@article{diethelm2002predictor,
  title={A predictor-corrector approach for the numerical solution of fractional differential equations},
  author={Diethelm, Kai and Ford, Neville J. and Freed, Alan D.},
  journal={Nonlinear Dyn.},
  volume={29},
  number={1-4},
  pages={3--22},
  year={2002},
  doi={10.1023/A:1016592219341},
  publisher={Springer}
}


@article{morin2012,
  title={A Quick Guide to Software Licensing for the Scientist-Programmer},
  author={Morin, A and Urban, J and Sliz P},
  journal={{PLoS Computational Biology}},
  volume={8},
  number={7},
  pages={e1002598},
  year={2012},
  doi={10.1371/journal.pcbi.1002598}}

@article{diethelm2004detailed,
  title={Detailed error analysis for a fractional {Adams} method},
  author={Diethelm, Kai and Ford, Neville J. and Freed, Alan D.},
  journal={Numer. Algorithms},
  volume={36},
  number={1},
  pages={31--52},
  year={2004},
  doi={10.1023/B:NUMA.0000027736.85078.be},
  publisher={Springer}
}

@Article{de2014review,
author={de Oliveira, Edmundo Capelas
and Tenreiro Machado, Jos{\'e} Ant{\'o}nio},
title={A Review of Definitions for Fractional Derivatives and Integral},
journal={Mathematical Problems in Engineering},
year={2014},
month={Jun},
day={10},
publisher={Hindawi Publishing Corporation},
volume={2014},
pages={238459},
abstract={This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.},
issn={1024-123X},
doi={10.1155/2014/238459},
url={https://doi.org/10.1155/2014/238459}
}


@article{garrappa2010linear,
  title={On linear stability of predictor--corrector algorithms for fractional differential equations},
  author={Garrappa, Roberto},
  journal={International Journal of Computer Mathematics},
  volume={87},
  number={10},
  pages={2281--2290},
  year={2010},
  doi={10.1080/00207160802624331},
  publisher={Taylor \& Francis}
}

@book{kilbas2006theory,
  title={Theory and applications of fractional differential equations},
  author={Kilbas, Anatoly Aleksandrovich and Srivastava, Hari M and Trujillo, Juan J},
  volume={204},
  year={2006},
  publisher={Elsevier}
}

@book{podlubny1998fractional,
  title={Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications},
  author={Podlubny, Igor},
  ISBN= {9780080531984},
  year={1998},
  publisher={Elsevier}
}

@article{KhalighiPlosCB,
    doi = {10.1371/journal.pcbi.1009396},
    author = {Khalighi, Moein AND Sommeria-Klein, Guilhem AND Gonze, Didier AND Faust, Karoline AND Lahti, Leo},
    journal = {PLOS Computational Biology},
    publisher = {Public Library of Science},
    title = {Quantifying the impact of ecological memory on the dynamics of interacting communities},
    year = {2022},
    month = {06},
    volume = {18},
    url = {https://doi.org/10.1371/journal.pcbi.1009396},
    pages = {1-21},
    abstract = {Ecological memory refers to the influence of past events on the response of an ecosystem to exogenous or endogenous changes. Memory has been widely recognized as a key contributor to the dynamics of ecosystems and other complex systems, yet quantitative community models often ignore memory and its implications. Recent modeling studies have shown how interactions between community members can lead to the emergence of resilience and multistability under environmental perturbations. We demonstrate how memory can be introduced in such models using the framework of fractional calculus. We study how the dynamics of a well-characterized interaction model is affected by gradual increases in ecological memory under varying initial conditions, perturbations, and stochasticity. Our results highlight the implications of memory on several key aspects of community dynamics. In general, memory introduces inertia into the dynamics. This favors species coexistence under perturbation, enhances system resistance to state shifts, mitigates hysteresis, and can affect system resilience both ways depending on the time scale considered. Memory also promotes long transient dynamics, such as long-standing oscillations and delayed regime shifts, and contributes to the emergence and persistence of alternative stable states. Our study highlights the fundamental role of memory in communities, and provides quantitative tools to introduce it in ecological models and analyse its 
    impact under varying conditions.},
    number = {6},

