Computational analysis of rabies and its solution by applying fractional operator

Abstract

Numerous novel concepts in fractional mathematics have been created to provide numerical models for a variety of real-world, engineering, and scientific challenges because of the kernel's memory and non-local effects. In this post, we have looked at a deadly illness known as rabies. For our analysis, we employed the Atangana–Baleanu fractional derivative in Caputo sense. Additionally, the mathematical answer was obtained by applying the Laplace transform. Our approach is distinct, and we illustrated the vital role immunizations play in limiting the spread of the illness using graphical data. Furthermore, in this article, we have shown that fractional order systems are preferable to integer order systems.

Keywords:

• Rabies
• Atangana – Baleanu fractional operator
• mathematical model
• Laplace transform
• existence and uniqueness

2010 Mathematics Classifications:

• Primary 92B05
• 92C60
• Secondary 26A33

1. Introduction

Rabies is a well-known disease and constitutes a serious infection. It can spread to people if they are bitten or scratched by a rabid animal. The symptoms of rabies infection are harsh movements of body parts, confusion, fear of water, overexcitement, loss of consciousness, and muscles begin to paralyze at the bite location. Then, a person may fall into a coma, and then finally, death may occur. Normally, it takes around three months for first symptoms to appear. Depending on how long the virus takes to go from peripheral nerves to central nervous system, this period might last anywhere from a week to more than a year. Although it is a curable infection, negligence of proper medical treatment may cause even death. According to some reports and data, Rabies is still in existence in warm-blooded animals in over 150 countries. Around 60,000 people die due to this infection per year across the globe. Most Rabies cases are reported in wildlife animals like bats, foxes, raccoons and skunks. This infection is also circulated in domesticated pets. Dogs and cats are the main carriers of human–animal transmission of the virus. This infection is still an important issue in human and wildlife relations. Vaccination and proper pet care have turned the chances of dog rabies at a minimal level. This virus can also be stopped by using antibodies before the symptoms appear. Infected area must be washed with soap or antibiotic liquid for 10–15 min to reduce the spread of pathogens. Hassel et al.[1] gave the case study of rabies in Namibia, while Meltzer el al. [2,3] gave the impact of disease on the economy.

Several mathematical models have been developed explaining Rabies [4–12] by various systems of differential equations. Four classes were considered in early models – susceptible, exposed, infected, and recovered. The dynamics of rabies were explained using a set of ordinary differential equations. Studying its dynamics is still curious for many mathematicians, researchers and biologists. Many researchers and mathematicians have already shown that fractional extension of integer-ordered mathematical models gives the natural facts in a precise way as compared to any other [13–23].

In this article, we are going to use Atangana–Baleanu derivative along with Laplace transform to get our results. Here one thing can be asked that why we have used only Atangana–Baleanu derivative? So the explanation to the mentioned issue that the Riemann and Caputo derivatives are with singular kernel, then Caputo–Fabrizio derivative came into existence with non-singular kernel. But for better outcome, we need non-singular and non-local derivative since it contains no singular point in the domain, which is found in Atangana–Baleanu derivative so we preferred this over others.

Researchers in modern mathematics are finding that mathematical modelling is an extremely helpful tool. After applying the primary techniques, we translate the results back into straightforward terms to forecast the objective, as it converts real-world problems into mathematical expressions. Every business makes plans to meet goals and make projections about the future. Sometimes, though, the results of fractional calculus better capture the model than the traditional one does [24–28].

Fractional calculus is used to model many technical, engineering, and physical situations these days. The field of fractional calculus has a rich 300-year history. The majority of systems exhibit memory and non-local effects, which make it difficult to model using integer order operators. This is the main reason why fractional models are used [29–34]. Numerous topics make use of fractional calculus and its operators. Primarily, early studies relied on the Riemann–Liouville fractional operator. However, over time, it was discovered that its kernel exhibited singularity; consequently, several new definitions were introduced that did not share this singularity. The Atangana Baleanu fractional operator is based on the Mittag–Leffler function, which produces the best outcomes. In addition, we have demonstrated the existence and uniqueness of the solution, which validates our new approach.

If we talk about the novelty of the paper, then our approach is distinct, and we illustrated the vital role of immunizations in limiting the spread of the illness. The population's daily impact of infection is increasing as a result of the lack of vaccines. Consequently, we may draw the conclusion that vaccination is essential for avoiding rabies based on the study.

