Fitting data with dynamical models: ten lessons on mathematical field work

Session 1 (Josep Sardanyés). Introduction to mathematical and computational tools to investigate dynamical processes.

Here we’ll start by providing a general catalogue of different modelling approaches for dynamical processes, including both mathematical and computational models (or both simultaneously) and explaining the strong and weak points, as well as the assumptions of each approach. This introduction to these mathematical and computational tools will cover a range of techniques used in the study of dynamical processes, including ODEs, PDES, and numerical simulations under deterministic and stochastic settings. Participants will learn how to design mathematical models to describe the behavior of dynamical systems and use them to investigate physical systems such as oscillators or biological systems such as species interactions in ecosystems.

Session 2 (Lluís Alsedà). Introduction to data fitting methods.

Specification of problems towards the comparison (error measuring) between the modeled and the observed data. Error measures and error norms (least squares and others). Parameters’ fitting error minimizers. Minimization techniques; deterministic versus heuristic.

Session 3 (Josep Sardanyés). Modelling examples in Biology: ODEs and cellular automata (CA) models.

This part of the course will provide participants with practical examples of how differential equations and CA models can be used to study dynamical processes in different fields such in Virology and Ecology. Differential equations are commonly used to model the reproduction cycles of RNA viruses, and participants will learn how to formulate and solve these equations. CA models, on the other hand, can be used to study the spatial distribution of populations, and participants will learn how to implement and analyze these models. Here, we will define useful and well-stablished theories behind different interactions among biological species such as cooperative, antagonistic, and eco-evolutionary phenomena.

Session 4 (Marc Jorba). Algorithms for the integration of ODEs.

Slides.

Session 5 (Lluís Alsedà). Deterministic minimisation algorithms with inequality restrictions: penalty and barrier methods.

Quasi-Newton algorithms and Levenberg-Marquardt. Advantages and disadvantages and when to use them.

Session 6 (Lluís Alsedà, continuation of session 5). Examples of deterministic minimisation methods.

Detailed solutions of concrete examples.

Session 7 (Lluís Alsedà). Heuristic optimization methods.

Genetic algorithms. How do they work and hints on their implementation.

Session 8 (Lluís Alsedà, continuation of session 7). Heuristic optimization methods.

Detailed solutions of concrete examples.

Session 9 (Josep Sardanyés). Using Macroevolutionary algorithms (MA) for data fitting: theory and examples.

MA [2,3] have been successfully used for real data fitting in a variety of fields, such as biology, ecology, and engineering. These algorithms simulate the process of natural selection by generating a population of candidate solutions and iteratively improving them through the application of selection, mutation, and macroextinctions. Macroevolutionary algorithms are particularly well-suited for complex, multi-dimensional optimization problems where traditional optimization techniques may fail. Here, we will provide some theoretical aspects of this method together with an example used to fit dynamical models to experiments about plant RNA viral strains’ competition at different temperatures [4].

Session 10 (Lluís Alsedà). Other heuristic optimization methods.

Simulated annealing and Montecarlo methods. Pseudo-random number generators. How do they work and hints on their implementation.

Session 11 (Lluís Alsedà & Josep Sardanyés). Open debate about pros and cons of the different modelling approaches and optimisation tools proposed in the course.

The aim of this session is to share experiences on modelling and fitting with the audience and discuss how to decide between different approaches depending on the data type and properties.

1. Oro D., Alsedà Ll., Hastings A., Genovart M., Sardanyés J. Social copying drives a tipping point for nonlinear population collapse. Proceedings of the National Academy of Sciences USA 120(11), e2214055120 (2023)

2. Marín J., Solé RV. Macroevolutionary algorithms: A new optimization method on fitness landscapes. IEEE Transactions on Evolutionary Computation, 3(4), 272-286

3. J. Sardanyés, C. Alcaide, P. Martínez, SF. Elena. Modelling temperature-dependent dynamics of single and mixed infections in a plant virus. Applied Mathematical Modelling 102, 695-705 (2022)

4. Marín J., Solé RV. Evolutionary Optimization through extinction dynamics. IEEE Transactions on Evolutionary Computation, 3(4), 272-286