Publications 2020

Book Chapters

29.  A new mathematical model for the efficiency calculation

Galindro, Aníbal and Santos, Micael and Torres, Delfim F. M. and Marta-Costa, 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 (non-parametric). The literature states this methodology also as the benchmark approach, since the techniques compare the sample upon a chosen “more-efficient” 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. | doi | Peer Reviewed

28.  Parametric identification of the dynamics of inter-sectoral 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


This work is devoted to modelling and identification of the dynamics of the inter-sectoral 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. | doi | Peer Reviewed

27.  An extension of the fractional Gronwall inequality

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

Advances in Non-Integer Order Calculus and Its Applications


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 ψ-fractional derivatives. | doi | Peer Reviewed

26.  A note on controlled invariance for behavioral nD systems

Pereira, Ricardo and Rocha, Paula

Algebraic and Symbolic Computation Methods in Dynamical Systems


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 to-be-controlled 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. | doi | Peer Reviewed

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


We revisit the SICA (Susceptible-Infectious-Chronic-AIDS) 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 stochastic-type SICA models. Two case studies, in Cape Verde and Morocco, are investigated. | doi | Peer Reviewed

24.  A survey on sufficient optimality conditions for delayed optimal control problems

Lemos-Paião, Ana P. and Silva, Cristiana J. and Torres, Delfim F. M.

Mathematical Modelling and Analysis of Infectious Diseases. Studies in Systems, Decision and Control


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 non-delayed problems. Such approach allows to use standard theorems that ensure sufficient optimality conditions for non-delayed optimal control problems. Examples are given with the purpose to illustrate the results. | doi | Peer Reviewed


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


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 second-order and fourth-order 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 forward-backward fourth-order 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. | doi | Peer Reviewed

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. | doi | Peer Reviewed

21.  On Hermite-Hadamard type inequalities for harmonical h-convex interval-valued functions

Dafang Zhao and Tianqing An and Guoju Ye and Torres, Delfim F. M.

Mathematical Inequalities and Applications


We introduce and investigate the concept of harmonical h-convexity for interval-valued functions. Under this new concept, we prove some new Hermite-Hadamard type inequalities for the interval Riemann integral. | doi | Peer Reviewed

20.  Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equations

Hyunsoo Kim and Sakthivel, Rathinasamy and Debbouche, Amar and Torres, Delfim F. M.

Chaos, Solitons and Fractals


In this article, exact traveling wave solutions of a Wick-type stochastic nonlinear Schrödinger equation and of a Wick-type stochastic fractional Regularized Long Wave-Burgers (RLW-Burgers) equation have been obtained by using an improved computational method. Specifically, the Hermite transform is employed for transforming Wick-type 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 Wick-type stochastic fractional partial differential equations. | doi | Peer Reviewed

19.  The stability and stabilization of infinite dimensional Caputo-time fractional differential linear systems

Zitane, Hanaa and Boutoulout, Ali and Torres, Delfim F. M.



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 Mittag-Leffler 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. | doi | Peer Reviewed

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 Marta-Costa, Ana

Discontinuity, Nonlinearity, and Complexity

L&H Scientific Publishing

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

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. | doi | Peer Reviewed

16.  Enlarged controllability and optimal control of sub-diffusion 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. | doi | Peer Reviewed

15.  Mathematical modeling of COVID-19 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


We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders 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 COVID-19 model for the outbreak that occurred in Wuhan, China. | doi | Peer Reviewed

14.  Synchronization and self-organization in complex networks for a tuberculosis model

Silva, Cristiana J. and Cantin, Guillaume

Mathematics in Computer Science


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 non-negative 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/GNU-Linux environment. | doi | Peer Reviewed

13.  On leader-following consensus in multi-agent systems with discrete updates at random times

Almeida, Ricardo and Girejko, Ewa and Hristova, Snezhana and Malinowska, Agnieszka



This paper studies the leader-following consensus problem in continuous-time multi-agent 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 leader-following 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. | doi | Peer Reviewed

12.  Optimal control of aquatic diseases: a case study of Yemen’s cholera outbreak

Lemos-Paião, Ana P. and Silva, Cristiana J. and Torres, Delfim F. M. and Venturino, Ezio

Journal of Optimization Theory and Applications


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 disease-free 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. | doi | Peer Reviewed

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. | Peer Reviewed

10.  Functional differential equations involving the ψ-Caputo fractional derivative

Almeida, Ricardo

Fractal and Fractional


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. | doi | Peer Reviewed

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. | Peer Reviewed

8.  A stochastic fractional calculus with applications to variational principles

Zine, Houssine and Torres, Delfim F. M.

Fractal and Fractional


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 Euler-Lagrange 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. | doi | Peer Reviewed

7.  Lyapunov functions for fractional-order 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


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. | doi | Peer Reviewed

6.  Corrigendum to "Mathematical Modeling of COVID-19 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


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 COVID-19 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. | doi | Peer Reviewed

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 non-conservative 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. | doi | Peer Reviewed

4.  Application of Bernoulli polynomials for solving variable-order fractional optimal control-affine problems

Nemati, Somayeh and Torres, Delfim F. M.



We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann--Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order 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 control-affine system. Using these approximations, and the Gauss--Legendre 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. | doi | Peer Reviewed

3.  Distributed-order non-local optimal control

Ndaïrou, Faïçal and Torres, Delfim F. M.



Distributed-order fractional non-local 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 distributed-order non-local 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 distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions. | doi | Peer Reviewed

2.  Mathematical modeling of Japanese encephalitis under aquatic environmental effects

Ndaïrou, Faïçal and Area, Iván and Torres, Delfim F. M.



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 well-posed and to have a biologically and ecologically meaningful disease-free equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed. | doi | Peer Reviewed

1.  A stochastic time-delayed model for the effectiveness of Moroccan COVID-19 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 (COVID-19) poses a great threat to public health and the economy worldwide. Currently, COVID-19 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 COVID-19 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 COVID-19 in Morocco after the deconfinement. To ensure the well-posedness 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. | doi | Peer Reviewed
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