Orientador: Prof. Dr. Norberto Anibal Maidana / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015. / O aumento do estudo de modelos matemáticos vem desempenhando um papel cada vez mais significativo para a compreensão e obtenção de resultados e previsões de fenômenos biológicos. Como no Brasil existe um alto risco de transmissão indireta de doenças por vetores, como mosquitos e outros insetos hematófagos, objetivou-se com este trabalho estudar a propagação da Doença de Chagas que é transmitida principalmente pelos triatomíneos. Para isso, propomos um modelo matemático para estudar e descrever a propagação desta doença na população humana. Primeiramente, discorreu-se sobre a doença e sobre suas principais características. A partir daí, por meio de um sistema de Equações Diferencias, modelamos a infecção e estudamos seu comportamento, calculando o Número Básico de Reprodução, verificando a estabilidade local dos pontos de equilíbrio e fazendo uma análise numérica dos resultados. Após, foi possível verificar a existência de soluções ondas viajantes e a velocidade mínima de propagação para o modelo geotemporal da doença, possibilitando avaliar estratégias de controle mediante os resultados obtidos através da modelagem. / The increase of the study of mathematical models has played a significant role in understanding and achieving results and predictions of biological phenomena. As in Brazil there is a high risk of indirect transmission of diseases by vectors such as mosquitoes and other blood sucking insects, we aim with this work is to study the spread of Chagas¿ Disease transmitted primarily by triatomines. For this, we propose a mathematical model to study and to describe the spread of this disease in the human population. First, we describe about the disease and its main features. Then, through a system of differential equations, we modelled the infection and studied its behaviour, calculating the basic reproduction number, checking the local stability of equilibrium points and making a numerical analysis of the results. After, it was possible to check the existence of travelling wave solutions and the wavefront speed for the spacial model, being able to evaluate control strategies using the results obtained by mathematical modelling.

Made available in DSpace on 2014-06-11T19:24:53Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-04-13Bitstream added on 2014-06-13T18:21:26Z : No. of bitstreams: 1 diniz_ln_me_rcla.pdf: 642209 bytes, checksum: b771fc2d2779ed3b064ee392fec7f231 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Nesta pesquisa, investiguei como os alunos utilizam as Tecnologias da Informação e Comunicação (TIC) nos Projetos de Modelagem Matemática. A pesquisa foi desenvolvida com alunos do primeiro ano do curso de Ciências Biológicas da Unesp, campus de Rio Claro, na disciplina Matemática Aplicada. O professor convidou os alunos para que, reunidos em grupos, investigassem um tema qualquer do cotidiano, escolhido por eles. Os dados foram coletados por meio da observação de aulas da disciplina e das reuniões em horário extra, das entrevistas com os grupos dos Projetos de Modelagem e dos documentos produzidos pelos alunos. Estabeleci esses procedimentos de pesquisa pois considero que estão em harmonia com a visão de conhecimento sintetizada pelo construto teórico do coletivo Seres-Humanos-com- Mídias, evidenciando o papel das TIC na produção de conhecimentos. A análise de dados aponta que os alunos utilizaram a Internet para realizar parte das suas pesquisas, com uma seleção, a priori, de sites, os quais, identifiquei como sendo, para eles, sites oficiais. Os estudantes também utilizaram softwares gráficos para realizar simulações, que nomeei como sendo positivas, negativas e nebulosas, possibilitando previsões para o passado e para o futuro. Também fiz um exercício inicial de reflexão sobre o uso do email feito pelos alunos, para que possibilitasse a discussão do trabalho e a produção do relatório escrito. Esses temas de análise possibilitaram engendrar possíveis aspectos sobre a combinação das perspectivas reorganização e a cidadania (relativas ao uso das TIC na Educação Matemática) com a perspectiva dos Projetos de Modelagem. Com isso, esta pesquisa gera novas reflexões para a área, compondo parte do Mosaico de pesquisas em Modelagem. / In this study, I investigated how students use Information and Communication Technologies (ICT) in Mathematical Modeling Projects. The research was developed with first-year Biology majors at Unesp, Rio Claro campus, participating in an Applied Mathematics course. The professor invited the students to form groups and choose any theme from their everyday lives to investigate in the class. The data were collected by means of: in-class observations as well as observations of meetings held outside of class; interviews with the Modeling Project groups; and analysis of the documents they produced. I established these research procedures because I considered them to be in harmony with the view of knowledge synthesized in the theoretical construct of the Humans-with-Media collective, providing evidence of the role of ICT in the production of knowledge. Data analysis indicated that students used the Internet to carry out part of their research, with an a priori selection of the sites, which I identified as being official sites for them. The students also used graphing software to conduct simulations, which I characterized as positive, negative, or nebulous, making possible predictions positive and negative. I also did an initial reflective exercise about students' use of e-mail to facilitate discussion of the work and production of the written report. These analytic themes made it possible to identify possible aspects of the combination of the perspectives of reorganization and citizenship (related to the use of ICT in Mathematics Education) with the perspective of modeling projects. The study generates new reflections for this field of study, composing part of the mosaic of research on Modeling.

