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This chapter presents the COVID-19 pandemic as a complex problem, which has resulted in multidisciplinary and interdisciplinary research of unprecedented magnitude at the international level. In particular, the important role that mathematical and computational modeling has played throughout the different phases of the evolution of the pandemic is analyzed. Some of the most notorious achievements and difficulties in obtaining results with the use of mathematical models and computational simulations are discussed, and some proposals are presented about relevant and still pending tasks.

In this paper, we investigate the convergence of the Faedo-Galerkin approximations, in a strong sense, to a strong T-periodic solution of the torso-coupled bidomain model where $T$ is the period of activation of the inner wall of heart. First, we define the torso-coupled bi-domain operator and prove some of its more important properties for our work. After, we define the abstract evolution system of equations associated with torso-coupled bidomain model and give the definition of strong solution. We prove that the Faedo-Galerkin's approximations have the regularity of a strong solution, and we find that some restrictions can be imposed over the initial conditions, so that this sequence of Faedo-Galerkin fully converge to a global strong solution of the Cauchy problem. Finally, this results are used for showing the existence a strong $T$-periodic solution., Comment: 22 pages

Introduction: In this work we discuss the relevance of the harmonic sources on the brain volume, which reproduce a given potential distribution on the scalp. These sources, apart from being a unicity class, they play a fundamental role in the resolution of the inverse problem of source identification with respect to any other sources class.Method: We make use of the volume conductor model for the head, in order to relate sources and reproduced measurements. The problem is rewritten as an operational formulation which allows to characterize the admissible measurements with respect to any considered sources class.Results: The admissible data set is characterized for the harmonic sources class on the brain volume. Also, the importance of this class in the context of the source estimation problem, with respect to any sources class, is shown. This is specifically illustrated considering the class of harmonic sources on a neighborhood of the cortex. Moreover, it is also shown the role the harmonic sources class on the brain plays when applying the Admissible Data Method (ADM) in order to get a general regularization scheme for the source estimation problem with respect to a unicity sources class.Conclusion: A general resolution methodology for the source estimation problem in the context of the inverse electroencephalographic problem is proposed, in which the harmonic sources class on the brain volume is crucial. Namely, given an arbitrary sources unicity class (for this inverse problem), a general method is developed for identifying the source in this class whose reproduced potential distribution best approximates a given potential measurement on the scalp. We consider sources classes in connection with the electrical activity near the cortex.

Background:The estimation of fiducial points is specially important in the analysis and automatic diagnose of Electrocardiographic (ECG) signals.Objective:A new algorithm which could be easily implemented is presented to accomplish this task.Methods:Its methodology is rather simple, and starts from some ideas available in the literature combined with new approachs provided by the authors. First, aQRScomplex detection algorithm is presented based on the computation of energy maxima in ECG signals which allow the measurement of cardiac frequency (in beats per minute) and the estimation of R peaks temporal positions (in number of samples). From these ones, an estimation of fiducial points Q, S, J, P and T waves onset and offset points are worked out, supported in a simple modified slope method with constraints.The location process of fiducial points is assisted with the help of the so called curvature filters, which allow to improve the accuracy in this task.Results:The procedure is simulated in Matlab and GNU Octave by using test signals from the MIT medical database, Cardiosim II equipment patterns and synthetic signals developed by the authors.Conclusion:One of the novelties of this work is the global strategy. Also, another significant innovation is the introduction of the curvature filters. We think this concept will prove to be a useful tool in signal processing, not only in ECG analysis.

El problema inverso electroencefalográfico (PIE) consiste en determinar a partir del Electroencefalograma sobre el cuero cabelludo (EEG), las fuentes de actividad bioeléctrica que lo producen. Este problema tiene características de mal planteamiento tales como la no unicidad de solución del problema y la inestabilidad numérica ante errores del EEG inherentes al proceso de medición. Para el estudio del PIE se analiza el llamado problema directo electroencefalográfico (PDE) que consiste en determinar el EEG sobre el cuero cabelludo, a partir de fuentes de actividad bioeléctrica en el cerebro conocidas. En este trabajo se analiza el caso de fuentes dipolares ubicadas en la corteza cerebral usando las técnicas de teoría de potencial y la teoría de funciones generalizadas (o distribuciones). Esto lleva a un sistema de ecuaciones integrales definido sobre un espacio de funciones generalizadas con la cual se puede proponer un algoritmo estable de solución del problema directo electroencefalográfico. Con estos resultados se encuentra un teorema de unicidad para el problema inverso.

