Modeling and analysis of fractional order Zika model

1.   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.

2.   Basic concepts of fractional operators

Definition 2.1. For a function $g\left(t\right)\in {W}_{2}^{1}\left(\mathrm{0, 1}\right), b > a\;and\;\sigma \in \left[\mathrm{0, 1}\right],$ the definition of Atangana-Baleanu derivative in the Caputo sense is given by

where

By using Sumudu transform (ST) for (1), we obtain

Definition 2.2. The Laplace transform of the Caputo fractional derivative of a function $g\left(t\right)$ of order $\sigma > 0$ is defined as

Definition 2.3. The Laplace transform of the function ${t}^{{\sigma }_{1}-1}{E}_{\sigma, {\sigma }_{1}}\left(\pm \mu {t}^{\sigma }\right)$ is defined as

Where ${E}_{\sigma, {\sigma }_{1}}$ is the two-parameter Mittag-Leffler function with $\sigma, {\sigma }_{1} > 0.$ Further, the Mittag-Leffler function satisfies the following equation ^[17].

Definition 2.4. Suppose that $g\left(t\right)$ is continuous on an open interval $(a, b),$ then the fractal-fractional integral of $g\left(t\right)$ of order $\sigma$ having Mittag-Leffler type kernel and given by

3.   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 $Sp$ and infected people $Ip$ so that $Np = Sp+Ip$. Same as we make two portions of a total number of mosquitoes $Nq$ into two groups: Susceptible mosquitoes Sq and infected mosquitoes $Iq$, so that $Nq = Sq+Iq$. 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 $q$, the effective contact rate human to human ${\beta }_{1}$, the effective contact rate of mosquitoes to human ${\beta }_{2}$, the effective contact rate human to mosquito's $\mu$, the natural death rate of human ${k}_{1}$, the natural death rate of mosquitoes ${k}_{2}$.

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 ${R}_{0}$ 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 ${R}_{+}^{4}.$

Proof.

The existence and uniqueness of the solution of the system (7) on the time interval $(0, \infty)$ can be obtained. Subsequently, we have to explain the non-negative region ${R}_{+}^{4}$ is a positively invariant region. From model (7) we find

If $({S}_{p}\left(0\right), {I}_{p}\left(0\right), {S}_{q}\left(0\right), {I}_{q}\left(0\right))\in {R}_{+}^{4},$ then according to Eq (7) the solution $\left[{S}_{p}\left(t\right), {I}_{p}\left(t\right), {S}_{q}\left(t\right), {I}_{q}\left(t\right)\right]$ cannot escape from the hyperplanes ${S}_{p} = 0, {I}_{p} = 0, {S}_{q} = 0\;and\;{I}_{q} = 0$. Also on each hyperplane bounding the non-negative orthant, the vector field points into ${R}_{+}^{4}$, i.e., the domain ${R}_{+}^{4}$ is a positively invariant set.

Theorem 3.2. The region $A = \left\{\right({S}_{p}\left(t\right), {I}_{p}\left(t\right), {S}_{q}\left(t\right), {I}_{q}\left(t\right))\in {R}_{+}^{4}|0 < {S}_{p}\left(t\right)+{I}_{p}\left(t\right)\le \frac{{\mathrm{\Lambda }}_{p}}{{\kappa }_{1}}, {S}_{q}\left(t\right)+{I}_{q}\left(t\right)\le \frac{{\mathrm{\Lambda }}_{q}}{{\kappa }_{1}}\}$ 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

where

we get

which further gives

We infer that if $({S}_{0p}, {I}_{0p})\in {R}_{+}^{4},$ then

Similarly, we can prove for ${\mathrm{N}}_{q}\left(t\right) = {S}_{q}\left(t\right)+{I}_{q}\left(t\right)$ that if ${\mathrm{N}}_{q}\left(t\right) = \frac{{\mathrm{\Lambda }}_{q}}{{\kappa }_{1}}$.

4.   Atangana-Baleanu Caputo sense

In this section, coniseder the system with Atangana-Baleanu fractional derivative (ABC) of order $\sigma$ and $\sigma \in \left(\left.\mathrm{0, 1}\right]\right.$ 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 (${X}_{2}, \left|\right|.\left|\right|)$ be a Banach space and M be a self-map of ${X}_{2}$ satisfying ||${K}_{X}-{K}_{Y}$||$\leq$C||X$-{K}_{X}|$|+C||X-Y|| for all x, y $\in {X}_{2}$ where 0$\le C, 0\le c < 1$.

Let consider that M is P-stable. Let us take into account the following recursive formula:

where $\frac{1-\vartheta }{B\left(\vartheta \right)\vartheta \mathrm{\Gamma }\left(\alpha +1\right){N}_{\vartheta }\left(-\frac{1}{1-\vartheta }{w}^{\vartheta }\right)}$ is the fractional Lagrange multiplier.

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 $H = {L}^{2}\left(\left(p, q\right)\times \left(0, T\right)\right)$ which is given as

In this regard, the following operators are considered

We establish that the inner product of

Where $\left({{S}_{p}}_{11}-{{S}_{p}}_{12}, {{I}_{p}}_{21}-{{I}_{p}}_{22}, {{S}_{q}}_{31}-{{S}_{q}}_{32}, {{I}_{q}}_{41}-{{I}_{q}}_{42}, \right),$ are the special solutions of the system. Taking into account the inner function and the norm, we have

Due to the large number of ${e}_{1}, {e}_{2}, {e}_{3}, {e}_{4}\;and\;{e}_{5},$ both solutions converge to the exact solution. Applying the topological idea, we have the very small positive five parameters $\left({\chi }_{{e}_{1}}, {\chi }_{{e}_{2}}, {\chi }_{{e}_{3}}, {\chi }_{{e}_{4}}\;and\;{\chi }_{{e}_{5}}\right).$

But, it is obvious that

where $‖{V}_{1}‖, ‖{V}_{2}‖, ‖{V}_{3}‖, ‖{V}_{4}‖, ‖{V}_{5}‖\ne 0$.

Therefore, we have

which yields that

Hence proved.

5.   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) in the form of a fractional integral equation by applying the fundamental theorem of fractional calculus.

At a given point ${t}_{n+1}, n = \mathrm{0, 1}, \mathrm{2, 3}, \dots,$ the above equation is reformulated as

Within the interval $\left[{t}_{k}, {t}_{k+1}\right],$ the function $f\left(\tau, A\left(\tau \right)\right),$ 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

where

We obtain the following for model (7)

6.   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 $k\left(t, y\left(t\right)\right) = {\alpha }_{2}{t}^{{\alpha }_{2}-1}f\left(t, y\left(t\right)\right),$ then system (21) becomes

At ${t}_{n+1} = \left(n+1\right)\Delta t,$ we have

Also, we have

Approximating the function $k\left(t, y\left(t\right)\right),$ 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)

7.   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, ${S}_{h}\left(t\right)$, ${S}_{m}\left(t\right)$ and ${I}_{h}\left(t\right)$ start increase steadily by decreasing the fractional values while ${I}_{m}\left(t\right)$ 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, ${S}_{h}\left(t\right)$, ${S}_{m}\left(t\right)$ and ${I}_{h}\left(t\right)$ start increase strictly by decreasing the fractional values while ${I}_{m}\left(t\right)$ 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, ${S}_{h}\left(t\right)$, ${S}_{m}\left(t\right)$ and ${I}_{h}\left(t\right)$ start increase strictly by decreasing the fractional values while ${I}_{m}\left(t\right)$ 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.

8.   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.

Acknowledgments

Research Supporting Project number (RSP-2021/167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors have no conflict of interest.