Orientador: Marília Freitas de Campos Tozoni-Reis / Resumo: Este estudo investigou, à luz de Vigotski, as possibilidades de aprendizagem da Modelagem Matemática através das atividades de elaboração de vídeos didáticos produzidos com e por estudantes numa perspectiva sociocrítica ambiental para sustentabilidade. Uma pesquisa qualitativa inspirada em metodologias participativas foi realizada com setenta e dois participantes, estudantes do Ensino Médio de uma escola pública em Minas Gerais. Para investigar os efeitos desta proposta no processo de aprendizagem, além de um estudo teórico a respeito da Modelagem Matemática e de temas relacionados a ela neste estudo, observamos a integração entre os participantes durante as atividades de campo, como palestra, aulas, diálogos em sessões plenárias, desenvolvimento de tarefas escritas, seminários, exibição de vídeos e filmes, oficinas, debates, elaboração de vídeos, questionários e entrevistas. Com base nos estudos teóricos e nos dados obtidos através da análise das entrevistas, dos questionários e das filmagens dos encontros, concluímos que os processos pedagógicos com Modelagem Matemática associados a questões ambientais, além de promover aprendizagem da Matemática contribuem para que os alunos tenham oportunidades de formação pela apropriação crítica de conhecimentos científicos, culturais, políticos e sociais. / Abstract: This study investigated, in the light of Vigotski, the possibilities of learning Mathematical Modeling through the activities of elaboration of didactic videos produced with and by students in a socio-critical environmental perspective for sustainability. A qualitative research inspired by participatory methodologies was carried out with seventy - two high school students from a public school in Minas Gerais-Brazil. In order to investigate the effects of this proposal in the learning process, besides a theoretical study about Mathematical Modeling and the themes related to it in this study, we observed the integration among the participants during the field activities, such as lectures, classes, dialogues in plenary sessions, development of written tasks, seminars, video and film shows, workshops, debates, videos, questionnaires and interviews. Based on the theoretical studies and the data obtained through the analysis of the interviews, questionnaires and the filming of meetings, we concluded that the pedagogic processes with Mathematical Modeling associated to the environmental questions, besides promoting apprenticeship of Mathematics it gives the students the opportunity of formation through the critical appropriation of scientific, cultural, political and social knowledges. / Doutor

Este trabalho aborda a modelagem matemática do processo de extração de óleos essenciais por arraste a vapor e a otimização deste processo em escala industrial. Utilizouse dados experimentais de óleos essenciais de alecrim e citronela obtidos em uma unidade industrial de extração por arraste a vapor. O modelo é constituído de um sistema de equações diferenciais parciais, que representam o balanço de massa unidimensional nas fases vapor e sólido. Para resolver o modelo desenvolvido, utilizouse o método das linhas com o uso da técnica de diferenças finitas para discretizar a coordenada espacial e suas derivadas, resultando em um sistema de equações diferenciais ordinárias implementado no simulador EMSO. Os parâmetros do modelo foram estimados pelo método dos poliedros flexíveis, utilizando dois conjuntos de dados experimentais e um terceiro para sua validação. O modelo ajustou-se bem aos dados experimentais industriais, resultando, para duas condições ambientais e geográficas das plantas aromáticas de alecrim, em coeficiente global de transferência de massa de 8,64 x 10 -4 s -1 e 9,10 x 10 -4 s -1, e a constante de equilíbrio de 2,72 x 10 -4 m3=kg e 1,07 x 10 -2m3/kg. Enquanto que para a citronela obteve-se coeficiente global de transferência de massa de 8,29 x 10 -4s -1 e 1,23 x 10 -3 s -1, e constante de equilíbrio de 1,25 x 10 -2 m3/kg e 1,52 x 10 -3 m3/kg. Foi proposto também um problema de otimização para determinar o tempo ótimo de extração e o número de bateladas por dia. Os resultados da otimização do processo indicam que se pode reduzir o tempo de extração, permitindo