Publications 2021

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47.  Programação matemática

Torres, Delfim Fernando Marado

UA Editora

O termo "programação matemática" refere-se ao estudo de problemas de otimização, em que se procura minimizar ou maximizar uma função através da escolha dos valores de variáveis dentro de um determinado conjunto admissível. Em problemas de engenharia, administração, logística, transporte, economia, biologia, medicina ou outras ciências, quando se consegue construir modelos matemáticos representativos dos respetivos sistemas dinâmicos em estudo, é possível aplicar as técnicas matemáticas de otimização para maximizar ou minimizar uma função previamente definida como índice de desempenho ou performance, visando encontrar uma "solução" do problema, isto é, os valores das variáveis que resultem no melhor desempenho possível do sistema, segundo o tal critério previamente definido. O livro "Programação Matemática" é uma obra introdutória, de natureza pedagógica, e que está escrito de uma forma sucinta, clara e rigorosa. Serve de suporte à unidade curricular com o mesmo nome do Departamento de Matemática da Universidade de Aveiro, que tem sido lecionada a alunos provenientes de várias licenciaturas (de Matemática, Física, Economia e Engenharia), oriundos de diversas universidades portuguesas, dos PALOP, Brasil e Timor-Leste, assim como de vários países europeus por intermédio do programa Erasmus. Pretende fornecer uma formação básica, mas sólida, em otimização não linear e, em simultâneo, estimular a utilização de tais modelos e resultados na resolução de problemas práticos. Estão incluídos os conceitos essenciais de programação matemática, que alguém que deseje prosseguir estudos na área de otimização deve conhecer e dominar. Os conteúdos são acompanhados de exemplos e exercícios, com o intuito de se desenvolver a capacidade de aplicação dos conceitos matemáticos envolvidos. As demonstrações dos resultados apresentados são dadas com todo o rigor, procurando-se estimular o desenvolvimento do raciocínio, essencial numa qualquer atividade profissional.

ria.ua.pt

46.  Analysis of infectious disease problems (Covid-19) and their global impact

Agarwal, Praveen and Nieto, Juan J. and Ruzhansky, Michael and Torres, Delfim F. M.

Springer

This book is a collection of selected research articles discussing the analysis of infectious diseases by using mathematical modelling in recent times. Divided into two parts, the book gives a general and country-wise analysis of Covid-19. Analytical and numerical techniques for virus models are presented along with the application of mathematical modelling in the analysis of their spreading rates and treatments. The book also includes applications of fractional differential equations as well as ordinary, partial and integro-differential equations with optimization methods. Probability distribution and their bio-mathematical applications have also been studied. This book is a valuable resource for researchers, scholars, biomathematicians and medical experts.

ria.ua.pt | doi

Book Chapters


45.  State-space estimation using the behavioral approach: a simple particular case

Ntogramatzidis, Lorenzo and Pereira, Ricardo and Rocha, Paula

CONTROLO 2020. Lecture Notes in Electrical Engineering

Springer

In this paper we apply the behavioral estimation theory developed in Ntogramatzidis et al. (2020) to the particular case of state-space systems. We derive new necessary and sufficient conditions for the solvability of the estimation problem in the presence of disturbances, and provide a method to construct an estimator in case the problem is solvable. This is a first step to investigate how our previous results, derived within the more general behavioral context, compare with the results from classical state space theory.

ria.ua.pt | doi | Peer Reviewed

44.  Optimal control of vaccination and plasma transfusion with potential usefulness for Covid-19

Couras, Juliana and Area, Iván and Nieto, Juan J. and Silva, Cristiana J. and Torres, Delfim F. M.