2. Pre-requisites

2.1. Atangan –Baleanu fractional derivative

Let $h\in H(0,1)$, then Atangana–Baleanu Fractional derivative (in Caputo) [35,36] of order α is: (1) $T_{\alpha }(h)(x)=\frac{B(\alpha )}{1-\alpha }{\int }_0^x{E}_{\alpha }\left[-\frac{\alpha }{1-\alpha }{(x-s)}^{\alpha }\right]h^{\prime }(s)\mathrm{d}s,$(1) here $B(\alpha )>0$ is normalizing function which agrees to $B(0)=B(1)=1$ and $E_{\alpha }$ is Mittag-Leffler function.

Figure 1. Graph showing no. of susceptible w.r.t. time t, for ξ= 0.3, 0.5, 0.7, 0.9 and 1.

Figure 2. Graph showing no. of infected w.r.t. time t, for ξ= 0.3, 0.5, 0.7, 0.9 and 1.

Figure 3. Graph showing total population w.r.t. time t, for ξ= 0.3, 0.5, 0.7, 0.9 and 1.

2.2. Atangana –Baleanu fractional integral operator (in caputo)

Atangana–Baleanu fractional integral operator (in Caputo) of function f and order β is given like (2) ${}_0^{\mathrm{ABC}}{I}_t^{\beta }f(t)=\frac{1-\beta }{B(\beta )}f(t)+\frac{\beta }{B(\beta )\mathrm{\Gamma \beta }}{\int }_0^{t}f(y){(t-y)}^{\beta -1}\mathrm{d}y.$(2)

2.3. Laplace transform

The Laplace transformation is one of the important transformation in mathematics. It usually converts the system to an algebraic system, which is easily solvable. The Laplace transform of $f(t)$ is represented by $L\{f(t)\}$ and is explained as follows: (3) $L\{f(t)\}={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{st}}f(t)\mathrm{d}t,s>0.$(3)

2.3.1. Laplace transform of Atangana–Baleanu fractional differential operator

Let ${}^{\mathrm{ABC}}{D}^{\alpha }f(t)$ be the Atangana–Baleanu fractional differential operator of any function $f(t)$, then the Laplace transform of Atangana-Baleanu fractional differential operator is defined as (4) $L\{{}^{\mathrm{ABC}}{D}^{\alpha }f(t)\}=\frac{M(\alpha )}{1-\alpha }\frac{p^{\alpha }L\{f(t)\}-p^{\alpha -1}f(0)}{p^{\alpha }+\frac{\alpha }{1-\alpha }}.$(4) The layout of the article is given here. The presented paper has eight segments. Part 1 is an introductory part and brief literature review of the article. Section 2 deals with the prerequisites, while Section 3 concerns the mathematical model of Rabies. The following segments are the existence and uniqueness of the approach and result. Segment 6 is a solution to the model that uses the Laplace transform. The next section, with the numerical and graphical explanation of the model, and the last findings are discussed in the last part of the article, named as a conclusion. At the end of the article, we have listed references to earlier work on which our article is based.

3. Mathematical model of rabies

Rabies is still a curious field to investigate. Many scientists and researchers have been working on it. In connection with this investigation, Demirci [4] analysed a mathematical system describing the growth of disease of an integer order. Here, we have considered this system but with a unique and different fractional calculus line of action. In this system, the total population of pets is split into three subclasses: vulnerable animal $S(t)$ , contaminated pets $I(t)$ and retrieved pets $R(t)$ at time t. This is a chronic disease, and animals die yearly at a considerable amount, and vaccination is the only way to stop the fatalities. Consider the model given below: (5) $\begin{array}{rl}{\displaystyle \frac{\mathrm{d}S}{\mathrm{d}t}}& =\mathrm{lN}+\mathrm{mR}-d_{N}S-\mathrm{qS}-\mathrm{lI}-\mathrm{xSI},\\ {\displaystyle \frac{\mathrm{d}I}{\mathrm{d}t}}& =\mathrm{lI}+\mathrm{xSI}-d_{N}I-\mathrm{\epsilon I},\\ {\displaystyle \frac{\mathrm{d}R}{\mathrm{d}t}}& =\mathrm{qS}-d_{N}R-\mathrm{mR},\end{array}\}$(5) where the meanings of every parameter are explained below:

l: per capita birth rate,

m: reduced immunity,

$d_N$: per capita death function rate depending upon population size ($d_N=.01+.004N$),

q: vaccination rate,

x: disease transferral rate, and

ε: dying rate in disease.

