Understanding the seasonal/periodic reoccurrence of influenza will be
very helpful in designing successful vaccine programs and introducing
public health interventions. However, the reasons for seasonal/periodic
influenza epidemics are still not clear even though various explanations
have been proposed. In this talk, we present an age-structured type evolutionary epidemiological model of influenza A drift, in which the susceptible class is continually replenished because the pathogen changes genetically and immunologically from one epidemic to the next, causing previously immune hosts to become susceptible. Applying our recent established center manifold theory for semilinear equations with non-dense domain, we show that Hopf bifurcation occurs in the model. This
demonstrates that the age-structured type evolutionary epidemiological model of influenza A drift has an intrinsic tendency to oscillate due to the evolutionary and/or immunological changes of the influenza viruses.
(based on joint work with Pierre Magal).

Many systems in the engineering and biology involve moving interfaces or boundaries. Front tracking method is one of the most accurate and efficient computational approaches for studying such systems. A main challenge of developing front tracking algorithms is to capture the interface topological changes. In this talk I shall introduce an improved three-dimensional front tracking method and consider an application for turbulent mixing driven by Rayleigh-Taylor instability, which shows an excellent agreement with the experiments. For the second part of the talk, I will present a computational analysis of cell signaling in biology and medicine. Scaffold, a class of proteins, plays many important roles in signal transduction. Through studying various models of scaffold, I will show novel regulations induced by its spatial location and switch-like responses due to scaffold. To efficiently compute the models, we introduce a new fast numerical algorithm incorporated with adaptive mesh refinement for solving the stiff systems with spatial dynamics.