We use mean field analysis to study bursting in networks of identical, pulse-coupled neurons. The individual neurons are represented using a class of two-dimensional integrate and fire model. The mean field model is a system of switching ordinary differential equations and the transition to bursting involves both standard and nonsmooth bifurcations. The results of the mean field analysis are compared with numerical simulations of large networks. This is joint work with Wilten Nicola.

A relatively simple dynamic model that incorporated several different attributes of an stewardship program(i.e., diminished administration of antimicrobials to uninfected individuals; limited duration of therapy; reduced use of antipseudomonal antimicrobials) was successful in reproducing observed declines in the mean risk of colonization or infection with resistant P. aeruginosa before and after the implementation of an antimicrobial stewardship program. Using this approach we were able to determine the effects of the different elements of the antimicrobial stewardship program and to identify reduced prescribing of antimicrobials as the attribute of an antimicrobial stewardship program that would most substantially reduce antimicrobial resistance in P. aeruginosa in the future.

This is based on a joint work with Amy Hurford (Biology and Mathematics, MUN), Andres Morris (Infectious Disease and Internal Medicine, MSH) and David Fisman (Public Health, Toronto).

Delay differential equations (DDEs) can be formulated as infinite dimensional dynamical systems, and the infinite dimensionality can lead to very interesting dynamics even in simple looking equations. We consider the model DDE $$\dot{u}(t)=-\gamma u(t)-\sum_{i=1}^2\kappa_i u(t-a_i-c_iu(t))$$ which has two linearly state-dependent delays, but apart from the state-dependency of the delays the DDE has no other nonlinearity. Nevertheless it displays very complex dynamics and bifurcation structures including bi-stability of periodic orbits, torus bifurcations and period doubling. We explore in depth the dynamics associated with the stable invariant tori of the DDE including double Hopf bifurcations (and their unfolding), phase locking, Arnold tongues, and torus break up. The study also raises several numerical analysis issues in the computation of invariant manifolds and rotation numbers which we will touch upon.

By using the singular perturbation theory, especially on canard cycles, we study the canard phenomenon for a class of epidemic models with nonlinear incidence. We discuss and prove that the cyclicity of any possible slow-fast cycle is at most two, that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the slow-fast cycle by small perturbations. I will also present the results to the study of transmission and control of trachoma. This is a joint work with Chengzhi Li, Jianquan Li and Zhien Ma.