Analysis of infectious disease problems (Covid-19) and their global impact

Springer

The SEIR model is a compartmental model used to simulate the dynamics of an epidemic. In this chapter, we introduce two control functions in the compartmental SEIR model representing vaccination and plasma transfusion. Optimal control problems are proposed to study the effects of these two control measures, on the reduction of infected individuals and increase of recovered ones, with minimal costs. Up to our knowledge, the plasma transfusion treatment has never been considered as a control strategy for epidemics mitigation. The proposed vaccination and treatment strategies may have a real application in the challenging and hard problem of controlling the Covid-19 pandemic.

ria.ua.pt | doi | Peer Reviewed

43.  Modeling the spread of Covid-19 pandemic in Morocco

Zine, Houssine and Lotfi, El Mehdi and Mahrouf, Marouane and Boukhouima, Adnane and Aqachmar, Yassine and Hattaf, Khalid and Torres, Delfim F. M. and Yousfi, Noura

Analysis of infectious disease problems (Covid-19) and their global impact

Springer

Nowadays, coronavirus disease 2019 (Covid-19) poses a great threat to public health and economy worldwide. Unfortunately, there is yet no effective drug for this disease. For this, several countries have adopted multiple preventive interventions to avoid the spread of Covid-19. Here, we propose a delayed mathematical model to predict the epidemiological trend of Covid-19 in Morocco. Parameter estimation and sensitivity analysis of the proposed model are rigorously studied. Moreover, numerical simulations are presented in order to test the effectiveness of the preventive measures and strategies that were imposed by the Moroccan authorities and also help policy makers and public health administration to develop such strategies.

ria.ua.pt | doi | Peer Reviewed

Articles


42.  A matrix based list decoding algorithm for linear codes over integer residue rings

Napp, Diego and Pinto, Raquel and Saçıkara, Elif and Toste, Marisa

Linear Algebra and its Applications

Elsevier

In this paper we address the problem of list decoding of linear codes over an integer residue ring Zq, where q is a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Zpr , called the standard form, and the p-adic expansion of the to-be-decoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Zpr . From the parity-check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p^{r−1}Zpr . This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal.

ria.ua.pt | doi | Peer Reviewed

41.  Fractional model of COVID-19 applied to Galicia, Spain and Portugal

Ndaïrou, Faïçal and Area, Iván and Nieto, Juan J. and Silva, Cristiana J. and Torres, Delfim F. M.

Chaos, Solitons & Fractals

Elsevier

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models.

ria.ua.pt | doi | Peer Reviewed

40.  Stability analysis and optimal control of a fractional HIV-AIDS epidemic model with memory and general incidence rate

Boukhouima, Adnane and Lotfi, El Mehdi and Mahrouf, Marouane and Rosa, Silvério and Torres, Delfim F. M. and Yousfi, Noura

The European Physical Journal Plus

Springer Verlag; EDP Sciences; Società Italiana di Fisica

We investigate the celebrated mathematical SICA model but using fractional differential equations in order to better describe the dynamics of HIV-AIDS infection. The infection process is modelled by a general functional response, and the memory effect is described by the Caputo fractional derivative. Stability and instability of equilibrium points are determined in terms of the basic reproduction number. Furthermore, a fractional optimal control system is formulated and the best strategy for minimizing the spread of the disease into the population is determined through numerical simulations based on the derived necessary optimality conditions.

ria.ua.pt | doi | Peer Reviewed

39.  Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets

Nemati, S. and Lima, Pedro M. and Torres, Delfim F. M.

Numerical Algorithms

Springer

We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss–Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is first transformed into the interval [− 1, 1], by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss–Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme.

ria.ua.pt | doi | Peer Reviewed

38.  Focus point: cancer and HIV/AIDS dynamics: from optimality to modelling

Debbouche, Amar and Nieto, Juan J. and Torres, Delfim F. M.

