Mathematical models II.
Description:
The aim of the course to give an introduction to the basics of mathematical modelling with deterministic and simulational approaches.
The first part of the course considers one-dimensional ordinary differential equations: after getting familiar with the basics concepts (initial value problem, autonomous equations, equilibrium, direction field), we solve simple types of equations (separable variables, linear equation). In the rest of this part we investigate problems from biology, chemistry and physics writing down the modelling equations and answering simple questions regarding the considered phenomenon. First we solve the problems by hand, later we implement interactive programs for analyzing solutions, stability of equilibria and dependency on the parameters.
The second part uses simulational approach to model real-world time-dependent systems: we focus on agent-based and cellular automata models. For programming the models we use a Python framework called Mesa, in which we build models applying object-oriented disciplines enabling simulation and visualization of the models.
In the last part we return to the deterministic models and have a brief introduction to higher-dimensional and higher-order models applied in physics and epidemiology.
During the whole course we mix the by-hand and programming way of analysis for analyzing the underlying models, improving skills both on quantitative (finding analytical solution) and qualitative (parameter dependency) sides. Additionally we use state-of-the-art tools (Python, Google Colaboratory, PyCharm) in our work and we aim to give industry-ready knowledge to the students.
Curriculum:
- Modelling with differential equations 1.
- theory of one-dimensional first-order ordinary differential equations (existence and uniqueness of the solution, equilibrium and stability in autonome equations)
- solution of special 1D equations: separable and linear differential equations
- application in biology, chemistry, physics
- implementing interactive analysis of the models in Python
- Modelling with simulation approach
- introduction to agent-based (ABM) and cellular automata (CA) models (neighborhood, update rules)
- introduction to Mesa modelling framework, implementation of a ABM and a CA model
- Modelling with differential equations 2.
- introduction to higher dimensional differential equations, SIR model as an example
- implementing solution of higher dimensional models
Evaluation:
- written test about solving exercises on differential equations (25 points)
- writing a mini project about a modelling problem (using differential equations) (25 points)
- implementing and presenting a cellular automaton/agent-based model (50 points)
VZS 