Mathematical models of epidemics and other biological processes
The course fee for 1-week on-site courses - that includes tuition fee, meals (coffee break and lunch), local transport and the cost of the leisure time programs - is 490 EUR. All applicants are required to pay 90 EUR (out of this 490) as registration fee at registration. The registration fee is non-refundable.
Credits: 4 EC
Our course offers ECTS points, which may be accepted for credit transfer by the participants' home universities. Those who wish to obtain these credits should inquire about the possible transfer at their home institution prior to their enrolment. The International Strategy Office will send a transcript to those who have fulfilled all the necessary course requirements and request one.
Venue: ELTE Institute of Mathematics, Department of Applied Analysis and Computational Mathematics
APPLICATION:
Please pay the registration fee and fill out this form: https://www.elte.hu/en/mathematical-models-bsu2024
COURSE DESCRIPTION
In this course, the participants will be able to gain knowledge about population dynamics and epidemic models. The students will construct adequate numerical models and run computer simulations. The course has no special mathematical preliminary requirements since the mathematical background will be discussed.
Course schedule
Day 1
Lecture 1: Introduction: mathematical background (given by B. Takács).
In this introductory lecture the mathematical basics needed for the course are discussed: the meaning of continuous functions, the application of derivatives and their role in mathematical modeling. The operations on matrices are also defined, and the method to calculate their eigenvalues is shown. Moreover, a brief introduction to numerical sequences will also be gi-ven.
Lecture 2: Initial-value problems for ordinary differential equations (given by I. Faragó).
We give a short introduction to the theory of the ordinary differential equations (ODE). Well posedness of problems will be discussed. Different examples will be shown, including se-veral continuous population models.
Day 2
Lecture 3: Stability theory, part 1 (given by G. Svantnerné Sebestyén).
The stability theory of differential equations will be discussed. We give an introduction on the basics of Lyapunov-stability. We review the equilibrium points and their stability analysis in the case of several differential equations. We also define the two-dimensional phase portraits of systems of differential equations.
Lecture 4: Stability theory, part 2 (given by G. Svantnerné Sebestyén).
The stability analysis of nonlinear equations and systems of differential equations will be int-roduced. We review the most common methods: the theory of linearization and Lyapunov-functions. We also discuss the existence of periodic orbits. The theory will be illustrated by examples.
Lecture 5: Introduction to discretization methods (given by I. Faragó).
We introduce the numerical solution of ODE problems. We explain the basic concepts and by using the explicit Euler method we construct discrete models. We show the conver-gence of the numerical solution of the discretized problem.
Day 3
Lecture 6: Discretization methods for ODE (given by I. Faragó).
We define the different discretization methods, including one-step methods. The well-known Runge-Kutta method will be investigated in detail. We show their application to the conti-nuous population models.
Lecture 7: General notions of numerical analysis (given by I. Faragó).
In the first part of the lecture we will investigate the multistep discretization methods. The ab-solute stability of the different numerical methods will be also discussed. Finally, we define the basic notions of numerical analysis on an abstract level: consistency, stability and con-vergence. Their relation will be also given (Lax theorem).
Lecture 8: Models of population dynamics (given by G. Svantnerné Sebestyén).
After a short summary of the history of population dynamics, we discuss the single species and multispecies models. We particularly study the common population dynamics models, like the Malthus-, or the Lotka-Volterra model.
Day 4
Lecture 9: Epidemic models, part 1 (given by B. Takács)
A brief look into the history of epidemic models is given: the Ross- and the Kermack-Mckend-rick models, SIR and SIS models. Some extensions are also discussed (e.g. the SEIR mo-del, models with vaccination, equations defined on graphs).
Lecture 10: Epidemic models, part 2 (given by B. Takács)
Some properties of the previous models are discussed: equilibrium points and their stability, the meaning of the R_0 number, persistence, periodic orbits. We might also give a brief out-look into chaotic systems.
Lecture 11: Models of discrete dynamics (given by G. Svantnerné Sebestyén).
The theory of discrete dynamical models will be discussed. We give an introduction to linear and nonlinear dynamical systems. We review the equilibrium points and the stability analysis of these systems. The theory will be illustrated by examples from biology.
Day 5
Lecture 12: Introduction to Matlab (given by G. Svantnerné Sebestyén).
The basics of Matlab will be introduced. We provide a tutorial guide to Matlab. The students will get to know and study the most common built-in functions which are important in biologi-cal models.
Lecture 13: Computational modeling of population dynamics (given by B. Takács)
In this session the students can model the previously discussed continuous models by using the Matlab functions introduced before. Discrete models are also constructed and simulated by computers.
Lecture 14: Computational modeling of epidemics (given by B. Takács)
The last session of this course consists of the computational modeling of epidemic models. Numerical models for a multiple-region epidemic model are investigated and simulated by using Matlab.