}

@Article{Garrappa2018,
AUTHOR = {Garrappa, Roberto},
TITLE = {Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial},
JOURNAL = {Mathematics},
VOLUME = {6},
YEAR = {2018},
NUMBER = {2},
ARTICLE-NUMBER = {16},
URL = {https://www.mdpi.com/2227-7390/6/2/16},
ISSN = {2227-7390},
ABSTRACT = {Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.},
DOI = {10.3390/math6020016}
}


@article{HarmonicMotion,
title = {A comparative study of the fractional oscillators},
journal = {Alexandria Engineering Journal},
volume = {59},
number = {4},
pages = {2649-2676},
year = {2020},
note = {New trends of numerical and analytical methods for engineering problems},
issn = {1110-0168},
doi = {https://doi.org/10.1016/j.aej.2020.04.029},
url = {https://www.sciencedirect.com/science/article/pii/S1110016820301794},
author = {Azhar Ali Zafar and Grzegorz Kudra and Jan Awrejcewicz and Thabet Abdeljawad and Muhammad Bilal Riaz},
keywords = {Linear oscillator, Fractional derivative, Power law kernel, Non-singular kernel, Phase portraits},
abstract = {In this work, we have investigated the fractional differential equation to describe the motion of a linear oscillator using fractional derivative operators with or without singular kernels. In order to be consistent with the physical systems the value of the fractional parameter that characterizes the existence of fractional structures in the system, lies within unit interval. The solutions of the non-integer order differential equation are obtained and expressed in terms of generalized G function depending upon the fractional parameter. The classical cases could be recovered by making the limit of fractional parameter approaches to unity. Moreover, we will analyse and compare the behaviour of the oscillator with different definitions of the fractional operators via graphical illustrations, phase portraits, and Poincare maps.}
}

@article{Saeedian2017PhysRevE,
  title = {Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model},
  author = {Saeedian, Meghdad and Khalighi, Moein and Azimi-Tafreshi, Nahid and Jafari, Gholamreza and Ausloos, Marcel},
  journal = {Physical Review E},
  volume = {95},
  issue = {2},
  pages = {022409},
  numpages = {9},
  year = {2017},
  doi={10.1103/PhysRevE.95.022409},
  publisher = {American Physical Society}
}

@article{Kermack,
 title={A contribution to the mathematical theory of epidemics},
 author={Kermack, William Ogilvy and McKendrick, Anderson G.},
 journal={Proceedings of the royal society of london. Series A, Containing papers of a mathematical and physical character},
 volume={115},
 number={772},
 pages={700--721},
 year={1927},
 publisher={The Royal Society London}
}

URL = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1927.0118},
eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1927.0118}
,
    abstract = { (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection. }
}



@Article{KhalighiSymmetry,
AUTHOR = {Khalighi, Moein and Eftekhari, Leila and Hosseinpour, Soleiman and Lahti, Leo},
TITLE = {Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators},
JOURNAL = {Symmetry},
VOLUME = {13},
YEAR = {2021},
NUMBER = {3},
ARTICLE-NUMBER = {368},
URL = {https://www.mdpi.com/2073-8994/13/3/368},
ISSN = {2073-8994},
ABSTRACT = {In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.},
DOI = {10.3390/sym13030368}
}


@software{SciFracx,
  author       = {Qingyu Qu and
                  Jadon Clugston},
  title        = {SciFracX/FractionalDiffEq.jl: v0.2.11},
  month        = aug,
  year         = 2022,
  publisher    = {Zenodo},
  version      = {v0.2.11},
  doi          = {10.5281/zenodo.7026703},
  url          = {https://doi.org/10.5281/zenodo.7026703}
}


@article{Gokhale2021PlosCB,
    doi = {10.1371/journal.pcbi.1009431},
    author = {Gokhale, Chaitanya S. AND Giaimo, Stefano AND Remigi, Philippe},
    journal = {PLOS Computational Biology},
    publisher = {Public Library of Science},
    title = {Memory shapes microbial populations},
    year = {2021},
    volume = {17},
    url = {https://doi.org/10.1371/journal.pcbi.1009431},
    pages = {1-16},
    abstract = {Correct decision making is fundamental for all living organisms to thrive under environmental changes. The patterns of environmental variation and the quality of available information define the most favourable strategy among multiple options, from randomly adopting a phenotypic state to sensing and reacting to environmental cues. Cellular memory—the ability to track and condition the time to switch to a different phenotypic state—can help withstand environmental fluctuations. How does memory manifest itself in unicellular organisms? We describe the population-wide consequences of phenotypic memory in microbes through a combination of deterministic modelling and stochastic simulations. Moving beyond binary switching models, our work highlights the need to consider a broader range of switching behaviours when describing microbial adaptive strategies. We show that memory in individual cells generates patterns at the population level coherent with overshoots and non-exponential lag times distributions experimentally observed in phenotypically heterogeneous populations. We emphasise the implications of our work in understanding antibiotic tolerance and, in general, bacterial survival under fluctuating environments.},
    number = {10},

