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 systems biology, Artificial Intelligence, epidemiology, 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 pririty, thus at the end of the course the students will be conscius of these aspects and we wil 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 Python and scientific Python.
Basic Calculus
Basic Newtonian Mechanics
Basic Knowledge of Differential Equations Basic bifurcations and stability
Knowledge of numerical analysis (linear algebra solvers, optimization,
ODE solvers).
Basic probability theory

1. Discrete state spae
stochastic modelling. Markov chains in discrete and
continuous time, discrete event simulation, population models. Continuous space-discrete time MArkov Processes
2. Continuous stochastic modeling: Stochastic Differential Equations
(SDEs), Numerical Algorithms for SDEs, Fokker Planck Equation, Ito and
Stratonovich SDEs, Noise-Induced Transitions, Bounded Stochastic
Processes, Nonlinear Fokker-Planck Equation as Model of Phase
Transitions
3. Stochastic approximations: mean field, ,
Langevin approximation, hybrid approximations.
4. Parameter estimation, ABC method and system design.
Examples from systems biology, epidemiology, statistical physics,
performance of computer networks, ecology.

Most of the material will be provided through online e-learning platforms
as pdf. Main books to consult:.
Crispin Gardiner Stochastic Methods: A Handbook for the Natural and
Social Sciences (4th Edition) Springer 2009
R. Durrett, Essentials of Stochastic Processes. Springer, 2012.
D. J Wilkinson, Stochastic Modelling for Systems Biology. Chapman & Hall,
2006.
Livi & Politi, Nonequilibrium Statistical Physics, A Modern Perspective,
Cambridge University Press (only the First Chapter)
A Papoulis “Probability, Random variables and Stochastic Processes” 2nd
Edition
Other books on Stochastic Modeling:
J. R. Norris, Markov chains. Cambridge, UK; New York: Cambridge
University Press, 1998.
G.A. Pavliotis - Stochastic Processes and Applications, Springer
(reasonably mathematical)
Vulpiani Appunti di Meccanica Statistica del Non Equilibrio (Chapters 1-5)
Suggested Textbooks On prerequisites:
Hale and Koçak, Dynamics and Bifurcations, Springer
Glass and Kaplan, Understanding Nonlinear systems, Springer
Boffetta & Vulpiani, Probabilità per Fisici, Springer Italy

1. Discrete stochastic modelling. Markov chains in discrete and
continuous time, discrete event simulation, population models. Continous-state discrete-time MArkov Processes
2. Continuous stochastic modeling: Stochastic Differential Equations
(SDEs), Numerical Algorithms for SDEs, Fokker Planck Equation, Ito and
Stratonovich SDEs, Noise-Induced Transitions, Bounded Stochastic
Processes, Nonlinear Fokker-Planck Equation as Model of Phase
Transitions
3. Stochastic approximations;
4Langevin approximation, hybrid approximations.
5 Parameter estimation, ABC and system design.
6. Bounded Stochastic Processes
7. Large dimension systems of SDEs and nonlinear Fokker Planck
equations as model of Phase Transitions
Examples from systems biology, epidemiology, statistical physics,
performance of computer networks, ecology.
Texts:
Lecture Notes provided by “the Professore”
Crispin Gardiner Stochastic Methods: A Handbook for the Natural and
Social Sciences (4th Edition) Springer 2009
R. Durrett, Essentials of Stochastic Processes. Springer, 2012.
D. J Wilkinson, Stochastic Modelling for Systems Biology. Chapman & Hall,
2006.
Livi & Politi, Nonequilibrium Statistical Physics, A Modern Perspective,
Cambridge University Press (only the First Chapter)
A Papoulis “Probability, Random variables and Stochastic Processes” 2nd
Edition
G.A. Pavliotis - Stochastic Processes and Applications, Springer
Vulpiani Appunti di Meccanica Statistica del Non Equilibrio

Frontal lectures and hands on sessions, both individual and in groups.
Ideally, each lecture will have a part of frontal teaching and a part of
hands-on training, which will range from simple implementation of the
simulation algorithms, to use of existing tools and libraries, to the
modelling and simulation of concrete case studies. Hands on exercises
will be given throughout the course.

Bring your own laptop.

The exam will be a project work. Depending on the complexity of the
project, this can be an individual or a group work. Each individual/ group
will have a certain number of tasks to perform, ranging from reproducing
results from literature, to build and analyze new models, to
implementation tasks.
To complete the exam, a brief presentation explaining the work done has
to be given and, upon request, (commented) code has to be provided.
During the presentation, few questions will be asked to assess the
individual contributions and preparation on the topics of the course. In
case of group projects, questions will be asked to each member of the
group individually.

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