On rotavirus infectious disease model using piecewise modified $ABC$ fractional order derivative

1.   Introduction

Rotavirus is an acute gastrointestinal pathogen that mainly affect infants and young children under the age of five throughout the world. The virus of the said disease was first identified by Bishop ^[1], when he found wheel-shaped particles in bowel specimens of a child. It can cause acute gastroenteritis and diarrhea ^[2,3]. According to the data on the peak incidence, which occurs between four and 36 months of age, about 95 percent of children worldwide have contracted the disease ^[4]. Its name comes from the fact that the virus resembles a wheel when viewed under a microscope. Its symptoms usually appear about two days after a person is exposed to rotavirus and remain for eight days ^[5,6,7]. Symptoms include fever, vomiting, pains in the abdomen, nausea, and recurrent watery diarrhea. Recurrent watery diarrhea causes severe dehydration that can be dangerous, especially for infants and young children. Therefore, limiting dehydration is essential for disease control. The oral-fecal pathway, hand-to-mouth contact, infected objects and surfaces, and potentially the respiratory route are the main ways that the aforementioned virus is spread ^[8,9,10,11]. The virus takes around two days to incubate ^[12,13,14].

It has been noted that the rate of rotavirus infection is comparable in industrialized and underdeveloped nations, indicating that improvement to sanitation, hygiene, or water supply may not be sufficient to realize adequate control measures. Therefore, it is advised to develop, test, and utilize rotavirus vaccines widely in order to prevent severe and also fatal rotavirus sickness ^[15]. The World Health Organization recommended in June 2009 that rotavirus vaccination shall be necessitated in every country's immunization program on the basis of recent, persuasive data on the illness load of rotavirus and the ability of vaccinations to stop it in situations with limited resources ^[16].

Recent studies clarify that mathematical modeling can be used to obtain a better understanding of the dynamics of rotavirus transmission, predict its effects in particular countries, and evaluate the potential benefits of interventions. Epidemiology is being studied for treatment, cure, control, and other outcomes for a number of infectious diseases because of the technological advancements in the field. Consequently, throughout the past few decades, there has been a significant advancement in the mathematical modeling of infectious diseases (see ^[17,18]). Mathematical models can be used in secure public health systems to effectively control a variety of diseases, including those listed in a previous paper ^[19]. Both the dynamic behavior of infections and spatial temporal patterns can be studied by using these mathematical models. Over the past three years, researchers have examined rotavirus sickness from a number of perspectives by utilizing and exploiting the importance of mathematical models. Many approaches are being used by researchers in this field to provide workable protocols to manage this sickness (see ^[20,21]). Some researchers ^[22] have analyzed at a mathematical model that was designed to look into how vaccinations and breastfeeding affect the rotavirus pandemic.

The mentioned infectious disease has been studied through the use of mathematical modeling recently. In addition, recent studies that also examined state-of-the-art techniques for diagnosis and treatment have identified risk factors for severe rotavirus infection. Sustained investigation in these domains is crucial in the fight against rotavirus and the mitigation of fatalities resulting from this pathogen. Focusing on the aforementioned disease, a group of authors studied the following compartmental model ^[23]:

Fractional calculus has garnered much attention by researchers in the last few decades. The mentioned area has been utilized very well to study various problems in the real world.

Here, we remark that researchers^[24] studied various biological problems by using fractional calculus to investigate the bifurcation and control analysis. In the same way, other researchers ^[25] have studied some problems related to neural networks by using fractional derivatives. Another group of authors ^[26] used tools of fractional calculus to investigate a mixed-controller neural network problem. In addition, regarding numerous problems in neural network models, researchers have used the fractional-order derivatives to investigate bifurcation analysis in detail. In this regard, some recent contributions can be found in ^[27]. There are several definitions available in the current literature about the fractional calculus. The first notable definition was given by Reimann and Liouville, and it has been used in large numbers of articles. Also, Caputo, Grünwald-Letnikov, Hadamard, Hilfer, and Riesz have defined fractional-order differentiations with their own concepts. For some historical improvement in the mentioned area, readers should read ^[28]. Recently, in 2015 and 2016, two new differential operators were introduced on the basis of exponential and Mittag-Leffler kernels, and they were called non-singular fractional differential operators. The derivative with an exponential kernel was given by Caputo and Fabrizio ^[29]. In the same way, Atangana and Baleanu introduced their derivatives on the basis of the Mittag-Leffler kernel ^[30]. The said nonlocal operators have various applications to the real world problems. The first definition given by Atangana and Baleanu in the Caputo sense is called $ABC$ ^[31]. The said derivative suffers from the lack of initialization condition. Therefore, some modification was needed to upgrade the said operator. Other authors ^[32] modified the former $ABC$ derivative to an updated operator which circumvents the mentioned drawback. The modified ABC $(mABC)$ derivative has been used in many research articles for applied analysis. Here, we refer the readers to ^[33,34].

