A fractional order model for Hepatitis B virus with treatment via Atangana–Baleanu derivative

Shah, Syed İnayat Ali; Khan, Muhammad Altaf; Farooq, Muhammad; Ullah, Saif; Alzahrani, Ebraheem O.

doi:10.1016/j.physa.2019.122636

Cited by 43 publications

(24 citation statements)

References 35 publications

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“…Atangana-Baleanu operator has a non-singular and non-local (that is, Mittag-Leffler) kernel while the Caputo operator does not [41,42]. From the biological meanings, a model with Atangana-Baleanu operator gives better results in applying memory effects [43][44][45]. Nevertheless, the Caputo operator has more complex analytical tools in investigating the dynamics of the model such as the local stability [46][47][48][49][50], the global stability [51,52], bifurcation theory [52][53][54], and so on.…”

Section: Variables and Parameters Description X(t)mentioning

confidence: 99%

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Panigoro

Suryanto

Kusumawinahyu

et al. 2020

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The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator.

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Section: Variables and Parameters Description X(t)mentioning

confidence: 99%

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Panigoro

Suryanto

Kusumawinahyu

et al. 2020

Axioms

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“…Research related Atangana-Baleanu derivative and their applications to various models arising in science and engineering can be seen in [4][5][6][7][8]. Some other related papers to the applications of Atangana-Baleanu derivative can be seen in [9][10][11][12][13][14] We consider the reported cases since January 21, 2020 till January 28, 2020, which are higher than the initial days and formulate a mathematical model to parameterize the model. We give brief details about the mathematical modeling of the novel corona virus in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative

Khan

Atangana

2020

Alexandria Engineering Journal

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628

459

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The present paper describes the mathematical modeling and dynamics of a novel corona virus (2019-nCoV). We describe the brief details of interaction among the bats and unknown hosts, then among the peoples and the infections reservoir (seafood market). The seafood marked are considered the main source of infection when the bats and the unknown hosts (may be wild animals) leaves the infection there. The purchasing of items from the seafood market by peoples have the ability to infect either asymptomatically or symptomatically. We reduced the model with the assumptions that the seafood market has enough source of infection that can be effective to infect people. We present the mathematical results of the model and then formulate a fractional model. We consider the available infection cases for January 21, 2020, till January 28, 2020 and parameterized the model. We compute the basic reproduction number for the data is R 0 % 2:4829. The fractional model is then solved numerically by presenting many graphical results, which can be helpful for the infection minimization.Ó 2020 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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“…The model formulation is shown initially using integer order and then the model is generalized to obtain the fractalfractional model. The fractional models and their applications to biological and physical problems are numerous in the literature; see [21][22][23][24][25]. We provided above comprehensive details on the mathematical modeling of the coronavirus infection and its background results.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of COVID-19 with quarantined and isolation

et al. 2020

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In the present paper, we formulate a new mathematical model for the dynamics of COVID-19 with quarantine and isolation. Initially, we provide a brief discussion on the model formulation and provide relevant mathematical results. Then, we consider the fractal-fractional derivative in Atangana-Baleanu sense, and we also generalize the model. The generalized model is used to obtain its stability results. We show that the model is locally asymptotically stable if R 0 < 1. Further, we consider the real cases reported in China since January 11 till April 9, 2020. The reported cases have been used for obtaining the real parameters and the basic reproduction number for the given period, R 0 ≈ 6.6361. The data of reported cases versus model for classical and fractal-factional order are presented. We show that the fractal-fractional order model provides the best fitting to the reported cases. The fractional mathematical model is solved by a novel numerical technique based on Newton approach, which is useful and reliable. A brief discussion on the graphical results using the novel numerical procedures are shown. Some key parameters that show significance in the disease elimination from the society are explored.

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A fractional order model for Hepatitis B virus with treatment via Atangana–Baleanu derivative

Cited by 43 publications

References 35 publications

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative

The dynamics of COVID-19 with quarantined and isolation

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