Mathematical Models of Biology and Medical Sciences

elective courses

Ordinary differential equations I 1000-114aRRZa

The lecture is devoted to the widely understood mathematical modelling in biology and medicine. We mainly focus on ecological models which are built using differential and difference equations. We also consider models of immune reactions and those of classical genetics (Mendel theory) based on Markov chains.

The lecture is devoted to the widely understood mathematical modelling in biology and medicine. We mainly focus on the ecological models which are built on the basis of differential and difference equations, however we also consider models of immune reactions and the classical genetics (the Mendel theory) in the context of Markov chains.

The following problems are discussed:

• Simple ecological models - continuous and discrete in time. Birth and death processes with migrations. Saturated growth - the logistic model, comparison between continuous and discrete version of the logistic equation. Age dependent models (Leslie matrices in discrete time and delay models in continuous version - the logistic equation with time delay.
• The prey - predator system. The Lotka - Volterra model (the rule of mean densities and the effect of fishery). models with hiding-places and carrying capacity for preys (stabilisation effect). The Kolmogorov model - limit cycles.
• The model of competiton.
• The Nicholson-Bailey model for parazite and its host. Simple epidemiological models (SIS model, the Kermack - McKendrick model).
• Models of immune system
• Graph theory and food chains.
• Markov chains and the Mendel theory.
• Game theory and evolutionary stable strategy. Reaction - diffusion models.
• Microscopic and macroscopic models

• Nicolas Bacaër, A Short History of Mathematical Population Dynamics, Springer London 2011
• J. Banasiak, Introduction to mathematical methods in population theory
• J. Banasiak, M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhäuser Basel 2014
• Fred Brauer, Carlos Castillo-Chavez, Zhilan Feng, Mathematical Models in Epidemiology, Springer New York
• Fred Brauer, Carlos Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer New York 2012
• Nicholas F. Britton, Essential Mathematical Biology, Springer, 2003.
• Karl Peter Hadeler, Topics in Mathematical Biology, Springer, Cham 2017
• F.Roberts. Discrete mathematical models with applications to social, biological and environmental problems. Prentice Hall, Englewood Cliffs, NJ, 1976
• Horst R. Thieme, Mathematics in Population Biology, Princeton University Press 2003

Knowledge and skills: she / he

1. is able to describe basic population processes, such as reproduction, mortality, migration, competition, in therms of difference and differential equations;
2. can analyze the dynamics of solutions of a single differential equation and can formulate appropriate conclusions regarding the described population (or other biological process);
3. can analyze the dynamics of a single differential equation (analytical and graphic methods) and formulate appropriate conclusions regarding the described population (or other biological process);
4. understands the differences in the dynamics of solutions that appear as a result of the application of various types of mathematical description, namely: difference or differential equation, can describe these differences using the example of a logistic equation;
5. knows how the delay can affect the dynamics of the population;
6. can describe various types of interactions between populations in the terms of ordinary differential equations;
7. on the basis of phase portrait analysis of two ordinary differential equations, can describe changes in population dynamics over time;
8. understands the difference between local and global stability and the resulting biological consequences;
9. knows what a food chain is, can describe it in the language of graph theory;
10. knows what the trophic status is, can calculate it for a given species in a given food chain;
11. understands what the diffusion describes in the case of population dynamics models and for models of biochemical reactions;
12. is able to check whether diffusion instability occurs in a given model and explain the biological consequences of this;
13. can describe simple interactions between two species in the language of game theory.

Social competence: understanding of the importance of mathematical modeling in explaining natural phenomena.

score system and a written exam