Bifurcations and chaos in a discrete SI epidemic model with fractional order

Abdelaziz, Mahmoud A. M.; Ismail, Ahmad Izani Md.; Abdullah, Farah Aini; Mohd, Mohd Hafiz

doi:10.1186/s13662-018-1481-6

Cited by 34 publications

(35 citation statements)

References 49 publications

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“…Let S 1 = s m mΓ(m) and S * be a perturbation in the bifurcation parameter S 1 , where |S * | << 1. Then a perturbation form of model (14) can be represented as [35] x n+1 = x n + (…”

Section: Hopf Bifurcation and Its Stabilitymentioning

confidence: 99%

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Mondal¹,

Biswas

Bairagi

2020

J. Appl. Math. Comput.

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This paper is focused on local and global stability of a fractional-order predator-prey model with habitat complexity constructed in the Caputo sense and corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions in fractional-order model is presented. Conditions for local and global stability of different equilibrium points are proved. It is shown that there may exist fractional-order-dependent instability through Hopf bifurcation for both fractional-order and corresponding discrete systems. Dynamics of the discrete fractional-order model is more complex and depends on both step length and fractional-order. It shows Hopf bifurcation, flip bifurcation and more complex dynamics with respect to the step size. Several examples are presented to substantiate the analytical results.This model says that the prey population x grows logistically with intrinsic growth rate r to its carrying capacity K. Predator y captures the prey at a maximum rate α in absence of any habitat complexity (c = 0). In presence of complexity, predation rate decreases to α(1 − c), where the dimensionless parameter c is called the degree or strength of complexity. The value of c ranges from 0 to 1. In particular, c = 0.4 implies that predation rate decreases by 40% due to habitat complexity. If c = 0, i.e. if there is no habitat complexity then the system (1) reduces to well known Rosenzweig-MacArthur model [25]. However, if c = 1 then y → 0 as t → ∞ and the prey population grows logistically to its maximum value K. The parameter θ (0 < θ < 1) is the conversion efficiency, measuring the number of newly born predators for each captured prey and d is the death rate of predator. All parameters are assumed to be positive. For construction and more explanation of the model, readers are referred to [26].Considering the fractional derivatives in the sense of Caputo, we have the following fractional-order model corresponding to the integer order model (1): Existence and uniquenessHere we study the existence and uniqueness of the solution of our system (2). We have the following Lemma due to Li et al [30]. Lemma 2.3 Consider the system . If f (t, x) satisfies the locally Lipschitz condition with respect to x then there exists a unique solution of the above system on [t 0 , ∞) × Ω.We study the existence and uniqueness of the solution of system (2) in the region Ω × [0, T ], where Ω = {(x, y) ∈ ℜ 2 | max{|x|, |y|} ≤ M}, T < ∞ and M is large. Denote X = (x, y),X = (x,ȳ). Consider a mapping H : Ω → ℜ 2 such that H(X) = (H 1 (X), H 2 (X)), where

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“…Let S 1 = s m mΓ(m) and S * be a perturbation in the bifurcation parameter S 1 , where |S * | << 1. Then a perturbation form of model (14) can be represented as [35] x n+1 = x n + (…”

Section: Hopf Bifurcation and Its Stabilitymentioning

confidence: 99%

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Mondal¹,

Biswas

Bairagi

2020

J. Appl. Math. Comput.

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“…Fractional models can describe complex physics problems more clearly and concisely, especially the nonlinear model and their physical meaning. Research has shown that the fractional order equation o ers a possibility for the situation that the traditional integer order equation cannot model [3]. Most importantly, the most prominent feature of the body's immune system is about its memory.…”

Section: Introductionmentioning

confidence: 99%

Fractal Control and Synchronization of the Discrete Fractional SIRS Model

2020

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SIRS model is one of the most basic models in the dynamic warehouse model of infectious diseases, which describes the temporary immunity after cure. The discrete SIRS models with the Caputo deltas sense and the theories of fractional calculus and fractal theory provide a reasonable and sensible perspective of studying infectious disease phenomenon. After discussing the fixed point of the fractional order system, controllers of Julia sets are designed by utilizing fixed point, which are introduced as a whole and a part in the models. Then, two totally different coupled controllers are introduced to achieve the synchronization of Julia sets of the discrete fractional order systems with different parameters but with the same structure. And new proofs about the synchronization of Julia sets are given. The complexity and irregularity of Julia sets can be seen from the figures, and the correctness of the theoretical analysis is exhibited by the simulation results.

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“…Mathematical modeling for various biological problems is considered to be an exciting research area in the discipline of applied mathematics. In the literature, many biological phenomena were modeled and formulated mathematically [1][2][3][4][5][6]. For example, Liu and Xiao established a predator-prey system in discrete time to analyze the local stability and the bifurcation of solutions around the positive equilibrium point [4].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation and Stability Analysis of a System of Fractional-Order Differential Equations for a Plant–Herbivore Model with Allee Effect

Yousef

2019

Mathematics

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This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according to a low population density of the plant population, we incorporate the Allee function into the model. Considering the center manifold theorem and bifurcation theory, we show that the model shows flip bifurcation. Finally, the simulation results agree with the theoretical studies.

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Bifurcations and chaos in a discrete SI epidemic model with fractional order

Cited by 34 publications

References 49 publications

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Fractal Control and Synchronization of the Discrete Fractional SIRS Model

Bifurcation and Stability Analysis of a System of Fractional-Order Differential Equations for a Plant–Herbivore Model with Allee Effect

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