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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.

Aedes aegypti female mosquitoes are the principal transmitters of dengue and other vector-borne infections. This species is closely associated with human habitation, due to its blood-feeding habits and the presence of breeding sites widely available around households. In this paper, we introduce a mathematical model for the life cycle of Aedes aegypti mosquitoes comprising two stages, aerial and aquatic, that reflects seasonal changes in the mosquito abundance. This model is further amended by three season-dependent control actions. Two coercive actions are introduced during the hot seasons characterized by higher abundance and enhanced growth rates of mosquitoes. They consist in the application of two chemical substances, insecticide and larvicide, acting upon the aerial and aquatic mosquito stages, respectively. During the cool seasons, characterized by the slower growth rates of mosquitoes and abundance of quiescent unhatched eggs, we introduce a preventive vector control measure consisting in mechanical elimination of mosquito breeding sites. Using the framework of optimal control in combination with the cost-benefit approach and epidemiological assessment, we identify the most efficient strategy capable of essentially reducing the population of adult and immature mosquitoes during both seasons and provide a sketch for its modus operandi.

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

The basic reproduction number R0 is an index worldwide commonly used by public health organizations as a key estimator of the severity of a given epidemic. In this work we use a Lagrangian approach to model vector-borne diseases (SIR-SI) into a metapopulation network in order to derive an expression of the basic reproduction number and we analyze its dependency on human mobility. We prove that this index can be computed by evaluating the spectral radius of the risk matrix W, whose entries Wij are the number of secondary cases in patch j produced by the inclusion of a single infected human in patch i. Based on the risk matrix, we propose a risk index which locally describes the epidemic vulnerability, while R0 give us an estimation of the global vulnerability. Further, we numerically analyze the effect of human mobility over the values of R0 in a system composed of two and three patches, and for a network connected in a star topology configuration.

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.

Dengue fever is increasing in geographical range, spread by invasion of its vector mosquitoes. The trade in second-hand tires has been implicated as a factor in this process as they act as mobile reservoirs of mosquito eggs and larvae. Regional transportation of tires can create linkages between rural areas with dengue to disease-free urban areas, potentially giving rise to outbreaks even in areas with strong local control measures. In this work we sought to model the dynamics of mosquito transportation via the tire trade, in particular to predict its role in causing unexpected dengue outbreaks through vertical transmission of the virus across generations of mosquitoes. We also aimed to identify strategies for regulating the trade in second-hand tires, improving disease control. We created a mathematical model which captures the dynamics of dengue between rural and urban areas, taking into account the movement, storage time of tires, and mosquito diapause. We simulate a series of scenarios. First a mosquito population is introduced to a dengue-free area via movement of tires, either as single or multiple events, increasing the likelihood of a dengue outbreak. An endemic state can be induced regardless of whether urban conditions for an outbreak are met, and an existing endemic state can be enhanced by vector input. Finally we assess the potential for regulation of tire processing as a means of reducing the transmission of dengue fever using a specific case study from Puerto Rico. Our work demonstrates the importance of the second-hand tire trade in modulating the spread of dengue fever across regions, in particular its role in introducing dengue to disease-free areas. We propose that regulation of tire storage and movement can play a crucial role in containing outbreaks and dengue spread., 24 pages, 7 figures

Aedes aegypti is the most common and important vector that transmits dengue disease. It has been observed that its abundance is one of the main factors influencing dengue incidence and hence the importance of studying its population dynamics. By means of a dynamical model, in this work, we study the effects temperature and precipitation in the abundance of the mosquito Aedes aegypti . We also analyze the correlation between mosquito and dengue outbreaks using historical data of 8 Mexican regions and the role of diapause in seasonality outbreaks. We found that the joint action of different mechanisms can enhance the mosquito abundance, but also inhibit it when they act in an asynchronous way. For the studied regions, the seasonality of the low temperature rather than mean temperature is the main driving force promoting mosquito outbreaks and hence dengue outbreaks. We conclude that this simpler model rather than more complex dengue models can also be used to predict the risk of dengue outbreaks with good accuracy. In addition to this, the model can be used to discover the underlying mechanism of dengue outbreaks in different regions and thus help to apply targeted control measures.

In this paper we propose a model of the cardiovascular sys- tem, where stressed blood volume is a model parameter instead of an initial condition. Stressed blood volume (SBV) is an important indica- tor of uid responsiveness, i.e., this term can help to classify patients between responders and non-responders to uid therapy. We study a six compartment model, then using the conservation of total blood volume, one dierential equation of the original model is omitted and a new model is obtained. Comparing the haemodynamic signals of the previous model and the new version, we show that the simulations are qualitatively sim- ilar. One important dierence with the original model is that the initial conditions for solving the proposed model are arbitrary. This allows us to perform a sensitivity analysis using automatic dierentiation of the re- duced model for all model parameters without excluding any parameter a priori, unlike the authors of the original formulation have done. This analysis showed that two parameters, the heart period and the stressed blood volume, have a major eect on the performance of the model.

In this study, we consider the inverse problem of determining a source term in an elliptic problem based on boundary measurements. The boundary measurements allow the unique determination of the harmonic component of the source, and thus a priori information is required for its complete identification. Using this a priori information, we determine the compactness of the class of sources and a uniqueness theorem for its identification from Cauchy data, thereby allowing us to propose a stable algorithm for finding “approximate solutions” of the inverse problem. The proposed procedure is demonstrated by individual cases, where we know the “geometry” of the inner region and the source takes a constant value.

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

In this work the problem of recovering bioelectrical sources on the cerebral volume, from measurement of the potential generated by these sources on the scalp, is studied. This problem is an ill posed problem, since given a measurement on the scalp, there are different bioelectrical sources that produce this measurement and small variations in the input data can produce significant variations in the source localization. The problem is studied through a boundary value problem, which is obtained using a model that describes the head as a conductive layer medium. This model allows relationships between the characteristics of the bioelectrical activity and the EEG to be established. We find in this work conditions under which the inverse solution is unique and we give an algorithm to find it. In the simple case, when the head is modeled through two concentric circles, we give a regularization strategy.

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.

The limbic circuit, involving the prefrontal cortex, hippocampus and certain subcortical structures plays a determinant role in the emotional activity and for understand psychopathologies like schizophrenia and sensitization to certain psychostimulants.In this work, we constructed a non-linear network representing the interaction between seven important nuclei ofthe limbic system.This model is a first approach that allows to simulate different activities ofthe circuit, associated with the dopamine sensitization and the neurodevelopmental hypothesis for the neuropathology of schizophrenia.

A model with the brain and the other shell of the head like conductors medium with differents conductivities has been used for to estudy the inverse electroencephalographic problem. Technics of the potential theory has been used for to transform the model in a operational problem which under some conditions gives the uniqueness of recuperation of cortical neurons aggregate (sources) in the cerebral cortex from measurement of the potential in the scalp.

In view of the serious health problems concerning infectious diseases in heavily populated areas, we followed the strategy of lead compound diversification to evaluate the near-by chemical space for new organic compounds. To this end, twenty derivatives of nitazoxanide (NTZ) were synthesized and tested for activity against Entamoeba histolytica parasites. To ensure drug-likeliness and activity relatedness of the new compounds, the synthetic work was assisted by a quantitative structure-activity relationships study (QSAR). Many of the inherent downsides - well-known to QSAR practitioners - we circumvented thanks to workarounds which we proposed in prior QSAR publication. To gain further mechanistic insight on a molecular level, ligand-enzyme docking simulations were carried out since NTZ is known to inhibit the protozoal pyruvate ferredoxin oxidoreductase (PFOR) enzyme as its biomolecular target.