Sem3 UAQ Social Sciences

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Applications  @  UAQ  30 ECTS credits

Mathematical models in social sciences

The specialization track “Mathematical models in social sciences” will take place at the University of L’Aquila. The proposed curriculum covers newly developed mathematical modelling approaches, aiming at realizing practical solutions for smart and healthy communities. UAQ has longstanding collaborations with the Imperial College of London (UK) and with the University of Muenster (Germany) on many-agent modelling and mean-field limits in several branches of social sciences such as opinion formation, emergence of collective motion, and the social behaviour of largely crowded communities (the latter also in collaboration with the Universities of Catania and Ferrara, in Italy); the UAQ team has a longstanding expertise on "nonlocal aggregation models", an emerging subject in applied mathematical research with applications in particular to the modelling of criminal behaviour in urban areas. Important achievements have been also carried out at UAQ in the area of nonlinear conservation laws with applications to traffic modeling and population dynamics.

 

Below you can find information about the subjects for this semester.

  • Advanced analysis [6 credits]

    Advanced analysis

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      The main objectives of the course are as follows: to provide knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics; to apply this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about nonlinear equations (traffic flow, gas dynamics, fluid dynamics).

    • Topics

       

      Measure and integration theory. Functions of bounded variation. Distributions theory. Fourier transform. Sobolev spaces. Application to the study of partial differential equations: elliptic equations of second order, parabolic equations of second order, hyperbolic systems of first-order equations, nonlinear conservation laws. An outline of semigroup theory.


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  • Mathematical fluid dynamics [6 credits]

    Mathematical fluid dynamics

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      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.

    • Topics

       

      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.


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  • Mathematical models for collective behaviour [6 credits]

    Mathematical models for collective behaviour

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      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.

    • Topics

       

      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.


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Pick 12 credits

  • Deterministic modelling in population dynamics and epidemiology [6 credits]

    Deterministic modelling in population dynamics and epidemiology

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      • Enabling the student to formulate "ad-hoc" deterministic models, such as ODEs, PDEs, interacting particle systems, that describe the dynamics of an epidemics in specific situations.
      • Providing analytical and numerical techniques allowing to determine the qualitative behaviour of those models.
      • Complement the models with "control" terms in order to plan specific "containment strategies".
    • Topics

       

      • Introduction to epidemic modelling.
      • SIR models and their variants.
      • Multi group modelling.
      • Simulation of SIR and SEIR models.
      • Control methods based on drug therapy.
      • Models with time delaly, asymptotic behavior and stability.
      • Spatial models for the spread of an epidemic.
      • Control strategies based on wave propagation.
      • Interactin particle systems for the evolution of epidemics.
      • Discrete vs Continuum modeling using integro-differential equations.
      • Simulation of multi-agent systems.
      • Computation of Rt from numerical simulations.
      • Control and optimisation strategies based on lockdown and drug therapy.
    • Prerequisites

       

      • Dynamical systems, numerical methods for ordinary differential equations.
      • Linear partial differential equations of diffusive type.
    • Books

       

      • James D. Murray; Mathematical biology I: An introduction; Springer.
      • James D. Murray; Mathematical biology II: Spatial Models and biomedical applications; Springer.
      • Fred Brauer, Pauline van den Driessche, Jianhong Wu; Mathematical Epidemiology; Lecture notes in mathematics; Springer.

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  • Mathematics for decision making [6 credits]

    Mathematics for decision making

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      The goal of this course is to describe some mathematical models of strategic interaction among rational decision-makers.

      We provide knowledge of the main mathematical tools of nonlinear and set-valued analysis which are crucial for studying the existence and the stability of the solutions of particular equilibrium problems such as variational and quasi-variational inequalities and non-cooperative games.

    • Topics

       

      We cover aspects of pure mathematics such as continuity notions and fixed point theorems for set-valued maps (Browder, Kakutani, Fan-Glicksberg) and we study the the existence of solutions for problems arising in behavioral and social science, among which Nash equilibria and Ky Fan minimax inequalities.


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  • Time series and prediction [6 credits]

    Time series and prediction

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      The course is an introduction to Time Series Analysis and Forecasting. The level is the first-year graduate in Mathematics with a prerequisite knowledge of basic inferential statistical methods. The aim of the course is to present important concepts of time series analysis (Stationarity of stochastic processes, ARIMA models, forecasting etc.). At the end of the course, the student should be able to select an appropriate ARIMA model for a given time series.

    • Topics

       

      Stochastic processes (some basic concepts). Stationary stochastic processes. Autocovariance and autocorrelation functions. Ergodicity of a stationary stochastic process. Estimation of moment functions of a stationary process. ARIMA models. Estimation of ARIMA models. Building ARIMA models. Forecasting from ARIMA models

    • Books

       

      Time Series Analysis Univariate and Multivariate Methods, 2nd Edition, W. W. Wei, 2006, Addison Wesley.
      Time Series Analysis, J. Hamilton, 1994, Princeton University Press.
      Time Series Analysis and Its Applications with R Examples, Shumway, R. and Stoffer, D., 2006, Springer.
      Introduction to Time Series and Forecasting. Second Edition, P. Brockwell and R. Davis, 2002, Springer.


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  • Mathematical modeling in cellular biology [3 credits]

    Mathematical modeling in cellular biology

    • ECTS credits 3
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      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.

    • 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.

    • Prerequisites

       

      Mathematical analysis, linear algebra, numerical analysis, Matlab.

    • Books

       

      - 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.


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  • Numerical methods for stochastic modelling [3 credits]

    Numerical methods for stochastic modelling

    • ECTS credits 3
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      The Aim of this course is to provide the student with the knowledge of numerical modeling for stochastic problems and the ability to analyze theoretical properties and design mathematical software based on the proposed schemes.

      On successful completion of this module, the student should
      - have profound knowledge and understanding of the most relevant numerical methods for the approximation of stochastic differential problems and the design of accurate and efficient mathematical software;
      - demonstrate skills in choosing the most suitable discretization in relation to the problem to be solved and ability to provide theoretical analysis and mathematical software based on the proposed schemes;
      - demonstrate capacity to read and understand other texts on the related topics.

    • Topics

       

      - Discretized Brownian motion.
      - Ito and Stratonovich stochastic integrals.
      - Stochastic differential equations: motivation, modeling, existence and uniqueness, strong and weak solutions.
      - Ito formula.
      - Explicit methods: Euler-Maruyama and Milstein.
      - Implicit methods: stochastic theta-methods.
      - Strong and weak convergence.
      - Mean-square and asymptotic linear stability.
      - Nonlinear stability analysis.
      - Stochastic geometric numerical integration.

    • Prerequisites

       

      Basic Numerical Analysis, differential equations and stochastic processes.

    • Books

       

      An Introduction to the Numerical Simulation of Stochastic Differential Equations, D.J. Higham and P. E. Kloeden, SIAM, 2021.


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