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Delfim F. M. Torres

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

At least 163 records · Page 9Linked to original sources
arXiv2018

Fractional Order Version of the HJB Equation

We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

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arXiv2018

Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection

A human respiratory syncytial virus surveillance system was implemented in Florida in 1999, to support clinical decision-making for prophylaxis of premature newborns. Recently, a local periodic SEIRS mathematical model was proposed in [Stat. Optim. Inf. Comput. 6 (2018), no.1, 139--149] to describe real data collected by Florida's system. In contrast, here we propose a non-local fractional (non-integer) order model. A fractional optimal control problem is then formulated and solved, having treatment as the control. Finally, a cost-effectiveness analysis is carried out to evaluate the cost and the effectiveness of proposed control measures during the intervention period, showing the superiority of obtained results with respect to previous ones.

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arXiv2018

A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.

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arXiv2018

A cholera mathematical model with vaccination and the biggest outbreak of world's history

We propose and analyse a mathematical model for cholera considering vaccination. We show that the model is epidemiologically and mathematically well posed and prove the existence and uniqueness of disease-free and endemic equilibrium points. The basic reproduction number is determined and the local asymptotic stability of equilibria is studied. The biggest cholera outbreak of world's history began on 27th April 2017, in Yemen. Between 27th April 2017 and 15th April 2018 there were 2275 deaths due to this epidemic. A vaccination campaign began on 6th May 2018 and ended on 15th May 2018. We show that our model is able to describe well this outbreak. Moreover, we prove that the number of infected individuals would have been much lower provided the vaccination campaign had begun earlier.

q-bio.PE↗
arXiv2018

New inequalities for $η$-quasiconvex functions

The class of $η$-quasiconvex functions was introduced in 2016. Here we establish novel inequalities of Ostrowski type for functions whose second derivative, in absolute value raised to the power $q\geq 1$, is $η$-quasiconvex. Several interesting inequalities are deduced as special cases. Furthermore, we apply our results to the arithmetic, geometric, Harmonic, logarithmic, generalized log and identric means, getting new relations amongst them.

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arXiv2018

Some inequalities for interval-valued functions on time scales

We introduce the interval Darboux delta integral (shortly, the $ID$ $Δ$-integral) and the interval Riemann delta integral (shortly, the $IR$ $Δ$-integral) for interval-valued functions on time scales. Fundamental properties of $ID$ and $IR$ $Δ$-integrals and examples are given. Finally, we prove Jensen's, Hölder's and Minkowski's inequalities for the $IR$ $Δ$-integral. Also, some examples are given to illustrate our theorems.

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arXiv2018

A Mathematical Model for Vineyard Replacement with Nonlinear Binary Control Optimization

Vineyard replacement is a common practice in every wine-growing farm since the grapevine production decays over time and requires a new vine to ensure the business sustainability. In this paper, we formulate a simple discrete model that captures the vineyard's main dynamics such as production values and grape quality. Then, by applying binary non-linear programming methods to find the vineyard replacement trigger, we seek the optimal solution concerning different governmental subsidies to the target producer.

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arXiv2018

Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis

Oncolytic virotherapy (OV) has been emerging as a promising novel cancer treatment that may be further combined with the existing therapeutic modalities to enhance their effects. To investigate how OV could enhance chemotherapy, we propose an ODE based model describing the interactions between tumour cells, the immune response, and a treatment combination with chemotherapy and oncolytic viruses. Stability analysis of the model with constant chemotherapy treatment rates shows that without any form of treatment, a tumour would grow to its maximum size. It also demonstrates that chemotherapy alone is capable of clearing tumour cells provided that the drug efficacy is greater than the intrinsic tumour growth rate. Furthermore, OV alone may not be able to clear tumour cells from body tissue but would rather enhance chemotherapy if viruses with high viral potency are used. To assess the combined effect of OV and chemotherapy we use the forward sensitivity index to perform a sensitivity analysis, with respect to chemotherapy key parameters, of the virus basic reproductive number and the tumour endemic equilibrium. The results from this sensitivity analysis indicate the existence of a critical dose of chemotherapy above which no further significant reduction in the tumour population can be observed. Numerical simulations show that a successful combinational therapy of the chemotherapeutic drugs and viruses depends mostly on the virus burst size, infection rate, and the amount of drugs supplied. Optimal control analysis was performed, by means of Pontryagin's principle, to further refine predictions of the model with constant treatment rates by accounting for the treatment costs and sides effects.

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arXiv2018

Analysis of fractional integro-differential equations of thermistor type

We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: with Riemann-Liouville, Caputo, and time-scale fractional operators. Existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.

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arXiv2018

Regional enlarged observability of Caputo fractional differential equations

We consider the regional enlarged observability problem for fractional evolution differential equations involving Caputo derivatives. Using the Hilbert Uniqueness Method, we show that it is possible to rebuild the initial state between two prescribed functions only in an internal subregion of the whole domain. Finally, an example is provided to illustrate the theory.

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arXiv2018

The Variable-Order Fractional Calculus of Variations

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints.

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arXiv2018

A survey on fractional variational calculus

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and compute approximated solutions based on truncated Grünwald--Letnikov approximations of Caputo derivatives.

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arXiv2018

Optimal control of non-autonomous SEIRS models with vaccination and treatment

We study an optimal control problem for a non-autonomous SEIRS model with incidence given by a general function of the infective, the susceptible and the total population, and with vaccination and treatment as control variables. We prove existence and uniqueness results for our problem and, for the case of mass-action incidence, we present some simulation results designed to compare an autonomous and corresponding periodic model, as well as the controlled versus uncontrolled models.

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arXiv2018

Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

This paper presents three direct methods based on Grünwald-Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald-Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCP are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.

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arXiv2018

A stochastic SICA epidemic model for HIV transmission

We propose a stochastic SICA epidemic model for HIV transmission, described by stochastic ordinary differential equations, and discuss its perturbation by environmental white noise. Existence and uniqueness of the global positive solution to the stochastic HIV system is proven, and conditions under which extinction and persistence in mean hold, are given. The theoretical results are illustrated via numerical simulations.

q-bio.PE↗
arXiv2018

A sufficient optimality condition for non-linear delayed optimal control problems

We prove a sufficient optimality condition for non-linear optimal control problems with delays in both state and control variables. Our result requires the verification of a Hamilton-Jacobi partial differential equation and is obtained through a transformation that allow us to rewrite a delayed optimal control problem as an equivalent non-delayed one.

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