The initial conditions are $S(0)=S_0,I(0)=I_0,R(0)=R_0.$ Now, since N = S + I + R so using this relationship in our existing model, the model becomes, (6) $\begin{array}{rl}{\displaystyle \frac{\mathrm{d}S}{\mathrm{d}t}}& =\mathrm{lN}+m(N-S-I)-d_{N}S-\mathrm{qS}-\mathrm{lI}-\mathrm{xSI},\\ {\displaystyle \frac{\mathrm{d}I}{\mathrm{d}t}}& =\mathrm{lI}+\mathrm{xSI}-d_{N}I-\mathrm{\epsilon I},\\ {\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}}& =\mathrm{lN}-d_{N}N-\mathrm{\epsilon I}\end{array}\}$(6) with initial conditions are $S(0)=S_0,I(0)=I_0,N(0)=N_0.$

Converting integer-ordered mathematical system to a fractional ordered model [37–51] by introducing the Atangana-Baleanu fractional operator. All these articles show the importance of fractional operators in various issues of real world. Now, the system reduced to (7) $\begin{array}{rl}{}^{\mathrm{ABC}}{D}^{\xi }S(t)& =\mathrm{lN}+m(N-S-I)-d_{N}S-\mathrm{qS}-\mathrm{lI}-\mathrm{xSI},\\ {}^{\mathrm{ABC}}{D}^{\xi }I(t)& =\mathrm{lI}+\mathrm{xSI}-d_{N}I-\mathrm{\epsilon I},\\ {}^{\mathrm{ABC}}{D}^{\xi }N(t)& =\mathrm{lN}-d_{N}N-\mathrm{\epsilon I}.\end{array}\}$(7) Here in the system (7(7) $\begin{array}{rl}{}^{\mathrm{ABC}}{D}^{\xi }S(t)& =\mathrm{lN}+m(N-S-I)-d_{N}S-\mathrm{qS}-\mathrm{lI}-\mathrm{xSI},\\ {}^{\mathrm{ABC}}{D}^{\xi }I(t)& =\mathrm{lI}+\mathrm{xSI}-d_{N}I-\mathrm{\epsilon I},\\ {}^{\mathrm{ABC}}{D}^{\xi }N(t)& =\mathrm{lN}-d_{N}N-\mathrm{\epsilon I}.\end{array}\}$(7) ), the RHS of the equations has dimensions $\mathrm{tim}{e}^{-1}$, but because the order of the equations has been altered to ξ, the dimension of LHS has changed to $\mathrm{tim}{e}^{-\xi }$. Therefore, we must alter the parameters in RHS's dimensions to balance the dimensions on both sides, resulting in the following system: (8) $\begin{array}{rl}{}^{\mathrm{ABC}}{D}^{\xi }S(t)& =l^{\xi }N+m^{\xi }(N-S-I)-d_N^{\xi }S-q^{\xi }S-l^{\xi }I-x^{\xi }\mathrm{SI},\\ {}^{\mathrm{ABC}}{D}^{\xi }I(t)& =l^{\xi }I+x^{\xi }\mathrm{SI}-d_N^{\xi }I-{\epsilon }^{\xi }I,\\ {}^{\mathrm{ABC}}{D}^{\xi }N(t)& =l^{\xi }N-d_N^{\xi }N-{\epsilon }^{\xi }I.\end{array}\}$(8) For sake of convenience, we may take $L=l^{\xi }$, $M=m^{\xi }$, $D_N=d_N^{\xi }$, $Q=q^{\xi }$, $X=x^{\xi }$ and $E={\epsilon }^{\xi }$ then the system reduces to (9) $\begin{array}{rl}{}^{\mathrm{ABC}}{D}^{\xi }S(t)& =\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI},\\ {}^{\mathrm{ABC}}{D}^{\xi }I(t)& =\mathrm{LI}+\mathrm{XSI}-D_{N}I-\mathrm{EI},\\ {}^{\mathrm{ABC}}{D}^{\xi }N(t)& =\mathrm{LN}-D_{N}N-\mathrm{EI},\end{array}\}$(9) here $0<\xi \le 1$ with $S(0)=S_0,I(0)=I_0,N(0)=N_0$.

