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Dengue and Zika incidence data and the latest research have raised questions about how dengue vaccine strategies might be impacted by the emergence of Zika virus. Existing antibodies to one virus might temporarily protect or promote infection by the other through antibody-dependent enhancement (ADE). With this condition, understanding the dynamics of propagation of these two viruses is of great importance when implementing vaccines. In this work, we analyze the effect of vaccination against one strain, in a two-strain model that accounts for cross-immunity and ADE. Using basic and invasion reproductive numbers, we examined the dynamics of the model and provide conditions to ensure the stability of the disease-free equilibrium. We provide conditions on cross-immunity, ADE and vaccination rate under which the vaccination could ensure the global stability of the disease-free equilibrium. The results indicate scenarios in which vaccination against one strain may improve or worsen the control of the other, as well as contribute to the eradication or persistence of one or both viruses in the population., Comment: Corrected typos. Revised figures, results unchanged

We collected individualized (unidentifiable) and aggregated openly available data from various sources related to suspected/confirmed SARS-CoV-2 infections, vaccinations, non-pharmaceutical government interventions, human mobility, and levels of population inequality in Brazil. In addition, a data structure allowing real-time data collection, curation, integration, and extract-transform-load processes for different objectives was developed. The granularity of this dataset (state- and municipality-wide) enables its application to individualized and ecological epidemiological studies, statistical, mathematical, and computational modeling, data visualization as well as the scientific dissemination of information on the COVID-19 pandemic in Brazil., Comment: 12 pages, 6 figures, 2 tables

We study solutions of bifurcation problems with periodic boundary conditions, with periods in an $n+1$-dimensional lattice and their projection into $n$-dimensional space through integration of the last variable. We show that generically the projection of a single mode solution is a mode interaction. This can be applied to the study of black-eye patterns.

In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. We are concerned with functions in $R^3$ that are invariant under the action of a crystallographic group and the symmetries of their projections into a function defined on a plane. We obtain a list of the crystallographic groups for which the projected functions have a hexagonal lattice of periods. The proof is constructive and the result may be used in the study of observed patterns in thin domains, whose symmetries are not expected in 2-dimensional models, like the black-eye pattern., Comment: corrected initials of second author