Delay Equations and Asymptotic Methods in Biomathematics 1000-1M25ROA

The plan of the lecture is as follows.

1. Theory of ordinary differential equation with discrete delay
   (i.e. the right-hand side of a differential equation depends not only on the
   variables at time but also on one specific moment in the past, where is an arbitrary constant). With simple examples we will illustrate the influence of the delay on the behaviour of the solutions to differential equations. We will prove theorems, analogous to those for ordinary differential equations, about existence and uniqueness of the solutions, extension of the solutions, stability of steady states, Hopf bifurcation. We will present criterions for the stability of steady states. To illustrate the theory we will use mathematical models: logistic with time delay, illnesses with periodic dynamics, regulation of haematopoiesis process, dynamic of whales populations, dynamic of HIV, and possibly other models of biomedical phenomena.
   
2. Asymptotic methods. Many times we are unable to find the exact solution to some differential equation (or a system of differential equations). It is common that we are interested in the behaviour of solutions of a differential equation near a specific point (ex. we are interested in knowing the speed of the convergence of the solution to a steady state or in the behaviour of the solution near a limit cycle). Due to asymptotic methods we may find an approximate solution and observe its behaviour without using computers. In this manner, we may also find parameters of the model which are relevant for specific behaviour. This theory will be illustrated with mathematical models of biochemical reactions: B-Z reaction, activator-inhibitor reaction, Michaelis-Menten system, systems of weakly coupled oscillators.
   

We plan 10 hours of clases in computer laboratory. We will teach numerical methods solving delay differential equations using Julia or Python.