Well-posedness and qualitative analysis of a SEIR model with spatial diffusion for COVID-19 spreading José Paulo Carvalho dos Santos, Evandro Monteiro, José Claudinei Ferreira, Nelson Henrique Teixeira Lemes, Diego Samuel Rodrigues Biomath, 2023 In this paper, we study the well-posedness and the qualitative behavior of equilibria of a SEIR epidemic models with spatial diffusion for the spreading of COVID-19. The well-posedness of the model is proved using both the Semigroup Theory of sectorial operators and existence results for abstract parabolic differential equations. The asymptotical local stability of both disease-free and endemic equilibria are established using standard linearization theory, and confirmed by illustrative numerical simulations. The asymptotical global stability of both disease-free and endemic equilibria are established using a Lyapunov function.
Reproducing properties of differentiable Mercer-like kernels José C. Ferreira, Valdir A. Menegatto Mathematische Nachrichten, 2012 Let X be an open subset of \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {R}^d$\\end{document} and ν the restriction of the usual Lebesgue measure of \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {R}^d$\\end{document} to X. In this paper, we investigate properties of the range of positive integral operators on L2(X, ν), in connection with the reproducing kernel Hilbert space of the generating kernel. Assuming differentiability assumptions on the kernel, we deduce smoothness properties for the functions in the range of the operator and also properties of the so‐called inclusion map. The results are deduced when the assumptions are defined via both, weak and partial derivatives. Further, assuming the generating kernel has a Mercer‐like expansion based on sufficiently smooth functions, we deduce results on the term‐by‐term differentiability of the series and reproducing properties for the derivatives of the functions in the reproducing kernel Hilbert space.
