- Research
- Open Access
A non-integer order dengue internal transmission model
- Zain Ul Abadin Zafar^{1, 2}Email author,
- Muhammad Mushtaq^{1} and
- Kashif Rehan^{3}
https://doi.org/10.1186/s13662-018-1472-7
© The Author(s) 2018
- Received: 28 September 2017
- Accepted: 3 January 2018
- Published: 18 January 2018
Abstract
A non-linear mathematical model with non-integer order γ, \(0 < \gamma \leq 1\), is used to analyze the dengue virus transmission in the human body. Both disease-free \(\mathcal{F}_{0}\) and endemic \(\mathcal{F}^{*}\) equilibria are calculated. Their stability is also described using the stability theorem of non-integer order. The threshold parameter \(\mathcal{R}_{0}\) demonstrates an important behavior in the stability of a considerable model. For \(\mathcal{R}_{0} < 1\), the disease-free equilibrium (DFE) \(\mathcal{F}_{0}\) is an attractor. For \(\mathcal{R}_{0} > 1\), \(\mathcal{F}_{0}\) is not stable, the endemic equilibrium (EE) \(\mathcal{F}^{*}\) exists, and it is an attractor. Numerical examples of the proposed model are also proven to study the behavior of the system.
Keywords
- dengue model
- fractional derivatives
- stability
- predictor-corrector method
- Grunwald-Letnikov method
1 Introduction
Dengue viral diseases are a standout amongst the supreme critical mosquito-borne maladies these days. They create problems like dengue fever (DF), dengue hemorrhagic fever (DHF), and dengue stun disorder (DSS) or dengue hemorrhagic fever (DHF). Lately, the frequency of DHF has expanded significantly. Dengue may be caused by one of the serotypes DEN-1 to DEN-4. For the most part, septicity with one serotype presents upcoming defensive resistance against that specific serotype yet not against different serotypes. When anyone is infected for the second time with various serotypes, a serious disease will occur [1]. After an infected mosquito bites, the virus enters the human body and repeats inside the cell of the mononuclear phagocyte ancestry (monocytes, macrophages, and B cell). The incubation time frame is 7-10 days. Then a viremia stage, where the patient is plainly febrile and infective, takes place. From that point, the infected human body may either recuperate or advance to the leakage stage, prompting DHF and/or potentially DSS [2]. To calculate the span of viremia, analysts’ assumed that noticeable viremia began on the eve of onset of ailment, and after the recognition of the disease in the human body, it vanishes soon.
Non-integer calculus characterizes a speculation of conventional integration, differentiation to the fractional number and complex order. For instance, control theory, viscoelasticity, electricity, heat conduction, fractals, chaos, etc. searching real frameworks examples portray by the non-integer derivative is an open problem in the field of non-integer calculus [3]. The generalization of differential calculus to the fractional order of derivatives can be followed back to Leibnitz. It can help us to decrease inaccuracies emerging from the ignored parameters in the modeling of real-life problems. Different applications, like in the anomalous electron transport in amorphous materials, the reaction kinetics of proteins, the irregular electron transference in undefined materials, the di-electrical or mechanical relation of polymers, the demonstrating of glass framing fluids, and in many other fields, are effectively achieved in various articles [4]. The principle purpose behind utilizing non-fractional order models was the non-appearance of arrangement techniques for non-integer DEs. It is an evolving field in the area of mathematical physics and applied mathematics, like chemistry, biology, economics, and image and signal dispensation. It has several uses in numerous fields of engineering and science. The calculus of variations is broadly used for some disciplines such as pure mathematics, engineering, and applied mathematics. In addition, the analysts have lately revealed that the physical frameworks with dissipation can be unmistakably modeled more precisely by utilizing non-integer representations [5].
