Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

1.   Introduction

Infectious diseases are mainly caused by four microorganisms: bacteria, parasites, viruses, and fungi; they can be transmitted in different ways and can cause approximately fifty thousand deaths daily. In the past various epidemics worldwide resulted in millions of deaths. Very recently, a severe outbreak of respiratory illness was begun at the end of 2019 in China, Wuhan; this novel corona-virus was later called named SARS-CoV-2 (COVID-19). It was first detected in January, 2020. Note that the first transmission source of the virus was an animal; however, it rapidly spread through human interaction and such that, by 5th March 2021, a total of 116,216,580 cases were globally reported with 2,581,649 deaths. In every disease controlling, the control scenario, vaccination, etc., play a key role. Vaccines are important weapons in the fight against SARS-CoV-2, and the fact that so many vaccines are being developed and are proving to be effective is extremely promising. Due to the hard work of researchers and scientists vaccines that can save lives and put an end to the pandemic are now available. According to World Health Organization (WHO), safe and reliable vaccinations will be a game-changer. However, the need to wear masks, maintaining social distancing, and avoiding crowds still necessary because being vaccinated does not mean that we can ignore caution and endanger ourselves and others, especially since the extent to which vaccines can protect against not only the disease but also infection and transmission is still unknown.

Mathematical models play an important role in exploring the transmission dynamics of disease and predicting its future spread. Based on these models effective control strategies have been provided for compiling useful guidelines for the health officials and taking various steps towards disease control and eradication. Fractional calculus is a rapidly growing area in the field of mathematics that is used to catch the inherited axioms and memory of different natural and physical phenomena that occur in natural and physical science, technology, and engineering. Numerous classical epidemiological models have proved to be less accurate at predicting the dynamics of a system for the future due to their local nature. On the other hand, models with fractional-order are very usefully subjected to allocating and detaining the missing detail in models of classical case. More specifically, the classical derivative (integer-order) does not explore the dynamics between two distinct points. Various concepts have been developed for overcoming the limitations in case of the classical differentiation (see for more detail ^[1,2,3,4]). Moreover, numerous of researchers used different classical as well as fractional order epidemic models to forecast the long-term behaviors of SARS-CoV-virus and other diseases, and as well suggest some control measures. Particularly the stability analysis of various integer order epidemic models have been reported by different authors, (see for detail ^{[5,6,7,8,9,10]}). Due to the developing of fractional calculus many authors formulated fractional order epidemiological models using various fractional operators, e.g., Atangana proposed a mathematical model by using $ABC$-fractional operator ^[11]. Another study has been reported to investigate the transmission dynamics of novel corona virus disease ^[12]. Similarly the dynamics of novel corona virus with control analysis are discussed in ^{[13,14,15,16]}. In which comparison of the fractional and integer order, epidemiological models show that generally the integer-order models do not explore the dynamics more accurately.

The literature reveals that different dynamical systems are analyzed by different fractional derivative e.g., Riemann and Liouville, Hadamard and Caputo, etc., ^{[17,18,19,20]}. Many numerical and iterative approaches have been developed to solve the Caputo fractional order epidemic models ^{[21,22,23,24]}, while the complications arise due to a singular kernel. Therefore, a novel idea has been reported by Caputo and Fabrizio to the fractional-order derivative based upon non-singular kernel subject to various important results for the Caputo-Fabrizio fractional integral ^[25]. Consequently, this operator has a lot of applications in material and thermal sciences ^[26,27,28]. Further, the novel idea of fractional Caputo-Fabrizio derivative has been utilized frequently in physical and biological sciences to describe the temporal dynamics of communicable diseases ^{[29,30,31,32,33,34,35]}.

