A mathematical theoretical study of a particular system of Caputo–Fabrizio fractional differential equations for the Rubella disease model

*Correspondence: rezapourshahram@yahoo.ca 3Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran 4Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we study the rubella disease model with the Caputo–Fabrizio fractional derivative. The mathematical solution of the liver model is presented by a three-step Adams–Bashforth scheme. The existence and uniqueness of the solution are discussed by employing fixed point theory. Finally some numerical simulations are showed to underpin the effectiveness of the used derivative.


Introduction
Rubella was first described in the mid-eighteenth century. Friedrich Hoffmann made the first clinical description of rubella in 1740, which was confirmed by de Bergen in 1752 and Orlow in 1758 [1]. Rubella, also known as German measles or three-day measles, is an infection caused by the rubella virus and has symptoms that are similar to those of flu. However, the primary symptom of rubella virus infection is the appearance of a rash (exanthem) on the face which spreads to the trunk and limbs and usually fades after three days [2]. It usually spreads through the air via coughs of people who are infected. People are infectious during the week before and after the appearance of the rash [3]. This disease is often mild with half of people not realizing that they are infected [4]. Rubella is a common infection in many areas of the world, and each year about 100,000 cases of congenital rubella syndrome occur [5].
The mathematical model of measles and rubella has been studied by a number of mathematicians (see, for example, [6][7][8][9][10]). It has been demonstrated by many scientists and mathematicians that fractional extensions of mathematical models of integer order represent the natural fact in a very systematic way such as in the approach of Caputo [11], Podlubny [12], Baleanu et al. [13], Haq et al. [14], Atangana et al. [15], Erturk et al. [16], Kilbas et al. [17], Zafar et al. [18][19][20][21][22]. In a very recent attempt, Caputo and Fabrizio [11] propounded a novel fractional derivative having exponential kernel; in addition, Losada and Nieto [23] analyzed the properties of a newly presented fractional derivative. The classical fractional derivatives, especially the Caputo and Riemann derivatives, have their own limitation because their kernel is singular. Since the kernel is employed to describe the memory effect of the physical system, it is obvious that due to this weakness, both derivatives cannot precisely describe the full effect of the memory. Recently, many works related to the fractional equations and applications have been published (see, for example, [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). Therefore, we use the novel Caputo-Fabrizio (CF) fractional derivative to study the rubella disease model and explain this problem in a better and more efficient manner. We recall some fundamental notions. The Caputo fractional derivative of order α for a continuous function f is defined by Our second notion is a fractional derivative without singular kernel introduced by Caputo and Fabrizio [11,34]. Let b > 0, u ∈ H 1 (a, b), and α ∈ (0, 1). The Caputo-Fabrizio derivative of order α for a function f is defined by where t > 0 and M(α) is a normalization constant depending on α such that M(0) = M(1) = 1. It is well known that the Laplace transform plays an important role in the study of ordinary differential equations [23]. Let α ∈ (0, 1) and n ≥ 1. The Laplace transform of CF D is defined by (for n = 1). The Riemann-Liouville fractional integral of order 0 < α < 1 is defined by [12] whenever the integral exists. Also, the fractional integral of Caputo-Fabrizio is defined by [23] Losada and Nieto gave an explicit formula for M(α) as M(α) = 2 2-α (for 0 < α < 1). Thus, the fractional Caputo-Fabrizio derivative of order 0 < α < 1 for a function u is given by CF For n ≥ 1 and α ∈ (0, 1), the fractional derivatives CF D α+n of order n + α are defined by CF D α+n u(t) := CF D α (D n u(t)) [26].

Mathematical model for the spread of rubella disease
In this section, we investigate the rubella disease model discussed by Koca [8]. He considered the model of rubella disease by employing the Atangana-Baleanu fractional derivative as follows: where S(t), E(t), I(t), R(t), V (t) are susceptible, latent, infectious, recovered, and vaccinated parameters respectively. P(a) is a parameter for which immunized by vaccination and λ(a, t) is the force of infection of age a at time t and σ is the latent rate and β is the infection rate [40]. In this section, we moderate the system by substituting the timederivative by the newly introduced Caputo-Fabrizio derivative [11] for α ∈ (0, 1) given by with initial conditions In system (2), the right-hand sides of the equations have dimension (time) -1 . When we change the order of the equations to α, the dimension of the left-hand side would be (time) (-α) . In order to have the dimensions match, we should change the dimensions of the parameters σ , β, and the system we obtain eventually is The system state is made up with S, E, I, R, V .

