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- A Consistent Discrete Version of a Nonautonomous SIRVS ModelPublication . Mateus, Joaquim; Silva, César M.; Vaz, SandraA family of discrete nonautonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens's nonstandard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. Concerning extinction and persistence, the consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (permanence) for the continuous model, we also have extinction (permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.
- Nonuniform behavior and stability of Hopfield neural networks with delayPublication . Bento, António; Oliveira, José J.; Silva, César M.Based on a new abstract result on the behavior of nonautonomous delayed equations, we obtain a stability result for the solutions of a general discrete nonautonomous Hopfield neural network model with delay. As an application we improve some existing results on the stability of Hopfield models.
- Optimal Control Of Non-Autonomous Seirs Models With Vaccination And TreatmentPublication . Mateus, Joaquim; Rebelo, Paulo; Rosa, Silvério; Silva, César M.; Torres, Delfim F. M.We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.
- Cosmic infinity: a dynamical system approachPublication . Bouhmadi Lopez, Mariam; Marto, João; Morais, João; Silva, César M.Dynamical system techniques are extremely useful to study cosmology. It turns out that in most of the cases, we deal with finite isolated fixed points corresponding to a given cosmological epoch. However, it is equally important to analyse the asymptotic behaviour of the universe. On this paper, we show how this can be carried out for 3-forms model. In fact, we show that there are fixed points at infinity mainly by introducing appropriate compactifications and defining a new time variable that washes away any potential divergence of the system. The richness of 3-form models allows us as well to identify normally hyperbolic non-isolated fixed points. We apply this analysis to three physically interesting situations: (i) a pre-inflationary era; (ii) an inflationary era; (iii) the late-time dark matter/dark energy epoch.