Keywords

1 Introduction

There is no denying fact that alcohol consumption not only effects in the area of disorders but also influences the incidences of chronic diseases, injuries, and few health problems. This is because alcoholic beverages have become a part of many cultures for thousands of years. The impact of alcohol consumption is categorized in three factors, namely (i) the quality of alcohol consumed, (ii) the volume of alcohol consumed and (iii) the consumption pattern, on rare occasions. In context with the above three categories, alcohol consumption has become a detrimental and beneficial health impact. For instance few epidemiological and animal studies suggested that excessive alcohol consumption depresses cardiac function and causes cardiomyopathy or cardiomyopathy injury. The heavy alcohol consumption not only depresses cardiac function, but also it includes being thirsty, tired, sleepy, drowsy, weak and nauseaus as weil as having dry mouth and headache and several other concentration problems [1,2,3,4,5,6,7,8]. On the other hand, the World Health Organization has suggested a terrifying report, the harmful utilization of alcohol causes approximately \(5.1 \%\) of the global burden of disease is which as attributable to alcohol consumption and 3.3 million deaths every year (or \(5.9 \%\) of all the global deaths) [9]. The non-integer order derivatives have attracted many researchers and scientists due to its several significant applications in science and engineering; these derivatives model the various dynamical processes and they carry information regarding their present as well as past states (memory effects). In order to characterize memory property of complex systems, one need to employ the non-integer order derivatives because these operators give a complete description of different physical processes with dissipation and long-range interaction. On the basis of non-integer order derivatives, several researchers have utilized these derivatives among different physical aspects. For instance, pharmacokinetics [10], anomalous diffusion [11,12,13,14], control theory [15], electromagnetism [16, 17], rheological fluids [18,19,20,21], electrical engineering [22, 23], and heat transfer [24, 25]. Ludwin in [26] investigated the blood alcohol content as a function of time by employing direct integration method for the exact solutions of the concentrations of alcohol in stomach and alcohol in the blood respectively. Here, an experimental data was depicted with exact solutions of the concentrations of alcohol in stomach and alcohol in the blood and it was suggested that the average accuracy of the model was found to be \(94.4 \%\). Almeida et al. [27] compared integer and fractional models versus real experimental data. Liouville–Caputo fractional derivative was considered to solve the fractional order differential equations that describe the process studied. A numerical optimization approach based on least squares approximation was used to determine the order of the fractional derivative that better describes real experimental data, as well as other related parameters. Aqsa et al. [28] traced out the mathematical modeling of \(CD4+T-\) cells through analytical technique Optimal Variational Iteration Method (OVIM) on the system of governing nonlinear differential equations.

In this paper, our aim is to analyze the blood alcohol model via Caputo–Fabrizio and Atangana–Baleanu fractional derivatives in the Liouville–Caputo sense. The governing ordinary differential equations of blood alcohol model have been converted in terms of non-integers order derivatives. The analytic calculations of the concentrations of alcohol in stomach \((C_1(t))\) and the concentrations of alcohol in the blood \((C_2(t))\) have been investigated by applying Laplace transform method. The general solutions of the concentrations of alcohol in stomach \((C_1(t))\) and the concentrations of alcohol in the blood \((C_2(t))\) are expressed in terms of wright function \(\varPhi (a,b;c)\).

2 Fractional Modeling of Blood Alcohol Model

The governing ordinary differential equations of blood alcohol model can be written as [27]

$$\begin{aligned} \frac{dC_1(t)}{dt}+R_1C_1(t)=0, \end{aligned}$$
(1)
$$\begin{aligned} \frac{dC_2(t)}{dt}-R_1C_1(t)+R_2C_2(t)=0, \end{aligned}$$
(2)

where \(C_1(t)\) and \(C_2(t)\) represent the concentrations of alcohol in stomach and alcohol in the blood respectively. \(R_1\) and \(R_2\) are non-zero constants. While the corresponding initial conditions are taken into consideration for Eqs. (1) and (2) which satisfy the analytic solutions for validations which are defined as

$$\begin{aligned} C_1(0)=C_0,\qquad \qquad \qquad \qquad C_2(0)=0. \end{aligned}$$
(3)