The current pandemic of SARS-CoV-2 and its vaccination remains a challenging issue and therefore requires significant attention. Very few studies have examined to the best of our knowledge that the novel dynamics of SARS-CoV-2 under the effect of vaccination, whereas few of them are the combination of statistical and computational models ^{[36,37,38,39,40,41,42]}. Very recently a mathematical model has been reported by Rahman et al., to study the dynamics of corona-virus transmission using the aforesaid operator ^[43]. However many characteristics that can influence the transmission of SARC-CoV-2 have been ignored. Particularly the impact of social behavior, mobility, symptomatic and asymptomatic classification, and vaccination, etc., are not collectively proposed which can influence the dynamics of the human population in ways that are not fully understood ^{[44,45,46,47]}. Motivated by the fact that fractional-order epidemiological models are more appropriate than the classical order for describing the real-world problem in a true sense as well as to fill the gap in the above-reported study, we propose a model to investigate the dynamics of SARS-COV-2 with vaccination in the frame of Caputo-Fabrizio-Caputo fractional operator by taking into account the vaccination of the susceptible population. We also classify the infected compartment into two groups of symptomatic and asymptomatic according to the SARS-CoV-2 characteristics.

We formulate the new fractional-order epidemiological model by dividing the total population into various population groups of susceptible, asymptomatic, symptomatic, and recovered individuals. Because the asymptomatic and symptomatic population take an important part in the spreading of SARS-CoV-2 virus transmission. Further, we assume that the permanent immunity of the susceptible group by getting vaccinated. We show that the solution of the proposed fractional-order epidemiological model exists and unique with the utilization of the theory of fixed point. We also prove that positivity and boundedness make the problem biologically feasible. The steady-state of the model will be calculated to discuss the stability analysis. We find the basic reproductive number and discuss the sensitivity analysis to find the role of the epidemic parameter in the spreading of the disease. We also use the Ordinary Least Square (OLS) method to parameterize the model and estimate the value of model parameters from the real data of SARS-CoV-2 virus reported in Oman. Finally, we present some graphical representations to support our analytical work and to show the effectiveness of the fractional-order epidemiological model.

The organization of the manuscript as follows: In Section 2 we present some basic definitions that will be used in our analysis, while Section 3 is devoted to the formulation of the model. The detailed analysis of the proposed fractional-order epidemiological model in terms of existence and uniqueness, positivity, and bounded-ness are discussed in Section 4. In Section 5, we perform the stability analysis and discuss the sensitivity of the basic reproductive number. All the theoretical results are supported with numerical simulation in Section 6 and concluding our work in the last Section.

2.   Preliminaries

We give some fundamental concepts that will be helpful to use in the upcoming sections. Let $\phi(t)$ be a function and $\phi\in H^{1}(0, T), T > 0$, assume that $\alpha > 0$ and $n-1 < \alpha < n$, $n\in \mathbb{N}$, then the fractional order derivative in Caputo and Caputo-Fabrizio-Caputo sense are respectively defined as:

and

where $C$ and $CFC$ stands for Caputo and Caputo-Fabrizio respectively, while $t > 0$ and $M(\alpha)$ represents the normilization function, such that $M(1) = 0 = M(0)$. We assume that $0 < \alpha < 1$ and $\phi(t)$, varies with $t$, then the Riemann–Liouville fractional integral of order $\alpha$ is determined by

while the Caputo-Fabrizio-Caputo (CFC) integral having order $\alpha$ takes the form

where $t\geq 0$.

3.   Model formulation

We explore a model using the Caputo-Fabrizio derivative to discuss the SARS-CoV-2 dynamics by categorizing the entire population into susceptible, symptomatic, asymptomatic and recovered groups. These population groups are symbolized respectively with $S$, $A$, $C$, and $R$. Moreover, rather than direct representation, we must place some constraints by making the following assumptions:

a. The constants, parameters, and variables are taken to be non-negative for the proposed problem.

b. The total population is subdivided into four groups and represented by $N(t)$.

c. The infected groups are assumed to be asymptomatic and symptomatic: as asymptomatic individuals become the major source of disease transmission.

d. Vaccination of susceptible individuals will result in permanent immunity.

e. The transmission of the disease is taken to be probability-based i.e., if $p$ is the probability that the interaction of susceptible with infected may lead to the asymptomatic class, then $(1-p)$ portion of the infected individuals automatically may go to the symptomatic class.

f. It is also noted that some of the individuals in case of SARS-CoV-virus got recovered without showing any symptoms while some may produce complications and causes the symptom of the disease, therefore this has been also proposed.

