BIOMATHEMATICS
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- ANDREA CORLI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The aim of the course consists in showing some concrete application of mathematics (namely, ordinary differential equations) to biology. Conversely, some important mathematical results will be motivated by the applications. Some lectures are devoted to the implementation on a pc of the applications taken into consideration.
During the course, the students shall learn several techniques about dynamical systems and some basic notion on partial differential equations of evolution type. A particular care shall be devoted to the construction of a mathematical model from a biological phenomenon.
At the end of the course the students should be able to tackle some simple modeling problems and to find the corresponding theoretical and numerical solutions.
The course is suitable for students interested in Pure Mathematics or Applied Mathematics. Prerequisites
- Differential and integral calculus for functions of one and several variables, basic notions of ordinary differential equations and systems, some basic notions of MatLab. Linear algebra. No previous knowledge of biology is required.
Course programme
- The course consists in 48 teaching hours, some of them of which concerning computer lab. Here follows the content of the course.
Some reminds on ordinary differential equations and introduction to dynamical systems (about 10 hours). Characterization of linear system of two equations. Stability theory.
Networks of chemical reactions (about 22 hours). Reversible and irreversible reactions. Arrhenius' law. Reactions by transition. The law of mass action. Qualitative properties and equilibria for reactions by transitions. Nonlinear reactions.
In alternative to the last part of the previous section: Enzyme kinetics (about 6 hours). Kinetics of Michaelis-Menten. The quasi-stationary state assumption. Singular perturbations.
Deterministic models for epidemiology (about 16 hours). The common hypotheses of the models; incidence relations. Some models for the diffusion of an epidemy: the models SI and SIS, either in the epidemic or in the endemic case. The models SIR, SEIR with a detailed study of equilibria and stability. The SIR model with exponential increasing or decreasing of the population.
Computer simulations using MatLab of the principal models (included in the above hours, topic by topic). Introduction to standard MatLab routines for solving ordinary differential equations. Didactic methods
- The course is organized as follows.
Theoretical lessons in classroom about all topics of the course (Matlab excepted).
Computer elaborations with Matlab lessons. The software Matlab can be freely downloaded from Unife site for Unife students. Learning assessment procedures
- The final exam consists of an oral discussion about the theoretical and applicative contents of the course.
In particular, he will be asked to explain some significant and representative topics of the course, of his choice, therefore concerning the general theory of dynamical systems, reaction kinetics, mathematical epidemiology. The optional preparation of additional material, agreed in advance with the teacher, will be taken into consideration.
Scores are assigned in the following way: understanding of modeling aspects (10), understanding of mathematical aspects (10), capacity of solving simple exercises (5), capacity of using MatLab to solve simple applicative problems (5).
Exams will be "in presence". On-line exams are admitted only for students in quarantine because of Covid. Reference texts
- C. Mascia, E. Montefusco, A. Terracina: Biomat 1.0. Edizioni La Dotta (2018).
L. Perko: Differential equations and dynamical systems, Springer, 2001.
F. Brauer, C. Castillo-Chavez: Mathematical models in population biology and epidemiology, Springer (2000).
M.Iannelli, A. Pugliese: An introduction to mathematical population dynamics, Springer (2014).
J.D. Murray: Mathematical Biology. Springer (1989).