}

@Article{Power2015,
author={Power, Daniel A.
and Watson, Richard A.
and Szathm{\'a}ry, E{\"o}rs
and Mills, Rob
and Powers, Simon T.
and Doncaster, C. Patrick
and Czapp, B},
title={What can ecosystems learn? Expanding evolutionary ecology with learning theory},
journal={Biology Direct},
year={2015},
volume={10},
number={1},
pages={69},
abstract={The structure and organisation of ecological interactions within an ecosystem is modified by the evolution and coevolution of the individual species it contains. Understanding how historical conditions have shaped this architecture is vital for understanding system responses to change at scales from the microbial upwards. However, in the absence of a group selection process, the collective behaviours and ecosystem functions exhibited by the whole community cannot be organised or adapted in a Darwinian sense. A long-standing open question thus persists: Are there alternative organising principles that enable us to understand and predict how the coevolution of the component species creates and maintains complex collective behaviours exhibited by the ecosystem as a whole?},
issn={1745-6150},
doi={10.1186/s13062-015-0094-1}
}


@article {simsek2019PNAS,
	author = {{\c S}im{\c s}ek, Emrah and Kim, Minsu},
	title = {Power-law tail in lag time distribution underlies bacterial persistence},
	volume = {116},
	number = {36},
	pages = {17635--17640},
	year = {2019},
	doi = {10.1073/pnas.1903836116},
	publisher = {National Academy of Sciences},
	abstract = {Persisters are antibiotic-tolerant cells that can evade antibiotic killing by maintaining long lag phase. They complicate antibiotic treatment, leading to treatment failure. Extensive studies in the field found that a myriad of molecular mechanisms leads to persisters. However, we still do not quantitatively understand, nor can we predict, the population dynamics of persistence, e.g., how the number of persisters changes over time. To address this issue, we characterized the lag time distribution of persisters. We observed a power-law decay in the tail of the distribution. We showed that this feature can quantitatively account for the population dynamics of persistence without ad hoc parameters. Additionally, we showed how this simple quantitative feature can emerge due to complicated molecular mechanisms.Genetically identical microbial cells respond to stress heterogeneously, and this phenotypic heterogeneity contributes to population survival. Quantitative analysis of phenotypic heterogeneity can reveal dynamic features of stochastic mechanisms that generate heterogeneity. Additionally, it can enable a priori prediction of population dynamics, elucidating microbial survival strategies. Here, we quantitatively analyzed the persistence of an Escherichia coli population. When a population is confronted with antibiotics, a majority of cells is killed but a subpopulation called persisters survives the treatment. Previous studies have found that persisters survive antibiotic treatment by maintaining a long period of lag phase. When we quantified the lag time distribution of E. coli cells in a large dynamic range, we found that normal cells rejuvenated with a lag time distribution that is well captured by an exponential decay [exp(-kt)], agreeing with previous studies. This exponential decay indicates that their rejuvenation is governed by a single rate constant kinetics (i.e., k is constant). Interestingly, the lag time distribution of persisters exhibited a long tail captured by a power-law decay. Using a simple quantitative argument, we demonstrated that this power-law decay can be explained by a wide variation of the rate constant k. Additionally, by developing a mathematical model based on this biphasic lag time distribution, we quantitatively explained the complex population dynamics of persistence without any ad hoc parameters. The quantitative features of persistence demonstrated in our work shed insights into molecular mechanisms of persistence and advance our knowledge of how a microbial population evades antibiotic treatment.},
	issn = {0027-8424},
	URL = {https://www.pnas.org/content/116/36/17635},
	eprint = {https://www.pnas.org/content/116/36/17635.full.pdf},
	journal = {PNAS}
}


 
@article{gonze2017multi,
  title={Multi-stability and the origin of microbial community types},
  author={Gonze, Didier and Lahti, Leo and Raes, Jeroen and Faust, Karoline},
  journal={The ISME Journal},
  volume={11},
  number={10},
  pages={2159},
  year={2017},
  doi= {10.1038/ismej.2017.60},
  publisher={Nature Publishing Group}
}