4. Validity of solution

Next, we will check the validity of our new approach in investigating the Rabies model. In this process, we have used fixed point theory and Lipschitz condition [22,23] to obtain our objective. Applying fundamental theorem of calculus, we have (10) $\begin{array}{rl}S(t)-S(0)& =\frac{1-\xi }{B(\xi )}\left[\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right]+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}\\ & \times {\int }_0^t{\left(t-y\right)}^{\xi -1}\left[\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right]\mathrm{d}y.\end{array}$(10) Similarly, we obtain. (11) $\begin{array}{rl}I(t)-I(0)& =\frac{1-\xi }{B(\xi )}\left[\mathrm{LI}+\mathrm{XSI}-D_{N}I-\mathrm{EI}\right]\\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{(t-y)}^{\xi -1}\left[\mathrm{LI}+\mathrm{XSI}-D_{N}I-\mathrm{EI}\right]\mathrm{d}y,\end{array}$(11) (12) $\begin{array}{rl}N(t)-N(0)& =\frac{1-\xi }{B(\xi )}\left[\mathrm{LN}-D_{N}N-\mathrm{EI}\right]\\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{(t-y)}^{\xi -1}\left[\mathrm{LN}-D_{N}N-\mathrm{EI}\right]\mathrm{d}y.\end{array}$(12) Here, we define the following kernels, (13) $\begin{array}{rl}{M}_1(S,t)& =\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI},\\ M_2(I,t)& =\mathrm{LI}+\mathrm{XSI}-D_{N}I-\mathrm{EI},\\ M_3(N,t)& =\mathrm{LN}-D_{N}N-\mathrm{EI}.\end{array}\}$(13) Further, suppose S and $S_1$ be two functions then, $\begin{array}{rl}\Vert M_1(S,t)-M_1(S_1,t)\Vert & =\Vert \left(\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right)\\ & -\left(\mathrm{LN}+M(N-S_1-I)-D_N{S}_1-Q{S}_1-\mathrm{LI}-X{S}_{1}I\right)\Vert ,\\ & =\Vert \left(M+Q+\mathrm{XI}+D_N\right)\left(S_1-S\right)\Vert \\ & \le \Vert \left(M+Q+\mathrm{XI}+D_N\right)\Vert \Vert (S_1-S.)\Vert \end{array}$Let $M+Q+\mathrm{XI}+D_N=H_1<1$ then, $\Vert M_1(S,t)-M_1(S_1,t)\Vert \le H_1\Vert \left(S_1-S\right)\Vert$In the same way, we can show that $M_2\mathrm{and}{M}_3$ satisfy Lipschitz condition. Now, using the above kernels, Equations (10(10) $\begin{array}{rl}S(t)-S(0)& =\frac{1-\xi }{B(\xi )}\left[\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right]+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}\\ & \times {\int }_0^t{\left(t-y\right)}^{\xi -1}\left[\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right]\mathrm{d}y.\end{array}$(10) )–(12(12) $\begin{array}{rl}N(t)-N(0)& =\frac{1-\xi }{B(\xi )}\left[\mathrm{LN}-D_{N}N-\mathrm{EI}\right]\\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{(t-y)}^{\xi -1}\left[\mathrm{LN}-D_{N}N-\mathrm{EI}\right]\mathrm{d}y.\end{array}$(12) ) become, (14) $\begin{array}{rl}S(t)& =S(0)+\frac{1-\xi }{B(\xi )}{M}_1(S,t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_1(S,y)\mathrm{d}y,\end{array}$(14) (15) $\begin{array}{rl}I(t)& =I(0)+\frac{1-\xi }{B(\xi )}{M}_2(I,t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_2(I,y)\mathrm{d}y,\end{array}$(15) and (16) $N(t)=N(0)+\frac{1-\xi }{B(\xi )}{M}_3(N,t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_3(N,y)\mathrm{d}y.$(16) We give the following recurrence relation for Equations (14(14) $\begin{array}{rl}S(t)& =S(0)+\frac{1-\xi }{B(\xi )}{M}_1(S,t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_1(S,y)\mathrm{d}y,\end{array}$(14) )–( 16(16) $N(t)=N(0)+\frac{1-\xi }{B(\xi )}{M}_3(N,t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_3(N,y)\mathrm{d}y.$(16) ): (17) $\begin{array}{rl}{S}_n(t)& =\frac{1-\xi }{B(\xi )}{M}_1(S_{n-1},t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_1(S_{n-1}y)\mathrm{d}y,\end{array}$(17) (18) $\begin{array}{rl}{I}_n(t)& =\frac{1-\xi }{B(\xi )}{M}_2(I_{n-1},t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_2(I_{n-1}y)\mathrm{d}y,\end{array}$(18) and (19) $N_n(t)=\frac{1-\xi }{B(\xi )}{M}_3(N_{n-1},t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_3(N_{n-1}y)\mathrm{d}y.$(19) Now, assume that the difference between two successive terms for Equation (17(17) $\begin{array}{rl}{S}_n(t)& =\frac{1-\xi }{B(\xi )}{M}_1(S_{n-1},t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_1(S_{n-1}y)\mathrm{d}y,\end{array}$(17) ) is denoted by ${\gamma }_n$ and is given as ${\gamma }_n=S_n(t)-S_{n-1}(t)$. or, (20) $\begin{array}{rl}{\gamma }_n& =\frac{1-\xi }{B(\xi )}\left\{M_1(S_{n-1},t)-M_1(S_{n-2},t)\right\}\\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}\left\{M_1(S_{n-1},y)-M_1(S_{n-2},y)\right\}\mathrm{d}y\end{array}$(20) It is obvious that $S_n(t)=\sum _{i=0}^n{\gamma }_i$. Now taking the norm to both sides, we have (21) $\begin{array}{rl}\Vert {\gamma }_n\Vert & =\Vert S_n(t)-S_{n-1}(t)\Vert =\frac{1-\xi }{B(\xi )}\Vert M_1(S_{n-1},t)-M_1(S_{n-2},t)\Vert \\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}\Vert {\int }_0^t{\left(t-y\right)}^{\xi -1}\left\{M_1(S_{n-1},y)-M_1(S_{n-2},y)\right\}\mathrm{d}y\Vert .\end{array}$(21) Since $M_1$ satisfies the Lipschitz condition, so (22) $\Vert {\gamma }_n\Vert \le \frac{1-\xi }{B(\xi )}{H}_1\Vert S_{n-1}-S_{n-2}\Vert +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{H}_1{\int }_0^t{\left(t-y\right)}^{\xi -1}\Vert S_{n-1}-S_{n-2}\Vert \mathrm{d}y,$(22) or (23) $\Vert {\gamma }_n\Vert \le \frac{1-\xi }{B(\xi )}{H}_1\Vert {\gamma }_{n-1}(t)\Vert +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{H}_1{\int }_0^t{\left(t-y\right)}^{\xi -1}\Vert {\gamma }_{n-1}(y)\Vert \mathrm{d}y.$(23) Similarly, we can get the other expressions too like (24) $\Vert {\varphi }_n\Vert \le \frac{1-\xi }{B(\xi )}{H}_2\Vert {\varphi }_{n-1}(t)\Vert +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{H}_2{\int }_0^t{\left(t-y\right)}^{\xi -1}\Vert {\varphi }_{n-1}(y)\Vert \mathrm{d}y,$(24) and (25) $\Vert {\mathrm{\Omega }}_n\Vert \le \frac{1-\xi }{B(\xi )}{H}_3\Vert {\mathrm{\Omega }}_{n-1}(t)\Vert +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{H}_3{\int }_0^t{\left(t-y\right)}^{\xi -1}\Vert {\mathrm{\Omega }}_{n-1}(y)\Vert \mathrm{d}y.$(25)