que outras extrações sejam realizadas por dia, maximizando o volume de óleo extraído ao fim do dia. / This work has as objective the development of a mathematical model of essential oils extraction by steam distillation and the optimization of this process on an industrial scale. We used experimental data obtained from rosemary and citronella in an industrial steam distillation unit. The model consists a system of partial differential equations, which represent the one-dimensional mass balance in steam and solid phases. To solve the developed model, we used the method of lines with the finite differences technique to discretize the spatial coordinate and its derivatives, resulting in a system of ordinary differential equations implemented in the EMSO simulator. The model parameters were estimated by the method of flexible polyhedra, using two experimental runs and a third experimental run to test the model. The model fitted to the experimental data industry, resulting, for two environmental and geographical conditions of rosemary, in mass transfer coefficients of 8,64 x 10 -4 s -1 and 9,10 x 10 -4 s -1, and the equilibrium constant of 2,72 x 10 -4 m3/kg and 1,07 x 10 -2m3/kg. For citronella, the obtained mass transfer coefficient of 8,29 x 10 -4 s -1 and 1,23 x 10 -3 s -1 and equilibrium constant of 1,25 x 10 -2 m3/kg and 1,52 x 10 -3 m3/kg. It was also proposed an optimization problem to determine the optimum extraction time and number of batches per day. The optimization results indicate that the process can reduce the extraction time, allowing other extractions are performed each day, maximizing the essential oil volume extracted in the day.

abstract: Cancer is a major health problem in the world today and is expected to become an even larger one in the future. Although cancer therapy has improved for many cancers in the last several decades, there is much room for further improvement. Mathematical modeling has the advantage of being able to test many theoretical therapies without having to perform clinical trials and experiments. Mathematical oncology will continue to be an important tool in the future regarding cancer therapies and management. This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically. As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice. Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

abstract: The immune system plays a dual role during neoplastic progression. It can suppress tumor growth by eliminating cancer cells, and also promote neoplastic expansion by either selecting for tumor cells that are fitter to survive in an immunocompetent host or by establishing the right conditions within the tumor microenvironment. First, I present a model to study the dynamics of subclonal evolution of cancer. I model selection through time as an epistatic process. That is, the fitness change in a given cell is not simply additive, but depends on previous mutations. Simulation studies indicate that tumors are composed of myriads of small subclones at the time of diagnosis. Because some of these rare subclones harbor pre-existing treatment-resistant mutations, they present a major challenge to precision medicine. Second, I study the question of self and non-self discrimination by the immune system, which is fundamental in the field in cancer immunology. By performing a quantitative analysis of the biochemical properties of thousands of MHC class I peptides, I find that hydrophobicity of T cell receptors contact residues is a hallmark of immunogenic epitopes. Based on these findings, I further develop a computational model to predict immunogenic epitopes which facilitate the development of T cell vaccines against pathogen and tumor antigens. Lastly, I study the effect of early detection in the context of Ebola. I develope a simple mathematical model calibrated to the transmission dynamics of Ebola virus in West Africa. My findings suggest that a strategy that focuses on early diagnosis of high-risk individuals, caregivers, and health-care workers at the pre-symptomatic stage, when combined with public health measures to improve the speed and efficacy of isolation of infectious individuals, can lead to rapid reductions in Ebola transmission. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

