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Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed...
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Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 July 2022) | Viewed by 38128

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Natália Martins

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Guest Editor
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: time scale calculus; fractional calculus; calculus of variations; optimal control; consensus theory
Special Issues, Collections and Topics in MDPI journals
Dr. Ricardo Almeida

E-Mail Website1 Website2
Guest Editor
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: fractional calculus; calculus of variations
Special Issues, Collections and Topics in MDPI journals
Dr. Cristiana João Soares da Silva

E-Mail Website
Guest Editor
1. Department of Mathematics, ISTA, Iscte-Instituto Universitário de Lisboa, Av. das Forças Armadas, 1649-026 Lisboa, Portugal
2. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Interests: optimal control; calculus of variations; optimization; mathematical models for infectious diseases; application of optimal control to epidemiology; ordinary differential equations; stability; qualitative investigation and simulation of models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Moulay Rchid Sidi Ammi

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AMNEA Group, Department of Mathematics, Faculty of Sciences and Technics, Moulay Ismail University, B.P. 509, Errachidia 52000, Morocco
Interests: PDEs; fractional calculus; time scale theory; epidemiology

Special Issue Information

Dear Colleagues,

This Special Issue of Axioms is dedicated to Professor Delfim F. M. Torres on the occasion of his 50th birthday, as recognition of his significant contribution in the field. Professor Torres is a Distinguished University Professor, a Highly Cited Researcher in Mathematics (2015, 2016, 2017 and 2019), and Lifetime Member of The American Mathematical Society. He is one of the founders of fractional variational analysis and fractional optimal control, and has made tremendous contributions to the theory of variational analysis with applications to many other fields such as optimization, optimal control, time-scale analysis, and mathematical epidemiology and biology. Professor Delfim F. M. Torres is the recipient of several international awards, including the 325 Years of Fractional Calculus Award, in Testimony of the High Regard for His Achievements in the Area of Fractional Calculus and Its Applications, is included in the World's Top 2% Scientists 2020 by Stanford University, both as a Career-long Highly Cited Researcher and as a single calendar year Highly Cited Researcher, was considered by Thomson Reuters as one of the World's Most Influential Scientific Minds, won a Publons Peer Review Award as a world's top peer reviewer, was recognized in the top 1% of reviewers, a Sentinel of Science Award, and so on. Many of his research works have been considered top papers in the area, served as research highlights, and been awarded prizes. Besides being a great scholar, he is also a great teacher who has already educated twenty Ph.D. students from all over the world. 

This Special Issue covers many of Professor Torres' research interests, which include several areas of pure and applied mathematical sciences, such as approximations and expansions, biology and other natural sciences, calculus of variations and optimal control, optimization, difference and functional equations, fluid mechanics, functional analysis, game theory, economics, social and behavioral sciences, general measure and integration, mechanics of deformable solids, number theory, numerical analysis, operations research, mathematical programming, ordinary differential equations, partial differential equations, quantum theory, real functions, systems and control theory, fractional calculus and its applications, integral equations and transforms, higher transcendental functions and their applications, q-series and q-polynomials, inventory modeling and optimization, dynamic equations on time scales, and mathematical modeling. 

Short Biography 

Delfim Fernando Marado Torres was born 16 August 1971 in Nampula, Mozambique. He is, since March 2015, a Full Professor of Mathematics at University of Aveiro (UA), Director of the R&D Unit CIDMA, the largest Portuguese research center in Mathematics, and Coordinator of its Systems and Control Group. He obtained a PhD in Mathematics from UA in 2002, and Habilitation in Mathematics, UA, in 2011. His main research area is calculus of variations and optimal control; optimization; fractional derivatives and integrals; dynamic equations on time scales; and mathematical biology. Torres has written outstanding scientific and pedagogical publications. In particular, he has co-authored two books with Imperial College Press and three books with Springer. He has strong experience in graduate and post-graduate student supervision and teaching in mathematics. Moreover, he has been team leader and member in several national and international R&D projects, including EU projects and networks. He is, since 2013, the Director of the Doctoral Programme Consortium in Mathematics and Applications (MAP-PDMA) of Universities of Minho, Aveiro, and Porto. Prof. Torres is married since 2003, and has one daughter and two sons.