The European Physical Journal Plus

Springer

Human cancer is a multistep process involving acquired genetic mutations, each of which imparts a particular type of growth advantage to the cell and ultimately leads to the development of a malignant phenotype. It is also a generic term for a group of diseases and figures as a leading cause of death globally; it lays a significant burden on healthcare systems and continues to be among the major health problems worldwide. The consequences of mutations in tumor cells include alterations in cell signaling pathways that result in uncontrolled cellular proliferation, insensitivity to growth inhibitory signals, resistance to apoptosis, development of cellular immortality, angiogenesis, tissue invasion and metastasis.

ria.ua.pt | doi | Peer Reviewed

37.  Modeling and forecasting of COVID-19 spreading by delayed stochastic differential equations

Mahrouf, Marouane and Boukhouima, Adnane and Zine, Houssine and Lotfi, El Mehdi and Torres, Delfim F. M. and Yousfi, Noura

Axioms

MDPI

The novel coronavirus disease (COVID-19) pneumonia has posed a great threat to the world recent months by causing many deaths and enormous economic damage worldwide. The first case of COVID-19 in Morocco was reported on 2 March 2020, and the number of reported cases has increased day by day. In this work, we extend the well-known SIR compartmental model to deterministic and stochastic time-delayed models in order to predict the epidemiological trend of COVID-19 in Morocco and to assess the potential role of multiple preventive measures and strategies imposed by Moroccan authorities. The main features of the work include the well-posedness of the models and conditions under which the COVID-19 may become extinct or persist in the population. Parameter values have been estimated from real data and numerical simulations are presented for forecasting the COVID-19 spreading as well as verification of theoretical results.

ria.ua.pt | doi | Peer Reviewed

36.  Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal

Silva, Cristiana J. and Cruz, Carla and Torres, Delfim F. M. and Muñuzuri, Alberto P. and Carballosa, Alejandro and Area, Iván and Nieto, Juan J. and Fonseca-Pinto, Rui and Passadouro, Rui and Santos, Estevão Soares dos and Abreu, Wilson and Mira, Jorge

Scientific Reports

Nature Research

The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to "normal life" and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.

ria.ua.pt | doi | Peer Reviewed

35.  Control of COVID-19 dynamics through a fractional-order model

Bushnaq, Samia and Saeed, Tareq and Torres, Delfim F. M. and Zeb, Anwar

Alexandria Engineering Journal

Elsevier

We investigate, through a fractional mathematical model, the effects of physical distance on the SARS-CoV-2 virus transmission. Two controls are considered in our model for eradication of the spread of COVID-19: media education, through campaigns explaining the importance of social distancing, use of face masks, etc., towards all population, while the second one is quarantine social isolation of the exposed individuals. A general fractional order optimal control problem, and associated optimality conditions of Pontryagin type, are discussed, with the goal to minimize the number of susceptible and infected while maximizing the number of recovered. The extremals are then numerically obtained.

ria.ua.pt | doi | Peer Reviewed

34.  Analysis of Hilfer fractional integro-differential equations with almost sectorial operators

Karthikeyan, Kulandhaivel and Debbouche, Amar and Torres, Delfim F. M.

Fractal and Fractional

MDPI

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.

ria.ua.pt | doi | Peer Reviewed

33.  On the necessary optimality conditions for the fractional Cucker–Smale optimal control problem

Almeida, Ricardo and Kamocki, Rafał and Malinowska, Agnieszka B. and Odzijewicz, Tatiana

Communications in Nonlinear Science and Numerical Simulation

Elsevier

This paper develops a sparse flocking control for the fractional Cucker–Smale multi-agent model. The Caputo fractional derivative, in the equations describing the dynamics of a consensus parameter, makes it possible to take into account in the self-organization of group its history and memory dependency. External control is designed based on necessary conditions for a local solution to the appropriate optimal control problem. Numerical simulations demonstrate the effectiveness of the control scheme.

ria.ua.pt | doi | Peer Reviewed

32.  Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input

Almeida, R. and Hristova, S. and Dashkovskiy, S.