@article{Pathogen,
 title={Pathogen.jl: Infectious Disease Transmission Network Modeling with Julia},
 volume={104},
 url={https://www.jstatsoft.org/index.php/jss/article/view/v104i04},
 doi={10.18637/jss.v104.i04},
 abstract={&amp;lt;p&amp;gt;We introduce Pathogen.jl for simulation and inference of transmission network individual level models (TN-ILMs) of infectious disease spread in continuous time. TN-ILMs can be used to jointly infer transmission networks, event times, and model parameters within a Bayesian framework via Markov chain Monte Carlo (MCMC). We detail our specific strategies for conducting MCMC for TN-ILMs, and our implementation of these strategies in the Julia package, Pathogen.jl, which leverages key features of the Julia language. We provide an example using Pathogen.jl to simulate an epidemic following a susceptible-infectious-removed (SIR) TN-ILM, and then perform inference using observations that were generated from that epidemic. We also demonstrate the functionality of Pathogen.jl with an application of TN-ILMs to data from a measles outbreak that occurred in Hagelloch, Germany, in 1861 (Pfeilsticker 1863; Oesterle 1992).&amp;lt;/p&amp;gt;},
 number={4},
 journal={Journal of Statistical Software},
 author={Angevaare, Justin and Feng, Zeny and Deardon, Rob},
 year={2022},
 pages={1–30}
}

@article{JAHANSHAHI,
title = {A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19},
journal = {Chaos, Solitons \& Fractals},
volume = {143},
pages = {110632},
year = {2021},
issn = {0960-0779},
doi = {https://doi.org/10.1016/j.chaos.2020.110632},
url = {https://www.sciencedirect.com/science/article/pii/S0960077920310237},
author = {Hadi Jahanshahi and Jesus M. Munoz-Pacheco and Stelios Bekiros and Naif D. Alotaibi},
keywords = {COVID-19, Fractional-order, SIRD model, Variable memory, Multifractional},
abstract = {COVID-19 is a novel coronavirus affecting all the world since December last year. Up to date, the spread of the outbreak continues to complicate our lives, and therefore, several research efforts from many scientific areas are proposed. Among them, mathematical models are an excellent way to understand and predict the epidemic outbreaks evolution to some extent. Due to the COVID-19 may be modeled as a non-Markovian process that follows power-law scaling features, we present a fractional-order SIRD (Susceptible-Infected-Recovered-Dead) model based on the Caputo derivative for incorporating the memory effects (long and short) in the outbreak progress. Additionally, we analyze the experimental time series of 23 countries using fractal formalism. Like previous works, we identify that the COVID-19 evolution shows various power-law exponents (no just a single one) and share some universality among geographical regions. Hence, we incorporate numerous memory indexes in the proposed model, i.e., distinct fractional-orders defined by a time-dependent function that permits us to set specific memory contributions during the evolution. This allows controlling the memory effects of more early states, e.g., before and after a quarantine decree, which could be less relevant than the contribution of more recent ones on the current state of the SIRD system. We also prove our model with Italy’s real data from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University.}
}

@article{Covid_Model_Main,
title = {Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan},
journal = {Chaos, Solitons \& Fractals},
volume = {135},
pages = {109846},
year = {2020},
issn = {0960-0779},
doi = {https://doi.org/10.1016/j.chaos.2020.109846},
url = {https://www.sciencedirect.com/science/article/pii/S0960077920302460},
author = {Faïçal Ndaïrou and Iván Area and Juan J. Nieto and Delfim F.M. Torres},
keywords = {Mathematical modeling of COVID-19 pandemic, Wuhan case study, Basic reproduction number, Stability, Sensitivity analysis, Numerical simulations},
abstract = {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.}
}


@Article{Covid-Model-Frac-Analysis,
AUTHOR = {Ndaïrou, Faïçal and Torres, Delfim F. M.},
TITLE = {Mathematical Analysis of a Fractional COVID-19 Model Applied to Wuhan, Spain and Portugal},
JOURNAL = {Axioms},
VOLUME = {10},
YEAR = {2021},
NUMBER = {3},
ARTICLE-NUMBER = {135},
URL = {https://www.mdpi.com/2075-1680/10/3/135},
ISSN = {2075-1680},
ABSTRACT = {We propose a qualitative analysis of a recent fractional-order COVID-19 model. We start by showing that the model is mathematically and biologically well posed. Then, we give a proof on the global stability of the disease free equilibrium point. Finally, some numerical simulations are performed to ensure stability and convergence of the disease free equilibrium point.},
DOI = {10.3390/axioms10030135}
}