abstract: Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). I demonstrate that the inverse problem of parameter estimation might be too complicated and simply relying on data fitting can give incorrect conclusions, since there is a large error in parameter values estimated and parameters might be unidentifiable. I provide confidence intervals to give estimate forecasts using data assimilation via an ensemble Kalman Filter. Using the ensemble Kalman Filter, I perform dual estimation of parameters and state variables to test the prediction accuracy of the models. Finally, I present a novel model with time delay and a delay-dependent parameter. I provide a geometric stability result to study the behavior of this model and show that the inclusion of time delay may improve the accuracy of predictions. Also, I demonstrate with clinical data that the inclusion of the delay-dependent parameter facilitates the identification and estimation of parameters. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2017

Submitted by Ranulfo Acir de Oliveira Resende (acir_r@yahoo.com.br) on 2018-07-10T16:42:19Z No. of bitstreams: 1 Tese_Acir.pdf: 2525445 bytes, checksum: 3a47b07dbbf7f67ec6d03ba9b352e27b (MD5) / Approved for entry into archive by Maria Marlene Zaniboni null (zaniboni@bauru.unesp.br) on 2018-07-11T13:20:13Z (GMT) No. of bitstreams: 1 resende_rao_dr_bauru.pdf: 2525445 bytes, checksum: 3a47b07dbbf7f67ec6d03ba9b352e27b (MD5) / Made available in DSpace on 2018-07-11T13:20:13Z (GMT). No. of bitstreams: 1 resende_rao_dr_bauru.pdf: 2525445 bytes, checksum: 3a47b07dbbf7f67ec6d03ba9b352e27b (MD5) Previous issue date: 2018-05-25 / Esta Tese de Doutorado é composta pelo estudo da aplicação de simulações numéricas automatizadas para solucionar problemas de controle ótimo (PCO) não convexos. Um algoritmo para tratar PCOs não convexos é apresentado como a principal contribuição da Tese. Outra contribuição, a estrutura ASAF - Automated Simulation and Analysis Framework, codificada em linguagem Python e baseada em modernas ferramentas de modelagem, simulação, JModelica.org, e otimização, Interior-Point Optimizer (IPOPT), é utilizada como auxílio para aplicação do referido algorítmo. O algoritmo proposto foi testado de forma metodológica com escala métrica e quatro PCOs não convexos com soluções analíticas exatas conhecidas. A viabilidade de aplicação do algoritmo foi verificada por clássicos métodos estatísticos aplicados aos dados das simulações, alcançando-se os resultados esperados e fundamentando-se a possibilidade de se tratar PCOs não convexos de soluções desconhecidas. Assim, esta Tese apresenta uma contribuição relevante para o tratamento de PCOs não convexos, ao propor um algoritmo para, com uso eficaz dos crescentes recursos computacionais de metaprogramação e automatização, obter solução subótima recomendada aceitável. A mediana da efetividade do algoritmo é estimada, a partir de testes estatísticos, em 93%, com significância de 0,05 (95% de confiança). / This doctoral Thesis is comprised by the study of the application of automated numerical simulations to solve nonconvex optimal control problems (OCP). An algorithm to treat nonconvex OCPs is presented as the main contribution of the Thesis. Another contribution of the Thesis, the ASAF-Automated Simulation and Analysis Framework, coded in Python language and based on modern tools of modeling, simulation, JModelica.org, and optimization, Interior-Point Optimizer (IPOPT), is used as an aid in the application of this algorithm. The proposed algorithm was tested by a methodological procedure with a metric scale and four nonconvex OCPs with known exact analytical solutions. The feasibility of using the algorithm was verified by classic statistical methods applied to the simulations data, achieving the expected outcome and grounding the possibility to treat nonconvex OCPs of unknown solutions. Thus, this Thesis presents a relevant contribution to the treatment of nonconvex OCPs, by proposing an algorithm to, upon the effective use of the increasing computer resources of metaprogramming and automation, obtain an acceptable sub-optimal recommended solution. The median of the algorithm effectiveness is estimated, from statistical tests, in 93%, with a significance of 0.05 (95% confidence).