Prof. Dr. Natália Martins
Prof. Dr. Ricardo Almeida
Dr. Cristiana João Soares da Silva
Prof. Dr. Moulay Rchid Sidi Ammi
Guest Editors

Manuscript Submission Information

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Keywords

  • Optimization
  • Fractional calculus
  • Systems biology
  • Optimal control
  • Calculus of variations
  • Time scales calculus
  • Numerical mathematics
  • Mathematical modeling
  • Dynamical systems
  • Nonlinear analysis
  • Applied mathematics

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Published Papers (16 papers)

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Editorial

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5 pages, 192 KiB  
Editorial
Editorial for the Special Issue of Axioms “Calculus of Variations, Optimal Control and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday”
by Natália Martins, Ricardo Almeida, Cristiana J. Silva and Moulay Rchid Sidi Ammi
Axioms 2023, 12(2), 110; https://doi.org/10.3390/axioms12020110 - 20 Jan 2023
Viewed by 1454
Abstract
This publication is an editorial for the Special Issue of Axioms “Calculus of Variations, Optimal Control and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F [...] Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)

Research

Jump to: Editorial

33 pages, 1026 KiB  
Article
A Note on an Epidemic Model with Cautionary Response in the Presence of Asymptomatic Individuals
by Francesca Acotto and Ezio Venturino
Axioms 2023, 12(1), 62; https://doi.org/10.3390/axioms12010062 - 6 Jan 2023
Cited by 1 | Viewed by 1539
Abstract
We analyse a simple disease transmission model accounting for demographic features and an illness appearing in two forms, asymptomatic and symptomatic. Its main feature is the epidemic-induced fear of the population, for which contacts are reduced, responding to increasing symptomatic numbers. We find [...] Read more.
We analyse a simple disease transmission model accounting for demographic features and an illness appearing in two forms, asymptomatic and symptomatic. Its main feature is the epidemic-induced fear of the population, for which contacts are reduced, responding to increasing symptomatic numbers. We find that in the presence of asymptomatic individuals, if the progression rate to symptomatic is high, protection measures may prevent the whole population becoming infected. The results also elucidate the importance of assessing transmission rates as quickly as possible. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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14 pages, 311 KiB  
Article
Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces
by Zoubida Ech-chaffani, Ahmed Aberqi, Touria Karite and Delfim F. M. Torres
Axioms 2022, 11(8), 379; https://doi.org/10.3390/axioms11080379 - 31 Jul 2022
Cited by 1 | Viewed by 1354
Abstract
We investigate a class of fractional neutral evolution equations on Banach spaces involving Caputo derivatives. Main results establish conditions for the controllability of the fractional-order system and conditions for existence of a solution to an optimal control problem of minimum energy. The results [...] Read more.
We investigate a class of fractional neutral evolution equations on Banach spaces involving Caputo derivatives. Main results establish conditions for the controllability of the fractional-order system and conditions for existence of a solution to an optimal control problem of minimum energy. The results are proved with the help of fixed-point and semigroup theories. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
24 pages, 4156 KiB  
Article
Fractional Dynamics of a Measles Epidemic Model
by Hamadjam Abboubakar, Rubin Fandio, Brandon Satsa Sofack and Henri Paul Ekobena Fouda
Axioms 2022, 11(8), 363; https://doi.org/10.3390/axioms11080363 - 26 Jul 2022
Cited by 25 | Viewed by 2131
Abstract
In this work, we replaced the integer derivative with Caputo derivative to model the transmission dynamics of measles in an epidemic situation. We began by recalling some results on the local and global stability of the measles-free equilibrium point as well as the [...] Read more.