@article{Covid-Model-Fractional,
title = {Fractional model of COVID-19 applied to Galicia, Spain and Portugal},
journal = {Chaos, Solitons \& Fractals},
volume = {144},
pages = {110652},
year = {2021},
issn = {0960-0779},
doi = {https://doi.org/10.1016/j.chaos.2021.110652},
url = {https://www.sciencedirect.com/science/article/pii/S0960077921000059},
author = {Faïçal Ndaïrou and Iván Area and Juan J. Nieto and Cristiana J. Silva and Delfim F.M. Torres},
keywords = {Mathematical modelling of COVID-19 pandemic, Galicia, Spain and Portugal case studies, Fractional differential equations, Numerical simulations},
abstract = {A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models.}
}

@article{Data-Covid,
title = {An interactive web-based dashboard to track COVID-19 in real time},
journal = {The Lancet Infectious Diseases},
volume = {20},
number = {5},
pages = {533-534},
year = {2020},
issn = {1473-3099},
doi = {https://doi.org/10.1016/S1473-3099(20)30120-1},
url = {https://www.sciencedirect.com/science/article/pii/S1473309920301201},
author = {Ensheng Dong and Hongru Du and Lauren Gardner}
}

@software{optim,
  author       = {Patrick Kofod Mogensen and
                  John Myles White and
                  Asbjørn Nilsen Riseth and
                  Tim Holy and
                  Miles Lubin and
                  Christof Stocker and
                  Andreas Noack and
                  Antoine Levitt and
                  Christoph Ortner and
                  Blake Johnson and
                  Dahua Lin and
                  Kristoffer Carlsson and
                  Yichao Yu and
                  Christopher Rackauckas and
                  Josua Grawitter and
                  Alex Williams and
                  Alexey Stukalov and
                  Ben Kuhn and
                  Benoît Legat and
                  Jeffrey Regier and
                  cossio and
                  Michael Creel and
                  Ron Rock and
                  Thomas R. Covert and
                  Benoit Pasquier and
                  Takafumi Arakaki and
                  Andrew Clausen and
                  Arno Strouwen},
  title        = {JuliaNLSolvers/Optim.jl: Release 1.7.3},
  month        = sep,
  year         = 2022,
  publisher    = {Zenodo},
  version      = {v1.7.3},
  doi          = {10.5281/zenodo.7068265},
  url          = {https://doi.org/10.5281/zenodo.7068265}
}

@Article{Diethelm2020Nonsingular,
author={Diethelm, Kai
and Garrappa, Roberto
and Giusti, Andrea
and Stynes, Martin},
title={Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used},
journal={Fractional Calculus and Applied Analysis},
year={2020},
month={Jun},
day={01},
volume={23},
number={3},
pages={610-634},
abstract={In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.},
issn={1314-2224},
doi={10.1515/fca-2020-0032},
url={https://doi.org/10.1515/fca-2020-0032}
}

@article{Matlab-Review-FDEs,
author = {Zhuo Li and Lu Liu and Sina Dehghan and YangQuan Chen and Dingyü Xue},
title = {A review and evaluation of numerical tools for fractional calculus and fractional order controls},
journal = {International Journal of Control},
volume = {90},
number = {6},
pages = {1165-1181},
year  = {2017},
publisher = {Taylor & Francis},
doi = {10.1080/00207179.2015.1124290},

URL = { 
        https://doi.org/10.1080/00207179.2015.1124290
    
},
eprint = { 
        https://doi.org/10.1080/00207179.2015.1124290
    
}

}




@article{Bezanson_Julia,
author = {Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B.},
title = {Julia: A Fresh Approach to Numerical Computing},
journal = {SIAM Review},
volume = {59},
number = {1},
pages = {65-98},
year = {2017},
doi = {10.1137/141000671},

