Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Hattaf, Khalid

doi:10.1155/2021/8608447

Cited by 14 publications

(11 citation statements)

References 19 publications

(22 reference statements)

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“…Other works such as a multi-strain SEIR models with optimal control [5,6], multi-strain SEIR models with saturated and general incidence rates [5,15], SIRD [45] and SEIPAHRF model [46] with Caputo fractional derivative, SCIRP model incorporating media influence [47], and statistical analysis [48] have also been considered in describing the transmission of the virus and its strains. We direct the readers to the work of Hattaf et al [49,50] for more recent information about the fractional differential equations and its generalization.…”

Section: Introductionmentioning

confidence: 99%

Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19

Otunuga¹

2022

PLoS ONE

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In this work, an innovative multi-strain SV EAIR epidemic model is developed for the study of the spread of a multi-strain infectious disease in a population infected by mutations of the disease. The population is assumed to be completely susceptible to n different variants of the disease, and those who are vaccinated and recovered from a specific strain k (k ≤ n) are immune to previous and present strains j = 1, 2, ⋯, k, but can still be infected by newer emerging strains j = k + 1, k + 2, ⋯, n. The model is designed to simulate the emergence and dissemination of viral strains. All the equilibrium points of the system are calculated and the conditions for existence and global stability of these points are investigated and used to answer the question as to whether it is possible for the population to have an endemic with more than one strain. An interesting result that shows that a strain with a reproduction number greater than one can still die out on the long run if a newer emerging strain has a greater reproduction number is verified numerically. The effect of vaccines on the population is also analyzed and a bound for the herd immunity threshold is calculated. The validity of the work done is verified through numerical simulations by applying the proposed model and strategy to analyze the multi-strains of the COVID-19 virus, in particular, the Delta and the Omicron variants, in the United State.

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Section: Introductionmentioning

confidence: 99%

Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19

Otunuga¹

2022

PLoS ONE

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“…Remark 4. Theorem 4 generalizes the result concerning the Mittag-Leffler stability presented in Theorem 3 of [28]. It suffices to take ψ(z) = z q , where z ∈ [0, +∞) and q is an arbitrary positive constant.…”

Section: Stability Of Fdes With the Ghf Derivativementioning

confidence: 60%

“…a α = N(α) + µ(1 − α). Now, we apply the numerical scheme presented in (28) in order to approximate the solution of (35). For all numerical simulations, we chose A = 0.01, µ = 0.01, and the normalization function as follows:…”

Section: Application To Biologymentioning

confidence: 99%

“…The stability in the sense of Ulam-Hyers of FDEs with GHF derivative was studied in [33] using a new version of the Gronwall inequality. However, this paper extends the fractional comparison principle to the GHF derivative and the results related to Mittag-Leffler stability given in [28] using class K functions. Furthermore, the present paper establishes new interesting results concerning the stability, as well as the asymptotic stability of FDEs with the GHF derivative by means of the Lyapunov direct method and class K functions.…”

Section: Introductionmentioning

confidence: 97%

“…In [27], the authors focused on the Ulam stability of a generalized delayed differential equation of fractional order. A more recent study presented in [28] discussed the Mittag-Leffler stability of FDEs using the new generalized Hattaf fractional (GHF) derivative [29], which includes many fractional derivatives available in the literature such as the Caputo-Fabrizio fractional derivative [30], the Atangana-Baleanu fractional derivative [31], and the weighted Atangana-Baleanu fractional derivative [32]. The stability in the sense of Ulam-Hyers of FDEs with GHF derivative was studied in [33] using a new version of the Gronwall inequality.…”

Section: Introductionmentioning

confidence: 98%

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On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

Hattaf

2022

Computation

Self Cite

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The fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

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Nonlinear analysis and dynamics of COVID‐19 mathematical model with optimal control strategies

Muthukumar

Myilsamy

Balakumar

et al. 2023

Optim Control Appl Methods

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Efficacy of the healthcare system and illumination (awareness) activities control COVID‐19. To defend public health, the spreading pandemic of COVID‐19 disease necessitates social distancing, wearing masks, personal cleanliness, and precautions. Due to inadequate awareness programs, COVID‐19 rapidly increases in India. The primary goal of this research is to investigate the spreading behavior of the COVID‐19 virus in India when people are aware of the disease. We find the optimum value of disease transmission rate and detection of the unidentified asymptomatic and symptomatic populations. An optimal control problem is designed with limited resource allocation to improve the recovered individuals. A stability analysis presents for emphasizes the relevance of disease awareness in preventing the spread of the disease. The control parameters are used to explore the increase and decrease of the infected individual with and without control in optimal control analysis. The model is simulated using the Hattaf‐fractional derivative to study the memory effect in the epidemic. To adapt the model to the total number of reported COVID‐19 cases in India, we collected data from March 20, 2021 to September 30, 2021. According to the simulation results, the pandemic would spread faster if awareness campaigns were improperly carried out.

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Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

Cited by 14 publications

References 19 publications

Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19

Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

Nonlinear analysis and dynamics of COVID‐19 mathematical model with optimal control strategies

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