International Journal of Robust and Nonlinear Control

Wiley

The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time-varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system’s dynamic response, and demonstrate the effectiveness of our theoretical results.

ria.ua.pt | doi | Peer Reviewed

31.  Optimal leader-following consensus of fractional opinion formation models

Almeida, Ricardo and Kamocki, Rafał and Malinowska, Agnieszka B. and Odzijewicz, Tatiana

Journal of Computational and Applied Mathematics

Elsevier

This paper deals with a control strategy enforcing consensus in a fractional opinion formation model with leadership, where the interaction rates between followers and the influence rate of the leader are functions of deviations of opinions between agents. The fractional-order derivative determines the impact of the memory during the opinion evolution. The problem of leader-following consensus control is cast in the framework of nonlinear optimal control theory. We study a finite horizon optimal control problem, in which deviations of opinions between agents and with respect to the leader are penalized along with the control that is applied only to the leader. The existence conditions for optimal consensus control are proved and necessary optimality conditions for the considered problem are derived. The results of the paper are illustrated by some examples.

ria.ua.pt | doi | Peer Reviewed

30.  Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

Cruz, Fátima and Almeida, Ricardo and Martins, Natália

AIMS Mathematics

AIMS Press

In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler–Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.

ria.ua.pt | doi | Peer Reviewed

29.  Global stability of a Caputo fractional SIRS model with general incidence rate

Ammi, Moulay Rchid Sidi and Tahiri, Mostafa and Torres, Delfim F. M.

Mathematics in Computer Science

Springer

We introduce a fractional order SIRS model with non-linear incidence rate. Existence of a unique positive solution to the model is proved. Stability analysis of the disease free equilibrium and positive fixed points are investigated. Finally, a numerical example is presented.

ria.ua.pt | doi | Peer Reviewed

28.  A generalization of a fractional variational problem with dependence on the boundaries and a real parameter

Almeida, Ricardo and Martins, Natália

Fractal and Fractional

MDPI

In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.

ria.ua.pt | doi | Peer Reviewed

27.  Pest control using farming awareness: impact of time delays and optimal use of biopesticides

Abraha, Teklebirhan and Al Basir, Fahad and Obsu, Legesse Lemecha and Torres, Delfim F. M.

Chaos, Solitons & Fractals

Elsevier

We investigate a mathematical model in crop pest management, considering plant biomass, pest, and the effect of farming awareness. The pest population is divided into two compartments: susceptible pest and infected pest. We assume that the growth rate of self-aware people is proportional to the density of healthy pests present in the crop field. Impacts of awareness is modeled via a saturated term. It is further assumed that self-aware people will adopt biological control methods, namely integrated pest management. Susceptible pests are detrimental to crops and, moreover, there may be some time delay in measuring the healthy pests in the crop field. A time delay may also take place while becoming aware of the control strategies or taking necessary steps to control the pest attack. In agreement, we develop our model incorporating two time delays into the system. The existence and the stability criteria of the equilibria are obtained in terms of the basic reproduction number and time delays. Stability switches occur through Hopf-bifurcation when time delays cross critical values. Optimal control theory has been applied for the cost-effectiveness of the delayed system. Numerical simulations illustrate the obtained analytical results.

ria.ua.pt | doi | Peer Reviewed

26.  New variational problems with an action depending on generalized fractional derivatives, the free endpoint conditions, and a real parameter

Almeida, Ricardo and Martins, Natália

Symmetry

MDPI

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.

ria.ua.pt | doi | Peer Reviewed

25.  A new spectral method based on two classes of hat functions for solving systems of fractional differential equations and an application to respiratory syncytial virus infection

Nemati, Somayeh and Torres, Delfim F. M.

Soft Computing

Springer

We propose a new spectral method, based on two classes of hat functions, for solving systems of fractional differential equations. The fractional derivative is considered in the Caputo sense. Properties of the basis functions, Caputo derivatives and Riemann–Liouville fractional integrals, are used to reduce the main problem to a system of nonlinear algebraic equations. By analyzing in detail the resulting system, we show that the method needs few computational efforts. Two test problems are considered to illustrate the efficiency and accuracy of the proposed method. Finally, an application to a recent mathematical model in epidemiology is given, precisely to a system of fractional differential equations modeling the respiratory syncytial virus infection.

ria.ua.pt | doi | Peer Reviewed

24.  Cauchy’s formula on nonempty closed sets and a new notion of Riemann–Liouville fractional integral on time scales

Torres, Delfim F. M.