La conductividad en las lesiones cerebrales cambia de acuerdo a la patología, estas pueden ser: edemas, calcificaciones, tumores, esto se debe al tipo de lesión que se tenga, ya que unos presentan una mayor o menor conductividad e incluso en algunos casos el llamado silencio eléctrico. En este trabajo se estudia el problema de identificar de manera única alguna de las diferentes patologías mencionadas. Como problema de identificación es importante, ya que la mayoría de estos problemas no tienen solución única. Este problema inverso se plantea por medio del EEG medido sobre el cuero cabelludo, y por medio de este se quiere identificar el daño ocurrido. Para ello se utiliza un modelo que considera a la cabeza dividida en capas conductoras que lleva a su vez a dos problemas de contorno que se obtienen de diferentes consideraciones. Para cada uno de ellos se analiza el problema de la unicidad.

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn+1domain Ω. The piecewise regular boundary of Ω is defined as the union∂Ω = Γ1∪ Γ0∪ Σ, where Γ1and Γ0are disjoint, regular, andn-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0corresponding to an harmonic function inC2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in theL2-norm from the measured Cauchy data to the subset of admissible data characterized by givena prioriinformation, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

A mathematical model for a non-linear continuous stochastic process is presented. Analysis of the stability by uncoupling system is realized.

There are economic and physical limitations when applying prevention and control strategies for urban vector borne diseases. Consequently, there are increasing concerns and interest in designing efficient strategies and regulations that health agencies can follow in order to reduce the imminent impact of viruses like Dengue, Zika and Chikungunya. That includes fumigation, abatization, reducing the hatcheries, picking up trash, information campaigns. A basic question that arise when designing control strategies is about which and where these ones should focus. In other words, one would like to know whether preventing the contagion or decrease vector population, and in which area of the city, is more efficient. In this work, we propose risk indexes based on the idea of secondary cases from patch to patch. Thus, they take into account human mobility and indicate which patch has more chance to be a corridor for the spread of the disease and which is more vulnerable, i.e. more likely to have cases?. They can also indicate the neighborhood where hatchery control will reduce more the number of potential cases. In order to illustrate the usefulness of these indexes, we run a set of numerical simulations in a mathematical model that takes into account the urban mobility and the differences in population density among the areas of a city. If we label by i a particular neighborhood, the transmission risk index (TRi) measures the potential secondary cases caused by a host in that neighborhood. The vector transmission risk index (VTRi) measures the potential secondary cases caused by a vector. Finally, the vulnerability risk index (VRi) measures the potential secondary cases in the neighborhood. Transmission indexes can be used to give geographical priority to some neighborhoods when applying prevention and control measures. On the other hand, the vulnerability index can be useful to implement monitoring campaigns or public health investment., Author summary A disease is called vector-borne when it is not transmitted directly among humans, but in a human-vector-human way. Examples of major importance, due to its epidemiological magnitude, are Dengue, Chikungunya and Zika tropical diseases, for which the main vector is the Aedes aegypti mosquito. Usually, some indexes are used to measure the potential damage that such a disease could cause, which concern infectivity and entomological issues, human mobility and geographical density, etc. We consider a mathematical epidemiological model which takes specifically into account these spatial factors (mobility, density), and propose a set of new risk indexes which give information about how the epidemic spread could occur, and where the outbreaks could take place. These indexes come from a preventive perspective, and they also pay attention to the epidemic dynamics. Consequently, they potentially allow to proceed in the correct places, and before troubles have arisen. We think these new tools could eventually help health agencies in the design of efficient and effective strategies on prevention and control of vector-borne diseases epidemics.

The linear stability problem of the rotational motion of a top around a fixed point containing an inner cavity filled up with a viscous fluid is considered. The effect of the viscosity in the stability problem is studied.