In this course you will learn principles and methods of mathematical and computational modelling of population processes and stochastic differential equations, which have applications in several disciplines, including Artificial Intelligence, epidemiology,biology, socioeconomic sciences, statistical physics, computer networks, ecology. Knowledge and understanding: you will learn the foundations of stochastic models with both discrete and continuous state space and of stochastic approximations. You will learn how to simulate such models to analyze them and understand their behaviours, and how to estimate parameters from experimental data. Applying knowledge and understanding: you will be capable of building a model of a complex system, by capturing the key features to be modelled and by understanding what kind of experimental data and information is available and which level of model complexity they support. You will learn how to simulate a model efficiently, and to use approximations, judging the best one in the light of the system and the property to be analyzed, and the computational resources available. Making judgments: At the end of the course the students will have the ability to integrate knowledge aount modeling and simulation of complex stochastic natural and virtual systems,and formulate judgements with the information coming from data and from qualitative knowledge . This information is by nature incomplete or, more frequently, limited information. During the course ethics of scientific research and the impact of research will be a priority, thus at the end of the course the students will be conscious of these aspects and we will able, in principle, to deal with them Communication skills: being able to explain the basic ideas and communicate the results to experts and to non-experts. Learning skills: being capable of exploring literature, find related and alternative approaches, and combine them to solve complex problems.

Basic knowledge of scientific Python. Linear Algebra Basic Newtonian Mechanics Basic Knowledge of Differential Equations Basic bifurcations and stability Basic Knowledge of numerical analysis (especially: simulatiosn of ODEs) Basic applied probability

0. This is **not** a course in Stochastic Calculus. This is a course on the modelling and simulation of stochastic phenomena. 1. Stochastic processes and Markov stochastic processes 2. Stochastic processes with continuous time and continuous state space 3. Discrete Time Markov Chains 4. Continuous Time Markov Chains 5. Simulation of stochastic processes 6. Bayesian parameter inference

Most of the material will be provided through UIniTS online e-learning platforms as pdf. Main books to consult: The professor’s lecture notes Crispin Gardiner Stochastic Methods: A Handbook for the Natural and Social Sciences (4th Edition) Springer 2009 Livi & Politi, Nonequilibrium Statistical Physics, A Modern Perspective, Cambridge University Press (only the First Chapter) Vulpiani Appunti di Meccanica Statistica del Non Equilibrio (Chapters 1-5) Textbooks On math prerequisites: Hale and Koçak, Dynamics and Bifurcations, Springer Glass and Kaplan, Understanding Nonlinear systems, Springer Boffetta & Vulpiani, Probabilità per Fisici, Springer Italy A Papoulis “Probability, Random variables and Stochastic Processes”

Important: this is not a stochastic calculus course, this is a modeling and simulation course. Our main aim is to give a introduction focusing on meaning of concepts far more than on mathematical properties of equations. Formulas are explained but with a heuristic approach. Each topic will be introduced by means of examples from systems biology, biology, epidemiology, statistical physics, ecology, sociophysics etc... Detailed List of topics: Short recap of some key concepts on nonlinear dynamical systems Continuous time – continuous state space modeling:: Stochastic Differential Equations (SDEs) Numerical Algorithms for SDEs Stochastic Equilibria and Stochastic Local Stability Ito Formula Fokker Planck Equation Effect of noise on multi-stable system Noise-Induced Transitions Stochastic Integration Ito and Stratonovich SDEs Bounded Stochastic Processes Nonlinear Fokker-Planck Equation for large dimensional SDE models and Phase Transitions Spatiotemporal Stochastic Processes: order out of noise Stochastic Resonance. Going beyond diffusion models: the Master Equation A first introduction to Continuous and Discrete Time Markov Chains starting from the Master Equation Continuous Time Random Walks Levy Processes and Levy SDE Discrete Time Markov Chains: main properties. The concept of Ergodicity Continuous Time Markov Chains (CTMC) and the Master Equation The Gillespie Stochastic Simulation Algorithm Doubly Stochastic CTMC and their simulation The Population Continuous Time Markov Chain and the Chemical Master Equation From CTMC to ODE models The Tau-Leaping approximate algorithm The Cjemical Langevin Equation Parameter estimation: Approximated Bayesian Computation Method

Frontal lectures, illustration of scientific papers and hands on sessions. Hands-on training will include simple implementation of the simulation algorithms on key models, with use of existing tools and libraries. Hands on exercises will be given throughout the course.

Bring your own laptop.

The exam will be split in two parts. In the first part, the student must choose a scientific paper, and after the approval by the professor she/he will have to work on that paper. The work can range from reproducing the results of the paper, to make new simulation, up to extend the modesl proposed in the paper. Then the student will have to prepare a presentation on the work done, which will be the first part of the exam. IMPORTANT: from the above description it should be clear theat the project and presentation must not be a simple extended abstract of the chosen paper. The second part of the exam consist in a number of easy questions, both theoretical and practical on the program of the course.

The course introduces students to modern techniques of analysis and modeling of stochastic systems in the natural and social sciences, as well as the related “heuristic mathematical” and simulation techniques. Stochastic mathematical and computational modeling of natural and social phenomena is one of the pillars of sustainable development, and all the techniques learned in this course can be applied in this context.