Equation (3) is the imposed condition for Eqs. (1) and (2) respectively. Converting Eqs. (1) and (2) in terms of non-integer time derivatives of Atangana–Baleanu–Caputo type, we have [29,30,31]

$$\begin{aligned} ^{ABC}\Bigg (\frac{d^{\alpha _1}C(t)}{dt^{\alpha _1}}\Bigg )=\int _{0}^{t}E_{\alpha _1}\Bigg [-\frac{\alpha _1 (z-t)^{\alpha _1}}{1-\alpha _1}\Bigg ]\frac{C'(t)}{1-\alpha _1}dt,\qquad 0<\alpha _1 \le 1. \end{aligned}$$
(4)

Meanwhile, in the Caputo–Fabrizio–Caputo sense, we get [32,33,34,35]

$$\begin{aligned} ^{CFC}\Bigg (\frac{d^{\beta _1}C(t)}{dt^{\beta _1}}\Bigg )=\int _{0}^{t}\exp \Bigg [-\frac{\beta _1 (z-t)^{\beta _1}}{1-\beta _1}\Bigg ]\frac{C'(t)}{1-\beta _1}dt,\qquad 0<\beta _1 \le 1. \end{aligned}$$
(5)

Using Eqs. (4) and (5), we arrive at the generalized equations in fractional form as

$$\begin{aligned} \frac{d^{\alpha _1}C_1(t)}{dt^{\alpha _1}}+R_1C_1(t)=0, \end{aligned}$$
$$\begin{aligned} \frac{d^{\alpha _2}C_2(t)}{dt^{\alpha _2}}-R_1C_1(t)+R_2C_2(t)=0, \end{aligned}$$
(6)

and

$$\begin{aligned} \frac{d^{\beta _1}C_1(t)}{dt^{\beta _1}}+R_1C_1(t)=0, \end{aligned}$$
$$\begin{aligned} \frac{d^{\beta _2}C_2(t)}{dt^{\beta _2}}-R_1C_1(t)+R_2C_2(t)=0, \end{aligned}$$
(7)

where the fractional derivatives are in Atangana–Baleanu–Caputo and Caputo–Fabrizio–Caputo sense, respectively.

The system of fractional differential equations (6) and (7) can be solved by the technique of Laplace transforms with inversions. Even fractional differential equations (6) and (7) can be explained in principle by enormous methodologies and their effectiveness is usually subjective by the domain of definition.

3 Solution of the Fractional Blood Alcohol Model

Calculation of the Problem via Atangana–Baleanu–Caputo Fractional Operator

Applying Laplace transform on fractional differential equations (6) and using imposed condition for Eq. (3), we obtain

$$\begin{aligned} \frac{s^{\alpha _1}\eta _1C_1(s)-C_0}{s^{\alpha _1}+\alpha _1\eta _1}+R_1C_1(s)=0, \end{aligned}$$
$$\begin{aligned} \frac{s^{\alpha _2}\eta _2C_2(s)}{s^{\alpha _2}+\alpha _2\eta _2}-R_1C_1(s)+R_2C_2(s)=0, \end{aligned}$$
(8)

where \(\eta _1=\frac{1}{1-\alpha _1}\) and \(\eta _2=\frac{1}{1-\alpha _2}\) represent the letting parameters. Simplifying Eq. (8), we get

$$\begin{aligned} C_1(s)=\frac{C_0\lambda _0\eta _1}{s^{\alpha _1}+\lambda _1}, \end{aligned}$$
$$\begin{aligned} C_2(s)=\frac{C_0R_1\lambda _0\lambda _2(s^{\alpha _2}+\alpha _2\eta _2)}{(s^{\alpha _1}+\lambda _1)(s^{\alpha _2}+\lambda _3)}, \end{aligned}$$
(9)