Applied Mathematics Letters

Elsevier

We prove Cauchy’s formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.

ria.ua.pt | doi | Peer Reviewed

23.  A dynamically-consistent nonstandard finite difference scheme for the SICA model

Vaz, Sandra and Torres, Delfim F. M.

Mathematical Biosciences and Engineering

AIMS Press

In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible–Infected–Chronic–AIDS) model and analyze the dynamical properties of the discretized system. We prove that the discretized model is dynamically consistent with the continuous, maintaining the essential properties of the standard SICA model, namely, the positivity and boundedness of the solutions, equilibrium points, and their local and global stability.

ria.ua.pt | doi | Peer Reviewed

22.  A behavioral approach to estimation in the presence of disturbances

Pereira, Ricardo and Rocha, Paula and Ntogramatzidis, Lorenzo

IEEE Transactions on Automatic Control

IEEE

In this article, we study the problem of estimation in the presence of disturbances within the context of the behavioral approach developed by J.C. Willems. For this purpose, we use the behavioral theory of observers introduced by Valcher, Willems, Trentelman, and Trumpf, combined with the notions of behavioral invariance, conditioned invariance, and behavioral detectability subspaces. With these tools, we provide necessary and sufficient conditions for the solvability of the aforementioned problem together with the construction of an estimator.

ria.ua.pt | doi | Peer Reviewed

21.  Farming awareness based optimum interventions for crop pest control

Abraha, Teklebirhan and Al Basir, Fahad and Obsu, Legesse Lemecha and Torres, Delfim F. M.

Mathematical Biosciences and Engineering

AIMS Press

We develop a mathematical model, based on a system of ordinary differential equations, to the upshot of farming alertness in crop pest administration, bearing in mind plant biomass, pest, and level of control. Main qualitative analysis of the proposed mathematical model, akin to both pest-free and coexistence equilibrium points and stability analysis, is investigated. We show that all solutions of the model are positive and bounded with initial conditions in a certain significant set. The local stability of pest-free and coexistence equilibria is shown using the Routh–Hurwitz criterion. Moreover, we prove that when a threshold value is less than one, then the pest-free equilibrium is locally asymptotically stable. To get optimum interventions for crop pests, that is, to decrease the number of pests in the crop field, we apply optimal control theory and find the corresponding optimal controls. We establish existence of optimal controls and characterize them using Pontryagin's minimum principle. Finally, we make use of numerical simulations to illustrate the theoretical analysis of the proposed model, with and without control measures.

ria.ua.pt | doi | Peer Reviewed

20.  Mathematical analysis of a fractional COVID-19 model applied to Wuhan, Spain and Portugal

Ndaïrou, Faïçal and Torres, Delfim F. M.

Axioms

MDPI

We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point.

ria.ua.pt | doi | Peer Reviewed

19.  List decoding of convolutional codes over integer residue rings

Lieb, Julia and Napp, Diego and Pinto, Raquel

Finite Fields and Their Applications

Elsevier

A convolutional code over is a -submodule of where stands for the ring of polynomials with coefficients in . In this paper, we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword when some of its coefficients have been erased. We do that using the p-adic expansion of w and particular representations of the parity-check polynomial matrix of the code. From these matrix polynomial representations we recursively select certain equations that w must satisfy and have only coefficients in the field . We exploit the natural block Toeplitz structure of the sliding parity-check matrix to derive a step by step methodology to obtain a list of possible codewords for a given corrupted codeword w, that is, a list with the closest codewords to w.

ria.ua.pt | doi | Peer Reviewed

18.  Minimal state-space representation of convolutional product codes

Climent, Joan-Josep and Napp, Diego and Pinto, Raquel and Requena, Verónica

Mathematics

MDPI

In this paper, we study product convolutional codes described by state-space representations. In particular, we investigate how to derive state-space representations of the product code from the horizontal and vertical convolutional codes. We present a systematic procedure to build such representation with minimal dimension, i.e., reachable and observable.