URL = { 
        https://doi.org/10.1137/141000671
    
},
eprint = { 
        https://doi.org/10.1137/141000671
    
}
,
    abstract = { Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast and questions notions generally held to be “laws of nature" by practitioners of numerical computing: \beginlist \item High-level dynamic programs have to be slow. \item One must prototype in one language and then rewrite in another language for speed or deployment. \item There are parts of a system appropriate for the programmer, and other parts that are best left untouched as they have been built by the experts. \endlist We introduce the Julia programming language and its design---a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, which is what good computation is really about, recognizes what remains the same after differences are stripped away. Abstractions in mathematics are captured as code through another technique from computer science, generic programming. Julia shows that one can achieve machine performance without sacrificing human convenience. }
}


@article{GARRAPPA201596,
title = {Trapezoidal methods for fractional differential equations: Theoretical and computational aspects},
journal = {Mathematics and Computers in Simulation},
volume = {110},
pages = {96-112},
year = {2015},
note = {7th edition of the workshop "Structural Dynamical Systems: computational aspects},
issn = {0378-4754},
doi = {https://doi.org/10.1016/j.matcom.2013.09.012},
url = {https://www.sciencedirect.com/science/article/pii/S0378475413002802},
author = {Roberto Garrappa},
keywords = {Fractional differential equations, Multistep methods, Trapezoidal method, Stability, Computational aspects},
abstract = {The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation.}
}

@Article{Diethelm-FFT,
AUTHOR = {Diethelm, Kai and Garrappa, Roberto and Stynes, Martin},
TITLE = {Good (and Not So Good) Practices in Computational Methods for Fractional Calculus},
JOURNAL = {Mathematics},
VOLUME = {8},
YEAR = {2020},
NUMBER = {3},
ARTICLE-NUMBER = {324},
URL = {https://www.mdpi.com/2227-7390/8/3/324},
ISSN = {2227-7390},
ABSTRACT = {The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results.},
DOI = {10.3390/math8030324}
}




@ARTICLE{Cuda,
  author={Besard, Tim and Foket, Christophe and De Sutter, Bjorn},
  journal={IEEE Transactions on Parallel and Distributed Systems}, 
  title={Effective Extensible Programming: Unleashing Julia on GPUs}, 
  year={2019},
  volume={30},
  number={4},
  pages={827-841},
  doi={10.1109/TPDS.2018.2872064}}

@article{DifferentialEquations,
  title={Differentialequations. jl--a performant and feature-rich ecosystem for solving differential equations in julia},
  author={Rackauckas, Christopher and Nie, Qing},
  journal={Journal of open research software},
  volume={5},
  number={1},
  year={2017},
  publisher={Ubiquity Press}
}

@misc{DiffEqFlux,
  doi = {10.48550/ARXIV.1902.02376},
  
  url = {https://arxiv.org/abs/1902.02376},
  
  author = {Rackauckas, Chris and Innes, Mike and Ma, Yingbo and Bettencourt, Jesse and White, Lyndon and Dixit, Vaibhav},
  
  keywords = {Machine Learning (cs.LG), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences},
  
  title = {DiffEqFlux.jl - A Julia Library for Neural Differential Equations},
  
  publisher = {arXiv},
  
  year = {2019},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}



@InProceedings{Turing,
  title = 	 {Turing: A Language for Flexible Probabilistic Inference},
  author = 	 {Ge, Hong and Xu, Kai and Ghahramani, Zoubin},
  booktitle = 	 {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics},
  pages = 	 {1682--1690},
  year = 	 {2018},
  editor = 	 {Storkey, Amos and Perez-Cruz, Fernando},
  volume = 	 {84},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {09--11 Apr},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v84/ge18b/ge18b.pdf},
  url = 	 {https://proceedings.mlr.press/v84/ge18b.html},
  abstract = 	 {Probabilistic programming promises to simplify and democratize probabilistic machine learning, but successful probabilistic programming systems require flexible, generic and efficient inference engines. In this work, we present a system called Turing for building MCMC algorithms for probabilistic programming inference. Turing has a very simple syntax and makes full use of the numerical capabilities in the Julia programming language, including all implemented probability distributions, and automatic differentiation. Turing supports a wide range of popular Monte Carlo algorithms, including Hamiltonian Monte Carlo (HMC), HMC with No-U-Turns (NUTS), Gibbs sampling, sequential Monte Carlo (SMC), and several particle MCMC (PMCMC) samplers. Most importantly, Turing inference is composable: it combines MCMC operations on subsets of variables, for example using a combination of an HMC engine and a particle Gibbs (PG) engine.  We explore several combinations of inference methods with the aim of finding approaches that are both efficient and universal, i.e. applicable to arbitrary probabilistic models. NUTS—a popular variant of HMC that adapts Hamiltonian simulation path length automatically, although quite powerful for exploring differentiable target distributions, is however not universal. We identify some failure modes for the NUTS engine, and demonstrate that composition of PG (for discrete variables) and NUTS (for continuous variables) can be useful when the NUTS engine is either not applicable, or simply does not work well. Our aim is to present Turing and its composable inference engines to the world and encourage other researchers to build on this system to help advance the field of probabilistic machine learning. }
}