Les gliomes des bas-grade sont des tumeurs cérébrales lentement évolutives, affectant principalement les jeunes adultes, qui peuvent rester des années sans symptôme. Les patients peuvent être opérés, ou traités par radiothérapie ou chimiothérapie, avec deux thérapies possibles : le PCV et le témozolomide (TMZ).Nous souhaitons montrer dans ces travaux de thèse que la modélisation mathématique, à travers l'approche de population, peut permettre l'amélioration des traitements en termes de durée et d'amplitude de décroissance pour les gliomes de bas-grade traités par chimiothérapie (PCV et TMZ).Dans un premier temps, nous nous concentrons sur la possibilité de modifier le protocole d'administration du PCV, au niveau de la population, afin de prolonger la durée de décroissance tumorale. Nous concluons qu'espacer les cycles de traitement permet de repousser de manière significative le moment de recroissance de la tumeur.Dans un second temps, nous étudions l'évolution des gliomes de bas-grade traités par TMZ. Sur la base des données de tailles tumorales de 77 patients, ainsi que d'informations génétiques, nous développons un modèle K-PD à effets mixtes permettant de décrire la dynamique tumorale avant, et suite au traitement. Nous évaluons ensuite les capacités du modèle à prédire la durée et l'amplitude de la réponse tumorale, à partir de mesures précoces de tailles de la tumeur ainsi que des informations génétiques. Ces prédictions pourraient être utilisées pour aider les cliniciens à déterminer si le traitement doit être prolongé ou non, pour un patient donné.Enfin, nous nous intéressons plus particulièrement au phénomène de résistance au traitement par TMZ. A partir des mêmes données de tailles tumorales que précédemment, nous construisons un modèle PK-PD à effets mixtes décrivant l'apparition des cellules résistantes au sein de la tumeur. Ce modèle reproduit plus précisément l'évolution du TMZ dans l'organisme et son impact sur la tumeur. Il est utilisé pour optimiser le protocole thérapeutique au niveau individuel. A l'aide d'un algorithme d'optimisation, nous déterminons l'intervalle entre chaque cycle et la dose à administrer afin de prolonger la durée de décroissance tumorale tout en limitant l'émergence de résistance. Les protocoles ainsi optimisés sont évalués à l'aide d'une approche stochastique, permettant de tester la robustesse du modèle et de l'optimisation.A travers les différents travaux de cette thèse, nous montrons l'utilité de la modélisation mathématique pour aider à l'amélioration des traitements chimiothérapiques pour les patients souffrant de gliomes de bas-grade. Nous croyons que ces résultats peuvent être transposés à d'autres types de cancers. / Low-grade gliomas are slow-growing brain tumors, mainly affecting young adults who may remain without any symptoms for years. Patients can undergo surgery, or receive radiotherapy or chemotherapy with two different treatments: PCV of temozolomide (TMZ).In our different projects, we aim to show that mathematical modeling, and population approach, can allow to improve treatments, in terms of response duration and amplitude, for low-grade gliomas treated with chemotherapy (PCV and TMZ).In a first part, we focus on the possibility to modify PCV administration protocol, on a population level, to prolong tumor decrease duration. We claim that prolonging time interval between cycles enables us to significantly postpone the time to tumor regrowth.In a second part, we study the evolution of low-grade gliomas treated with TMZ. We analyze tumor size observations of 77 low-grade glioma patients, as well as genetic information, to develop a K-PD mixed-effects model describing tumor evolution before and after treatment onset. We then evaluate model capacity to predict tumor response duration and amplitude, on the base of early tumor sizes and genetic information. These predictions could be used to help clinicians to determine if they should prolong the treatment or not, for a given patient.In a last part, we more particularly focus on the phenomenon of resistance to TMZ. We build a PK-PD mixed-effects model describing the emergence of resistant tumor cells, using the same tumor size observations as previously. This model more accurately reproduces the evolution of TMZ in the body and its effect on the tumor. It is then used to optimize TMZ therapeutic protocol, on an individual level. Using an optimization algorithm, we determine the time interval between TMZ cycles, and the dose to administer, to prolong tumor decrease duration while limiting the emergence of resistance. The optimized protocols are evaluated with a stochastic approach, allowing to test the robustness of the model and the optimization.Through these different projects, we show the utility of mathematical modeling to help to improve chemotherapy treatments of low-grade glioma patients. We believe that these results could be transposed to other types of cancers.

