Since my Master’s thesis my interests have been focusing mainly on numerical methods for evolution models described by delay equations.
Delay equations are rules for extending (in one direction) a function that is a priori defined on an interval.
The above definition allows to include different classes of equations of retarded type subjects of my research, that is delay differential equations, Volterra integro-differential equations, Volterra integral equations (also known as renewal equations), and partial retarded differential equations. Their distinguishing feature is that the time evolution depends on the past history.
The introduction of the delay, either discrete or distributed, can provide us with more realistic models of many real-world phenomena in various fields such as Biology, Medicine, Engineering, Chemistry and Financial Mathematics. In particular delay equations play a crucial role in mathematical modelling of population dynamics and epidemic phenomena.
But on the other hand with respect to ordinary differential equations numerical integration methods for delay equations raise new questions and challenges. In my research I have dealt with the development of accurate and efficient numerical methods (of Runge–Kutta type) equipped with continuous extensions, with a particular attention to their stability properties and to the derivation of suitable test equations.
From a dynamical point of view, delay equations describe infinite-dimensional dynamical systems and, as a consequence, the stability and bifurcation analyses of equilibria and periodic solutions need numerical methods. More recently my interests have been concentrated on the development of efficient direct numerical methods to face these issues. The basic idea consists in discretizing the involved operators by the pseudospectral approach. Next, the analysis of stability of equilibria or periodic solutions is performed by approximating the spectra of the resulting linear operators, that is the infinitesimal generators or the solution operators, while for the bifurcation analysis a system of ordinary differential equations has been derived and analyzed by using the available bifurcation software.
In the context of epidemiology and populations dynamics, the attention is on structured populations and epidemics models, and on the computation of the stability indicators and of the basic reproduction number R0. It is worth recalling that R0 plays a crucial role in the study of infectious disease dynamics.
Besides mathematical models and numerical methods, another crucial question is the specification of the model parameters due to some limitations in experimental data available. The parameters are often represented as random quantities in a suitable probabilistic framework. In this context Polynomial Chaos Expansions (PCEs), coupled with suitable numerical methods, furnish important tools for uncertainty quantification and the sensitivity analysis of the random model response. We have applied this approach to study the stability of delay differential equations with uncertain parameters and to describe how the variability of R0 is affected by the variability of the input parameters of structured population models, through the evaluation of Sobol’ indices.
Keywords
numerical analysis, delay equations, Runge–Kutta methods, collocation, continuous extensions, numerical stability, numerical bifurcation, population dynamics, epidemics models, R0, structured population models, uncertainty and sensitivity analysis, polynomial chaos expansions, PC-based methods