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- Dynamics of non-autonomous SEIRS models with general incidencePublication . Mateus, Joaquim Manuel Pereira; Silva, César Augusto Teixeira Marques daWe consider SEIRS models with general incidence functions depending on the susceptibles, the infectives and the total population, and we analyze this models in several scenarios: autonomous, general non-autonomous and periodic. In all this settings, we discuss the strong persistence and the extinction of the disease. Additionally, we address the following problems: in the autonomous setting, we obtain results on the existence and global stability of disease-free and endemic equilibriums; in the periodic setting, we obtain the global stability of disease-free periodic solution when the basic reproductive number is less than one, and, using the wellknown Mawhin continuation theorem, we discuss the existence of endemic periodic solutions; in the general non-autonomous setting, we prove that our conditions for strong persistence and extinction are robust, in the sense that they are unchanged by su ciently small perturbations of the parameters and the incidence functions. Finally, we consider a version of our model with two control variables, vaccination and treatment, and study the existence and uniqueness of solution of the optimal control model considered. Some computational experiences illustrate our 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.
- 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.