Made available in DSpace on 2014-06-11T19:22:35Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-05-31Bitstream added on 2014-06-13T20:09:50Z : No. of bitstreams: 1 guelfi_r_me_ilha.pdf: 667745 bytes, checksum: 4cf02e10ba92af5e81fd3f18fda6f6cd (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / No presente trabalho é apresentado um método para o cálculo de volumes determinados por K-Hiperbolóides de Carga e Potência, de modo a reduzir os gastos de empresas consideradas, em relação ao fator de potência. A redução de gastos é obtida alterando-se o valor máximo do fator potência que se pretende chegar, do fator de potência ideal 1, para o fator de potência de referência 0,92, ou mantendo-o quando já estiver acima de 0,92. Quando o fator de potência é maior ou igual a 0,92 ele já está eficiente, assim, não há necessidade de se chegar ao fator de potência ideal. É realizada uma comparação entre um método existente na literatura e o método proposto no presente trabalho, para determinar os pontos em que estes métodos diferem no cálculo dos volumes determinados pelos K-Hiperbolóides de Carga e Potência e mostrar a eficácia do método proposto. Estes métodos são equiparados, com relação a redução obtida no faturamento da conta de energia elétrica. A redução no faturamento da conta de energia elétrica que deve ser pago à concessionária é obtida por meio da diminuição da demanda contratada pela empresa; por sua vez a diminuição desta demanda é alcançada através da fórmula do fator de carga. Considerando-se a demanda média e o maior fator de carga obtido no período em estudo, encontra-se assim, uma nova demanda máxima menor que a demanda atual utilizada pela empresa. Logo, esta demanda máxima passa a ser a demanda que será contratada pela empresa, assim, esta nova demanda resulta em uma nova fatura que deve ser paga a concessionária de energia elétrica, menor que a fatura atual. / The present work presents a method for the calculation of volumes determined for K- Load and Power Hyperboloid, in order to reduce the expenses of considered companies, in relation to the power factor. The reduction of expenses is gotten by changing the maximum value of the wished power factor, of the ideal power factor1, to the reference power factor 0,92,or keeping it when it s above 0,92. When the power factor is higher or equal 0,92, it s already efficient, thus, it is not necessary to achieve the ideal power factor. A comparison is carried between an existing method in the literature and the considered method in the present work, to determine the points where these methods differ from each other in the calculation of the volumes determined for the K-Load and Power Hyperboloid and to show the effectiveness of the considered method. These methods are equalized, regarding the reduction gotten in the invoicing of the electric energy account. The reduction in the invoicing of the account of electric energy that must be paid to the concessionaire is gotten through the reduction of the contracted demand by the company; in turn the reduction of this demand is reached through the formula of the load factor. Considering the average demand and the highest load factor gotten in the period in study, this way a new demand, lower than the current demand used by the company, is gotten. Therefore, this maximum demand starts to be the demand that will be contracted by the company, thus, this new demand results in a new invoice which must be paid to the concessionaire of electric energy, lower than the current invoice.