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- Open Access
A fractional order pine wilt disease model with Caputo–Fabrizio derivative
- Muhammad Altaf Khan^{1}Email author,
- Saif Ullah^{2},
- K. O. Okosun^{3} and
- Kamil Shah^{1}
https://doi.org/10.1186/s13662-018-1868-4
© The Author(s) 2018
- Received: 22 May 2018
- Accepted: 30 October 2018
- Published: 7 November 2018
Abstract
A Caputo–Fabrizio type fractional order mathematical model for the dynamics of pine wilt disease (FPWD) is presented. The basic properties of the model are investigated. The existence and uniqueness of the solution for the proposed FPWD model are given via the fixed point theorem. The numerical simulations for the model are obtained by using particular parameter values. The non-integer order derivative provides more flexible and deeper information about the complexity of the dynamics of the proposed FPWD model than the integer order models established before.
Keywords
- Caputo–Fabrizio (CF) fractional derivative
- Pint wilt disease
- Mathematical model
- Fixed point theorem
- Numerical simulation
1 Introduction
Mathematical models in epidemiology are used widely in order to understand better the dynamics of an infectious disease [1, 2]. The application of the mathematical models is not limited to only human diseases, but they are also widely applied in other phenomena of biological sciences, such as ecology, forest, etc. In the human life, forest has an important role, therefore, it is necessary to ensure the safety strategies to protect it from being infected with diseases. The forest provides greenery to the environment and pleasant atmosphere for humans. The pine wilt disease (PWD) infects pine trees and is one of the main threats to the ecosystem and forest. The PWD is considered to be the most destructive disease which damages pine trees in a short period of time, that is, a year or sometimes in a few months. The initial symptoms of the PWD include discoloration of needles, which turn from yellow to green then to reddish brown. The main agent of the disease is small worms, known as pinewood nematode (bursaphelenchus xylophilus), causing tree decline [3]. As the trees begin to die, they are attacked by insects, known as sawyers, which are species that transfer the nematode to healthy trees, which is one of the causes of pine wilt disease [4–6]. Native to North America, the PW nematode was introduced while the first epidemic of the PWD was accursed in Japan in 1905 [7], and it has spread in southern China, Korea, Taiwan, and other regions of Europe since the early 1980s [4–6, 8]. The PWD has three main organisms: the gymnosperm host, the pine wood nematode, and the insect vector. At the stage of primary transmission, dauer juveniles (JIV stage) of bursaphelenchus xylophilus are carried vertically in the tracheae of their beetle host to young twigs of susceptible trees, where they enter through resin canals in wounds made during maturation feeding by the insect [8].
Recently, some mathematical models have been presented to explore the dynamics of PWD consisting of a system of nonlinear differential equations. Lee and Kwang [9] explored the stability analysis of PWD and proposed some suitable controlling strategies for this disease. Khan et al. [10] introduced a model on PWD and its optimal control. A mathematical model with variable population and suggested optimal control was developed in [11]. Most recently, in [12] the dynamics of PWD with saturated incidence rate was explored. All of the above PWD models are restricted to classical integer order differential equations. In the present paper we consider a PWD model with saturated incidence rate in fractional environment using the CF derivative. First, we give an overview of recently published papers on fractional mathematical models using the CF derivative.
Fractional order models are more reliable and helpful in the real phenomena than the classical models due to hereditary properties and the description of memory [13, 14]. Also, in the real world explanation, the integer order derivative does not explore the dynamics between two different points. To deal with such failures of classical local differentiation, different concepts on differentiation with non-local or fractional orders have been developed in the existing literature. For instance, Riemann and Liouville introduced the concept of fractional orders differentiation in [14]. Recently, Caputo and Fabrizio [15] introduced a new derivative with fractional order based on the exponential kernel. The new CF fractional order derivative has been used successfully in modeling of various real phenomena. For example, a fractional Adams–Bashforth technique via the CF derivative was presented in [16]. A study of magnetohydrodynamic electroosmotic flow of Maxwell fluids with CF derivatives was carried out by Abdulh et al. [17]. In [18], the CF fractional derivative was used for numerical approach of the Fokker–Planck equation using Ritz approximation. A mathematical comparative analysis of RL and RC electrical circuits using AB and CF fractional derivatives was recently done in [19]. Mustafa et al.[20] explored the dynamics of the cancer treatment model with the CF fractional derivative. Recently, a new fractional model of hepatitis B virus in the CF derivative sense was presented in [21].
The classical integer order mathematical model is useful for a local dynamic system with no external forces. These models cannot therefore replicate the complexity of the dynamics of the communicable disease like PWD as the model can sometimes have a crossover behavior and this cannot be handled by the classical differential operators. Further, in the literature fractional order models provide a better fit to the real data for different diseases and other experimental work in fluid mechanics. For example, Diethelm [22] provided a good agreement to the real data of the 2009 dengue outbreak in Cape Verde using a non-integer order biological model instead of the ordinary one. A fractional order model for Ebola epidemic was applied to provide a suitable approximation to the real data on Ebola virus [23]. Makris et al. [24] used a fractional order Maxwell model to attain a better fit to the experimental work.
Therefore, motivated by the above work, in this paper, we aim to extend the recently published PWD model [12] to a fractional case by using the newly established derivative known as CF derivative of order \(\tau\in(0,1]\). The details of the remaining sections of this paper are as follows: The basic definition and results of fractional order derivative are stated in Sect. 2. In Sect. 3, we explore the model formulation, model equilibria, and the basic reproduction number. Section 4 deals with the existence of solution in the spread PWD disease model via the fixed point theorem. Also, the uniqueness of a model solution is obtained. Numerical simulations are presented in Sect. 5. Finally, the concluding remarks are given in Sect. 6.
2 Preliminaries
Here, we give some basic definitions of the fractional calculus that will be used in the onward analysis of the model.
Definition 1
Remark 1
Definition 2
Remark 2
The CF derivative [15], given in the above definitions, has been recently used in the mathematical modeling of HBV [21], Maxwell fluid with slip effects [26], and diabetes model [27].
3 Model formulation
3.1 Equilibria and basic reproduction number
Theorem 1
The FPWD model (9) has a unique endemic equilibrium if \(\mathcal{R}_{0}>1\).
4 Existence and uniqueness of FPWD model
Theorem 2
Proof
Now we state the theorem below.
Theorem 3
Proof
Theorem 4
Proof
5 Numerical results
6 Conclusion
In the present work, we extended the PWD model [12] to fractional order using the Caputo–Fabrizio fractional derivative. The model equilibria and basic reproduction number are explored. The existence and uniqueness of the solution for the FPWD model with CF derivative are proved in detail. Some numerical simulations are carried out to explore the effect of fractional order. From numerical simulations one can see that when fractional order of derivative τ decreases, the CF derivative provides more biologically feasible behavior about the dynamic of pine wilt disease. Therefore, we concluded that the newly fractional derivative is very useful for modeling such phenomena. Also, from the graphical behavioral we conclude that the proposed fractional order model provides richer and more flexible results when compared with the corresponding integer order PWD model. Thus, we suggest that this new fractional model gives more effective results which could be implemented for the elimination of the pine tree infection.
Declarations
Acknowledgements
The authors are thankful to the handling editor and respected reviewers for the careful reading and useful comments that greatly improved the presentation of this paper.
Availability of data and materials
The data is available without any restrictions.
Funding
Not applicable.
Authors’ contributions
MAK and KS formulated the fractional order model and obtained the numerical results. SU obtained the existence results for the model. MAK and SU wrote the introduction, abstract, and conclusion. MAK, KOO revised the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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