Derivatives of fractional order involve more information about the system in study than integer order derivatives, which are local operators. The physical and geometrical importance of the fractional integral containing the complex and real conjugate power-law exponent has been proposed. One physical connotation of the non-integer order in non-integer derivatives is that of the file (index) of memory [6]. The memory property is very useful in modeling of several phenomena. The state of infection at a given moment t depends on the state before t, namely \(t-1, t-2, \ldots\) . In this sense fractional calculus may help to distinguish distinct routes in (dengue) infection in different patients. We note that, for smaller value of γ, the variable approaches the corresponding asymptotic values faster [7].
Additionally, non-integer calculus shows a vital part of superdiffusive and subdiffusive procedures, which make it a helpful instrument in the study of disease transmission [8]. Since integer order differential conditions cannot decisively portray the exploratory and field estimation information, alternative tactic fractional order differential equation models are presently being widely applied [9, 10]. The upside of non-integer order differential equation systems over ordinary differential equation frameworks is that they allow more noteworthy degrees of flexibility and incorporate memory effect in the model. In other words, they give a magnificent device for the portrayal of memory and traditional chattels which were not considered in the established non-fractional order prototypes [11]. Fractional calculus has heretofore been applied in epidemiological investigations [12–15]. Also, fractional order models possess memory, fractional order differential equations give us a more realistic way to model the dengue system. Lately, they have been utilized to analyze a dengue plague prototype [16]. Regardless of the way that the operator of the non-integer is more complex than the traditional one, there occur numerical strategies for cracking systems of DEs which are nonlinear [17].
Purohit and Kalla [18] discussed the generalized non-integer PDEs containing the Liouville space non-integer derivatives and the Caputo time non-integer derivatives. The elucidations of these equations were attained using the Laplace and Fourier transforms. Also, Purohit [19] discussed the generalized non-integer partial differential including the Hilfer time non-integer derivative and the space non-integer generalized Laplace operators occurring in quantum mechanics. Chouhan et al. [20] presented the technique for developing the result of the generalized forms of non-integer DE and Volterra-type DE. Nisar et al. [21] discussed a generalized non-integer kinetic equation containing a generalized Bessel function of the first kind. Also, some of the interesting nonlinear models and fractional models were discussed in [22–35].
Lately, the majority of the dynamical frameworks based on the non-fractional order calculus have been changed into the non-integer order domain. Because of the additional degrees of opportunity and the adaptability which can be utilized to decisively fit the test information much more superior to anything in the integer order modeling, the concept of fractional calculus has become an alternative mathematical method to describe models with non-local behavior. These models represented by fractional differential equations contain the historical memory and global information of physical problems. In fact, the fractional approach generalizes the classical models of the dengue model. The purpose of this modification is to have better understanding and prediction of epidemic patterns and intervention measures.
The fractional order models are said to be useful in distinguishing distinct patterns in patients’ disease progression and possibly provide better fit data. Clinicians may use the information from the general fractional order system to devise new treatments to each individual in particular by fitting his/her data with the most appropriate fixed index. We analyze the model’s behavior for distinct values of the order of the fractional derivative γ and for biologically relevant parameters, namely the ones related with the infection and treatment [7]. Also biological systems have fractal structures and very close ties with fractional equations. Thus using the fractional differential equation for this system can produce natural results. A more reliable model can be obtained by choosing a relevant fractional index according to available real data.
This article is organized into five sections. The introduction is the first section in which we elaborate on some history of fractional calculus. In Section 2, we give notations related to the concept of FDEs. In Section 3, we ponder on the fractional order model linked with the dynamics of dengue model. Qualitative dynamics of the considerable system is resolute using an elementary reproduction number. We provide a comprehensive investigation of the global asymptotical stability of the DFE point and the native asymptotical stability of the EE point. In Section 4, numerical imitations are offered to validate the main outcomes, and conclusion is drawn in Section 5.
2 Inceptions
Definition 1
([22])
We improve a generalized inequality, in which the core appraisal system is a vector fractional order system.
