Modeling and analysis of fractional order Zika model

1 Department of Mathematics and Statistics, University of Lahore, Lahore 54590, Pakistan 2 Art and Science Faculty, Department of Mathematics, Siirt University, Siirt 56100, Turkey 3 Department of Statistics and Operations Research, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand 5 Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy


Introduction
The virus name Zika is first time found in 1947 in monkeys and the first patient was reported in Uganda in 1952. The genus of Zika is also found in dengue, yellow fever, and West Nile virus [1]. Since the 1950s, it has been found in a very narrow area. During 2007-2016 it spread across the Pacific Ocean to the Americas which cause 2015-16 Zika virus epidemics [2]. Zika virus often causes only mild symptoms which are very similar to dengue. There is no specific treatment for this virus [3,4].
Mathematicians and biologist's main theme is to study disease. Many mathematicians tried to represent the mathematical model in a very natural way such as in the approach of Baleanu et al. [5][6][7]. In recent years, fractional calculus has fascinated the attention of researchers and the various features of that study under investigation. This is because genetic mutations are an important tool for defining the dynamic function of various body systems. The power of these component operators is their non-local features that are not in the integer separator operator. Separated features of differentiated statistics that define the memory and transfer structures of many mathematical models. As a fact that fractional-order models are more realistic and useful than classical integer-order models. Fractional order findings produce a greater degree of freedom in these models. Unnecessary order outsourcing is a powerful tool for understanding the dynamic behavior of various bio objects and systems. The most repetitive feature of these models is in their global (non-local) features that are not in the old order models. Fractional calculus has acquired great rating and significance over the last few years in various branches of science and engineering. Effective systematic and statistical techniques have been established but they still require special care. This distinctive problem aims to create an assemblage of articles showing the advances in mathematics and the branch of fractional calculus and to explore the applications in applied science [8][9][10]. Caputo [11] presented from the group that allows for common initial and borderline conditions related to a real-world problem. Baleanu et al. [13] stated advanced techniques in the field of fractional calculus and nanotechnology using monographs. Kailas et al. [14] obtainable basic ideas of equation differences including their uses are explained. Bulut et al. [15] studied the differential measurement of the orderly application of analytical methods and some related details are given in [16][17][18][19][20][21]. In recent years researchers have been using some mathematical models to simulate the transmission of the Zika virus [22][23][24][25].
The common SEIR model is generalized in order to show the dynamics of COVID-19 transmission taking into account the ABO blood group of the infected people. Fractional order Caputo derivative are used in the proposed model [28]. New system is confirmed to have chaotic behaviors by calculating its Lyapunov exponents [29]. the analytical solution using the Reduced differential transforms method (RDTM) for the nonlinear ordinary mathematical smoking model [30]. Complete synchronization between two chaotic systems means complete symmetry between them, but phase synchronization means complete symmetry with a phase shift. In addition, the proposed method is applied to the synchronization of two identical chaotic Lorenz models [31]. Important and adequate conditions to ensure the presence and singularity of the arrangements of the control issue are assumed [32]. The existence and uniqueness of stable solution of the proposed fractional order COVID-19 SEIASqEqHR paradigm are proved. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced [33,34]. SDM is considered as a mixture of Adomian decomposition method and the Sumudu transform method. several vital characteristics and features of this model are investigated, such as its hamiltonian, symmetry, signal flow graph, dissipation, equilibriums and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams, and chaotic behavior [35][36][37] and some others applications of fractional order also given in [38][39][40].
In this paper, Section 1 consists an introduction and some basic definition of fractional-order derivatives to solve the epidemiological model respectively. Sections 3 and 4 consists of the generalized solution of the fractional-order model, consist of the uniqueness and stability of the model. Fractal fractional techniques with exponential decay kernel and Mittag-Leffler kernel are applied for suitable results in Section 5. Results and conclusion are discussed in Sections 6 and 7 respectively.

Basic concepts of fractional operators
where By using Sumudu transform (ST) for (1), we obtain Definition 2.2. The Laplace transform of the Caputo fractional derivative of a function ( ) of order > 0 is defined as Definition 2.3. The Laplace transform of the function 1 −1 , 1 (± ) is defined as Where , 1 is the two-parameter Mittag-Leffler function with , 1 > 0. Further, the Mittag-Leffler function satisfies the following equation [17].
Definition 2.4. Suppose that ( ) is continuous on an open interval ( , ), then the fractal-fractional integral of ( ) of order having Mittag-Leffler type kernel and given by 0,

Material and method
In this portion, we give a mathematical model for the transmission of the Zika virus using the Atangna-Baleanue in Caputo sense of fractional order. We make two portions of the Human population: Susceptible people and infected people so that = + . Same as we make two portions of a total number of mosquitoes into two groups: Susceptible mosquitoes Sq and infected mosquitoes , so that = + . To explain the method of the spread of the Zika virus given in [26], we consider the compartmental mathematical model as follows: with the initial conditions The model parameters are: The recruitment rate of human population p, the recruitment rate of mosquito population , the effective contact rate human to human 1 , the effective contact rate of mosquitoes to human 2 , the effective contact rate human to mosquito's , the natural death rate of human 1 , the natural death rate of mosquitoes 2 .

