Fractional derivative models for the spread of diseases

This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution.This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Differential Models, Numerical Simulations and Applications

This Special Issue includes 12 high-quality articles containing original research findings in the fields of differential and integro-differential models, numerical methods and efficient algorithms for parameter estimation in inverse problems, with applications to biology, biomedicine, land degradation, traffic flows problems, and manufacturing systems

Cálculo fraccionario, geometría fractal y modelos de crecimiento tumoral

El cálculo fraccionario estudia la posilidad de extender los operadores de derivación e integración clásicos a operadores de órdenes no enteros. El carácter no local de estos nuevos operadores fraccionarios ofrece nuevas perspectivas para formular modelos matemáticos en ramas muy diversas. En este trabajo presentamos una introducción teórica a las definiciones y propiedades más importantes utilizadas en cálculo fraccionario y en el estudio de las ecuaciones diferenciales fraccionarias. Introducimos el método numérico que surge de manera natural de la definición del operador fraccionario de Grünwald-Letnikov y lo aplicamos al estudio numérico de ecuaciones diferenciales fraccionarias que resultan de extender modelos clásicos de crecimiento tumoral. Finalmente, se definen algunos conceptos de cálculo fraccionario discreto y se propone un modelo de crecimiento fractal en el plano complejo, basado en el conjunto de Mandelbrot, que presenta aspectos que pueden compararse con el crecimiento de tumores reales

Um segundo estudo de invasão populacional dinâmica a partir da equação do telégrafo reativo e formulação de elementos de contorno - Um ensaio sobre o crescimento tumoral in vitro

This paper is a continuation of a study already carried out on the use of the reactive-telegraph equation to analyse problems of population dynamics based on a formulation of the boundary element method (BEM). In this paper, the numerical model simulates the evolution of a tumour as a problem of population density of cancer cells from different reactive terms coupled to the reactive-telegraph equation to describe the growth and distribution of the population, similar to the two-dimensional in vitro tumour growth experiment. The mathematical model developed is called D-BEM, uses a time independent fundamental solution and the finite difference method is combined with BEM to approximate the time derivative terms and the Gaussian quadrature is used to calculate the domain integrals. The solution of the system nonlinear of equations is based on the Gaussian elimination method and the stability of the proposed formulation was verified. As the telegraph equation has a wave behaviour, a phase change phenomenon that can lead to the appearance of negative population density may occur, an algorithm was developed to guarantee the solution's positivity. Important results were obtained and demonstrate the effect of the delay parameter on tumour growth. In one of the tested cases, the results indicated an oscillatory behaviour in the tumour growth when the delay parameter assumed increasing values. The results of numerical simulations that sought to represent tumour growth, as well as the entire formulation of the boundary elements are presented below.Este artigo é a continuação de um estudo já realizado sobre o uso da equação do telégrafo reativo para analisar problemas de dinâmica populacional a partir de uma formulação do método dos elementos de contorno (BEM). Neste artigo, o modelo numérico simula a evolução de um tumor como um problema de densidade populacional de células cancerosas a partir de diferentes termos reativos acoplados à equação do telégrafo reativo para descrever o crescimento e distribuição da população, semelhante ao experimento de crescimento do tumor in vitro. O modelo matemático desenvolvido é denominado D-BEM, usa uma solução fundamental independente do tempo e o método das diferenças finitas é combinado com o BEM para aproximar os termos de tempo derivativos e a quadratura Gaussiana é usada para calcular as integrais de domínio. A solução do sistema de equações é baseada no método de eliminação gaussiana e foi verificada a estabilidade da formulação proposta. Como a equação do telégrafo possui comportamento ondulatório, pode ocorrer um fenômeno de mudança de fase que pode levar ao aparecimento de densidade populacional negativa, para tanto, foi desenvolvido um algoritmo que garantir a positividade da solução. Resultados importantes foram obtidos e demonstram o efeito do parâmetro de atraso no crescimento do tumor. Em um dos casos testados, os resultados indicaram um comportamento oscilatório no crescimento tumoral quando o parâmetro de retardo assumiu valores crescentes. O importante resultado dessa antítese para o crescimento do tumor, bem como toda a formulação dos elementos de contorno são apresentados a seguir

Introducción al cálculo fraccionario y a los modelos de crecimiento tumoral clásicos y fraccionarios

Se presenta una recopilación de las ideas y definiciones esenciales del cálculo fraccionario y su aplicación a la resolución de ecuaciones diferenciales fraccionarias. Damos al cálculo fraccionario un contexto histórico, incluyendo la resolución del problema de Abel, definimos los diferentes operadores fraccionarios y los espacios funcionales donde vamos a trabajar, y por último introducimos la transformada de Laplace de los operadores definidos como herramienta para resolver ecuaciones diferenciales fraccionarias analíticamente y el método de Grünwald-Letnikov para la resolución numérica