A non-negative vector ν means that each constituent of ν is non-negative. We represent a non-negative vector by \(0 \leq\leq \nu\).
Theorem 1
Definition 2
([22])
“We say that \(\mathcal{F}\) is an equilibrium point of (13) if and only if \(g ( \mathcal{F} ) =0\)”.
Remark 1
When \(\varsigma \in ( 0, 1 )\), the fractional system \(D_{C}^{\varsigma} x ( t ) =g(x)\) has the identical equilibrium points as the arrangement \(\frac{dx(t)}{dt} =g(x)\).
Definition 3
([22])
“The equilibrium point \(\mathcal{F}\) of autonomous (13) is said to be stable if, for all \(\epsilon>0\), \(\varepsilon>0\) exists such that if \(\Vert x_{0}- \mathcal{F} \Vert <\varepsilon\), then \(\Vert x-\mathcal{F} \Vert <\epsilon\), \(t \geq0\); the equilibrium point \(\mathcal{F}\) of autonomous (13) is said to be asymptotically unwavering if \(\lim_{t\rightarrow\infty} x(t)=\mathcal{F}\)”.
Theorem 2
“The equilibrium points of system (13) are locally asymptotically stable if all eigenvalues \(\lambda_{{i}}\) of the Jacobian matrix J, calculated in the equilibrium points, satisfy \(\vert \arg ( \lambda_{{i}} ) \vert > \varsigma \frac{\pi}{2}\)”.
Here are a few newly developed various definitions of fractional derivative including one-parameter and two-parameter fractional derivatives:
Definition 4
([39])
Definition 5
([39])
Definition 6
([40])
Definition 7
([42])
Definition 8
([42])
It is obvious to see that 2-GC derivative of any order α knowing β is zero when \({u}({t})\) is constant and its kernel does not have any singularity at \({t}=\tau\). Moreover, the kernel is non-local due to the two-parameter Mittag-Leffler function \({E}_{\alpha, \beta} [ - \frac{\alpha\beta ({t}-\tau)^{\alpha}}{\beta-\alpha} ]\).
Definition 9
([42])
Definition 10
([43])
Note that \({T}_{\alpha} ( {f} ) ( {t} ) = {f}^{\alpha} ({t})\) denotes the conformable fractional derivatives of f of order α. If the conformable fractional derivative of f of order α exists, then we simply say that f is α-differentiable. Since it is given that \({T}_{\alpha} ( {t}^{{p}} ) ={p} {t}^{{p}-\alpha}\). Further, this definition coincides with the classical definitions of RL and of Caputo fractional derivatives on polynomials (up to constant multiple)”.
3 Mathematical model
3.1 Fractional order model
3.2 Equilibrium points and stability
3.3 \(\mathcal{R}_{0}\) sensitivity analysis
The following theorem defines the stability behavior of system (25) around the disease-free equilibrium point \(\mathcal{F}_{0}\).
Theorem 3
System (25) will be locally asymptotically stable around \(\mathcal{F}_{0}\) if \(\mathcal{R}_{0} < 1\), and it will not be stable if \(\mathcal{R}_{0} >1\).
Proof
Since all the parameters are positive and there is no negative term in \(q_{1}\), so \(q_{1} >0\). Then applying the Routh-Hurwitz criteria, we ensure that \(\mathcal{F}_{0}\) is locally asymptotically stable. If \(\mathcal{R}_{0} > 1\), then \(q_{2} < 0\), and there is one positive real root for Eq. (29), thus \(\mathcal{F}_{0}\) will be unstable. □
3.4 Global stability analysis of the disease-free equilibrium
Here, global stability is calculated for DFE for system (25). The condition for model (25) which guarantees the global stability of the disease-free state is the following.
Theorem 4
([7])
The fixed point \({P}=( {U}^{*},0)\) is globally asymptotically stable equilibrium of system (30) provided \(\mathcal{R}_{0} <1\) and that the assumptions in (31) are satisfied”.
