Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced. Global stability of the endemic equilibrium states of the models is thereby established

Characterising circulatory dysfunction and choosing a suitable treatment is often difficult and time consuming, and can result in a deterioration in patient condition, or unsuitable therapy choices. A stable minimal model of the human cardiovascular system (CVS) is developed with the ultimate specific aim of assisting medical staff for rapid, on site modelling to assist in diagnosis and treatment. Models found in the literature simulate specific areas of the CVS with limited direct usefulness to medical staff. Others model the full CVS as a closed loop system, but models were found to be very complex, difficult to solve, or unstable. This paper develops a model that uses a minimal

The growth of human cancers is characterised by long and variable cell cycle times that are controlled by stochastic events prior to DNA replication and cell division. Treatment with radiotherapy or chemotherapy induces a complex chain of events involving reversible cell cycle arrest and cell death. In this paper we have developed a mathematical model that has the potential to describe the growth of human tumour cells and their responses to therapy. We have used the model to predict the response of cells to mitotic arrest, and have compared the results to experimental data using a human melanoma cell line exposed to the anticancer drug paclitaxel. Cells were analysed for DNA content at multiple time points by flow cytometry. An excellent correspondence was obtained between predicted and experimental data. We discuss possible extensions to the model to describe the behaviour of cell populations in vivo.

A functional differential equation for the steady size distribution of a population is derived from the usual partial differential equation governing the size distribution, in the particular case where birth occurs by one individual of size x dividing into a new individuals of size x/a. This leads, in the case of constant growth and birth rate functions, to the functional differential equation y'{x) = -ay(x) + aay(ax) together with the integral condition J°° y(x) dx = 1. We first look at a number of properties that any solution of this equation and boundary condition must have, and then proceed to find the unique solution by the method of Laplace transforms. Results from number theory on the infinite product found in the solution are presented, and it is shown that y(x) tends to a normal distribution a s o -» l + .

Abstract-Critically ill patients are often hyperglycemic and extremely diverse in their dynamics. Consequently, fixed protocols and sliding scales can result in error and poor control. A two-compartment glucose-insulin system model that accounts for time-varying insulin sensitivity and endogenous glucose removal, along with two different saturation kinetics is developed and verified in proof-of-concept clinical trials for adaptive control of hyperglycemia. The adaptive control algorithm monitors the physiological status of a critically ill patient, allowing real-time tight glycemic regulation. The bolus-based insulin administration approach is shown to result in safe, targeted stepwise glycemic reduction for three critically ill patients.