Lecture Series in Phenomathematics, 2004

last update: December 15, 2004

Lecture 1, Next Lecture

Lecture 2: Mathematical analysis of an age-structured epidemic model for HIV infection
lecturer: Professor Hisashi Inaba (Graduate School of Mathematical Sciences, University of Tokyo)
date and time: January 12, 2005, 4:00 pm -- 5:30 pm
place: Room 803, Godoto Building
abstract:
In this lecture we consider an age-duration-structured population model for the spread of HIV infection. The basic model is formulated as a system of first order (McKendrick type) partial differential equation with nonlinear boundary condition. First we summarize some important features of HIV infection and discuss how to formulate mathematical models for HIV epidemic. Next we investigate the invasion problem to establish the basic reproduction ratio for the HIV epidemic model, which gives the invasion criteria whether the disease can invade into the completely susceptible host population. Subsequently we examine existence and uniqueness of endemic steady states. In contrast with the traditional epidemic models with the contact law of mass action type, our HIV model has a homogeneous (of degree one) force of infection, so we can prove that the backwardly bifurcating endemic solution could exist when the basic reproduction ratio crosses the unity. The presence of a backward bifurcation has practically important consequences for the control of infectious diseases, since if the disease is already endemic, in order to eradicate the disease, we have to reduce the basic reproduction ratio so far that it enters the region where the disease-free steady state is globally asymptotically stable and there is no endemic steady state. We also examine conditions for the local stability of the endemic steady states. Finally we discuss possible extensions of the basic formulation and open problems.

Lecture 1: Gene-protein network for circadian rhythm and temperature compensation
lecturer: Professor Yoh Iwasa (Department of Biology, Kyushu University)
date and time: June 23, 2004, 4:00 pm -- 5:30 pm
place: Room 803, Godoto Building
abstract:
Many organisms have an internal clock, by which they are able to show periodic activity even in a perfectly constant environment. Circadian rhythm is an oscillation generated by a feedback loop -- proteins produced by a gene (clock gene) move to nucleus in which they suppresses the expression of its own gene. [1] Rhythm-generating system often includes multiple clock genes, and have many structures, such as phosphorylation step required or nonlinear transport to nucleus. We show that these complications contribute to make the gene-protein network to produce a stable oscillation. [2] The saturation level of each reaction steps in the system has different effect to the oscillation. Saturation in Branch-reactions (e.g. decomposition) promotes oscillation, but saturation of an In-loop reaction (any reaction within the main feedback loop) would stop the oscillation. We found the saturation level of reactions is chosen according to the location of the rhythm system. [3] Circadian rhythm has a property named "temperature compensation" -- the period of the rhythm is unchanged when enhanced ambient temperature increases the rate of all the reaction steps in the network. We discuss the possible mechanisms for temperature compensation.

organized by
Izumi Takagi and Eiji Yanagida
Mathematical Institute, Tohoku University
Sendai, Japan 980-8578