Proof
Lemma 1
System (25) will be locally stable around \(\mathcal{F}_{0}\) if \(\mathcal{R}_{0} = 1\).
Proof
Since \(\mathcal{R}_{0} =1\), then \(q_{2} = 0\), \(q_{1} > 0\), then the roots of Eq. (29) will be \(\lambda_{2} = 0\), \(\lambda_{3} = - q_{1}\), so the system will be locally stable. □
Hence we will discuss the stability of the endemic equilibrium point \(\mathcal{F}^{*}\).
Definition 11
([48])
Theorem 5
Proof
For \(D(K) > 0\), \(c_{1} c_{2} > c_{3}\), then \(c_{1} > 0\), \(c_{3} > 0\), using the Routh-Hurwitz criteria, then \(\vert \arg ( \lambda ) \vert > \gamma \pi/2\), and the system will be locally asymptotically stable around \(\mathcal{F}^{*}\).
Since it is clear from \(c_{1} > 0\), \(c_{2} > 0\), and \(c_{1} c_{2} > c_{3}\), the conditions for stability of the non-integer order system are satisfied [49], and so \(\mathcal{F}^{*}\) is locally asymptotically stable. □
4 Numerical simulations
Here, we calculate different scenarios for different values of fractional exponent γ on the dynamics of dengue disease. The graphical view has been illustrated using the results of system (25). Here, three different techniques are considered using Matlab program.
Example
For the disease-free equilibrium, the threshold parameter \(\mathcal{R}_{0}\) has the value 0.04574, the value below one could help control the infection. The unique equilibrium point \(\mathcal{F}_{0} = (4.265, 0, 0)\) is asymptotically stable, this result enhances Theorem 1. For the endemic equilibrium point, the threshold parameter \(\mathcal{R}_{0}\) has the value 257.098, the value above one and the endemic equilibrium point \(\mathcal{F}^{*} = (0.5313,1.7432,10.50)\).
Values of disease-free equilibrium at different time periods
Values of disease-free equilibrium in 100 days | ||||
---|---|---|---|---|
\(\boldsymbol{\gamma_{i} =1}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.95}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.90}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.85}\) , i = 1,2,3 | |
Susceptible | ||||
Adams. | 4.265 | 6.597 | 10.37 | 16.02 |
Bin. coeff. | 4.265 | 4.398 | 4.491 | 4.514 |
GL coeff. | 4.265 | 4.398 | 4.491 | 4.514 |
Infected | ||||
Adams. | 6.181 × 10^{−12} | 0.004913 | 0.01693 | 0.0454 |
Bin. coeff. | 2.034 × 10^{−17} | 0.0002202 | 4.194 × 10^{−5} | 1.639 × 10^{−5} |
GL coeff. | 2.048 × 10^{−17} | 0.000225 | 4.365 × 10^{−5} | 1.742 × 10^{−5} |
Free Virus | ||||
Adams. | 3.334 × 10^{−11} | 0.02707 | 0.093 | 0.2486 |
Bin. coeff. | 1.112 × 10^{−16} | 0.001204 | 0.0002293 | 8.971 × 10^{−5} |
GL coeff. | 1.12 × 10^{−16} | 0.00123 | 0.0002386 | 9.533 × 10^{−5} |
Values of disease-free equilibrium in 75 days | ||||
---|---|---|---|---|
\(\boldsymbol{\gamma_{i} =1}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.95}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.90}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.85}\) , i = 1,2,3 | |
Susceptible | ||||
Adams. | 4.271 | 7.565 | 12.68 | 19.93 |
Bin. coeff. | 4.271 | 4.593 | 4.798 | 4.842 |
GL coeff. | 4.271 | 4.593 | 4.798 | 4.843 |
Infected | ||||
Adams. | 1.678 × 10^{−10} | 0.00964 | 0.03389 | 0.0938 |
Bin. coeff. | 3.055 × 10^{−12} | 0.0004264 | 7.839 × 10^{−5} | 2.973 × 10^{−5} |
