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- A dynamically-consistent nonstandard finite difference scheme for the SICA modelPublication . Vaz, Sandra; Torres, Delfim F. M.In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible-Infected-Chronic-AIDS) model and analyze the dynamical properties of the discretized system. We prove that the discretized model is dynamically consistent with the continuous, maintaining the essential properties of the standard SICA model, namely, the positivity and boundedness of the solutions, equilibrium points, and their local and global stability.
- Discrete-Time System of an Intracellular Delayed HIV Model with CTL Immune ResponsePublication . Vaz, Sandra; Torres, Delfim F. M.In [Math. Comput. Sci. 12 (2018), no. 2, 111--127], a delayed model describing the dynamics of the Human Immunodeficiency Virus (HIV) with Cytotoxic T Lymphocytes (CTL) immune response is investigated by Allali, Harroudi and Torres. Here, we propose a discrete-time version of that model, which includes four nonlinear difference equations describing the evolution of uninfected, infected, free HIV viruses, and CTL immune response cells and includes intracellular delay. Using suitable Lyapunov functions, we prove the global stability of the disease free equilibrium point and of the two endemic equilibrium points. We finalize by making some simulations and showing, numerically, the consistence of the obtained theoretical results.
- 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.
- Stable weakly shadowable volume-preserving systems are volume-hyperbolicPublication . Bessa, Mário; Lee, Manseob; Vaz, SandraWe prove that any C1-stable weakly shadowable volume-preserving diffeomorphism defined on a compact manifold displays a dominated splitting E ⊕ F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version of this result for divergence-free vector fields. As a consequence, in low dimensions, we obtain global hyperbolicity.
- Stable weak shadowable symplectomorphisms are partially hyperbolicPublication . Bessa, Mario; Vaz, SandraLet M be a closed, symplectic connected Riemannian manifold and f a symplectomorphism on M. We prove that if f is C1-stably weak shadowable on M, then the whole manifold M admits a partially hyperbolic splitting.
- A Discrete-Time Compartmental Epidemiological Model for COVID-19 with a Case Study for PortugalPublication . Vaz, Sandra; Torres, Delfim F. M.Recently, a continuous-time compartmental mathematical model for the spread of the Coronavirus disease 2019 (COVID-19) was presented with Portugal as case study, from 2 March to 4 May 2020, and the local stability of the Disease Free Equilibrium (DFE) was analysed. Here, we propose an analogous discrete-time model and, using a suitable Lyapunov function, we prove the global stability of the DFE point. Using COVID-19 real data, we show, through numerical simulations, the consistence of the obtained theoretical results.