APPLIED ANALYTICAL MODELS

Course objectives

Educational Goals General objectives: Acquire basic knowledge in modeling based on ordinary and partial differential equations, in the contexts presented in the program. In particular, he will be able to treat differential equations for networks of chemical reactions, the spread of epidemics, the kinetics of enzymes, the propagation of nerve impulses; in addition, he will be able to deal with models in which there is also dependence on space with diffusive terms. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results relating to some classes of ordinary differential equations and partial derivative equations useful for the description of models, mainly in the biochemical and epidemiological fields. Apply knowledge and understanding: at the end of the course the student will be able to present basic models in the biomathematic field, discussing their properties and characteristics. You will also be able to use the electronic calculator to perform basic numerical simulations of nonlinear differential equations using pre-existing libraries. Critical and judgmental skills: the student will have the bases to analyze the analogies and relationships between the topics covered and topics acquired in previous courses in the same field, critically recognizing their salient features. Communication skills: the student will have developed the ability to expose the contents in the oral part of the verification. Learning skills: the knowledge acquired will allow an individual and collective study of the subsequent LM courses that require modeling skills.

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LUCA ROSSI Lecturers' profile

Program - Frequency - Exams

Course program
INTRODUCTION What is Mathematical Modelling? Discrete, ODEs and PDEs Models. Fibonacci sequence. Graphical solution method. Transition from discrete to continuous. I ODEs Basic Properties: a) A-priori Estimates b) Existence c) Uniqueness d) Maximum Principle and Comparison Principle . I.1 Indefinite Integrals Fundamental Theorem of Calculus; boundary conditions for uniqueness; Comparison Principle; a-priori estimates and application to the Ascoli-Arzela Theorem. I.2 Linear ODEs I.3 Non-linear ODEs A-priori estimates; Cauchy-Lipschitz theorem using contractions; Comparison Principle between solutions and between sub/super solutions: Gronwall lemma; continuous dependence on initial conditions. I.4 Qualitative Study of Autonomous ODEs Consequences of the Comparison Principle: positivity and global existence; stationary solutions (or equilibria); stability, asymptotic stability and linear stability (e.g. harmonic oscillator); invariant sets: sufficient condition; Brouwer's theorem for finding equilibria; periodic solutions: sufficient condition, attractiveness and Poincaré Bendixson's theorem; Lorentz attractor; asymptotic behavior in dimension N=1. II ODE Models II.1 Networks of chemical reactions Simple and reversible transitions; general networks: existence of equilibria, their stability (Gerschgorin's theorem), comparison principle for cooperative systems, application: positive and bounded solutions; Perron-Frobenius theorem, convergence to the unique equilibrium. II.2 One-species population dynamics Fisher-KPP and weak and strong Allee effect. II.3 2-species population dynamics: Monotone (cooperation, competition) and non-monotone systems; the Lotka-Volterra prey/predator model. II.4 Epidemiological models SI model; SIR model; the R_0 number; reduction to the KPP equation; asymptotic behavior and total number of infections. III PDEs Basic Properties: a) A-priori Estimates b) Existence c) Uniqueness d) Maximum Principle and Comparison Principle III.1 Linear Parabolic Equations - Microscopic derivation of the diffusion equation - Heat equation: fundamental solution; existence for the Cauchy problem; equation with source (Duhamel's principle) and Cauchy problem (superposition principle) - Weak and Strong Maximum Principle, uniqueness for bounded solutions; counterexample to uniqueness - A priori estimates III .2 Semilinear Parabolic Equations - Weak and Strong Comparison Principle - Local existence and uniqueness - A priori estimates III .3 Parabolic Equations in bounded domains - Weak Maximum Principle - Reflection technique for reducing to the whole space IV PDE Models IV.1 Fisher-KPP reaction-diffusion equation - Bounded domain: Neumann boundary condition; complete invasion towards 1. - Bounded domain: Dirichlet boundary condition; uniqueness of the positive stationary solution; long-time behavior: extinction/persistence of the species depending on the size of the domain. - Unbounded domain: R; long-time behavior and hair-trigger effect; deduction of uniqueness of the positive stationary solution.
Prerequisites
The course requires familiarity with basic linear algebra, ordinary and partial differential equations
Books
Corrado Mascia, Eugenio Montefusco, Andrea Terracina, BioMat 1.0, Editrice LaDotta. Thomas Giletti, Parabolic PDE for population dynamics (notes)
Frequency
Attendance at lectures is recommended, but not mandatory
Exam mode
The examination consists of an optional written test and a mandatory oral test. An overall mark of no less than 18/30 is required to pass the examination
Bibliography
Martin Feinberg, Foundations of Chemical Reaction Network Theory, Springer. James Keener, James Sneyd, Mathematical Physiology, Springer. James D. Murray, Mathematical Biology I and II, Springer.
Lesson mode
The course lectures will be held in the classroom
  • Lesson code1031385
  • Academic year2025/2026
  • CourseApplied Mathematics
  • CurriculumMatematica applicata per le scienze
  • Year1st year
  • Semester1st semester
  • SSDMAT/05
  • CFU6