where \(\lambda _0=\frac{1}{\eta _1+R_1}\), \(\lambda _1=\frac{\eta _1R_1\alpha _1}{\eta _1+R_1}\), \(\lambda _2=\frac{1}{\eta _2+R_2}\) and \(\lambda _3=\frac{\eta _2 R_2 \alpha _2}{\eta _2+R_2}\) are the rheological parameters. Employing the fact of infinite series \(\frac{1}{1+x}=\sum _{n=0}^{\infty }(-x)^n\) [36] on Eq. (9), we investigated the equivalent form of Eq. (9) as

$$\begin{aligned} C_1(s)=\frac{C_0\lambda _0\eta _1}{\lambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\frac{1}{s^{-n\alpha _1}}, \end{aligned}$$
$$\begin{aligned} C_2(s)=\frac{C_0R_1\lambda _0\lambda _2}{\lambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Big (-\lambda _3\Big )^m \frac{1}{s^{-n\alpha _1+m\alpha _2}}+ \end{aligned}$$
$$\begin{aligned} +\frac{C_0R_1\lambda _0\lambda _2\alpha _2\eta _2}{\lambda _1\lambda _3}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Bigg (-\frac{1}{\lambda _3}\Bigg )^m\frac{1}{s^{-n\alpha _1-m\alpha _2}}. \end{aligned}$$
(10)

Applying inverse Laplace transform on Eq. (10), we have

$$\begin{aligned} C_1(t)=\frac{C_0\lambda _0\eta _1}{\lambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\frac{t^{-n\alpha _1-1}}{\varGamma (-n\alpha _1)}, \end{aligned}$$
$$\begin{aligned} C_2(t)=\frac{C_0R_1\lambda _0\lambda _2}{\lambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Big (-\lambda _3\Big )^m\frac{t^{-n\alpha _1+m\alpha _2-1}}{\varGamma (-n\alpha _1+m\alpha _2)}+ \end{aligned}$$
$$\begin{aligned} +\frac{C_0R_1\lambda _0\lambda _2\alpha _2\eta _2}{\lambda _1\lambda _3}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Bigg (-\frac{1}{\lambda _3}\Bigg )^m\frac{t^{-n\alpha _1-m\alpha _2-1}}{\varGamma (-n\alpha _1-m\alpha _2)}. \end{aligned}$$
(11)

In order to eliminate the Gamma function, Eq. (11) is expressed in terms of wright function \(\varPhi (a,b;c)=\sum _{n=0}^{\infty }\frac{(c)^n}{n!\varGamma (a-bn)}\), we obtain the final expression of concentrations of alcohol in stomach and concentrations of alcohol in the blood as

$$\begin{aligned} C_1(t)=\frac{C_0\lambda _0\eta _1}{\lambda _1 t}\varPhi \Bigg (0,-\alpha _1;-\frac{1}{\lambda _1 t^{\alpha _1}}\Bigg ), \end{aligned}$$
$$\begin{aligned} C_2(t)=\frac{C_0R_1\lambda _0\lambda _2}{\lambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n\varPhi \Big (-\alpha _1 n, \alpha _2;-\lambda _3 t^{\alpha _2}\Big )+ \end{aligned}$$
$$\begin{aligned} +\frac{C_0R_1\lambda _0\lambda _2\alpha _2\eta _2}{\lambda _1\lambda _3}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\lambda _1}\Bigg )^n \varPhi \Bigg (-\alpha _1 n, \alpha _2;-\frac{1}{\lambda _3 t^{\alpha _2}}\Bigg ). \end{aligned}$$
(12)

Equation (12) represents the final expressions for concentration of alcohol in stomach and concentration of alcohol in the blood in terms of Atangana–Baleanu fractional operator in Liouville–Caputo sense. It is also pointed out that one can retrieve classical concentration of alcohol in stomach and concentration of alcohol in the blood by taking \(\alpha _1=\alpha _2=1\) in Eq. (12).