ria.ua.pt | doi | Peer Reviewed

17.  Variational problems with time delay and higher-order distributed-order fractional derivatives with arbitrary kernels

Cruz, Fátima and Almeida, Ricardo and Martins, Natália

Mathematics

MDPI

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

ria.ua.pt | doi | Peer Reviewed

16.  State realizations of 2-periodic convolutional codes: a switching system approach

Fornasini, Ettore and Napp, Diego and Pereira, Ricardo and Pinto, Raquel and Rocha, Paula

IFAC-PapersOnLine

Elsevier; IFAC

In this work we investigate the realization problem of periodic convolutional codes. As convolutional codes are discrete linear systems over a finite field we use systems theory techniques to address our problem. In particular, we aim at deriving and studying state-space realizations of 2-periodic convolutional codes. Although one cannot expect, in general, to obtain a periodic state-space realization of a periodic convolutional code by means of the individual realizations of each of the associated time-invariant codes, we show that one can implement the periodic system switching periodically the output in each state system. Comments on the minimality of this realization are given.

ria.ua.pt | doi | Peer Reviewed

15.  On a non-Newtonian calculus of variations

Torres, Delfim F. M.

Axioms

MDPI

The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

ria.ua.pt | doi | Peer Reviewed

14.  Pontryagin maximum principle for distributed-order fractional systems

Ndaïrou, Faïçal and Torres, Delfim F. M.

Mathematics

MDPI

We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.

ria.ua.pt | doi | Peer Reviewed

13.  Local existence and uniqueness for a fractional SIRS model with Mittag-Leffler law

Sidi Ammi, Moulay Rchid and Tahiri, Mostafa and Torres, Delfim F. M.

General Letters in Mathematics

Refaad

In this paper, we study an epidemic model with Atangana-Baleanu-Caputo (ABC) fractional derivative. We obtain a special solution using an iterative scheme via Laplace transformation. Uniqueness and existence of a solution using the Banach fixed point theorem are studied. A detailed analysis of the stability of the special solution is presented. Finally, our generalized model in the ABC fractional derivative sense is solved numerically by the Adams-Bashforth-Moulton method.

ria.ua.pt | doi | Peer Reviewed

12.  Optimal control problems involving combined fractional operators with general analytic kernels

Ndaïrou, Faïçal and Torres, Delfim F. M.

Mathematics

MDPI

Fractional optimal control problems via a wide class of fractional operators with a general analytic kernel are introduced. Necessary optimality conditions of Pontryagin type for the considered problem are obtained after proving a Gronwall type inequality as well as results on continuity and differentiability of perturbed trajectories. Moreover, a Mangasarian type sufficient global optimality condition for the general analytic kernel fractional optimal control problem is proved. An illustrative example is discussed.

ria.ua.pt | doi | Peer Reviewed

11.  Analysis of a COVID-19 compartmental model: a mathematical and computational approach

Abreu, Zita and Cantin, Guillaume and Silva, Cristiana J.

Mathematical Biosciences and Engineering

AIMS Press

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.

