ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Corrado Lattanzio

Aims:

Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

Content:

• Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.;
• Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).;
• Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p n. Sobolev in- equalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H−1.;
• Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.;
  First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.

Pre-requisites:

Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Reading list:

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universi- text, Springer.;
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer.;
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.;
G. Gilardi, Analisi 3. McGraw–Hill.;
V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Raffaele D'Ambrosio

Aims:

Students will learn how to implement and apply the mathematical and statistical models and methods that at the moment are most successfully applied in epidemiology. Both spatial and temporal aspects are considered. The main, but not unique, framework is the stochastic one. 

Content:

• Informative and quantitative descriptive models, e.g. compartment models, random graphs, ETAS model. Spatiotemporal noninformative models, e.g. autoregressive models, nonparametric models. Intervention measures models.
• Bayesian statistics models.  Reproductive number Bayesian estimation.
• Statistical inference.
• Stochastic simulation and validation. Monte Carlo Simulation. Stochastic algorithms. Stochastic simulation with MatLab. Interpolation.  Prediction. Trends. Clustering. Features extraction. Hypothesis testing.

Pre-requisites:

Mathematical analysis, linear algebra, probability theory, scientific programming.

Reading list:

• Mathematical Biology - I. An Introduction | James D. Murray | Springer
• Lectures on Monte Carlo Methods. Neal Madras. Fields Institute Monograph
• Markov Chain Monte Carlo in Practice | David Spiegelhalter, Sylvia Richardson, W. R. Gilks. Taylor & Francis Group
• Selected recent papers in scientific literature

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Marco Di Francesco, Simone Fagioli

Aims:

• 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".

Content:

• 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.

Pre-requisites:

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

Reading list:

• 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.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Giordano Pola

Aims:

The aim of this course is to provide students with basics about modeling, analysis and control of networked and multi-agent systems through consensus techniques.

Content:

• Introduction to networked and multi-agent systems.
• Recalls of graph theory.
• The agreement protocol: the linear and nonlinear cases.
• Formation control.

Reading list:

Graph Theoretic Methods in Multiagent Networks, Princeton University Press, Mehran Mesbahi & Magnus Egerstedt, 2010

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Umberto Triacca

Aims:

Content:

Reading list: