Book Chapters
29.
A new mathematical model for the efficiency calculation
Galindro, Aníbal and Santos, Micael and Torres, Delfim F. M. and MartaCosta, Ana
Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Studies in Systems, Decision and Control
Springer Nature
During the past sixty years, a lot of effort has been made regarding the
productive efficiency. Such endeavours provided an extensive bibliography on this
subject, culminating in two main methods, named the Stochastic Frontier Analysis
(parametric) and Data Envelopment Analysis (nonparametric). The literature states
this methodology also as the benchmark approach, since the techniques compare
the sample upon a chosen “moreefficient” reference. This article intends to disrupt
such premise, suggesting a mathematical model that relies on the optimal input
combination, provided by a differential equation system instead of an observable
sample. A numerical example is given, illustrating the application of our model’s
features.
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28.
Parametric identification of the dynamics of intersectoral balance: modelling and forecasting
Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.
Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Studies in Systems, Decision and Control
Springer
This work is devoted to modelling and identification of the dynamics of the intersectoral balance of a macroeconomic system. An approach to the problem of specification and identification of a weakly formalized dynamical system is developed. A matching procedure for parameters of a linear stationary Cauchy problem with a decomposition of its upshot trend and a periodic component, is proposed. Moreover, an approach for detection of significant harmonic waves, which are inherent to real macroeconomic dynamical systems, is developed.
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27.
An extension of the fractional Gronwall inequality
Almeida, Ricardo and Malinowska, Agnieszka B. and Odzijewicz, Tatiana
Advances in NonInteger Order Calculus and Its Applications
Springer
In this work, we prove a generalization of the Gronwall type inequality. This relation can be used in the qualitative analysis of the solutions to fractional differential equations with the ψfractional derivatives.
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26.
A note on controlled invariance for behavioral nD systems
Pereira, Ricardo and Rocha, Paula
Algebraic and Symbolic Computation Methods in Dynamical Systems
Springer
In this chapter we extend the notion of invariance of nD behaviors introduced in Pereira and Rocha (European Control Conference 2013, ECC’13. ETH Zurich, Switzerland, pp. 301–305, 2013) [4], Rocha and Wood (Int. J. Appl. Math. Comput. Sci. 7(4):869–879, 1997) [7] to the controlsetting. More concretely, we introduce a notion which is the behavioral counterpart of classical controlled invariance, using the framework of partial interconnections. In such interconnections, the variables are divided into two sets: the variables tobecontrolled and the variables on which it is allowed to enforce restrictions (called control variables). In particular we focus on regular partial interconnection, i.e., interconnections in which the restrictions of the controller do not overlap with the ones already implied by the laws of the original behavior. For some particular cases, complete characterizations of controlled invariance and controller construction procedures are derived for both 1D and nD behaviors.
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25.
On SICA models for HIV transmission
Silva, Cristiana J. and Torres, Delfim F. M.
Mathematical Modelling and Analysis of Infectious Diseases. Studies in Systems, Decision and Control
Springer
We revisit the SICA (SusceptibleInfectiousChronicAIDS) mathematical model for transmission dynamics of the human immunodeficiency virus (HIV) with varying population size in a homogeneously mixing population. We consider SICA models given by systems of ordinary differential equations and some generalizations given by systems with fractional and stochastic differential operators. Local and global stability results are proved for deterministic, fractional, and stochastictype SICA models. Two case studies, in Cape Verde and Morocco, are investigated.
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24.
A survey on sufficient optimality conditions for delayed optimal control problems
LemosPaião, Ana P. and Silva, Cristiana J. and Torres, Delfim F. M.
Mathematical Modelling and Analysis of Infectious Diseases. Studies in Systems, Decision and Control
Springer
The aim of this work is to make a survey on recent sufficient optimality conditions for optimal control problems with time delays in both state and control variables. The results are obtained by transforming delayed optimal control problems into equivalent nondelayed problems. Such approach allows to use standard theorems that ensure sufficient optimality conditions for nondelayed optimal control problems. Examples are given with the purpose to illustrate the results.
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Articles
23.
Numerical optimal control of HIV transmission in Octave/MATLAB
Campos, Carlos and Silva, Cristiana J. and Torres, Delfim F. M.
Mathematical and Computational Applications
MDPI
We provide easy and readable GNU Octave/MATLAB code for the simulation of
mathematical models described by ordinary differential equations and for the solution of optimal
control problems through Pontryagin’s maximum principle. For that, we consider a normalized
HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva,
C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in
Cape Verde. Ecol. Complex. 2017, 30, 70–75), given by a system of four ordinary differential equations.
An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three
standard methods implemented by us in Octave/MATLAB: Euler method and secondorder and
fourthorder Runge–Kutta methods. Afterwards, a control function is introduced into the normalized
HIV model and an optimal control problem is formulated, where the goal is to find the optimal
HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least
HIV new infections and cost associated with the control measures. The optimal control problem is
characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed
numerically by implementing a forwardbackward fourthorder Runge–Kutta method. Complete
algorithms, for both uncontrolled initial value and optimal control problems, developed under the
free GNU Octave software and compatible with MATLAB are provided along the article.
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22.
Regional enlarged observability of Caputo fractional differential equations
Zouiten, Hayat and Boutoulout, Ali and Torres, Delfim F. M.
Discrete and Continuous Dynamical Systems  Series S
American Institute of Mathematical Sciences (AIMS)
We consider the regional enlarged observability problem for fractional
evolution differential equations involving Caputo derivatives. Using the
Hilbert Uniqueness Method, we show that it is possible to rebuild the initial
state between two prescribed functions only in an internal subregion of the
whole domain. Finally, an example is provided to illustrate the theory.
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21.
On HermiteHadamard type inequalities for harmonical hconvex intervalvalued functions
Dafang Zhao and Tianqing An and Guoju Ye and Torres, Delfim F. M.
Mathematical Inequalities and Applications
EleMath
We introduce and investigate the concept of harmonical hconvexity for intervalvalued functions. Under this new concept, we prove some new HermiteHadamard type inequalities for the interval Riemann integral.
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20.
Traveling wave solutions of some important Wicktype fractional stochastic nonlinear partial differential equations
Hyunsoo Kim and Sakthivel, Rathinasamy and Debbouche, Amar and Torres, Delfim F. M.
Chaos, Solitons and Fractals
Elsevier
In this article, exact traveling wave solutions of a Wicktype stochastic
nonlinear Schrödinger equation and of a Wicktype stochastic fractional
Regularized Long WaveBurgers (RLWBurgers) equation have been obtained by
using an improved computational method. Specifically, the Hermite transform is
employed for transforming Wicktype stochastic nonlinear partial differential
equations into deterministic nonlinear partial differential equations with
integral and fraction order. Furthermore, the required set of stochastic
solutions in the white noise space is obtained by using the inverse Hermite
transform. Based on the derived solutions, the dynamics of the considered
equations are performed with some particular values of the physical parameters.
The results reveal that the proposed improved computational technique can be
applied to solve various kinds of Wicktype stochastic fractional partial
differential equations.
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19.
The stability and stabilization of infinite dimensional Caputotime fractional differential linear systems
Zitane, Hanaa and Boutoulout, Ali and Torres, Delfim F. M.
Mathematics
MDPI
We investigate the stability and stabilization concepts for
infinite dimensional time fractional differential linear systems
in Hilbert spaces with Caputo derivatives.
Firstly, based on a family of operators generated by strongly
continuous semigroups and on a probability density function,
we provide sufficient and necessary conditions for the
exponential stability of the considered class of systems.
Then, by assuming that the system dynamics is symmetric
and uniformly elliptic and by using the properties of the
MittagLeffler function, we provide sufficient conditions
that ensure strong stability. Finally, we characterize
an explicit feedback control that guarantees the strong stabilization
of a controlled Caputo time fractional linear system
through a decomposition approach. Some examples are
presented that illustrate the effectiveness of our results.
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18.
A mathematical model for vineyard replacement with nonlinear binary control optimization
Galindro, Aníbal and Cerveira, Adelaide and Torres, Delfim F. M. and Matias, João and MartaCosta, Ana
Discontinuity, Nonlinearity, and Complexity
L&H Scientific Publishing
Vineyard replacement is a common practice in every winegrowing farm
since the grapevine production decays over time and requires a new vine
to ensure the business sustainability. In this paper, we formulate a simple
discrete model that captures the vineyard’s main dynamics such as production values and grape quality. Then, by applying binary nonlinear programming methods to find the vineyard replacement trigger, we seek the optimal solution concerning different governmental subsidies to the target producer.
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17.
Errata to "Modeling and optimal control of HIV/AIDS prevention through PrEP", Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 1, 119–141
Silva, Cristiana J. and Torres, Delfim F. M.
Discrete and Continuous Dynamical Systems  Series S
American Institute of Mathematical Sciences (AIMS)
No abstract available.
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16.
Enlarged controllability and optimal control of subdiffusion processes with Caputo fractional derivatives
Karite, Touria and Boutoulout, Ali and Torres, Delfim F. M.
Progress in Fractional Differentiation and Applications
Natural Sciences Publishing (NSP)
We investigate the exact enlarged controllability and optimal control of a
fractional diffusion equation in Caputo sense. This is done through a new
definition of enlarged controllability that allows us to extend available
contributions. Moreover, the problem is studied using two approaches: a reverse
Hilbert uniqueness method, generalizing the approach introduced by Lions in
1988, and a penalization method, which allow us to characterize the minimum
energy control.
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15.
Mathematical modeling of COVID19 transmission dynamics with a case study of Wuhan
Ndaïrou, Faïçal and Area, Iván and Nieto, Juan J. and Torres, Delfim F. M.
Chaos, Solitons and Fractals
Elsevier
We propose a compartmental mathematical model for the spread of the COVID19 disease with special focus on the transmissibility of superspreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID19 model for the outbreak that occurred in Wuhan, China.
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14.
Synchronization and selforganization in complex networks for a tuberculosis model
Silva, Cristiana J. and Cantin, Guillaume
Mathematics in Computer Science
Springer
In this work, we propose and analyze the dynamics of a complex network built with non identical instances of a tuberculosis (TB) epidemiological model, for which we prove the existence of nonnegative and bounded global solutions. A two nodes network is analyzed where the nodes represent the TB epidemiological situation of the countries Angola and Portugal. We analyze the effect of different coupling and intensity of migratory movements between the two countries and explore the effect of seasonal migrations. For a random complex network setting, we show that it is possible to reach a synchronization state by increasing the coupling strength and test the influence of the topology in the dynamics of the complex network. All the results are analyzed through numerical simulations where the given algorithms are implemented with the python 3.5 language, in a Debian/GNULinux environment.
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13.
On leaderfollowing consensus in multiagent systems with discrete updates at random times
Almeida, Ricardo and Girejko, Ewa and Hristova, Snezhana and Malinowska, Agnieszka
Entropy
MDPI
This paper studies the leaderfollowing consensus problem in continuoustime multiagent networks with communications/updates occurring only at random times. The time between two consecutive controller updates is exponentially distributed. Some sufficient conditions are derived to design the control law that ensures the leaderfollowing consensus is asymptotically reached (in the sense of the expected value of a stochastic process). The numerical examples are worked out to demonstrate the effectiveness of our theoretical results.
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12.
