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Delfim F. M. Torres

Publications and source records attributed to Delfim F. M. Torres.

At least 235 records · Page 13Linked to original sources
arXiv2016

A Hukuhara approach to the study of hybrid fuzzy systems on time scales

We introduce a new approach to study the practical stability of hybrid fuzzy systems on time scales in the Lyapunov sense. Our method is based on the delta-Hukuhara derivative for fuzzy valued functions and allow us to obtain new interesting stability criteria. We also show the validity of the results of [M. Sambandham: Hybrid fuzzy systems on time scales, Dynam. Systems Appl. 12 (2003), no. 1-2, 217-227], by embedding the space of all fuzzy subsets into a suitable Banach space.

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arXiv2016

Dynamics and optimal control of Ebola transmission

A major Ebola outbreak occurs in West Africa since March 2014, being the deadliest epidemic in history. As an infectious disease epidemiology, Ebola is the most lethal and is moving faster than in previous outbreaks. On 8 August 2014, the World Health Organization (WHO) declared the outbreak a public health emergency of international concern. Last update on 7 July 2015 by WHO reports 27 609 cases of Ebola with a total of 11 261 deaths. In this work, we present a mathematical description of the spread of Ebola virus based on the SEIR (Susceptible-Exposed-Infective-Recovered) model and optimal strategies for Ebola control. In order to control the propagation of the virus and to predict the impact of vaccine programmes, we investigate several strategies of optimal control of the spread of Ebola: control infection by vaccination of susceptible; minimize exposed and infected; reduce Ebola infection by vaccination and education.

math.OC↗
arXiv2016

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann-Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz's method. Examples show that the proposed approach is more accurate than recent methods available in the literature.

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arXiv2016

Direct and Inverse Variational Problems on Time Scales: A Survey

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation (Helmholtz's problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.

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arXiv2015

Optimal control strategies for the spread of Ebola in West Africa

The spread of Ebola virus in 2014 is unprecedented. The epidemic is still affecting West Africa, exacerbated by extraordinary socioeconomic disadvantages and health system inadequacies. With the aim of understanding, predicting, and control the propagation of the virus in the populations of affected countries, it is crucial to model the dynamics of the virus and study several strategies to control it. In this paper, we present a very simple mathematical model that describes quite well the spread of Ebola. Then, we discuss several strategies for the control of the propagation of this lethal virus into populations, in order to predict the impact of vaccine programmes, treatment, and the impact of educational campaigns.

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arXiv2015

Approximated Analytical Solution to an Ebola Optimal Control Problem

An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods.

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arXiv2015

Dengue disease: a multiobjective viewpoint

During the last decades, the global prevalence of dengue progressed dramatically. It is a disease that is now endemic in more than one hundred countries of Africa, America, Asia, and the Western Pacific. In this paper, we present a mathematical model for the dengue disease transmission described by a system of ordinary differential equations and propose a multiobjective approach to find the most effective ways of controlling the disease. We use evolutionary multiobjective optimization (EMO) algorithms to solve the resulting optimization problem, providing the performance comparison of different algorithms. The obtained results show that the multiobjective approach is an effective tool to solve the problem, giving higher quality and wider range of solutions compared to the traditional technique. The obtained trade-offs provide a valuable information about the dynamics of infection transmissions and can be used as an input in the process of planning the intervention measures by the health authorities. Additionally, a suggested hybrid EMO algorithm produces highly superior performance compared to five other state-of-the-art EMO algorithms, being indispensable to efficiently optimize the proposed model.

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arXiv2015

Predicting and controlling the Ebola infection

We present a comparison between two different mathematical models used in the description of the Ebola virus propagation currently occurring in West Africa. In order to improve the prediction and the control of the propagation of the virus, numerical simulations and optimal control of the two models for Ebola are investigated. In particular, we study when the two models generate similar results.

q-bio.PE↗
arXiv2015

Complex-valued fractional derivatives on time scales

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

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arXiv2015

Caputo derivatives of fractional variable order: numerical approximations

We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.

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arXiv2015

Noether's theorem for higher-order variational problems of Herglotz type

We approach higher-order variational problems of Herglotz type from an optimal control point of view. Using optimal control theory, we derive a generalized Euler-Lagrange equation, transversality conditions, a DuBois-Reymond necessary optimality condition and Noether's theorem for Herglotz's type higher-order variational problems, valid for piecewise smooth functions.

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arXiv2015

Coexistence of two dengue virus serotypes and forecasting for Madeira island

The first outbreak of dengue occurred in Madeira Island on 2012, featuring one virus serotype. Aedes aegypti was the vector of the disease and it is unlikely that it will be eliminated from the island. Therefore, a new outbreak of dengue fever can occur and, if it happens, risk to the population increases if two serotypes coexist. In this paper, mathematical modeling and numerical simulations are carried out to forecast what may happen in Madeira Island in such scenario.

q-bio.PE↗
arXiv2015

Multiobjective approach to optimal control for a dengue transmission model

During the last decades, the global prevalence of dengue progressed dramatically. It is a disease which is now endemic in more than one hundred countries of Africa, America, Asia and the Western Pacific. This study addresses a mathematical model for the dengue disease transmission and finding the most effective ways of controlling the disease. The model is described by a system of ordinary differential equations representing human and vector dynamics. Multiobjective optimization is applied to find the optimal control strategies, considering the simultaneous minimization of infected humans and costs due to insecticide application. The obtained results show that multiobjective optimization is an effective tool for finding the optimal control. The set of trade-off solutions encompasses a whole range of optimal scenarios, providing valuable information about the dynamics of infection transmissions. The results are discussed for different values of model parameters.

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arXiv2015

A Conformable Fractional Calculus on Arbitrary Time Scales

A conformable time-scale fractional calculus of order $α\in ]0,1]$ is introduced. The basic tools for fractional differentiation and fractional integration are then developed. The Hilger time-scale calculus is obtained as a particular case, by choosing $α= 1$.

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arXiv2015

Optimal Solutions to Relaxation in Multiple Control Problems of Sobolev Type with Nonlocal Nonlinear Fractional Differential Equations

We introduce the optimality question to the relaxation in multiple control problems described by Sobolev type nonlinear fractional differential equations with nonlocal control conditions in Banach spaces. Moreover, we consider the minimization problem of multi-integral functionals, with integrands that are not convex in the controls, of control systems with mixed nonconvex constraints on the controls. We prove, under appropriate conditions, that the relaxation problem admits optimal solutions. Furthermore, we show that those optimal solutions are in fact limits of minimizing sequences of systems with respect to the trajectory, multi-controls, and the functional in suitable topologies.

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