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A.Y. 2020 / 2021

Second semester
Type of Learning Activity 
Related/additional subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Teaching language 


Learning objectives 

In this course you will learn principles and methods of mathematical and computational modelling of population processes, which has applications in several disciplines, including systems biology, epidemiology, computer networks, ecology.

Knowledge and understanding: you will learn the foundations of stochastic models with a discrete 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.

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. Knowledge of numerical analysis (linear algebra solvers, optimization, ODE solvers). Basic calculus, basic probability theory.


1. Discrete stochastic modelling. Markov chains in discrete and continuous time, discrete event simulation, Petri Nets.
2. Stochastic approximations: mean field, linear noise, moment closure, Langevin approximation, hybrid approximations.
3. Parameter estimation and system design.
4. Formalization and verification of emergent behavioural properties (if time).
Examples from systems biology, epidemiology, performance of computer networks, ecology.

Teaching format 

Frontal lectures and hands on sessions, both individual and in groups. The
balance will be roughly 65% of frontal lectures and 35% of hands-on
sessions. 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.

Extended Programme 

1. Introduction to stochastic modelling of complex systems.
2. Markov chains: discrete time, continuous time Markov chains.
3. Simulation of MC: sampling and Markov Chain Monte Carlo, simulation of CTMC.
4. Discrete Event Simulation and Petri Nets
5. Mean Field approximations: fluid approximation, linear noise, moment closure.
6. Parameter Estimation and System Design.
— if there will be time —
7. Langevin approximation and Stochastic Differential Equations.
8. Hybrid approximations and hybrid simulation.
9. Formalization and verification of emergent behavioural properties.

The theoretical concepts will be illustrated via examples, most of them taken from systems biology (modelling of molecular interactions inside cells), though other domains may be covered as well (epidemiology, performance of computer networks, ecology). No preliminary knowledge on these areas is required.

End-of-course test 

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 short report has to be written, (commented) code used has to be provided, and a brief presentation explaining the work done has to be given. During the presentation, few questions will be asked to asses 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.

Other information 

Bring your own laptop.


Most of the material will be provided through online e-learning platforms as pdf. Some additional reference textbooks are the following.

D. J Wilkinson, Stochastic Modelling for Systems Biology. Chapman & Hall, 2006.R. Durrett, Essentials of Stochastic Processes. Springer, 2012.
J. R. Norris, Markov chains. Cambridge, UK; New York: Cambridge University Press, 1998.

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