Books
47.
Programação matemática
Torres, Delfim Fernando Marado
UA Editora
O termo "programação matemática" referese 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 TimorLeste, 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, procurandose estimular o desenvolvimento do raciocínio, essencial numa qualquer atividade profissional.
ria.ua.pt
46.
Analysis of infectious disease problems (Covid19) 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 countrywise analysis of Covid19. 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 integrodifferential equations with optimization methods. Probability distribution and their biomathematical applications have also been studied. This book is a valuable resource for researchers, scholars, biomathematicians and medical experts.
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Book Chapters
45.
Statespace 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 statespace 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.
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44.
Optimal control of vaccination and plasma transfusion with potential usefulness for Covid19
Couras, Juliana and Area, Iván and Nieto, Juan J. and Silva, Cristiana J. and Torres, Delfim F. M.
Analysis of infectious disease problems (Covid19) 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 Covid19 pandemic.
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Peer Reviewed
43.
Modeling the spread of Covid19 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 (Covid19) and their global impact
Springer
Nowadays, coronavirus disease 2019 (Covid19) 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 Covid19. Here, we propose a delayed mathematical model to predict the epidemiological trend of Covid19 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.
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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 padic expansion of the tobedecoded 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 paritycheck 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.
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Peer Reviewed
41.
Fractional model of COVID19 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 COVID19
disease is proposed. Special focus has been done on the transmissibility of
superspreaders 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.
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Peer Reviewed
40.
Stability analysis and optimal control of a fractional HIVAIDS 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 HIVAIDS 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.
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Peer Reviewed
39.
Numerical solution of a class of thirdkind 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 thirdkind 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.
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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.
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Peer Reviewed
37.
Modeling and forecasting of COVID19 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 (COVID19) pneumonia has posed a great threat to the world recent months by causing many deaths and enormous economic damage worldwide. The first case of COVID19 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 wellknown SIR compartmental model to deterministic and stochastic timedelayed models in order to predict the epidemiological trend of COVID19 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 wellposedness of the models and conditions under which the COVID19 may become extinct or persist in the population. Parameter values have been estimated from real data and numerical simulations are presented for forecasting the COVID19 spreading as well as verification of theoretical results.
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Peer Reviewed
36.
Optimal control of the COVID19 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 FonsecaPinto, Rui and Passadouro, Rui and Santos, Estevão Soares dos and Abreu, Wilson and Mira, Jorge
Scientific Reports
Nature Research
The COVID19 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
COVID19 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.
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Peer Reviewed
35.
Control of COVID19 dynamics through a fractionalorder 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 SARSCoV2 virus transmission. Two controls are
considered in our model for eradication of the spread of COVID19: 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.
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Peer Reviewed
34.
Analysis of Hilfer fractional integrodifferential 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 integrodifferential 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.
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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 multiagent
model. The Caputo fractional derivative, in the equations describing the dynamics of a consensus parameter, makes it possible to take into account in the selforganization 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.
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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 timevarying 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.
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Peer Reviewed
31.
Optimal leaderfollowing 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 fractionalorder derivative determines the impact of the memory during the opinion evolution. The problem of leaderfollowing 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.
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Peer Reviewed
30.
Optimality conditions for variational problems involving distributedorder 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: distributedorder 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.
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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 nonlinear 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.
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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 rightsided
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.
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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 selfaware 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 selfaware 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 Hopfbifurcation when time delays
cross critical values. Optimal control theory has been applied for the
costeffectiveness of the delayed system. Numerical simulations illustrate the
obtained analytical results.
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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 wellknown 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.
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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.
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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.
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Peer Reviewed
23.
A dynamicallyconsistent 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.
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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.
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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 pestfree 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 pestfree and coexistence equilibria is shown using the Routh–Hurwitz criterion. Moreover, we prove that when a threshold value is less than one, then the pestfree 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.
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Peer Reviewed
20.
Mathematical analysis of a fractional COVID19 model applied to Wuhan, Spain and Portugal
Ndaïrou, Faïçal and Torres, Delfim F. M.
Axioms
MDPI
We propose a qualitative analysis of a recent fractionalorder COVID19 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.
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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 padic expansion of w and particular representations of the paritycheck 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 paritycheck 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.
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Peer Reviewed
18.
Minimal statespace representation of convolutional product codes
Climent, JoanJosep and Napp, Diego and Pinto, Raquel and Requena, Verónica
Mathematics
MDPI
In this paper, we study product convolutional codes described by statespace representations. In particular, we investigate how to derive statespace 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.
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Peer Reviewed
17.
Variational problems with time delay and higherorder distributedorder 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 higherorder distributedorder fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributedorder 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 wellknown results can be obtained as particular cases.
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Peer Reviewed
16.
State realizations of 2periodic convolutional codes: a switching system approach
Fornasini, Ettore and Napp, Diego and Pereira, Ricardo and Pinto, Raquel and Rocha, Paula
IFACPapersOnLine
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 statespace realizations of 2periodic convolutional codes. Although one cannot expect, in general, to obtain a periodic statespace realization of a periodic convolutional code by means of the individual realizations of each of the associated timeinvariant 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.
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Peer Reviewed
15.
On a nonNewtonian 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 nonNewtonian 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 (nonNewtonian) 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 nonNewtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a firstorder 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.
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Peer Reviewed
14.
Pontryagin maximum principle for distributedorder fractional systems
Ndaïrou, Faïçal and Torres, Delfim F. M.
Mathematics
MDPI
We consider distributedorder nonlocal 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 needlelike 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 distributedorder fractional optimal control problems, illustrating its applicability with an example.
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Peer Reviewed
13.
Local existence and uniqueness for a fractional SIRS model with MittagLeffler 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 AtanganaBaleanuCaputo (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 AdamsBashforthMoulton method.
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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.
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11.
Analysis of a COVID19 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 COVID19. By using the free and opensource 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.
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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 posicionaremse 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.
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9.
Hybrid method for simulation of a fractional COVID19 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 COVID19, which continues to be a big source of threat for humanity. Our fractionalorder analysis is carried out using a nonsingular kernel type operator known as the AtanganaBaleanuCaputo (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.
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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 nonlinear 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 monotoneiterative technique.
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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 timevarying 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.
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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 ψCaputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leaderfollower consensus for fractional multiagent 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.
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5.
Global stability condition for the diseasefree 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
diseasefree 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
diseasefree equilibrium point of a fractional differential system is globally asymptotically stable.
We then exemplify the procedure with some epidemiological models: a fractionalorder SEIR model
with classical incidence function, a fractionalorder SIRS model with a general incidence function,
and a fractionalorder model for HIV/AIDS.
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4.
Evacuation by leaderfollower 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 leaderfollower 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 leaderfollowing 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.
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3.
Noninstantaneous 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 noninstantaneous
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.
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2.
Numerical solution of variableorder 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 variableorder fractional differential equations. The variableorder 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 variableorder derivatives. An operational matrix of variableorder 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 highaccuracy 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 stateoftheart methods.
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1.
A discretetime compartmental epidemiological model for COVID19 with a case study for Portugal
Vaz, Sandra and Torres, Delfim F. M.
Axioms
MDPI
Recently, a continuoustime compartmental mathematical model for the spread of the
Coronavirus disease 2019 (COVID19) 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 discretetime model and, using a suitable Lyapunov function, we prove the
global stability of the DFE point. Using COVID19 real data, we show, through numerical simulations,
the consistence of the obtained theoretical results.
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