Mathematical analysis of models describing infectious disease

Propagation of an infectious disease in a population, and the dynamics of density of pathogen in vivo are investigated by using mathematical models. Those mathematical models are consist of systems of functional equations, such as ordinary differential equations and partial differential equations. For example, in the case of spread of infectious disease, a susceptible becomes an infectious by a contact with an infectious, and the numbers of susceptible and infectious agents change. The numbers as unknown variables satisfy functional equations. If the population is uniformly mixing, then the equations are ordinary differential equations. If age-structure or spatial diffusion is considered, then the equations are partial differential equations. We investigate those models in a mathematical point of view, and develop the method of analysis for those models.

Study on Zonal Flow Formations in Two-Dimensional Turbulence on a Rotating Sphere

A two-dimensional incompressible flow on a rotating sphere governed by the two-dimensional Navier-Stokes equation is one of the simplest mathematical models for planetary atmospheres. Although this mathematical model is highly simplified and does not contain many physical processes that may be actively functioning in real atmospheres, the flow field shows rich and interesting phenomena, including a spontaneous formation of large-scale zonal flows. However, these phenomena are the results of very complicated interactions between many factors, and even the mechanisms of simple and typical flow dynamics are not necessarily clear. I am especially interested in the emergence and the development of large-scale structures in two-dimensional turbulence on a rotating sphere, and investigating these by both numerical and analytical methods.

Analysis on Characteristic Features/Structures of Creatures

Some creatures possess highly unique body structures and biological properties. In order to obtain knowledge and information of essential factors related to the appearance and formation of such characteristic features, I am performing two different kinds of investigations. One is data analysis of genetic and some biological data obtained by experiments (collaborative research with an experimentalist), and the other is based on mathematical modellings and numerical simulations.