Course title 
Mathematical modeling of nonlinear biosystems 
Assesment method 
Exam 
Hours per semester 
60 

Lect. 
Exercises 
Lab. 
Project 
ETCS 
4 


Hours/week 
2 
2 


Prerequisites 
Students should know basic notions from ordinary differential equations (ODEs),
such as Cauchy problem, solution of ODE, stationary solution (equilibrium, steady state), local/global stability.
Students should have some experience in using numerical software, preferably MATLAB®.

Course description 
The lecture presents a wide area of applications of nonlinear mathematical and computer modeling in biology, medical
science and other areas. Basic mathematical techniques for the description of system evolution in time, such as ordinary
differential equations (continuous time) and difference equations (discrete time), are discussed together with the analysis
of stability of equilibrium solutions and conditions for the loss of this stability, which yields transition to another
type of the dynamics.
Mathematical methods in the description of spatiotemporal processes based on nonlinear partial differential equations and
equations with delayed argument will be also discussed, focusing on the influence of time delays on the models dynamics, in
particular the appearance of oscillations caused by the delay, as well as traveling wave solutions of reactiondiffusion systems.

Course objectives 
In various applications linear mathematical models are the main tool used to describe, fit, explain,
and predict real phenomena. The main objective of this course is to present the role of mathematical modeling of biological and medical processes,
with the focus on nonlinear modeling, as it is obvious that phenomena observed in nature are nonlinear, and linear models can be only an approximation.
After the course student will understand the importance of mathematical modeling in explaining biological and medical phenomena.

Skills 
 student knows the role of Sshaped (logistic) function in the description of saturated processes
 student understands similarities and differences between continuous and discrete type of the mathematical description of natural phenomena
 student is able to mathematically describe various types of interactions, such as competition, mutualism, predation, etc
 student understands similarities and differences between ODE and DDE models, especially the role of delays in oscillatory patterns
 student understands connections between random walk and diffusion
 student understands the idea of the derivation of Greenspan solid tumor model
 student knows about the possibility of wave solutions of reactiondiffusion equations
 student knows how to describe immune reactions in various contexts
 student understands the role of Hill coefficient in mathematical modeling of natural phenomena
 student understands the ideas of mathematical modeling of spatial pattern formation

Grading 
 the student should prepare the project which will include analytical and numerical part of some real phenomena analysis
 in order to complete the whole course the student should additionally pass the final exam

Reference Texts and Software 
Literature:
Literature:
 J.D. Murray, Mathematical Biology: I. An Introduction, Springer, 2002
 J.D. Murray, Mathematical Biology: II. Springer, 2003
 N.F. Britton, Essential Mathematical Biology, Springer, 2003
 H. Smith, An Introduction to Delay Differential Equations with Application to the Life Sciences, Springer, 2010
 T. Erneux, Applied Delay Differential Equations, Springer, 2009
 U. Foryś, Matematyka w biologii, WNT, 2005
 R. Rudnicki, Modele i metody biologii matematycznej. Część 1. Modele deterministyczne, Instytut Matematyczny PAN, 2015
Software:
 Software: MATLAB®

Lecture Schedule 
1. 
Logistic equation in ODE version: the simplest way of nonlinear description of saturated growth of a population. Analysis of a single ODE.

2. 
Various processes associated with the dynamics of a single population described in the framework of ODE.

3. 
Discrete logistic equation: the simplest way to chaotic behavior.

4. 
Introduction to delay differential equations (DDE)  logistic equation with delay.

5. 
Derivation of diffusion equation from random walk. Animal dispersal models.

6. 
Introduction to simple solid tumor growth modeling: Greenspan and logistic equations.

7. 
Introduction to traveling waves – spreading of information.

8. 
Classic LotkaVolterra model describing preypredator interactions. The method of phase portraits for two ODEs.

9. 
Description of competition and mutualism for two and more populations.

10. 
Lorenz model of the weather forecast as a simple example of complex model dynamics.

11. 
Immune system: simple models of antigenantibody interactions.

12. 
Kuznetsov model of the immune reaction against tumors: sneaking through mechanism.

13. 
Hahnfeldt et al. model of tumor angiogenesis.

14. 
Hill function and coefficient. The role of Hill coefficient in the chaotic dynamics of MackeyGlass model of red blood cells production.

15. 
Spatial patterns formation models.