Optimal control of aquatic diseases: a case study of Yemen’s cholera outbreak
LemosPaião, Ana P. and Silva, Cristiana J. and Torres, Delfim F. M. and Venturino, Ezio
Journal of Optimization Theory and Applications
Springer
We propose a mathematical model for the transmission dynamics of some strains of the
bacterium Vibrio cholerae, responsible for the cholera disease in humans. We prove
that, when the basic reproduction number is equal to one, a transcritical bifurcation
occurs for which the endemic equilibrium emanates from the diseasefree point. A
control function is introduced into the model, representing the distribution of chlorine
water tablets for water purification. An optimal control problem is then proposed and
analyzed, where the goal is to determine the fraction of susceptible individuals who
should have access to chlorine water tablets in order to minimize the total number
of new infections plus the total cost associated with the distribution of chlorine water
tablets, over the considered period of time. Finally, we consider real data of the cholera
outbreak in Yemen, from April 27, 2017 to April 15, 2018, choosing the values of the
parameters of the uncontrolled model that fit the real data. Using our optimal control
results, we show, numerically, that the distribution of chlorine water tablets could
have stopped, in a fast way, the worst cholera outbreak that ever occurred in human
history. Due to the critical situation of Yemen, we also simulate the case where only
a small percentage of susceptible individuals has access to chlorine water tablets and
obtain an optimal control solution that decreases, substantially, the maximum number
of infective individuals affected by the outbreak.
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11.
Constructions of MDS convolutional codes using superregular matrices
Lieb, Julia and Pinto, Raquel
Journal of Algebra Combinatorics Discrete Structures and Applications
Jacodesmath Institute
Maximum distance separable convolutional codes are the codes that present best performance in
error correction among all convolutional codes with certain rate and degree. In this paper, we show
that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular
matrix, we obtain a column reduced generator matrix of an MDS convolutional code with a certain
rate and a certain degree. We then present two novel constructions that fulfill these conditions by
considering two types of superregular matrices.
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10.
Functional differential equations involving the ψCaputo fractional derivative
Almeida, Ricardo
Fractal and Fractional
MDPI
This paper is devoted to the study of existence and uniqueness of solutions for fractional
functional differential equations, whose derivative operator depends on an arbitrary function.
The introduction of such function allows generalization of some known results, and others can
be also obtained.
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9.
Dynamical analysis of a fractional SIR model with treatment and quarantine
Almeida, Ricardo
Chaotic Modeling and Simulation
We propose a fractional SIR model with treatment and quarantine policies, whose dynamics is described by the Caputo fractional derivative. Local stability of the equilibrium points is studied, and the threshold value R0 is found. Finally, some numerical simulations are presented for different values of the parameters.
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8.
A stochastic fractional calculus with applications to variational principles
Zine, Houssine and Torres, Delfim F. M.
Fractal and Fractional
MDPI
We introduce a stochastic fractional calculus.
As an application, we present a stochastic fractional calculus
of variations, which generalizes the fractional calculus
of variations to stochastic processes. A stochastic fractional
EulerLagrange equation is obtained, extending those available
in the literature for the classical, fractional,
and stochastic calculus of variations. To illustrate our main
theoretical result, we discuss two examples: one derived
from quantum mechanics, the second validated
by an adequate numerical simulation.
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7.
Lyapunov functions for fractionalorder systems in biology: methods and applications
Boukhouima, Adnane and Hattaf, Khalid and Lotfi, El Mehdi and Mahrouf, Marouane and Torres, Delfim F. M. and Yousfi, Noura
Chaos, Solitons & Fractals
Elsevier
We prove new estimates of the Caputo derivative of order α ∈ (0, 1] for some specific functions. The estimations are shown useful to construct Lyapunov functions for systems of fractional differential equations in biology, based on those known for ordinary differential equations, and therefore useful to determine the global stability of the equilibrium points for fractional systems. To illustrate the usefulness of our theoretical results, a fractional HIV population model and a fractional cellular model are studied. More precisely, we construct suitable Lyapunov functionals to demonstrate the global stability of the free and endemic equilibriums, for both fractional models, and we also perform some numerical simulations that confirm our choices.
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6.
Corrigendum to "Mathematical Modeling of COVID19 Transmission Dynamics with a Case Study of Wuhan" [Chaos Solitons Fractals 135 (2020), 109846]
Ndaïrou, Faïçal and Area, Iván and Bader, Georg and Nieto, Juan J. and Torres, Delfim F. M.