In this work, we replaced the integer derivative with Caputo derivative to model the transmission dynamics of measles in an epidemic situation. We began by recalling some results on the local and global stability of the measles-free equilibrium point as well as the local stability of the endemic equilibrium point. We computed the basic reproduction number of the fractional model and found that is it equal to the one in the integer model when the fractional order ν = 1. We then performed a sensitivity analysis using the global method. Indeed, we computed the partial rank correlation coefficient (PRCC) between each model parameter and the basic reproduction number R0 as well as each variable state. We then demonstrated that the fractional model admits a unique solution and that it is globally stable using the Ulam–Hyers stability criterion. Simulations using the Adams-type predictor–corrector iterative scheme were conducted to validate our theoretical results and to see the impact of the variation of the fractional order on the quantitative disease dynamics. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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10 pages, 477 KiB  
Article
Local Structure of Convex Surfaces near Regular and Conical Points
by Alexander Plakhov
Axioms 2022, 11(8), 356; https://doi.org/10.3390/axioms11080356 - 23 Jul 2022
Cited by 2 | Viewed by 1626
Abstract
Consider a point on a convex surface in Rd, d≥2 and a plane of support Π to the surface at this point. Draw a plane parallel with Π cutting a part of the surface. We study the limiting behavior [...] Read more.
Consider a point on a convex surface in Rd, d≥2 and a plane of support Π to the surface at this point. Draw a plane parallel with Π cutting a part of the surface. We study the limiting behavior of this part of the surface when the plane approaches the point, being always parallel with Π. More precisely, we study the limiting behavior of the normalized surface area measure in Sd−1 induced by this part of the surface. In this paper, we consider two cases: (a) when the point is regular and (b) when it is singular conical, that is the tangent cone at the point does not contain straight lines. In Case (a), the limit is the atom located at the outward normal vector to Π, and in Case (b), the limit is equal to the measure induced by the part of the tangent cone cut off by a plane. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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19 pages, 650 KiB  
Article
Pattern Formation Induced by Fuzzy Fractional-Order Model of COVID-19
by Abeer S. Alnahdi, Ramsha Shafqat, Azmat Ullah Khan Niazi and Mdi Begum Jeelani
Axioms 2022, 11(7), 313; https://doi.org/10.3390/axioms11070313 - 27 Jun 2022
Cited by 21 | Viewed by 1977
Abstract
A novel coronavirus infection system is established for the analytical and computational aspects of this study, using a fuzzy fractional evolution equation (FFEE) stated in Caputo’s sense for order (1,2). It is constructed using the FFEE formulated in Caputo’s meaning. The model consist [...] Read more.
A novel coronavirus infection system is established for the analytical and computational aspects of this study, using a fuzzy fractional evolution equation (FFEE) stated in Caputo’s sense for order (1,2). It is constructed using the FFEE formulated in Caputo’s meaning. The model consist of six components illustrating the coronavirus outbreak, involving the susceptible people Kℓ(ω), the exposed population Lℓ(ω), total infected strength Cℓ(ω), asymptotically infected population Mℓ(ω), total number of humans recovered Eℓ(ω), and reservoir Qℓ(ω). Numerical results using the fuzzy Laplace approach in combination with the Adomian decomposition transform are developed to better understand the dynamical structures of the physical behavior of COVID-19. For the controlling model, such behavior on the generic characteristics of RNA in COVID-19 is also examined. The findings show that the proposed technique of addressing the uncertainty issue in a pandemic situation is effective. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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12 pages, 283 KiB  
Article
Existence Results for a Multipoint Fractional Boundary Value Problem in the Fractional Derivative Banach Space
by Djalal Boucenna, Amar Chidouh and Delfim F. M. Torres
Axioms 2022, 11(6), 295; https://doi.org/10.3390/axioms11060295 - 16 Jun 2022
Viewed by 1717
Abstract
We study a class of nonlinear implicit fractional differential equations subject to nonlocal boundary conditions expressed in terms of nonlinear integro-differential equations. Using the Krasnosel’skii fixed-point theorem we prove, via the Kolmogorov–Riesz criteria, the existence of solutions. The existence results are established in [...] Read more.
We study a class of nonlinear implicit fractional differential equations subject to nonlocal boundary conditions expressed in terms of nonlinear integro-differential equations. Using the Krasnosel’skii fixed-point theorem we prove, via the Kolmogorov–Riesz criteria, the existence of solutions. The existence results are established in a specific fractional derivative Banach space and they are illustrated by two numerical examples. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
