the accurate identification of the radial distribution of vortices in that limit.

A threshold area ratio of organic to conventional agriculture causes recurrent pathogen outbreaks in organic agriculture.

Conventional agriculture uses herbicides, pesticides, and chemical fertilizers that have the potential to pollute the surrounding land, air and water. Organic agriculture tries to avoid using these and promotes an environmentally friendly approach to agriculture. Instead of relying on herbicides, pesticides and chemical fertilizers, organic agriculture promotes a whole system approach to managing weeds, pests and nutrients, while regulating permitted amendments. In this paper, we consider the effect of increasing the total area of agricultural land under organic practices, against a background of conventional agriculture. We hypothesized that at a regional scale, organic agriculture plots benefit from existing in a background of conventional agriculture, that maintains low levels of pathogens through pesticide applications. We model pathogen dispersal with a diffusive logistic equation in which the growth/death rate is spatially heterogeneous. We find that if the ratio of the organic plots to conventional plots remains below a certain threshold l(c), the pest population is kept small. Above this threshold, the pest population in the organic plots grows rapidly. In this case, the area in organic agriculture will act as a source of pest to the surrounding region, and will always infect organic plots as they become more closely spaced. Repeated localized epidemics of pest outbreaks threaten global food security by reducing crop yields and increasing price volatility. We recommend that regional estimates of this threshold are necessary to manage the growth of organic agriculture region by region.