ria.ua.pt | doi | Peer Reviewed

10.  Marketing Verde: comparando o consumo de produtos ecológicos nas gerações X e Y

Magalhães, Carla and Paço, Arminda and Alonso, Hugo and Oliveira, Marta

CBR - Consumer Behavior Review

Universidade Federal de Pernambuco

Este estudo analisa a influência de determinados estímulos de marketing (propaganda, informação veiculada e preço) no consumo de produtos ecológicos, comparando o comportamento dos consumidores portugueses das gerações X e Y. Através de uma pesquisa quantitativa, crosssectional, com base num questionário online, cujos resultados foram analisados com recurso ao software SPSS Statistics 25, concluímos que existem algumas semelhanças entre ambas as gerações, como a capacidade de identificação dos produtos ecológicos e a predisposição para a sua compra, o impacto positivo das campanhas de comunicação com apelo emocional e a perceção da importância dos rótulos dos produtos ecológicos. A variável que mais distingue o comportamento de ambas as gerações é o preço, pois a geração Y está mais predisposta a pagar um valor superior por um produto ecológico. Esta investigação contribui para a literatura sobre o comportamento do consumidor, especialmente no âmbito da variável “geração”, aplicado ao contexto do consumo de produtos ecológicos. Também ajuda as empresas a posicionarem-se melhor na relação com os consumidores de ambas as gerações analisadas. A definição de estratégias de targeting mais acuradas relativamente à promoção, preço e decisão de compra pode então tomar como ponto de partida os resultados deste estudo.

ria.ua.pt | doi | Peer Reviewed

9.  Hybrid method for simulation of a fractional COVID-19 model with real case application

Din, Anwarud and Khan, Amir and Zeb, Anwar and Ammi, Moulay Rchid Sidi and Tilioua, Mouhcine and Torres, Delfim F. M.

Axioms

MDPI

In this research, we provide a mathematical analysis for the novel coronavirus responsible for COVID-19, which continues to be a big source of threat for humanity. Our fractional-order analysis is carried out using a non-singular kernel type operator known as the Atangana-Baleanu-Caputo (ABC) derivative. We parametrize the model adopting available information of the disease from Pakistan in the period 9 April to 2 June 2020. We obtain the required solution with the help of a hybrid method, which is a combination of the decomposition method and the Laplace transform. Furthermore, a sensitivity analysis is carried out to evaluate the parameters that are more sensitive to the basic reproduction number of the model. Our results are compared with the real data of Pakistan and numerical plots are presented at various fractional orders.

ria.ua.pt | doi | Peer Reviewed

8.  Approximate iterative method for initial value problem of impulsive fractional differential equations with generalized proportional fractional derivatives

Agarwal, Ravi P. and Hristova, Snezhana and O’Regan, Donal and Almeida, Ricardo

Mathematics

MDPI

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.

ria.ua.pt | doi | Peer Reviewed

7.  Synchronization of Caputo fractional neural networks with bounded time variable delays

Almeida, Ricardo and Hristova, Snezhana and Tersian, Stepan

Open Mathematics

De Gruyter Open

One of the main problems connected with neural networks is synchronization. We examine a model of a neural network with time-varying delay and also the case when the connection weights (the influential strength of the jth neuron to the ith neuron) are variable in time and unbounded. The rate of change of the dynamics of all neurons is described by the Caputo fractional derivative. We apply Lyapunov functions and the Razumikhin method to obtain some sufficient conditions to ensure synchronization in the model. These sufficient conditions are explicitly expressed in terms of the parameters of the system, and hence, they are easily verifiable. We illustrate our theory with a particular nonlinear neural network.

ria.ua.pt | doi | Peer Reviewed

6.  On systems of fractional differential equations with the ψ‐Caputo derivative and their applications

Almeida, Ricardo and Malinowska, Agnieszka B. and Odzijewicz, Tatiana

Mathematical Methods in the Applied Sciences

Wiley

Systems of fractional differential equations with a general form of fractional derivative are considered. A unique continuous solution is derived using the Banach fixed point theorem. Additionally, the dependence of the solution on the fractional order and on the initial conditions are studied. Then the stability of autonomous linear fractional differential systems with order 0<α<1 of the ψ-Caputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leader-follower consensus for fractional multi-agent systems is presented. Sufficient conditions are given to ensure that the tracking errors asymptotically converge to zero. The results of the paper are illustrated by some examples.

ria.ua.pt | doi | Peer Reviewed

5.  Global stability condition for the disease-free equilibrium point of fractional epidemiological models

Almeida, Ricardo and Martins, Natália and Silva, Cristiana J.