Various evolutionary processes suffer from sudden changes in their state of rest or uniform motion. The multi-behavioral effect in dynamics as a result of these abrupt changes cannot be modeled by ordinary derivatives. As a result, a number of researchers have observed recently that piecewise differential equations can be used to design the aforementioned process more efficiently than integer-order equations. Piecewise operators of differentiations have received attention recently. For instance, a group of authors introduced the concepts in terms of fractional order (see ^[35]), where they used the traditional form of $ABC$ derivatives in the piecewise sense. Here, we remark that the piecewise form of the $mABC$ derivative has very rarely been used in epidemiological problems.

Therefore, motivated by the aforesaid work, for the analysis of crossover behavior, we chose to investigate the considered problem for qualitative, numerical, and stability analysis in the sense of the $mABC$ piecewise derivative. This article is focused on investigating the rotavirus disease model under the said operator, because the problem still has not been studied by using the piecewise version of $mABC$ derivatives. Here, we consider the model (1.1) with a vaccination class to analyze the considered rotavirus infection for $0 < \mathrm{p} < 1, \ t\in[0, T]$ by using the $mABC$ piecewise derivative as follows:

where $\mathrm{\mho} = (\mathcal{S}, \mathcal{V}, \mathcal{I}, \mathcal{R})$, and ${^{PABC}_{0}D_t}^\mathrm{p}$ represents the piecewise $ABC$ derivative, which can be defined in Eq (1.3) for the function $g$ as follows:

where $\mu_{\mathrm{p}} = \frac{\mathrm{p}}{1-\mathrm{p}}$, and $^{C}_0 D_t^\mathrm{p}$ and $^{mABC}_0 D_t^\mathrm{p}$ are the Caputo and $mABC$ derivatives, respectively. Further, the compartments, and parameters are described in Table 1 in detail.

In this case, we prove the positivity of the solution and the feasible region under the theory of the modified piecewise $ABC$ abbreviated as $mPABC$ fractional-order derivative. To realize the aforementioned outcomes, we chose to adhere the process described in ^[36,37]. Existence of a solution and uniqueness can be derived by using some tools of mathematical analysis ^[38]. Furthermore, we characterize the piecewise dynamics in order to observe crossover behaviors in the solution. The Lagrange's polynomial interpolation method^[39] is extended to simulate our results graphically. We use some real values for the parameters in the proposed model and present several graphical illustrations resulting from the numerical simulations.

The outline of this work is follows. A detailed introduction is given in Section 1. The basic results are given in Section 2. The qualitative analysis is given in Section 3. The numerical procedure is described in Section 4. Results and discussion are given in Section 5. Finally, the conclusion of the article is given in Section 6.

2.   Preliminaries

Here, we give some definitions that are needed in our work.

Definition 2.1. ^[32] The traditional ${mABC}$ derivative of a function $\phi\in L^1(0, T)$ with order $\mathrm{p}\in(0, 1)$ is defined in Eq (2.1) as follows:

where ${ABC}(\mathrm{p})$ satisfies, ${ABC}(0) = {ABC}(1) = 1$. Also, $\mathscr{E}_\mathrm{p}$ represents the Mittag-Leffler function. Then, $\Gamma(.)$ is defined by $\Gamma(\mathrm{p}) = \int_0^\infty t^{\mathrm{p}-1}\exp(-t)dt.$

Definition 2.2 ^[32] The Riemann-Liouville integral of fractional order $\mathrm{p}\in(0, 1)$ for the function $\phi\in L[0, t_1]$ as follows:

For $\phi\in L^1(t_1, T)$ with order $\mathrm{p}\in(0, 1),$ the modified Atangana-Baleanue $(mAB)$ integral is described in Eq (2.2) by

Lemma 2.3. ^[32] For $\phi\in C[0, T]$ and $\psi\in L^1[0, T],$ the solution of the problem with the $mABC$ derivative, i.e.,

is given by

Definition 2.4. ^[33]. The piecewise integral of a differentiable function $\phi$ is defined as follows:

Definition 2.5. ^[32,35]. From Definition 2.2 and the usual Riemann-Liouville integral of fractional-order for a function $\phi\in L[0, T],$ we define the piecewise integral in the $mAB$ sense as follows:

Definition 2.6. ^[32,35]. From Definition 2.1 and the usual Caputo fractional-order derivative, we define the piecewise $mABC$ derivative with order $\mathrm{p}\in(0, 1)$ as follows:

Lemma 2.7. ^[32,35]; Let $\phi\in L^1(0, T)$ and $\psi\in L^1[0, T];$ the solution of the problem with the $PABC$ derivative

is given by

Let us define a Banach space by $\mathbb{B} = C[0, T],$ with the norm described by $\|\mathrm{\mho}\| = \max_{t\in[0, T]}|\mathrm{\mho}(t)|.$

Theorem 2.8. ^[36] If the following hold true:

● $\mathbb{A}: \mathbb{B}\rightarrow \mathbb{B}$ is a compact operator;

● there exists a constant $\varepsilon > 0$ and $\mathrm{\mho} = \delta \mathbb{A}(\mathrm{\mho}), \delta\in (0, 1)$ such that $\| \mathbb{A}(\mathrm{\mho})\|\leq \varepsilon$;

then the operator equation $\mathbb{A}$ has at least one solution.

3.   Qualitative results

In this section, we establish some results about positivity, equilibrium points, the basic reproduction number, and its sensitivity. The mentioned results are investigated by using the $mABC$ derivative and integral transform.

Theorem 3.1. The set $\Upsilon = \{ (\mathcal{S}, \mathcal{V}, \mathcal{I}, \mathcal{R})\in \mathbb{R}^4_+: \ N\leq \frac{\mu}{\lambda}\}$ is positive- invariant. Additionally, every solution is drawn to $\mathbb{R}^4_+$.

Proof. From the model (1.2), and given that $N$ is the total population, we have

We can write Eq (3.1) as follows:

Taking the Laplace transform, and following ^[32] by using $\gamma_{\mathrm{p}} = ABC(\mathrm{p})+\lambda(1-\mathrm{p})$ and $\ N(0) = N_0,$ from Eq (3.2), the result follows:

If we apply $t\rightarrow \infty$ in Eq (3.3), we obtain that

Therefore, the region corresponds to that in which the solutions are bounded and $\Upsilon$ is positively invariant.□

Theorem 3.2. Let $\{(\mathcal{S}_0, \mathcal{V}_0, \mathcal{I}_0, \mathcal{R}_0)\geq 0 \} \in \mathbb{R}^4_+$; then, the set $\{(\mathcal{S}, \mathcal{V}, \mathcal{I}, \mathcal{R})\}$ of the solution of model (1.2) remains positive at all $t > 0$.

Proof. From the first equation of model (1.2), one has

Let us use $\omega = \kappa \mathcal{I}(t)-(\rho+\lambda)$ in Eq (3.4); we have

Now, we can write Eq (3.5) as follows:

We establish that

Taking the Laplace transform, one has

Now, consider the other case:

which leads to the Laplace transform by using the fact that $\gamma_\omega = ABC(\mathrm{p})+(1-\mathrm{p})\omega\ \text{and}\ \sigma = \frac{\mu_\mathrm{p}\omega(1-\mathrm{p})}{\gamma_\omega}$

Equation (3.6) implies that

Therefore, from Eq (3.7), we have that $\mathcal{S} > 0$ at every $t > 0$. Using the same process, for other compartments, we can prove that $\mathcal{V} > 0, \ \mathcal{I} > 0, \ \text{and}\ \mathcal{R} > 0.$ □

Remark 3.3. Furthermore, it can be easily seen that the disease-free equilibrium point is given by

Using the third equation of model (1.2), the basic reproduction number can be calculated by taking the following matrices as follows:

The Jacobian matrices at the trivial equilibrium point of $\mathbf{F}$ and $\mathbf{V}$ are respectively given by

We have

The total number of rotavirus infections that a single infected person can cause in the presence of vaccination is the basic reproduction number of the proposed model. Furthermore, $R_0$ indicates the basic reproduction number in the absence of such intervention. Using the next generation's matrix approach ^[32,35], we define $R_v$ as follows:

In the absence of vaccination, $\rho = \varsigma = \theta = 0$, we get the basic reproduction number as follows:

Using Eq (3.9), we can write Eq (3.8) as follows:

Since we have $0 < \zeta < 1,$ it follows from Eq (3.10) that $\frac{\lambda(1-\hat{\vartheta}+\zeta\hat{\vartheta})+\varsigma+\zeta\rho}{\rho+\lambda+\varsigma} < 1$, which means that $R_v < R_0$. Further, when $\zeta = 1$ or $\rho = \varsigma = \theta = 0$, $R_v = R_0$. The expression of $R_v$ demonstrates that vaccinations against new illnesses are beneficial for both susceptible individuals and those infected from birth. Keep in mind that $\zeta = 1$ suggests that vaccinations are not necessary, which is not the case.

Additionally, if $R_0 < 1$, the disease-free equilibrium will be asymptotically stable; if $R_0 > 1,$ it will be unstable. We present some 3D profiles for $R_0$ in Figure 1 by using different values of the parameters. This behavior indicates stabile behavior of the disease-free equilibrium point. We chose to take the parameter values from ^[23], as shown in Table 2.

We see that, for the given parameter values, $R_0 < 1,$ which indicates that the disease-free equilibrium point of the suggested model is asymptotically stable.

3.1.   Sensitivity analysis

Here, we present the sensitivity analysis for $R_v$ on the basis of the significance of parameters by using the procedure given in ^[40]. These indices emphasize the importance of each component in the development and spread of the disease. Sensitivity analysis has been used to evaluate how well the model's predictions hold up to various parameter values. In this way, we chose to use the following formula depends on the parameters to compute the sensitivity indices $p$:

Using Eq (3.11), we have

Here in Figure 2, we present the sensitivity indices based on the values in Eq (3.12).

3.2.   Existence of solution

Consider the following possible form for model (1.2)

Theorem 3.4. The solution of Eq (3.13) is given by

Proof. With the help of Lemma 2.7, we can easily obtain the solution. □

We describe some assumptions:

$(G_1)$ For the real values $\mathcal{L}_\Psi > 0$, and given that $\mathrm{\mho}, \mathrm{z}\in \mathbb{B}$, one has

$(G_2)$ For real values $\mathcal{M}_\Psi, \mathcal{N}_\Psi > 0$, one has

Theorem 3.5. In view of $(G_1)$, and if

holds, then the problem described by Eq (3.13) has a unique solution.

Proof. Let us define $\mathbb{A}: \mathbb{B}\rightarrow \mathbb{B}$ as follows:

For $\mathrm{\mho}, \mathrm{z}\in \mathbb{B}$, consider the following:

which further implies that

On further simplification of Eq (3.14), we have

Then, from Eq (3.14), we obtain the following:

Therefore, Eq (3.15) yields that $\mathbb{A}$ is a contraction operator. Hence, the problem described by Eq (3.13) has a unique solution. Consequently, the model (1.2) has a unique solution. □

Theorem 3.6. Using $(G_2)$, the problem described by Eq (3.13) has at least one solution.

Proof. We consider a bounded closed subset $\Omega = \{\mathrm{\mho}\in \mathbb{B}: \|\mathrm{\mho}\|\leq \varepsilon, \ \varepsilon > 0 \}\subseteq \mathbb{B}$. Then, we define $\mathbb{C}:\Omega\rightarrow \Omega$ such that $\mathrm{\mho}\in \Omega$ and we obtain the following:

from which we deduce that

Hence, using $(G_2)$, we get from Eq (3.16) that

Here, some parameters are presented for simplification:

Therefore, we can write Eq (3.17) as follows:

Assume that $\varepsilon\geq \max\bigg\{\frac{|\mathrm{\mho}_0|+t_1^{\mathrm{p}}\mathcal{M}_\Psi}{\Gamma(\mathrm{p}+1)-t_1^{\mathrm{p}}\mathcal{N}_\Psi}, \frac{\Delta_1}{1-\Delta_2} \bigg\};$ then, Eq (3.18) implies that

Hence, we have from Eq (3.19) $\mathbb{C}(\mathrm{\mho}\in \Omega$ which yields that $\mathbb{C} (\Omega)\subseteq \Omega$. Therefore, the operator $\mathbb{C}$ is bounded. We next aim to prove continuity. As we know that, $\Psi$ is continuous over the sub-intervals $[0, t_1]$ and $(t_1, T]$, $\mathbb{C}$ is also continuous over the same sub-intervals $[0, t_1]$ and $(t_1, T].$ To prove uniform continuity, let us take $t_k > t_j$ in the both sub-intervals; then, we have the following:

Upon simplification and using $(G_2)$, Eq (3.20) gives the following:

The right-handside of Eq (3.21) goes to zero when we apply $t_k\rightarrow t_j$ over both sub intervals of $[0, T]$. Hence, we conclude that $| \mathbb{C}(\mathrm{\mho}(t_k))- \mathbb{C}(\mathrm{\mho}(t_j))|\rightarrow 0$ with $t_k\rightarrow t_j$ over both sub-intervals of $[0, T].$ Also, $\mathbb{C}$ is a bounded and continuous. Hence, $\mathbb{C}$ is a uniformly continuous operator. Thus, $\mathbb{C}$ is compact. The conditions of Theorem 2.8 are satisfied. So, Eq (3.13) has at least one solution. In view of this theorem, our proposed model has at least one solution. □

4.   Numerical procedure

To derive the method for the numerical solution of the proposed model (1.2), let us consider the solution of Eq (3.13) as follows:

Here, we approximate the integral by taking the step size $h = t_{j+1}-t_j$, where $t_j = t_0+jh$ at $j = 0$, $t_0 = 0$. Replacing $t = t_{j+1}$ in Eq (4.1) gives the following:

Now, approximating the nonlinear function $\Psi(\chi, \mathrm{\mho}(\chi))$ by applying the Lagrange interpolation ^[39] with equally spaced arguments is achieved as follows:

Substituting Eq (4.3) in Eq (4.2), one obtains the following:

Evaluating the integrals in Eq (4.4) and simplifying, we get

where

In view of Eq (4.5), the numerical solution of the proposed model (1.2) is described as follows:

and, the last compartment is approximated as follows:

5.   Results and discussion

Here, we describe the use of the initial data in Table 3 and the parameters values given in Table 2 for the numerical analysis of model Eq (1.2).

In addition, we have divided the domain of $t$ into two sub-intervals, $[0, 10]$ and $[10, 50]$, and plotted the corresponding graphs for various fractional orders by using the relevant numerical scheme. Using the $mABC$ derivative of fractional order has significant application, as these type of operators do not force artificial singularities on any model; instead, they exhibit power law, stretched exponential, Brownian motion, Markovian, and non-Markovian properties simultaneously. Further, the derivative's probability distribution was found to be both Gaussian and non-Gaussian, and it can transition from Gaussian to non-Gaussian even without going through the steady state. The mean square displacement is a crossover from normal diffusion to sub-diffusion. This indicates that the non-singular kernel fractional derivatives are both stochastic and deterministic at the same time. Hence, the proposed model has been simulated in the following figures to investigate the mentioned properties.

We have presented the approximate solutions of the proposed model (1.2) by using three different sets of fractional- orders. We observed the crossover behaviors near the point $t_1 = 10$ have been presented for various compartments of the mentioned model. This interesting feature can be detected by using the piecewise derivative of fractional orders. The numerical solutions have been presented for various fractional-order values in Figures 3–5. We point out that the time-fractional derivative has a significant association with memory, and that it has been used in many scientific and technological domains, highlighting its enormous value in epidemiology and in other subjects. It is actually indicated that the memory function is the kernel of the fractional-order derivative, and the order of the time derivative emphasizes the memory rate. In this research work, we have effectively applied the idea of piecewise differentiation to capture the crossover behaviors in the transmission dynamics of the rotavirus disease. The population dynamics of the susceptible class and the infected class was found to decrease in the presence of the vaccination. Due to the vaccination and decline in the infection, the population density of the recovered class has been shown to increase. Here, near $t = 10$, we see the crossover behaviors in all classes of the proposed model, where a sudden change occurred and the dynamics show multiplicity in the behaviors.

6.   Conclusions

In this study, we examined a biological model of rotavirus by using vaccination guidelines. A few qualitative findings were obtained by using the Laplace transform under the $mABC$ derivative. Regarding the aforementioned model, the basic reproduction has been computed. We have also performed sensitivity analysis on $R_v$. Using the $mABC$ idea, we have implemented some updated piecewise derivative techniques. Piecewise derivatives can shed light on some of the sudden changes in the evolution of infectious diseases. In order to accommodate the necessary fractional differential operator, we have divided the time domain into two sub-intervals. We have simulated the findings for various fractional orders, $p$, by using Lagrange's polynomial interpolation criteria. In the same way, we have included a few graphs that show how the rotavirus vaccination model evolved in response to a piecewise derivative. In the future, we will analyze the aforementioned model by combining the first and second derivatives of the Lyapunov function method. Also, to predict the wave in the infection for the future, we will compute the strengthen number in next work. In addition, the aforementioned model will be studied by using the stochastic analysis for the aforementioned model.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank Prince Sultan University for APC and support through the TAS research lab.

Conflict of interest

The authors declare that there are no conflicts of interest.