@article{FractCovid-Hadamard,
author = {Cai,Min  and Em Karniadakis,George  and Li,Changpin },
title = {Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
volume = {32},
number = {7},
pages = {071101},
year = {2022},
doi = {10.1063/5.0099450},

URL = { 
        https://doi.org/10.1063/5.0099450
    
},
eprint = { 
        https://doi.org/10.1063/5.0099450
    
}

}


@article{matlob2019concepts,
  title={The concepts and applications of fractional order differential calculus in modeling of viscoelastic systems: a primer},
  author={Matlob, Mohammad Amirian and Jamali, Yousef},
  journal={Critical Reviews™ in Biomedical Engineering},
  volume={47},
  number={4},
  year={2019},
  publisher={Begel House Inc.}
}

@Article{Eftekhari2022,
author={Eftekhari, Leila
and Amirian, Mohammad M.},
title={Stability analysis of fractional order memristor synapse-coupled hopfield neural network with ring structure},
journal={Cognitive Neurodynamics},
year={2022},
month={Aug},
day={27},
abstract={A memristor is a nonlinear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To enhance the memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of n sub-network neurons, increasing the synchronization in the network. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the n-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.},
issn={1871-4099},
doi={10.1007/s11571-022-09844-9},
url={https://doi.org/10.1007/s11571-022-09844-9}
}

@article{SUN-reviewApplication,
title = {A new collection of real world applications of fractional calculus in science and engineering},
journal = {Communications in Nonlinear Science and Numerical Simulation},
volume = {64},
pages = {213-231},
year = {2018},
issn = {1007-5704},
doi = {https://doi.org/10.1016/j.cnsns.2018.04.019},
url = {https://www.sciencedirect.com/science/article/pii/S1007570418301308},
author = {HongGuang Sun and Yong Zhang and Dumitru Baleanu and Wen Chen and YangQuan Chen},
keywords = {Fractional calculus, Fractional differential equations, Anomalous diffusion, Visoelasticity, Control, Signal and image processing, Dynamic systems, Biology, Macroeconomic models, Environmental science},
abstract = {Fractional calculus is at this stage an arena where many models are still to be introduced, discussed and applied to real world applications in many branches of science and engineering where nonlocality plays a crucial role. Although researchers have already reported many excellent results in several seminal monographs and review articles, there are still a large number of non-local phenomena unexplored and waiting to be discovered. Therefore, year by year, we can discover new aspects of the fractional modeling and applications. This review article aims to present some short summaries written by distinguished researchers in the field of fractional calculus. We believe this incomplete, but important, information will guide young researchers and help newcomers to see some of the main real-world applications and gain an understanding of this powerful mathematical tool. We expect this collection will also benefit our community.}
}

@article{loverro2004fractional,
  title={Fractional calculus: history, definitions and applications for the engineer},
  author={Loverro, Adam and others},
  journal={Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering},
  pages={1--28},
  year={2004}
}


@article{AMIRIAN2020,
title = {Memory and mutualism in species sustainability: A time-fractional Lotka-Volterra model with harvesting},
journal = {Heliyon},
volume = {6},
number = {9},
pages = {e04816},
year = {2020},
issn = {2405-8440},
doi = {https://doi.org/10.1016/j.heliyon.2020.e04816},
url = {https://www.sciencedirect.com/science/article/pii/S2405844020316595},
author = {Mohammad M. Amirian and Isaac N. Towers and Zlatko Jovanoski and Andrew J. Irwin},
keywords = {Population modelling, Fractional calculus, Lotka-Volterra, Memory, Stability and harvesting, Mutualistic predation},
abstract = {We first present a predator-prey model for two species and then extend the model to three species where the two predator species engage in mutualistic predation. Constant effort harvesting and the impact of by-catch issue are also incorporated. Necessary sufficient conditions for the existence and stability of positive equilibrium points are examined. It is shown that harvesting is sustainable, and the memory concept of the fractional derivative damps out oscillations in the population numbers so that the system as a whole settles on an equilibrium quicker than it would with integer time derivatives. Finally, some possible physical explanations are given for the obtained results. It is shown that the stability requires the memory concept in the model.}
}

