arXiv subjects

Jonathan Wylie

Publications and source records attributed to Jonathan Wylie.

2 recordsLinked to original sources

arXiv2020

Uniformly accurate effective equations for disease transmission mediated by pair formation dynamics

We derive and asymptotically analyze mass-action models for disease spread that include transient pair formation and dissociation. Populations of unpaired susceptibles and infecteds are distinguished from the population of three types of pairs of individuals; both susceptible, one susceptible and one infected, and both infected. Disease transmission can occur only within a pair consisting of one susceptible individual and one infected individual. By considering the fast pair formation and fast pair dissociation limits, we use a perturbation expansion to formally derive a uniformly valid approximation for the dynamics of the total infected and susceptible populations. Under different parameter regimes, we derive uniformly valid effective equations for the total infected population and compare their results to those of the full mass-action model. Our results are derived from the fundamental mass-action system without implicitly imposing transmission mechanisms such as that used in frequency-dependent models. They provide a new formulation for effective pairing models and are compared with previous models.

q-bio.PE↗

arXiv2019

A Feynman-kac Formula Approach for Computing Expectations and Threshold Crossing Probabilities of Non-smooth Stochastic Dynamical Systems

We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman-Kac formula, where the underlying stochastic processes satisfy variational inequalities modelling elasto-plastic and obstacle oscillators. We then focus on solving them numerically. The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities.

math.PR↗