GL coeff. | 3.067 × 10^{−12} | 0.0004357 | 8.15 × 10^{−5} | 3.16 × 10^{−5} |
Free Virus | ||||
Adams. | 9.045 × 10^{−10} | 0.05296 | 0.1857 | 0.5125 |
Bin. coeff. | 1.67 × 10^{−11} | 0.002331 | 0.0004286 | 0.0001627 |
GL coeff. | 1.677 × 10^{−11} | 0.002381 | 0.000446 | 0.0001729 |
Values of disease-free equilibrium in 60 days | ||||
---|---|---|---|---|
\(\boldsymbol{\gamma_{i} =1}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.95}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.90}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.85}\) , i = 1,2,3 | |
Susceptible | ||||
Adams. | 4.307 | 8.69 | 15.15 | 23.73 |
Bin. coeff. | 4.307 | 4.982 | 5.31 | 5.331 |
GL coeff. | 4.306 | 4.981 | 5.31 | 5.332 |
Infected | ||||
Adams. | 5.115 × 10^{−9} | 0.0169 | 0.0611 | 0.1751 |
Bin. coeff. | 3.895 × 10^{−9} | 0.0007376 | 0.0001308 | 4.809 × 10^{−5} |
GL coeff. | 3.902 × 10^{−9} | 0.0007536 | 0.0001361 | 5.112 × 10^{−5} |
Free Virus | ||||
Adams. | 2.787 × 10^{−8} | 0.09263 | 0.3342 | 0.955 |
Bin. coeff. | 2.129 × 10^{−8} | 0.004032 | 0.0007151 | 0.0002632 |
GL coeff. | 2.133 × 10^{−8} | 0.004118 | 0.0007442 | 0.0002797 |
Values of endemic equilibrium at different time periods
Values of endemic equilibrium in 100 days | ||||
---|---|---|---|---|
\(\boldsymbol{\gamma_{i} =1}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.95}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.90}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.85}\) , i = 1,2,3 | |
Susceptible | ||||
Adams. | 0.5313 | 0.5185 | 0.5066 | 0.4969 |
Bin. coeff. | 0.5313 | 0.5266 | 0.5241 | 0.5239 |
GL coeff. | 0.5313 | 0.5264 | 0.5237 | 0.5233 |
Infected | ||||
Adams. | 1.743 | 2.612 | 4.006 | 6.139 |
Bin. coeff. | 1.743 | 1.757 | 1.763 | 1.761 |
GL coeff. | 1.743 | 1.758 | 1.765 | 1.762 |
Free Virus | ||||
Adams. | 10.5 | 15.74 | 24.14 | 37.01 |
Bin. coeff. | 10.5 | 10.58 | 10.62 | 10.61 |
GL coeff. | 10.5 | 10.59 | 10.63 | 10.62 |
Values of endemic equilibrium in 75 days | ||||
---|---|---|---|---|
\(\boldsymbol{\gamma_{i} =1}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.95}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.90}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.85}\) , i = 1,2,3 | |
Susceptible | ||||
Adams. | 0.5313 | 0.501 | 0.4817 | 0.4696 |
Bin. coeff. | 0.5313 | 0.5159 | 0.5068 | 0.5034 |
GL coeff. | 0.5313 | 0.5152 | 0.5055 | 0.5018 |
Infected | ||||
Adams. | 1.743 | 3.235 | 5.547 | 8.951 |
Bin. coeff. | 1.743 | 1.791 | 1.818 | 1.824 |
GL coeff. | 1.743 | 1.793 | 1.822 | 1.829 |
Free Virus | ||||
Adams. | 10.5 | 19.5 | 33.44 | 53.97 |
Bin. coeff. | 10.5 | 10.79 | 10.96 | 10.99 |
GL coeff. | 10.5 | 10.81 | 10.98 | 11.02 |
Values of endemic equilibrium in 60 days | ||||
---|---|---|---|---|
\(\boldsymbol{\gamma_{i} =1}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.95}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.90}\) , i = 1,2,3 | \(\boldsymbol{\gamma_{i} =0.85}\) , i = 1,2,3 | |
Susceptible | ||||
Adams. | 0.5313 | 0.5104 | 0.4945 | 0.4833 |
Bin. coeff. | 0.5313 | 0.5222 | 0.5168 | 0.5153 |
GL coeff. | 0.5313 | 0.5217 | 0.516 | 0.5142 |
Infected | ||||
Adams. | 1.743 | 2.961 | 4.761 | 7.528 |
Bin. coeff. | 1.743 | 1.771 | 1.786 | 1.786 |
GL coeff. | 1.743 | 1.773 | 1.788 | 1.790 |
Free Virus | ||||