@inbook{FOTF-Toolbox,
url = {https://doi.org/10.1515/9783110571745-011},
title = {FOTF Toolbox for fractional-order control systems},
booktitle = {Volume 6 Applications in Control},
author = {Dingyü Xue, Ivo Petráš},
publisher = {De Gruyter},
address = {Berlin, Boston},
pages = {237--266},
doi = {doi:10.1515/9783110571745-011},
isbn = {9783110571745},
year = {2019},
lastchecked = {2022-11-22}
}

@article{Khalighi_BEM,
author = {Khalighi, Moein and Amirianmatlob, Mohammad and Malek, Alaeddin},
title = {A new approach to solving multiorder time-fractional advection–diffusion–reaction equations using BEM and Chebyshev matrix},
journal = {Mathematical Methods in the Applied Sciences},
volume = {44},
number = {4},
pages = {2964-2984},
keywords = {boundary element method, Chebyshev operational matrix, multiorder fractional differential equations},
doi = {https://doi.org/10.1002/mma.6352},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.6352},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/mma.6352},
abstract = {In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two-dimensional multiorder time-fractional partial differential equations: nonlinear and linear in respect to spatial and temporal variables, respectively. Fractional derivatives are estimated by Caputo sense. Boundary element method is used to convert the main problem into a system of a multiorder fractional ordinary differential equation. Then, the produced system is approximated by Chebyshev operational matrix technique, and its condition number is analyzed. Accuracy and efficiency of the proposed hybrid scheme are demonstrated by solving three different types of two-dimensional time-fractional convection–diffusion equations numerically. The convergent rates are calculated for different meshing within the boundary element technique. Numerical results are given by graphs and tables for solutions and different type of error norms.},
year = {2021}
}


@article{Safdari_plosone,
    doi = {10.1371/journal.pone.0154983},
    author = {Safdari, Hadiseh AND Zare Kamali, Milad AND Shirazi, Amirhossein AND Khalighi, Moein AND Jafari, Gholamreza AND Ausloos, Marcel},
    journal = {PLOS ONE},
    publisher = {Public Library of Science},
    title = {Fractional Dynamics of Network Growth Constrained by Aging Node Interactions},
    year = {2016},
    month = {05},
    volume = {11},
    url = {https://doi.org/10.1371/journal.pone.0154983},
    pages = {1-13},
    abstract = {In many social complex systems, in which agents are linked by non-linear interactions, the history of events strongly influences the whole network dynamics. However, a class of “commonly accepted beliefs” seems rarely studied. In this paper, we examine how the growth process of a (social) network is influenced by past circumstances. In order to tackle this cause, we simply modify the well known preferential attachment mechanism by imposing a time dependent kernel function in the network evolution equation. This approach leads to a fractional order Barabási-Albert (BA) differential equation, generalizing the BA model. Our results show that, with passing time, an aging process is observed for the network dynamics. The aging process leads to a decay for the node degree values, thereby creating an opposing process to the preferential attachment mechanism. On one hand, based on the preferential attachment mechanism, nodes with a high degree are more likely to absorb links; but, on the other hand, a node’s age has a reduced chance for new connections. This competitive scenario allows an increased chance for younger members to become a hub. Simulations of such a network growth with aging constraint confirm the results found from solving the fractional BA equation. We also report, as an exemplary application, an investigation of the collaboration network between Hollywood movie actors. It is undubiously shown that a decay in the dynamics of their collaboration rate is found, even including a sex difference. Such findings suggest a widely universal application of the so generalized BA model.},
    number = {5},

}


@article{wilson2017good,
  title={Good enough practices in scientific computing},
  author={Wilson, Greg and Bryan, Jennifer and Cranston, Karen and Kitzes, Justin and Nederbragt, Lex and Teal, Tracy K},
  journal={PLoS computational biology},
  volume={13},
  number={6},
  pages={e1005510},
  year={2017},
  publisher={Public Library of Science}
}