MathMods :: Joint MSc

The course will cover some mathematical models currently used in the analysis of collective phenomena, such as vehicular and pedestrian traffic, flocking phenomena. Emphasis will be given to the mathematical treatment of specific problems coming from real world applications.

Macroscopic traffic models: LWR model, fundamental diagrams. The Riemann problem. Point-type constraints (the "Toll gate" problem). Second order models for traffic flow: Aw-Rascle model, shocks description, instabilities near vacuum. Basics on the theory of systems of conservation laws. Junction of roads, networks. Distribution rules along the roads, optimization of the flux. Solution to the Riemann problem at a junction. Pedestrian flow: normal and panic situation. Macroscopic models, conservation of "mass", eikonal equation. The Hughes model for pedestrian flow; in one space dimension: curve of turning points, Rankine-Hugoniot conditions. Macroscopic models for pedestrian flow that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain. Introduction to the theory of flocking. Cucker-Smale model for flocking, conservation of moment and decay of the kinetic energy. Asymptotic flocking. Singular communication rate. The kinetic limit. Introduction to synchronization: Kuramoto model, basic properties.

This course is designed to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type with a particular attention to biofluid dynamics. At the end of the course the students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for the analysis of other partial differential equations.

Derivation of the governing equations: Euler and Navier-Stokes. Eulerian and Lagrangian description of fluid motion; examples of fluid flows. Vorticity equation in 2D and 3D. Dimensional analysis: Reynolds number, Mach Number, Frohde number. From compressible to incompressible models. Existence of solutions for viscid and inviscid fluids. Modeling for biofluids mechanics: hemodynamics, cerebrospinal fluids, animal locomotion: swimming and flying; Bioconvection for swimming microorganisms.

Provide the mathematical tools suitable for the numerical solution of cellular dynamics models and the development of algorithms in a structured programming language.
At the end of the course, each learner should:
- have a good knowledge and understanding of the most widespread cellular models in the literature;
- be able to apply his knowledge to solve numerically, using a programming environment (Matlab), problems related to cellular dynamics;
- demonstrate critical spirit in the model construction phase, understanding the coherence between underlying biological hypotheses and their mathematical translation in the model;
- demonstrate ability in numerical reasoning, both in choosing the most suitable numerical methodology to solve the problem in question, based on its features, and in the details of the programming process;
- demonstrate ability to read and understand other texts on related topics.

Basic models: Malthus exponential growth, Verhlust logistic growth, Volterra model, prey-predator model.
Model of the growth of bacteria in the chemostat.
Growth of a structured population.
Study of the action of a drug.
Outline of the mathematical procedure for the identification of reaction constants.
In vitro model of the Ebola virus.
Model of immunosenescence and cancer cell proliferation.
CD4 + cell homeostasis model.
Deterministic and stoachastic cellular models of HIV.
Positivity preserving chemical Langevin equations.
Partitioning of the RNA graph for the research of modularity.
Non-negative matrix factorization applied to radiation-induced metabolic changes in human cancer cells.

Mathematical analysis, linear algebra, numerical analysis, Matlab.

- C. Molina-Paris and G. Lythe. Mathematical Models and Immune Cell Biology. Springer-Verlag New York (2011).
- L. A. Segel. Mathematical Models in Molecular and Cellular Biology. Cambridge University Press (1984).
-G.Monegato, Fondamenti di Calcolo Numerico (Seconda Edizione), Clut (1998).
-A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Matematica Numerica, Springer (2014).
-Bini, M. Capovani e O. Menchi, Metodi Numerici per l'Algebra Lineare, Zanichelli (1988).
-W. J. Palm III, Matlab 6 per l'Ingegneria e le Scienze. Mc Graw Hill. 2003.