Adams. | 10.5 | 17.57 | 28.7 | 45.39 |
Bin. coeff. | 10.5 | 10.67 | 10.76 | 10.76 |
GL coeff. | 10.5 | 10.68 | 10.78 | 10.78 |
- (i)
Adams-Bashforth-Moulton algorithm
Disease-free equilibrium (Figures 2-4).
Disease-free equilibrium (Figures 8-10).
5 Conclusion
A nonlinear mathematical dengue model with fractional order \(\gamma_{i}\), \(i=1,2,3\), is formulated. The stability of both DFE and EE points is discussed. Sufficient conditions for local stability of the DFE point \(\mathcal{F}_{0}\) are given in terms of the basic reproduction number \(\mathcal{R}_{0}\) of the model, where it is asymptotically stable if \(\mathcal{R}_{0} <1\). The positive infected equilibrium \(\mathcal{F}^{*}\) exists when \(\mathcal{R}_{0} >1\), and sufficient conditions that guarantee the asymptotic stability of this point are calculated. Besides this sensitivity analysis of the parameters involved, the threshold parameter (\(\mathcal{R}_{0}\)) is also discussed. Three fractional order techniques are used to check the best performance of the model. When simulating the model with all three algorithms, we have observed that all methods are converging to the disease-free and endemic equilibrium points through different paths and for different values of \(\gamma_{i}\), \(i=1,2,3\). The values are very close to each other in all three techniques as given in Tables 2 and 3. However, the time consumed (Core i5 laptop) by Grunwald-Letnikov (binomial coefficient) is \(3125.817~\mbox{sec} \approx 52~\mbox{min}\), by Grunwald-Letnikov (GL coefficient) is \(2483.031~\mbox{sec} \approx 41~\mbox{min}\), and by the Adams-Bashforth-Moulton algorithm is \(67326.743~\mbox{sec} \approx 18.7~\mbox{hrs}\), which indicates that the computational cost for Grunwald-Letnikov (GL coefficients) is cheaper than that for the other two. Dengue is one of the most rapidly spreading mosquito-borne viral diseases in the world, and it inflicts significant health, economics, and social burdens on populations. The main purpose of analyzing the dengue model with these techniques is that it helps the researchers and policy makers in targeting, prevention, and treatment resources for maximum effectiveness. Numerical simulations with different order show that the system decays to the equilibrium condition, e.g., power of \(t^{-\gamma}\). The result provides an important insight into the use of fractional order to model the dengue internal disease. The order of the fractional derivative γ may be associated with differences in individuals’ immune system, age, treatment compliance, treatment toxicities and other co-morbidities, amongst others. The fractional order may provide more ‘freedom’ to adjust the model to real data of specific patients. That is to say, the fractional order index contributes positively to better fit the patient data.
In the future, optimal control strategies can be incorporated into the proposed model for the control of virus inside the body.
Declarations
Acknowledgements
We would like to thank the referees for their valuable comments.
Authors’ contributions
The authors have achieved equal contributions. All authors read and approved the 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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