Calculation of the Problem via Caputo–Fabrizio–Caputo Fractional Operator

Applying Laplace transform on fractional differential equations (7) and using imposed condition for Eq. (3), we obtain

$$\begin{aligned} \frac{s\eta _3C_1(s)-C_0}{s+\beta _1\eta _3}+R_1C_1(s)=0, \end{aligned}$$
$$\begin{aligned} \frac{s\eta _4C_2(s)}{s+\beta _2\eta _4}-R_1C_1(s)+R_2C_2(s)=0, \end{aligned}$$
(13)

where \(\eta _3=\frac{1}{1-\beta _1}\) and \(\eta _4=\frac{1}{1-\beta _2}\) represent the letting parameters. Simplifying Eq. (13), we get

$$\begin{aligned} C_1(s)=\frac{C_0\varLambda _0\eta _3}{s+\varLambda _1}, \end{aligned}$$
$$\begin{aligned} C_2(s)=\frac{C_0R_1\varLambda _0\varLambda _2(s+\beta _2\eta _4)}{(s+\varLambda _1)(s+\varLambda _3)}, \end{aligned}$$
(14)

where \(\varLambda _0=\frac{1}{\eta _3+R_1}\), \(\varLambda _1=\frac{\eta _3R_1\beta _1}{\eta _3+R_1}\), \(\varLambda _2=\frac{1}{\eta _4+R_2}\) and \(\varLambda _3=\frac{\eta _4 R_2 \beta _2}{\eta _4+R_2}\) are the rheological parameters. Employing the fact of infinite series \(\frac{1}{1+x}=\sum _{n=0}^{\infty }(-x)^n\) [36] on Eq. (14), we investigated the equivalent form of Eq. (14) as

$$\begin{aligned} C_1(s)=\frac{C_0\varLambda _0\eta _3}{\varLambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\frac{1}{s^{-n}}, \end{aligned}$$
$$\begin{aligned} C_2(s)=\frac{C_0R_1\varLambda _0\varLambda _2}{\varLambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Big (-\varLambda _3\Big )^m \frac{1}{s^{-n+m}}+ \end{aligned}$$
$$\begin{aligned} +\frac{C_0R_1\varLambda _0\varLambda _2\beta _2\eta _4}{\varLambda _1\varLambda _3}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Bigg (-\frac{1}{\varLambda _3}\Bigg )^m\frac{1}{s^{-n-m}}. \end{aligned}$$
(15)

Applying inverse Laplace transform on Eq. (15), we have

$$\begin{aligned} C_1(t)=\frac{C_0\varLambda _0\eta _3}{\varLambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\frac{t^{-n-1}}{\varGamma (-n)}, \end{aligned}$$
$$\begin{aligned} C_2(t)=\frac{C_0R_1\varLambda _0\varLambda _2}{\varLambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Big (-\varLambda _3\Big )^m\frac{t^{-n+m-1}}{\varGamma (-n+m)}+ \end{aligned}$$
$$\begin{aligned} +\frac{C_0R_1\varLambda _0\varLambda _2\beta _2\eta _4}{\varLambda _1\varLambda _3}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\sum _{m=0}^{\infty }\Bigg (-\frac{1}{\varLambda _3}\Bigg )^m\frac{t^{-n-m-1}}{\varGamma (-n-m)}. \end{aligned}$$
(16)

In order to eliminate the Gamma function, Eq. (16) is expressed in terms of the wright function \(\varPhi (a,b;c)=\sum _{n=0}^{\infty }\frac{(c)^n}{n!\varGamma (a-bn)}\), we obtain the final expression of concentrations of alcohol in stomach and concentrations of alcohol in the blood as

$$\begin{aligned} C_1(t)=\frac{C_0\varLambda _0\eta _3}{\lambda _1 t}\varPhi \Bigg (0,-1;-\frac{1}{\varLambda _1 t}\Bigg ), \end{aligned}$$
$$\begin{aligned} C_2(t)=\frac{C_0R_1\varLambda _0\varLambda _2}{\varLambda _1}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n\varPhi \Big (-n,1;-\lambda _3 t\Big )+ \end{aligned}$$
$$\begin{aligned} +\frac{C_0R_1\varLambda _0\varLambda _2\beta _2\eta _4}{\varLambda _1\varLambda _3}\sum _{n=0}^{\infty }\Bigg (-\frac{1}{\varLambda _1}\Bigg )^n \varPhi \Bigg (-n,1;-\frac{1}{\varLambda _3 t}\Bigg ). \end{aligned}$$
(17)

Equation (17) represents the final expressions for concentration of alcohol in stomach and concentration of alcohol in the blood in terms of Atangana–Baleanu fractional operator in Liouville–Caputo sense. It is also pointed out that one can retrieve classical concentration of alcohol in stomach and concentration of alcohol in the blood by taking \(\alpha _1=\alpha _2=1\) in Eq. (17).

