Principles of Mathematical Modelling

Teacher: dr. Miljenko Marušić, professor
Semester: first
ECTS: 6
Required course

  1. To introduce students to the application of mathematical modelling in the analysis of biomedical systems including populations of molecules, cells and organisms.
  2. To show how mathematics, especially ordinary differential equations and computing can be used in an integrated way to analyse      biomedical systems.
  1. To have an enhanced knowledge and understanding of mathematical modelling and mathematical methods using differential equations for the analysis of biological/medical systems,
  2. To be able to assess biological/medical inferences that rest on mathematical arguments,
  3. To be aware of the use of differential equations and computers to assist them in studying biological/medical systems,
  4. To be able to formulate and analyse dynamical models using difference equations,
  5. To use differential equations to model population dynamics of single and multiple species and infectious diseases,
  6. To use qualitative theory of ordinary differential equations to derive conclusions in models used for biomedical systems.
  1. Introduction to continuous models.
  2. Population dynamics. Single-species populations. Malthus (exponential) model, Verhulst (logistic) model, Gompertz model. Mathematical models of tumour growth
  3. Modelling loss of population (death, harvesting). Growth under restriction. Monod model. Chemostat model.
  4. Parameter identification problem. Least squares method. Elements of numerical optimization.
  5. Numerical solution of ODE.
  6. Steady state solutions, stability, linearization. Systems of equations, phase-plane diagrams.
  7. Population dynamics. Multiple species populations. Predator-prey systems, Lotka-Volterra model. Competition models.
  8. Population biology of infectious diseases. SIR model.
  9. Linear difference equations with applications. Qualitative behaviour. Cell division, an insect population.
  10. Nonlinear difference equations with applications. Steady states, stability. Logistic difference equation. Density dependence, Nicholson-Bailey model.
  • Course objectives

    1. To introduce students to the application of mathematical modelling in the analysis of biomedical systems including populations of molecules, cells and organisms.
    2. To show how mathematics, especially ordinary differential equations and computing can be used in an integrated way to analyse      biomedical systems.
  • Expected learning outcomes

    1. To have an enhanced knowledge and understanding of mathematical modelling and mathematical methods using differential equations for the analysis of biological/medical systems,
    2. To be able to assess biological/medical inferences that rest on mathematical arguments,
    3. To be aware of the use of differential equations and computers to assist them in studying biological/medical systems,
    4. To be able to formulate and analyse dynamical models using difference equations,
    5. To use differential equations to model population dynamics of single and multiple species and infectious diseases,
    6. To use qualitative theory of ordinary differential equations to derive conclusions in models used for biomedical systems.
  • Course content

    1. Introduction to continuous models.
    2. Population dynamics. Single-species populations. Malthus (exponential) model, Verhulst (logistic) model, Gompertz model. Mathematical models of tumour growth
    3. Modelling loss of population (death, harvesting). Growth under restriction. Monod model. Chemostat model.
    4. Parameter identification problem. Least squares method. Elements of numerical optimization.
    5. Numerical solution of ODE.
    6. Steady state solutions, stability, linearization. Systems of equations, phase-plane diagrams.
    7. Population dynamics. Multiple species populations. Predator-prey systems, Lotka-Volterra model. Competition models.
    8. Population biology of infectious diseases. SIR model.
    9. Linear difference equations with applications. Qualitative behaviour. Cell division, an insect population.
    10. Nonlinear difference equations with applications. Steady states, stability. Logistic difference equation. Density dependence, Nicholson-Bailey model.
PMF
EU fondovi
UNI-ZG