Based on these assumptions and the disease characteristics, we suggest the below dynamics for the fractional-order epidemic model of SARS-CoV-2 as:

here $\Pi$ is the newborn rate and the transmission rate is $\beta$ for the disease. $\gamma$ is the reduced transmission rate, and $\nu$ is the vaccination rate of the susceptible group. The total natural and disease-related death rates are denoted by $\mu_0$, $\mu_1$, and $\mu_2$ respectively. Similarly, $\gamma_2$ and $\gamma_1$ are the recovery rates for the asymptomatic and symptomatic groups, respectively. Moreover, $p$ represents the probability of the asymptomatic individuals and $q$ is the probability of these people that recovers directly in the symptomatic stage. We assume some substitutions for the shake of simplicity i.e., $\rho_1 = \nu+\mu_0$, $\rho_2 = \mu_0+\mu_1+\gamma_1$ and $\rho_3 = \mu_0+\mu_2+\gamma_2$.

We draw up the fractional-order version of the proposed model as reported by Eq (3.1) in the development of fractional calculus using the Caputo-Fabrizio-Caputo (CFC) operator with the maintenance of dimension for each differential equations as given:

where $\alpha$ is the fractional order parameter.

4.   Existence and uniqueness

In this section, we will show that the solution of the fractional-order model reported by Eq (3.2) by analyzing the fixed point theory. We will also prove that the uniqueness of the solution. For this, first the proposed fractional order system can be transformed into an associated integral equation form as:

Upon the application of Caputo-Fabrizio-Caputo fractional order integration, one may obtain

We assume kernels as determined by

Theorem 4.1. The kernels $K_1$, $K_2$, $K_3$ and $K_4$ satisfies the Lipschitz axioms.

Proof. Let us assume that $S$ and $S_1$, $A$ and $A_1$, $C$ and $C_1$, $R$ and $R_1$ are respectively the two functions for the above kernels $K_1$, $K_2$, $K_3$ and $K_4$, so we establish the following system

With the employment of Cauchy's inequality to the above system, one may obtain

Recursively one may obtain

The application of norm with the concept of majorizing, the difference between successive terms implies

where

Moreover, the kernels $K_1, \ldots, K_4$ satisfy the Lipschitz property, so one can write

Theorem 4.2. The solution of the proposed fractional order model (3.2) exists under Caputo-Fabrizio-Caputo operator.

Proof. The employment of Eq (4.4) and the use of recursive scheme leads to the following system

Now to investigate that the functions in Eq (4.6) are solutions of the model (3.2) we make use of the following substitutions

where $\Pi_{1, n}(t)$, $\Pi_{2, n}(t)$, $\Pi_{3, n}(t)$, $\Pi_{4, n}(t)$ represent the remainder terms of the series solutions. Thus

Applying the norm on both sides with the application of Lipschitz axiom the above assertion yields

Upon the application of $\lim$ as $t$ approaches $\infty$ implies that

which proves the conclusion i.e., the above are solutions of the model as given by Eq (3.2).

Theorem 4.3. The fractional order epidemiological model as reported by Eq (3.2) posses a solution which is unique.

Proof. On the contradiction basis, we assume that $(S^{'}(t), A^{'}(t), C^{'}(t), R^{'}(t))$ is also the solution of the proposed fractional epidemiological model (3.2), thus

Upon the property of majorizing we may write the above system as

Using the results derived in Theorems 4.1 and 4.2, we obtain

The inequalities as reported in Eq (4.13) holds for every value of $n$, thus we obtain

The positivity and boundedness of the fractional-order model (3.2) will be proved to show the well possed-ness of the problem. Further, we discuss a certain region for the dynamics of the proposed problem which is invariant positively. For this, the Lemmas developed has been explored below.

Lemma 1. Let $(S(t), A(t), C(t), R(t))$ be the solution of model (3.2) and assume that possessing non-negative initial conditions, then the solutions are non-negative for all $t\geq 0$.

Proof. Let us consider a general fractional-order model of the Eq (3.2) becomes

where $G$ is the operator for fractional-order under consideration, while the parameter of fractional order is in $\Omega$. So the above system leads to

where $\kappa(\xi) = \big\{\xi = 0\; \rm{and}\; S, A, C, R$ contained in $C(R_{+}\times R_{+})\big\}$ and $\xi\in\{S, A, C, R\}$. Following the result in ^[48], it is therefore concluded that any solutions $(S(t), A(t), C(t), R(t))$ of model (4.15) are positive for all non-negative $t$.

