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: Simone Fagioli

Aims:

• To learn the basics in the mathematical modelling of population dynamics.
• To provide a mathematical description of ODE models in population dynamics and the interpretation of the qualitative behaviour of the solutions
• To get the basic notions in mathematical models in epidemiology and reaction kinetics.
• To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.
• To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.
• To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.

Content:

• Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.
• Continuous models for Interacting Populations. Predator-Prey Models: Lotka-Volterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis
• Reaction kinetics. Enzyme Kinetics: Basic Enzyme Reaction. Transient Time Estimates and Nondimensionalisation. Michaelis-Menten Quasi-Steady State Analysis
• Dynamics of Infectious Diseases: Epidemic Models and AIDS. Simple Epidemic Models (SIR, SI) and Practical Applications. Modelling Venereal Diseases. AIDS: Modelling the Transmission Dynamics of the Human Immunodeficiency Virus (HIV).
• Time-space dependent models: PDEs in biology. Diffusion equations. Diffusion and Random walk. The gaussian distribution. Smoothing and decay properties of the diffusion operator. Nonlinear diffusion-
• Reaction–diffusion models for one single species. Diffusive Malthus equation and critical patch size. Travelling waves. Fisher–Kolmogoroff equation.
• Reaction–diffusion systems. Multi species waves in Predator-Prey Systems. Turing instability and spatial patterns.
• Chemotaxis modelling. Diffusion vs. Chemotaxis: stability vs. instability. Diffusion vs. Chemotaxis: stability and blow–up. Chemotaxis with nonlinear diffusion. Models with maximal density
• Nonlocal interaction models in biology. Mathematical models of swarms. Approximation with interacting particle systems. Asymptotic behaviour.
• Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.

Pre-requisites:

Basic calculus and analysis (differential and integral calculus with functions of many variables).

Ordinary differential equations.

Basics in finite dimensional dynamical systems.

Elementary methods for the solution of linear partial differential equations (separation of the variables).

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

Unit Coordinator: Alessandra Tessitore, Daria Capece

Aims:

Tumor initiation, progression and invasion are complex processes involving multiple and different phenomena. Mathematical models and computer simulations can help to describe, schematize and comprehend them, to provide data which could be putatively used in clinical practice to prevent and/or more specifically treat cancer. In this context, a multidisciplinary approach is fundamental to reach this goal. The main objective of this course is to approach and understand the biological processes at the base of cancer, focusing on the most significant features of oncogenesis, with the aim to provide basic information which can be applied to mathematical modelling. On completion, the student should:

• know fundamentals about structure and functions of nucleic acids and proteins in eukaryotic cell;
• understand the significance of gene mutations and epigenetics alterations in diseases;
• identify tumor classification criteria;
• understand biological and functional mechanisms at the base of cancer initiation and progression; -
• know the most important databases for DNA mutation classification, microRNA and protein pathway analysis as well as on-line resources for acquiring datasets to be applied to big data analysis,
• know and understand the principles at the base of personalized therapy in cancer to predict the response to therapeutic schemes.

Content:

Fundamentals about the structure and the role of nucleic acids and proteins.
Genetic and epigenetic mechanism at the base of oncogenesis (gene mutations, DNA damage repair system failure, methylation, microRNA dysregulation). Features of cancer cells and tumor classification. Biological mechanisms of angiogenesis, invasion and metastasis. Molecular pathways involved in cell differentiation, proliferation and survival. Big data in cancer analysis.
Personalized therapy in cancer. Synopsis of cancer mathematical models for understanding tumor biology and predicting/optimizing therapy.

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

Unit Coordinator: Donatella Donatelli

Aims:

The aim of the course is 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 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.

Content:

• 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
• Fluid dynamic modeling in various fields: magnetohydrodynamics, combustion, astrophysics.
• Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Pre-requisites:

Basic notions of functional analysis and multi variable calculus, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

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

Unit Coordinator: Alessandro Borri

Aims:

Systems biology is an emerging research area, which aims at providing mathematical models helping to understand the dynamic interactions occurring within and among cells. This course provides the basic mathematical tools to model and analyze gene transcription as well as biochemical reaction networks: the most important network motifs are investigated exploiting both deterministic and stochastic approaches.

Content:

• Review of basic concepts of biology and probability; deterministic vs. stochastic approach.
• Transcription factors and transcription networks; gene expression: transcription and translation; separation of time scales and the Quasi-Steady-State Approximation (QSSA); activators and repressors; Hill and logic input functions; dynamics of simple regulation and switching production in ordinary differential equation (ODE) form; response time; mass action kinetics and conservation equation; the Michaelis-Menten kinetics; cooperativity; dependent and independent bindings; multi-dimensional input functions.
• Network Motifs: real and randomized networks; the Erdos-Renyi (ER) model; graph properties; negative autoregulation (NAR) and stability analysis; three-node network motifs: feedback loops (FBLs) and feedforward loops (FFLs); coherent and incoherent FFLs; sign-sensitive delay, output pulse generation, speed up of response time, biphasic response curves in I1-FFL; review of the other types of FFL; examples: Arabinose system, Galactose system, flagella motor genes in E. Coli.
• Introduction to Numerical Simulation in Systems Biology; MATLAB: introduction and basic notions; integration of non-stiff differential equations via ode45; randomized computation, Monte Carlo simulation, construction of randomized networks (E-R models) and digraphs.
• General Transcription and Interaction Networks: Single-Input Module (SIM), Multi-Output Feedforward Loop, LIFO and FIFO temporal programs, bi-fans and Densely Overlapping Regulons (DORs), interlocked FFLs; positive autoregulation (PAR) and response slow-down; two-node positive feedback: double-positive (lock-on) and double-negative (toggle-switch) circuits; three-node motives; protein-protein interaction (PPI) networks and hybrid motifs; network motifs in real-world networks: food webs, social networks, electrical circuits, neuronal networks. Examples: sporulation of bacterium Bacillus subtilis, phosphorylation, response to pain stimuli in humans.
• Biological Oscillations: the role of negative feedback and delay; time-separation, feedback strength and cooperativity; noise-induced and delay oscillations; bistability and hysteresis; robustness in biological circuits; models at a higher level of organization: tissues and organs; the glucose-insulin feedback loop.
• The stochastic approach in Systems Biology: motivation, Chemical Reaction Networks, Continuous-Time Markov Chains; the Chemical Master Equation (CME) and its properties, characterization of the CME stationary distribution, the macroscopic equation, one-step processes; stochastic simulation in Systems Biology, inverse sampling method, Gillespie Stochastic Simulation Algorithm (SSA); examples.
• Mesoscopic models and noise: extrinsic vs intrinsic noise, bursty reactions, the Chemical Langevin Equation, the Wiener Process, the Euler-Maruyama integration of Stochastic Differential Equations (SDEs).
• Biological and biomedical examples: metabolism, tumors, epidemics, pharmacokinetics and pharmacodynamics (PKPD), environmental pollution, ecology.

Pre-requisites:

Basic notions of mathematical analysis, dynamical systems and probability.

Reading list:

 Uri Alon, An introduction to systems biology: design principles of biological circuits, CRC press, 2019.

 E. Klipp, W. Liebermeister, C. Wierling, and A. Kowald, Systems biology: a textbook, John Wiley & Sons, 2016.