Resumen de Advances on uncertainty quantification techniques for dynamical systems: Theory and modelling
Uncertainty quantification collects different methods and computational techniques aimed at describing the randomness in real phenomena. These methods are useful in the modelling of different processes as biological, physical, natural or social, since they present some aspects that can not be determined exactly. For example, the contagious rate of a epidemiological disease or the growth factor of a tumour volume depend on genetic, environmental or behavioural factors. They may not always be fully described and therefore involve uncertainties that affects on the final result. The main objective of this PhD thesis is to extend techniques to quantify the uncertainty in two mathematical areas: fractional calculus and mathematical modelling.
Fractional derivatives allow us to model some behaviours that classical derivatives cannot, such as memory effects or the viscoelasticity of some materials. In this PhD thesis, from a theoretical point of view, fractional calculus is extended into the random framework, concretely in the mean square sense. Initial value problems will be studied. The calculus of the analytic solution, approximations for the mean and for the variance and the computation of the first probability density function are concepts we deal with them thought the following chapters. Nevertheless, it is not always possible to obtain the analytic solution of an initial value problem. Therefore, in this dissertation a chapter is addressed to describe a numerical procedure to approximate the solution for an initial value problem.
On the other hand, from a modelling point of view, computational techniques to quantify the uncertainty in mathematical models are developed. Merging these techniques with appropriate mathematical models, problems of biological dynamics are studied. Firstly, the carriers of meningococcus in Spain are determined using a competition Lotka-Volterra random fractional model. Then, the volume of breast tumours is modelled by a random logistic model. Finally, taking advantage of a mathematical model which describes the glucose level of a diabetic patient, a recommendation of insulin shots and carbohydrate intakes is proposed to a patient in order to maintain her/his glucose level in a healthy confidence range. An important observation is that to carry out these studies real data is required and they may include uncertainties contained in the measurements on the process to perform the corresponding study. This it is the reason why it is crucial to properly model the problem taking also into account the randomness of the data.