Lemma 2. Let $\Phi$ be the region (set) for considering the dynamics of the proposed model (3.2) within it, is invariant positively, then

Proof. Since $N$ symbolizes the total population, then $N = S+A+C+R$, which implies that

The solution of Eq (4.18) in the Caputo-Fabrizio sense leads to the assertion given by

In Eq (4.19) $E(.)$ represents the Mittag-Leffler function i.e., $E_{\Omega}(Z) = \sum_{n = 0}^{\infty}\frac{Z^n}{\Gamma(\Omega i+1)}$. It could be noted from the above Eq (4.19) that whenever time increases with no bound, then $N(t)\rightarrow \left(\frac{\Pi}{\rho_1}\right)^\Omega$. Hence, if $N(0)\leq \left(\frac{\Pi}{\rho_1}\right)^\Omega$, then $N(t)\leq\left(\frac{\Pi}{\rho_1}\right)^\Omega$ for all $t > 0$, while whenever $N(0) > \left(\frac{\Pi}{\rho_1}\right)^\Omega$, then $N$ goes into the feasible region $\Phi$, and will never leave. So it could be concluded that the fractional order model dynamics can be studied in the feasible region $\Phi$.

5.   Steady state analysis

The proposed epidemiological model (3.1) of the SARS-CoV-virus is examined for the equilibria: disease free and endemic states. Let $D_1$ be the disease free equilibrium point of the model, then for analyzing this point the population under consideration is assumed to be infection free. Thus the system reported by Eq (3.1) has a disease free equilibrium $D_1 = \left(S^0, A^0, C^0, R^0\right)$, where $S^0 = \frac{\Pi}{\rho_1}$, $A^0 = C^0 = 0$ and $R^0 = \frac{\nu\Pi}{\mu_0\rho_1}$. Now to calculate the basic reproductive number, we assume $X = (A, C)^T$ then system (3.1) yields

where

Therefore, the basic reproductive number is the spectral radius of $\rho(FV^{-1})$, i.e., $R_0 = R_1+R_2+R_3$, where

In a similar way it is assumed that the endemic state of the model (3.1) is $D_2 = (S^{*}, A^{*}, C^{*}, R^{*})$, then

The linearizable version of the proposed SARS-CoV-2 virus model (3.1) leads to a matrix given by

Two eigenvalues of the Jacobian matrix $J$ around the disease-free state are $-\mu_0$ and $-\rho_1$, while the remaining two are the roots of the quadratic equation is given by

It could be noted that that the roots of the above Eq (5.6) are negative if $R_1 < 1$, $R_3 < 1$ and $R_1+R_2 < 1$, which implies that the disease-free state $D_1$ is stable locally asymptotically whenever $R_0 < 1$. Similarly, it can be shown that the disease endemic state $D_2$ of the proposed model (3.1) is stable locally asymptotically whenever $R_0 > 1$.

To discuss the sensitivity of the basic reproductive number $(R_0)$ to each epidemic parameters of the model (3.1) we perform the following

Eq (5.7) describes that the basic reproductive number rises with the increasing of $\beta$ and $\gamma$, as depicted by Figure 1a, while reduces whenever the value of $\nu$ and $\gamma_2$ rises, as shown in Figure 1b.

6.   Numerical simulation

In this section, the numerical simulations are carried out to understand the temporal dynamical behavior corresponding with the SARS-CoV-virus fractional-order epidemiological model (3.2). We parameterize the proposed model against the real data of SARS-CoV-2 virus reported in the Sultanate of Oman from 1st January 2021 to 23rd May 2021. Based on reported data we estimate the epidemic parameters and then simulate the model for the long run. For this, the Ordinary Least Square (OLS) method is used. Using OLS to minimize the error terms for daily reported cases and the simulated data in Eq (6.1), and the associated relative error is used in the goodness of fit

