Seminar "Phương trình Vi phân và Hệ động lực"

Thời gian: Chiều thứ 2 hàng tuần, 14.00 -16.00 AM

Địa điểm: Phòng C2, Viện NCCC về Toán, Tầng 7, Thư viện Tạ Quang Bửu, số 1, Đại Cồ Việt, Hà Nội

Lịch báo cáo:

Thời gian: Ngày 27 tháng 9 năm 2015

Báo cáo viên: TS Ngô Hoàng Long, Đại học SƯ phạm Hà Nội

Tên báo cáo: New approaches to the study of the Euler-Maruyama approximation for SDEs with low regularity coefficients.

Tóm tắt báo cáo: This talk presents new approaches to the study of strong rate of convergence of the Euler-Maruyama approximation for stochastic differential equations (SDEs) with low regularity coefficients. In particular, we consider the SDEs whose drift coeffcient is discontinuous or Holder continuous.


Thời gian: Ngày 02 tháng 11 năm 2015 

Địa điểm: Phòng C2, Viện NCCC về Toán, Tầng 7, Thư viện Tạ Quang Bửu, số 1, Đại Cồ Việt, Hà Nội 

Tóm tắt báo cáo:

  1. Population dynamics as a point pattern dynamics
    Mathematical models of population dynamics are conventionally described as a dynamical system where population size (or population density) changes with time as a continuous "real-valued" variable; these are often given as differential or difference equation. In this approach, we implicitly assume infinitely large population thereby population size changes smoothly and deterministically. In reality, however, a population is a collection of a certain number of individuals each of which gives birth or dies with some stochasticity and the population size, as the number of individuals, is "integer-valued". In this talk, I first introduce a classical approach to deal with integer-valued dynamics as stochastic process. I then extend the stochastic process to be spatial, i.e., each individual is located on a space as a point and a population has a certain spatial structure as a point pattern. This spatial stochastic process can be readily realized by Gillespie algorithm and simulations are getting easier to carry out using the modern advanced computation power. A mathematical approach to understand stochastic spatial population dynamics in terms of moment dynamics are discussed.
  2. Spatial epidemic models as a point pattern dynamics
    Epidemic models have been conventionally described as a dynamical system where several compartments categorized by health status, e.g., susceptible, infectious, etc., change in population size with time. These models are described by ordinary differential equations when only population size does matter, or by integro-differential equations when each compartment has a certain structure such as age, or by partial differential equations when each compartment is continuously distributed over a space with diffusion. It has been widely recognized that non-spatial population dynamics where everything is well mixed can behave very differently when spatial distribution is explicitly considered. The role of “space” is indeed a critical issue also in mathematical epidemiology. In this talk, I present an approach to extend non-spatial epidemic models to be spatial as a point pattern dynamics. Different from spatial models described by PDEs, we treat each individual as a point on a continuous two dimensional space and each individual has a certain "mark" depending on its health status; a snapshot of individuals’ distribution over space is represented by a marked point pattern and this marked point pattern dynamically changes with time. This approach is very flexible, allows us to carry out simulations under various assumptions. As an example, the classical SIS and SIR models are extended as point pattern dynamics. I then discuss how these simulations can be mathematically understood in terms of moment dynamics. Quantification of a point pattern is the key question that is to be resolved in further study.