Axioms

MDPI

In this paper, we present a new result that allows for studying the global stability of the disease-free equilibrium point when the basic reproduction number is less than 1, in the fractional calculus context. The method only involves basic linear algebra and can be easily applied to study global asymptotic stability. After proving some auxiliary lemmas involving the Mittag–Leffler function, we present the main result of the paper. Under some assumptions, we prove that the disease-free equilibrium point of a fractional differential system is globally asymptotically stable. We then exemplify the procedure with some epidemiological models: a fractional-order SEIR model with classical incidence function, a fractional-order SIRS model with a general incidence function, and a fractional-order model for HIV/AIDS.

ria.ua.pt | doi | Peer Reviewed

4.  Evacuation by leader-follower model with bounded confidence and predictive mechanisms

Almeida, Ricardo and Girejko, Ewa and Machado, Luís and Malinowska, Agnieszka B. and Martins, Natália

Archives of Control Sciences

Polskiej Akademii Nauk

This paper studies an evacuation problem described by a leader-follower model with bounded confidence under predictive mechanisms. We design a control strategy in such a way that agents are guided by a leader, which follows the evacuation path. The proposed evacuation algorithm is based on Model Predictive Control (MPC) that uses the current and the past information of the system to predict future agents’ behaviors. It can be observed that, with MPC method, the leader-following consensus is obtained faster in comparison to the conventional optimal control technique. The effectiveness of the developed MPC evacuation algorithm with respect to different parameters and different time domains is illustrated by numerical examples.

ria.ua.pt | doi | Peer Reviewed

3.  Non-instantaneous impulsive fractional differential equations with state dependent delay and practical stability

Agarwal, Ravi and Almeida, Ricardo and Hristova, Snezhana and O’Regan, Donal

Acta Mathematica Scientia

Springer

Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay, depending on both the time and the state variable. The case when the lower limit of the Caputo fractional derivative is fixed at the initial time, and the case when the lower limit of the fractional derivative is changed at the end of each interval of action of the impulse are studied. Practical stability properties, based on the modified Razumikhin method are investigated. Several examples are given in this paper to illustrate the results.

ria.ua.pt | doi | Peer Reviewed

2.  Numerical solution of variable-order fractional differential equations using Bernoulli polynomials

Nemati, Somayeh and Lima, Pedro M. and Torres, Delfim F. M.

Fractal and Fractional

MDPI

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann–Liouville integral operator was used to give approximations for the unknown function and its variable-order derivatives. An operational matrix of variable-order fractional integration was introduced for the Bernoulli functions. By assuming that the solution of the problem is sufficiently smooth, we approximated a given order of its derivative using Bernoulli polynomials. Then, we used the introduced operational matrix to find some approximations for the unknown function and its derivatives. Using these approximations and some collocation points, the problem was reduced to the solution of a system of nonlinear algebraic equations. An error estimate is given for the approximate solution obtained by the proposed method. Finally, five illustrative examples were considered to demonstrate the applicability and high accuracy of the proposed technique, comparing our results with the ones obtained by existing methods in the literature and making clear the novelty of the work. The numerical results showed that the new method is efficient, giving high-accuracy approximate solutions even with a small number of basis functions and when the solution to the problem is not infinitely differentiable, providing better results and a smaller number of basis functions when compared to state-of-the-art methods.

ria.ua.pt | doi | Peer Reviewed

1.  A discrete-time compartmental epidemiological model for COVID-19 with a case study for Portugal

Vaz, Sandra and Torres, Delfim F. M.

Axioms

MDPI

Recently, a continuous-time compartmental mathematical model for the spread of the Coronavirus disease 2019 (COVID-19) was presented with Portugal as case study, from 2 March to 4 May 2020, and the local stability of the Disease Free Equilibrium (DFE) was analysed. Here, we propose an analogous discrete-time model and, using a suitable Lyapunov function, we prove the global stability of the DFE point. Using COVID-19 real data, we show, through numerical simulations, the consistence of the obtained theoretical results.

ria.ua.pt | doi | Peer Reviewed
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