Talks

Alexander Plakhov, CIDMA, University of Aveiro - Newton's problem of minimal resistance for convex bodies

Abstract: Isaac Newton (1687) posed the problem of finding the convex axisymmetric body of smallest aerodynamic drag. Therefore, we are looking for the optimal curve which is the generatrix of the body. In 1993, a similar problem was formulated in a wider class of convex (not necessarily symmetric) bodies. This task proved to be much more difficult: it is about finding the optimal surface. Open questions related to this problem will be discussed. 


Bárbara Rodrigues, PhD student at NOVAMath - Epidemiological models with Lagrangian approach

Abstract: Classical epidemic models are generally described by compartmental models that do not include spatial heterogeneity. There are mainly two distinct approaches to include spatial dependency that we will designate by Eulerian and Lagrangian approaches or Distributed-Infectives (DI) and Distributed-Contacts (DC), respectively. In the Eulerian approach, we follow the movement of individuals, assuming that the contacts are local. In the Lagrangian approach, we follow the individuals’ contacts and not their movement. Both these approaches can be implemented in discrete space using compartmental models or in continuous space, with integral-differential or with diffusion models. Based on the literature review of models with a Lagrangian approach, there are open results for the continuous case where we have integro-differential or diffusive models. The results we are interested in are the stability of the disease-free and endemic equilibria, as well as the definition of the basic reproduction number. In this way, we will focus on the Lagrangian models where we will use the SIR model with vital dynamics to exemplify the different open results. In future work, we intend to study these models and generalize the results to other operators.

David Julien, University of Nantes - Controlling an epidemiological model: an application of Statistical Model Checking

Laid Boudjellal, PhD Student at CMAT, University of Minho - Mathematical modeling and analysis of biological systems

Márcia Lemos, CIDMA, University of Aveiro - Exact Solution for a Discrete-Time SIR model

Abstract: We derive a nonstandard finite difference scheme for Bailey's Susceptible-Infected-Removed continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with simulations. This is a joint work with Sandra Vaz and Delfim F. M. Torres. 

Om Wanassi, CIDMA, University of Aveiro  - Modeling Blood Alcohol Concentration Using Fractional Differential Equations Based on the ψ-Caputo Derivative

Abstract: We propose a novel dynamical model for blood alcohol concentration that incorporates ψ-Caputo fractional derivatives. Using the generalized Laplace transform technique, we suc- cessfully derive an analytic solution for both the alcohol concentration in the stomach and the alcohol concentration in the blood of an individual. These analytical formulas provide us a straightforward numerical scheme, which demonstrates the efficacy of the ψ-Caputo derivative operator in achieving a better fit to real experimental data on blood alcohol levels available in the literature. In comparison to existing classical and fractional models found in the literature, our model outperforms them significantly. Indeed, by employing a simple yet non-standard kernel function ψ(t), we are able to reduce the error by more than half, resulting in an impressive gain improvement of 59 percent.

Paula Patrício and Paulo Doutor, NOVAMath  - Rational behaviour and Social Cost for imperfect vaccination

Abstract: In task 3, we are interested in studying the impact of individual behavior on vaccination efficacy for different scenarios. In this work, we consider vaccination for childhood diseases that can have more severe consequences for adults.  It is expected that the vaccination decreases the rate of infection, and, as a side effect, it increases the age at infection. This last effect is enhanced if the vaccine loses efficacy over time. Our aim is to compare the social and individual costs of vaccination, assuming that disease costs are age-dependent. We used a model coupling pathogen deterministic dynamics for a population consisting of juveniles and adults, both assumed to be rational agents. The parameter region for which vaccination has a positive social impact is fully characterized and the Nash equilibrium of the vaccination game is obtained. Now the key question is how to combine positive social impact with individual interest in vaccination. We can formulate this as an optimal control problem.