Chaos, Solitons and Fractals
Elsevier
We correct some numerical results of [Chaos Solitons Fractals 135 (2020), 109846], by providing the correct numbers and plots. The conclusions of the paper remain, however, the same. In particular, the numerical simulations show the suitability of the proposed COVID19 model for the outbreak that occurred in Wuhan, China. This time all our computer codes are provided, in order to make all computations reproducible. The authors would like to apologize for any inconvenience caused.
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5.
Fractional variational principle of Herglotz for a new class of problems with dependence on the boundaries and a real parameter
Almeida, Ricardo and Martins, Natália
Journal of Mathematical Physics
American Institute of Physics
The fractional variational problem of Herglotz type for the case where the Lagrangian depends on generalized fractional derivatives, the free endpoints conditions, and a real parameter is studied. This type of problem generalizes several problems recently studied in the literature. Moreover, it allows us to unify conservative and nonconservative dynamical processes in the same model. The dependence of the Lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard Herglotz’s variational problem into a problem of a different nature.
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4.
Application of Bernoulli polynomials for solving variableorder fractional optimal controlaffine problems
Nemati, Somayeh and Torres, Delfim F. M.
Axioms
MDPI
We propose two efficient numerical approaches for solving variableorder
fractional optimal controlaffine problems. The variableorder fractional
derivative is considered in the Caputo sense, which together with the
RiemannLiouville integral operator is used in our new techniques.
An accurate operational matrix of variableorder fractional integration
for Bernoulli polynomials is introduced. Our methods proceed as follows.
First, a specific approximation of the differentiation order of the
state function is considered, in terms of Bernoulli polynomials.
Such approximation, together with the initial conditions,
help us to obtain some approximations for the other existing
functions in the dynamical controlaffine system. Using these
approximations, and the GaussLegendre integration formula,
the problem is reduced to a system of nonlinear algebraic equations.
Some error bounds are then given for the approximate optimal state
and control functions, which allow us to obtain an error bound
for the approximate value of the performance index. We end by
solving some test problems, which demonstrate the high accuracy
of our results.
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3.
Distributedorder nonlocal optimal control
Ndaïrou, Faïçal and Torres, Delfim F. M.
Axioms
MDPI
Distributedorder fractional nonlocal operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributedorder nonlocal derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributedorder fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.
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2.
Mathematical modeling of Japanese encephalitis under aquatic environmental effects
Ndaïrou, Faïçal and Area, Iván and Torres, Delfim F. M.
Mathematics
MDPI
We propose a mathematical model for the spread of Japanese encephalitis with emphasis on the environmental effects on the aquatic phase of mosquitoes. The model is shown to be biologically wellposed and to have a biologically and ecologically meaningful diseasefree equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed.
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1.
A stochastic timedelayed model for the effectiveness of Moroccan COVID19 deconfinement strategy
Zine, Houssine and Boukhouima, Adnane and Lotfi, El Mehdi and Mahrouf, Marouane and Torres, Delfim F. M. and Yousfi, Noura
Mathematical Modelling of Natural Phenomena
EDP Sciences
Coronavirus disease 2019 (COVID19) poses a great threat to public health and
the economy worldwide. Currently, COVID19 evolves in many countries to a
second stage, characterized by the need for the liberation of the economy and
relaxation of the human psychological effects. To this end, numerous countries
decided to implement adequate deconfinement strategies. After the first
prolongation of the established confinement, Morocco moves to the deconfinement
stage on May 20, 2020. The relevant question concerns the impact on the
COVID19 propagation by considering an additional degree of realism related to
stochastic noises due to the effectiveness level of the adapted measures. In
this paper, we propose a delayed stochastic mathematical model to predict the
epidemiological trend of COVID19 in Morocco after the deconfinement. To ensure
the wellposedness of the model, we prove the existence and uniqueness of a
positive solution. Based on the large number theorem for martingales, we
discuss the extinction of the disease under an appropriate threshold parameter.
Moreover, numerical simulations are performed in order to test the efficiency
of the deconfinement strategies chosen by the Moroccan authorities to help the
policy makers and public health administration to make suitable decisions in
the near future.
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