Nonlinear dynamics perspective is an interesting approach to describe COVID-19 epidemics, providing information to support strategic decisions. This paper proposes a dynamical map to describe COVID-19 epidemics based on the classical susceptible-exposed-infected-recovered (SEIR) differential model, incorporating vaccinated population. On this basis, the novel map represents COVID-19 discretetime dynamics by adopting three populations: infected, cumulative infected and vaccinated. The map promotes a dynamical description based on algebraic equations with a reduced number of variables and, due to its simplicity, it is easier to perform parameter adjustments. In addition, the map description allows analytical calculations of useful information to evaluate the epidemic scenario, being important to support strategic decisions. In this regard, it should be pointed out the estimation of the number deaths, infection rate and the herd immunization point. Numerical simulations show the model capability to describe COVID-19 dynamics, capturing the main features of the epidemic evolution. Reported data from Germany, Italy and Brazil are of concern showing the map ability to describe different scenario patterns that include multi-wave pattern with bell shape and plateaus characteristics. The effect of vaccination is analyzed considering different campaign strategies, showing its importance to control the epidemics.

The mixed convection in a thin liquid film flow over a horizontal plate is investigated under finite Prandtl numbers. The gas-liquid interface is considered free, non-deformable and subject to surface tension gradients and convection, while gravity is assumed negligible. Therefore, Marangoni instead of buoyancy effects appear due to the unstable temperature stratification induced by the internal heating generated by viscous dissipation. A linear and modal stability analysis of this model is then performed to identify its convective/absolute nature. This is achieved by solving the resulting differential eigenvalue problem with a shooting method. Longitudinal rolls are the most unstable at the onset of instability for most parametric conditions. Otherwise, transverse rolls are the first to become convectively unstable. Finally, longitudinal rolls are absolutely stable. A transition to absolute instability occurs through transverse rolls, but only within a limited region in parametric space.

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This paper proposes a dynamical map to describe COVID-19 epidemics based on the classical susceptible-exposed-infected-recovered (SEIR) model. The novel map represents Covid-19 discrete-time dynamics standing for the infected, cumulative infected and vaccinated populations. The simplicity of the discrete description allows the analytical calculation of useful information to evaluate the epidemic stage and to support decision making. In this regard, it should be pointed out the estimation of the number death cases and the herd immunization point. Numerical simulations show the model capacity to describe Covid-19 dynamics properly representing real data and describing different scenario patterns. Real data of Germany, Italy and Brazil are of concern to verify the model ability to describe Covid-19 dynamics. The model showed to be useful to describe the epidemic evolution and the effect of vaccination, being able to predict different pandemic scenarios.

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