10 pages, 266 KiB  
Article
Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation
by Houssine Zine, El Mehdi Lotfi, Delfim F. M. Torres and Noura Yousfi
Axioms 2022, 11(4), 178; https://doi.org/10.3390/axioms11040178 - 15 Apr 2022
Cited by 2 | Viewed by 2098
Abstract
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with [...] Read more.
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
19 pages, 647 KiB  
Article
Modeling the Impact of the Imperfect Vaccination of the COVID-19 with Optimal Containment Strategy
by Lahbib Benahmadi, Mustapha Lhous, Abdessamad Tridane, Omar Zakary and Mostafa Rachik
Axioms 2022, 11(3), 124; https://doi.org/10.3390/axioms11030124 - 10 Mar 2022
Cited by 3 | Viewed by 2339
Abstract
Since the beginning of the COVID-19 pandemic, vaccination has been the main strategy to contain the spread of the coronavirus. However, with the administration of many types of vaccines and the constant mutation of viruses, the issue of how effective these vaccines are [...] Read more.
Since the beginning of the COVID-19 pandemic, vaccination has been the main strategy to contain the spread of the coronavirus. However, with the administration of many types of vaccines and the constant mutation of viruses, the issue of how effective these vaccines are in protecting the population is raised. This work aimed to present a mathematical model that investigates the imperfect vaccine and finds the additional measures needed to help reduce the burden of disease. We determine the R0 threshold of disease spread and use stability analysis to determine the condition that will result in disease eradication. We also fitted our model to COVID-19 data from Morocco to estimate the parameters of the model. The sensitivity analysis of the basic reproduction number, with respect to the parameters of the model, is simulated for the four possible scenarios of the disease progress. Finally, we investigate the optimal containment measures that could be implemented with vaccination. To illustrate our results, we perform the numerical simulations of optimal control. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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27 pages, 706 KiB  
Article
Determining COVID-19 Dynamics Using Physics Informed Neural Networks
by Joseph Malinzi, Simanga Gwebu and Sandile Motsa
Axioms 2022, 11(3), 121; https://doi.org/10.3390/axioms11030121 - 10 Mar 2022
Cited by 7 | Viewed by 3992
Abstract
The Physics Informed Neural Networks framework is applied to the understanding of the dynamics of COVID-19. To provide the governing system of equations used by the framework, the Susceptible–Infected–Recovered–Death mathematical model is used. This study focused on finding the patterns of the dynamics [...] Read more.
The Physics Informed Neural Networks framework is applied to the understanding of the dynamics of COVID-19. To provide the governing system of equations used by the framework, the Susceptible–Infected–Recovered–Death mathematical model is used. This study focused on finding the patterns of the dynamics of the disease which involves predicting the infection rate, recovery rate and death rate; thus, predicting the active infections, total recovered, susceptible and deceased at any required time. The study used data that were collected on the dynamics of COVID-19 from the Kingdom of Eswatini between March 2020 and September 2021. The obtained results could be used for making future forecasts on COVID-19 in Eswatini. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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15 pages, 327 KiB  
Article
Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints
by Aram V. Arutyunov, Dmitry Yu. Karamzin and Fernando Lobo Pereira
Axioms 2022, 11(2), 40; https://doi.org/10.3390/axioms11020040 - 20 Jan 2022
Cited by 2 | Viewed by 2227
Abstract
This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based [...] Read more.
This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an ε-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
31 pages, 486 KiB  
Article
Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces
by Antonio Leaci and Franco Tomarelli
Axioms 2022, 11(1), 30; https://doi.org/10.3390/axioms11010030 - 14 Jan 2022
Cited by 3 | Viewed by 2677
Abstract
We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional [...] Read more.
We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional bounded variation spaces of fractional order s, denoted by BVs(a,b). Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