In the above Eq (6.1), $C_i$ and $\hat{C}_i$ are the cumulative number of reported cases and the cumulative number of simulated cases. It could be noted that the crude birth rate (per 1000 population) is 25.2 and the total population of the Sultanate of Oman is 4.975 million, and our unit of time is the day, therefore the value of $\Pi$ is calculated as $\Pi = \left(4975000\times 25.2\right)/(1000\times 365)$. Moreover, the data fitting verse the proposed model is depicted in Figure 2. Thus the parameter's value is estimated and presented in Table 1. It is also is very important to show the feasibility of the reported work and investigate its validity of using large-scale numerical simulations. Unlike the traditional numerical analysis, there are not as many options to choose schemes for the numerical analysis of the fractional-order epidemiological model's simulations ^[49,50]. Thus, there is a need for extensive research to develop new schemes and techniques that are both convergent and robust in the field of fractional calculus. By following the numerical schemes as reported in ^[51,52,53], we assume $[0, t]$ interval of simulation and $h = 10^{-3}$ is the time step for integration, and $n = \frac{T}{h}$, $n\in \mathbb{N}$, and $u = 0, 1, 2, \ldots, n$. So the scheme may take the following structure:

This part of the study is specified to present the graphical illustrations of the proposed model. For this purpose, the value of the model parameters is considered from the Table 1, while moving on the same way the various order of $\alpha$ are taken to be $0.6$, $0.7$, $0.8$, $0.9$ and $1.0$ to demonstrate the difference between integer-order and fractional-order and its effect on the disease transmission. Consequently, the results are depicted in Figures 3–6, which represent the dynamical behaviors of the compartmental population of the Caputo-Fabrizio-Caputo fractional-order model (3.2). More precisely, different trajectories of Figure 3 visualize the dynamical behaviors of the susceptible population for the different values of $\alpha$. Similarly, the various trajectories of Figures 4–6 demonstrate the dynamical behaviors of the symptomatic, asymptomatic, and recovered population against the different values of the fractional-order parameter ($\alpha$). It is observed that the fractional-order parameter has a great influence on disease transmission. It can be also seen that there is an inverse relationship between the fractional-order parameter ($\alpha$) and the dynamics of the susceptible, asymptomatic, and symptomatic population i.e., increasing the value of $\alpha$ decreases the density of $S$, $A$ and $C$ are decreases as shown in Figures 3–5 respectively. On the other, a direct relation has been observed in the case of the dynamics of recovered population, and so increases the density of recovered population whenever the value of fractional parameter increasing as reported in Figure 6. This indicates that the Caputo-Fabrizio-Caputo, fractional-order reveals more valuable outputs regarding the model behavior which are usually could not be obtained in the case of the integer-order model.

7.   Conclusions

The research work carried out in this analysis consists of a new fractional-order epidemiological model related to the SARS-CoV-2 virus disease by using the $CFC$ operator. The proposed Caputo-Fabrizio model has been balanced dimensionally in respect of involved parameters. Upon the application of the theory of fixed point, it has been rigorously proved that the solution of the model under the $CFC$ operator exists and is unique. We also discussed the well possed-ness of the problem and showed that the solutions are bounded as well as positive. Steady-state analysis with sensitivity is also examined. Real data of SARS-CoV-2 virus are used and parameterized the proposed model. Finally, both the classical and fractional order Caputo-Fabrizio-Caputo model is simulated numerically and showed the feasibility and advantage of the obtained result. Thus the major findings of this work investigate that $CFC$ fractional-order operator is the best choice instead of classical order, where the long run of the models show that the SARS-CoV-2 virus infection decreasing asymptotically.

Nonetheless, the $CFC$ operator yielded interesting output in the reported work however there are some other operators are introduced very recently such as Atangana–Baleanu–Caputo, Atangana–Gomez, Atangana bi-order, Atangana–Koca, variable order, distributed orders and the fractal-fractional operator to capture much more information and complexities occurs in the real world problems, therefore in a near future, these operators are to be considered to analyze the epidemiological models of SARS-CoV-virus infection and other diseases.

Acknowledgement

This study has been supported by the Department of Computing Muscat College, Muscat Oman, Grant No. MC004/MoHERI/RG/2020. Also the 4th author (Jehad Alzabut) would like to thank OSTIM Technical University for their support of this work. We also would like to thank the anonymous reviewers for providing constructive comments, which have resulted in enhancing the paper. The usual disclaimer applies however.

Conflict of interest

The authors declare that there is no conflict of interest.