Fig. 1
figure 1

Comparative analysis of the concentrations of alcohol in the stomach \(C_1(t)\) via Atangana–Baleanu–Caputo and Caputo–Fabrizio–Caputo fractional derivatives for smaller and larger time when embedded parameter are \(C_0=225\), \(R_1=0.025\), \(R_2=0.031\), \(\alpha _1=\beta _1=[0,1]\)

Full size image
Fig. 2
figure 2

Comparative analysis of the concentrations of alcohol in the stomach \(C_2(t)\) via Atangana–Baleanu–Caputo and Caputo–Fabrizio–Caputo fractional derivatives for smaller and larger time when embedded parameter are \(C_0=225\), \(R_1=0.025\), \(R_2=0.031\), \(\alpha _1=\beta _1=[0,1]\)

Full size image

4 Results and Conclusions

Now, a comparative analysis of blood alcohol model via Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional derivatives is studied. The ordinary differential equations of blood alcohol model are generalized via non-integers order derivatives. The analytic calculations of the concentrations of alcohol in stomach \(C_1(t)\) and the concentrations of alcohol in the blood \(C_2(t)\) have been traced out by implementing a powerful technique namely Laplace transform method. The general solutions of the concentrations of alcohol in stomach \(C_1(t)\) and the concentrations of alcohol in the blood \(C_2(t)\) are expressed in the wright function \(\varPhi (a,b;c)\). The graphical illustrations are based on smaller time \(t=0.02\) s and larger time \(t=5\) s, for both types of concentrations. The influence of the fractional derivatives namely Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional operators on the concentrations of alcohol in stomach \(C_1(t)\) is depicted through Mathcad software as shown in Fig. 1a–f for smaller time \(t=0.02\) s and larger time \(t=5\) s. It can be seen from Fig. 1a–f that the comparative analysis for the concentrations of alcohol in stomach \(C_1(t)\) suggested that for smaller time \(t=0.02\) s, the concentration of alcohol in stomach \(C_1(t)\) has increasing behavior with Atangana–Baleanu–Caputo approach but decreasing with Caputo–Fabrizio–Caputo approach. While, for larger time \(t=5\) s, the concentration of alcohol in stomach \(C_1(t)\) with Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional derivatives have reciprocal behavior. On the other hand, for unit time \(t=1\)s, the concentrations of alcohol in stomach \(C_1(t)\) are identical in both fractional approaches. The rheological impacts of the fractional derivatives of Caputo–Fabrizio–Caputo and Atangana–Baleanu–Caputo fractional type on the concentrations of alcohol in blood \(C_2(t)\) are showed in Fig. 2a–f for smaller time \(t=0.02\) s and larger time \(t=5\) s. It is observed that in the Atangana–Baleanu–Caputo fractional operators decrease the concentration of alcohol in blood \(C_2(t)\) for smaller time \(t=0.02\) s but for larger time \(t=5\) s, the concentration of alcohol in blood \(C_2(t)\) increases in the Atangana–Baleanu–Caputo fractional derivative. Meanwhile, an opposite trend is perceived in the Caputo–Fabrizio–Caputo sense for the concentration of alcohol in blood \(C_2(t)\). Finally, the comparative analysis of both fractional types of concentration of alcohol level in blood decay faster for higher fractional order. It is also pointed out that the comparison of fractional model verses ordinary/classical model can also be depicted as well for both concentrations of alcohol in stomach \(C_1(t)\) and in blood \(C_2(t)\).