17 pages, 1137 KiB  
Article
Hybrid Method for Simulation of a Fractional COVID-19 Model with Real Case Application
by Anwarud Din, Amir Khan, Anwar Zeb, Moulay Rchid Sidi Ammi, Mouhcine Tilioua and Delfim F. M. Torres
Axioms 2021, 10(4), 290; https://doi.org/10.3390/axioms10040290 - 1 Nov 2021
Cited by 12 | Viewed by 2357
Abstract
In this research, we provide a mathematical analysis for the novel coronavirus responsible for COVID-19, which continues to be a big source of threat for humanity. Our fractional-order analysis is carried out using a non-singular kernel type operator known as the Atangana-Baleanu-Caputo (ABC) [...] Read more.
In this research, we provide a mathematical analysis for the novel coronavirus responsible for COVID-19, which continues to be a big source of threat for humanity. Our fractional-order analysis is carried out using a non-singular kernel type operator known as the Atangana-Baleanu-Caputo (ABC) derivative. We parametrize the model adopting available information of the disease from Pakistan in the period 9 April to 2 June 2020. We obtain the required solution with the help of a hybrid method, which is a combination of the decomposition method and the Laplace transform. Furthermore, a sensitivity analysis is carried out to evaluate the parameters that are more sensitive to the basic reproduction number of the model. Our results are compared with the real data of Pakistan and numerical plots are presented at various fractional orders. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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16 pages, 332 KiB  
Article
On Periodic Fractional (p, q)-Integral Boundary Value Problems for Sequential Fractional (p, q)-Integrodifference Equations
by Jarunee Soontharanon and Thanin Sitthiwirattham
Axioms 2021, 10(4), 264; https://doi.org/10.3390/axioms10040264 - 19 Oct 2021
Cited by 1 | Viewed by 1396
Abstract
We study the existence results of a fractional (p, q)-integrodifference equation with periodic fractional (p, q)-integral boundary condition by using Banach and Schauder’s fixed point theorems. Some properties of (p, q)-integral are also presented [...] Read more.
We study the existence results of a fractional (p, q)-integrodifference equation with periodic fractional (p, q)-integral boundary condition by using Banach and Schauder’s fixed point theorems. Some properties of (p, q)-integral are also presented in this paper as a tool for our calculations. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
17 pages, 1101 KiB  
Article
Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models
by Ricardo Almeida, Natália Martins and Cristiana J. Silva
Axioms 2021, 10(4), 238; https://doi.org/10.3390/axioms10040238 - 25 Sep 2021
Cited by 5 | Viewed by 2897
Abstract
In this paper, we present a new result that allows for studying the global stability of the disease-free equilibrium point when the basic reproduction number is less than 1, in the fractional calculus context. The method only involves basic linear algebra and can [...] Read more.
In this paper, we present a new result that allows for studying the global stability of the disease-free equilibrium point when the basic reproduction number is less than 1, in the fractional calculus context. The method only involves basic linear algebra and can be easily applied to study global asymptotic stability. After proving some auxiliary lemmas involving the Mittag–Leffler function, we present the main result of the paper. Under some assumptions, we prove that the disease-free equilibrium point of a fractional differential system is globally asymptotically stable. We then exemplify the procedure with some epidemiological models: a fractional-order SEIR model with classical incidence function, a fractional-order SIRS model with a general incidence function, and a fractional-order model for HIV/AIDS. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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15 pages, 597 KiB  
Article
On a Non-Newtonian Calculus of Variations
by Delfim F. M. Torres
Axioms 2021, 10(3), 171; https://doi.org/10.3390/axioms10030171 - 29 Jul 2021
Cited by 11 | Viewed by 3091
Abstract
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. [...] Read more.
The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given. Full article
(This article belongs to the Special Issue Calculus of Variations, Optimal Control, and Mathematical Biology: A Themed Issue Dedicated to Professor Delfim F. M. Torres on the Occasion of His 50th Birthday)
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