Existence and uniqueness of a system of solutions of rubella model
We examine the existence of the system of solutions by applying the fixed point theorem.
Employing the fractional integral operator due to Nieto and Losada [23] on equation (3), (4) By using the notation presented by Nieto and Losada [23], we get For clarity, we write

Theorem 1 The kernel P 1 satisfies the Lipschitz condition and contraction if the following inequality holds:
Proof Let S and S 1 be two functions, then we assess the following: Taking γ 1 = λ(a, t) + P(a) + μ(a) are bounded functions, we get Hence the Lipschitz condition is satisfied for P 1 . If additionally then it is also a contraction for P 1 .
Similarly, the kernels P 2 , P 2 , P 3 , P 4 , P 5 satisfy the Lipschitz condition given as follows: On consideration of the aforesaid kernels, equation (5) becomes Now, we present the following recursive formula on consideration of the aforesaid kernels, equation (5) becomes The initial conditions are given as follows: Now we present the difference between the successive terms in the following manner: It is worth noticing that On the other hand, By using the triangular inequality, we get Since the kernel satisfies the Lipschitz condition, we have then we get Similarly, we get the following results: By taking the above results, we can present the following theorem.

Theorem 2 Fractional rubella model (3) has a system of solutions under the conditions
that we can find t 0 such that Proof We have considered that the functions S(t), E(t), I(t), R(t), V (t) are bounded. Additionally, we have proven that the kernels satisfy the Lipschitz condition, hence by taking the results of equations (19) and (20) and by employing the recursive method, we derive the succeeding relation as follows: Therefore, the system of functions (15) exists and is smooth. To show that the above functions are a system of solutions of the system of equation (3), we assume Therefore, we get By using this process recursively, it yields Now, taking the limit on equation (25) as n tends to infinity, we get Similarly, we have This completes the proof.
To prove the uniqueness of a system of solutions of equation (3), we present the following theorem.

Theorem 3 The system of equations (3) has a unique system of solutions if the following condition holds:
Proof Let there exist another system of solutions of (3) Applying the norm on equation (29), we get By employing the Lipschitz conditions of the kernel, we get It gives On the other hand, (28) holds, then We conclude from (31) and (32) that Then we get S(t) = S 1 (t). Similarly, we have This completes the proof.

Equilibrium points of the model and asymptotic stability
To determine the equilibrium points of fractional order system (3) To investigate the stability of equilibrium point, first consider the fractional-order linear system as follows: where u(t) ∈ R n , T ∈ R n×n , 0 < α < 1. The Jacobian matrix associated with system (3) is given as follows: Thus the characteristic equation of system (3) is Proof According to the above Jacobian matrix, we have J = J(E * ). Therefore, it is sufficient that we compute the roots of equation (39). We have By solving this algebraic equation, we get s 1 = -α(λ(a, t) + P(a) + μ(a)) 1 + (1α)(λ(a, t) + P(a) + μ(a)) , , μ(a)) , Since α ∈ (0, 1), then s 1 , s 2 , s 3 , s 4 are negative. Hence by using Theorem 5, the equilibrium point E * of model (3) is asymptotically stable.

Numerical method and simulations
In this section, using the Adams-Bashforth scheme, we present a numerical solution for the rubella model (3). Owolabi and Atangana introduced the three-step Adams-Bashforth scheme with the Caputo-Fabrizio fractional derivative [42]. We use this method to find three step Adams-Bashforth scheme for fractional order system (3). Consider the fractional differential equation with Caputo-Fabrizio derivative fined from system (3) as follows:

t, x(t)) = B(a) -[λ(a, t) + P(a) + μ(a)]S(t),
(46) We write system (3) in the vector form as follows: Using equation (44), we obtain the solution of system (3) as the following iterative formula: , then Thus, we obtain the iterative formulas Plots of all variables in model (3) with α = 0.9  Fig. (1) shows the plots of the solutions of model (3) for α = 0.9. As can be seen, the system is stable at equilibrium point. Figures 2-4 show plots for S(t), E(t), I(t), R(t), V (t) in model (3) for the fractional orders α = 0.9, 0.8, 0.7, 0.6, respectively. We can observe from these plots that the curves of each variable have the same trend when α is changed. However, their values are slightly different. We can observe from Fig. 2 that the curves of S(t) are decreasing, and they finally converge to the equilibrium point S * = 90.9. Figure 2 shows that all the graphs of E(t) increase with time and tend to the equilibrium point E * = 49.24.

Conclusion
In this paper, we have investigated a Caputo-Fabrizio fractional differential equation model for the spread of rubella disease. Using fixed point theory, we have demonstrated the existence of a unique solution. Also, we have determined the equilibrium point of the model and investigated its stability. We have used a three-step fractional Adams-Bashforth scheme to obtain numerical results of the fractional system of rubella model. Eventually, we have presented the numerical simulations for different values of the    fractional-order α = 0.9, 0.8, 0.7, 0.6 and have compared the numerical results of the standard derivative with two fractional derivatives of rubella model for α = 0.95.