Equilibrium points
In this section, we will discuss the equilibrium points of the given Zika Virus model (7). Equilibrium points have two types namely disease-free equilibrium and endemic equilibrium. We obtained these points by putting the right-hand side of the system (7) is zero. We suppose that E ' represents disease-free equilibrium and endemic equilibrium is represented by E*. We take our both equilibriums by, we have Reproductive number 0 given in [26], we have Theorem 3.1. The solution of the proposed fractional-order model (7) along initial conditions is unique and bounded in + 4 .
} is a positively invariant set for the system (7).
Proof. For the proof of the theorem, firstly we use the first to equations of system (7). So which further gives We infer that if ( 0 , 0 ) ∈ + 4 , then Similarly, we can prove for N ( ) = ( )

Atangana-Baleanu Caputo sense
In this section, coniseder the system with Atangana-Baleanu fractional derivative (ABC) of order and ∈ (0,1] for sytem (7), we By applying the definition (2) of sumudu transform in ABC sense, we have Rearranging, we get Now taking the inverse Sumudu transform on both sides of the Eq (10) we get We next attain the following recursive formula: And solution of above is Uniqueness and stability of the iterative scheme Theorem 4.1. Let ( 2 , ||. ||) be a Banach space and M be a self-map of 2 satisfying || − || C||X− ||+C||X-Y|| for all x, y ∈ 2 where 0≤ , 0 ≤ < 1.
Let consider that M is P-stable. Let us take into account the following recursive formula:

Theorem 4.2. Define M be a self-map is given by
Proof. In the first step we will show that M is a fixed point Applying the properties of the norm and also taking into account the triangular inequality, we obtain K fulfills the conditions associated with Theorem (4.1), when Hence system is Picard P-Stable. Theorem 4.3. Prove that system (11) has a special solution is unique.
Proof. Let Hilbert space = 2 �( , ) × (0, )� which is given as In this regard, the following operators are considered (0,0,0,0,0), = We establish that the inner product of , �, are the special solutions of the system. Taking into account the inner function and the norm, we have Due to the large number of 1 , 2 , 3 , 4 5 , both solutions converge to the exact solution. Applying the topological idea, we have the very small positive five parameters  Hence proved.

Numerical scheme with Atangana-Toufik
In this section we consider the Atangana-Toufik technique given in [27] for fractional derivative model (7). For this purpose, we suppose that We express the Eq (8)  At a given point +1 , = 0,1,2,3, …, the above equation is reformulated as Within the interval [ , +1 ], the function � , ( )�, using the two-steps Lagrange polynomial interpolation can be approximate as follows: The above approximation can therefore be included in equation (18) to produce For simplicity, we let and also , By using Eqs (23) and (24) we obtain We obtain the following for model (7)

Zika model with fractal fractional operator
In this section, we consider the Zika virus model (7) with fractal-fractional in ABC sense. We have The fractal-fractional Zika virus model algorithm for (19), we need to generalize the system and present steps by considering the Cauchy problem as below: after integrating the above equation, we get: Let � , ( )� = 2 2 −1 � , ( )�, then system (21) becomes At +1 = ( + 1)∆ , we have Also, we have Approximating the function � , ( )�, using the Newton polynomial, we have Using Eq (32) into system (31), we have Rearranging the above system, we have Writing further system (34), we have Now, calculating the integrals in the system (35), we get Inserting them into system (35), we get Finally, we have the following approximation: We obtain the following for system (26)

Numerical results and discussions
To identify the potential effectiveness of Zika virus transmission in the Community, We consider the following parameters values and initial conditions [18] for our simulations: The mechanical features of the fractional-order model are identified by the various numerical methods with the time-fractional parameters. We demonstrate our results using different techniques in Figures 1-12 to check the efficiency of obtained solutions. The results of the nonlinear system memory were also detected with the help of fractional value. It provides a better way of understanding to control the disease without defining other parameters. Figures 1-4 represents the dynamical behavior of the Zika virus by using ABC derivative, ℎ ( ), ( ) and ℎ ( ) start increase steadily by decreasing the fractional values while ( )start decreasing by decreasing the fractional values. Figures 5-8 represents the dynamical behavior of the Zika virus by using fractal fractional derivative with dimensions 0.9, ℎ ( ), ( ) and ℎ ( ) start increase strictly by decreasing the fractional values while ( ) start decreasing strictly by decreasing the fractional values. Figures 9-12 represents the dynamical behavior of the Zika virus by using fractal fractional derivative with dimensions 0.8, ℎ ( ), ( ) and ℎ ( ) start increase strictly by decreasing the fractional values while ( )start decreasing strictly by decreasing the fractional values. Similar behavior can be seen with both techniques, but fractal fractional gives results fastly with minor effects of dimensions according to steady state. Moreover, it provides better results by decreasing the fractional value.

Conclusions
A fractional order differential equation model has been investigated in this article for the Zika virus. By using the fixed point theory, stability and uniqueness of the Zika virus model have been investigated. The arbitrary derivative of fractional order has been taken in the Attangana Baleeno in Caputo sense with no singular kernel and fractal fractional with Mittag-Leffler kernel respectively to analyses the Zika virus. Theoretical results are investigated for the fractional-order model, which proved the efficiency of the developed schemes. Numerical simulation has been made to check the actual behavior of the Zika virus outbreak. Such type of study will be helpful in future to understand the outbreak of this epidemic and to control the disease in a community.