Theorem

If we can locate $t_0$ s.t. (26) $\frac{1-\xi }{B(\xi )}{H}_1+\frac{{t_0}^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}{H}_1<1,$(26) then, we have the solution of the model.

Proof.

Because kernels meet the Lipschitz constraints and $S(t)$, $I(t)$ and $N(t)$ are bounded. So, (27) $\begin{array}{rl}\Vert {\gamma }_n\Vert & \le \Vert S(0)\Vert {\left[\frac{1-\xi }{B(\xi )}{H}_1+\frac{t^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}{H}_1\right]}^n,\end{array}$(27) (28) $\begin{array}{rl}\Vert {\varphi }_n\Vert & \le \Vert I(0)\Vert {\left[\frac{1-\xi }{B(\xi )}{H}_2+\frac{t^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}{H}_2\right]}^n,\end{array}$(28) and (29) $\Vert {\mathrm{\Omega }}_n\Vert \le \Vert A(0)\Vert {\left[\frac{1-\xi }{B(\xi )}{H}_3+\frac{t^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}{H}_3\right]}^n.$(29) We will now demonstrate that the aforementioned functions are system solutions. So, take $\begin{array}{rl}S(t)-S(0)& =S_n(t)-G_n(t),\\ I(t)-I(0)& =I_n(t)-P_n(t),\\ N(t)-N(0)& =N_n(t)-Q_n(t).\text{■}\end{array}$

Then, (30) $\begin{array}{rl}\Vert G_n(t)\Vert & =\Vert \frac{1-\xi }{B(\xi )}\left\{M_1(S,t)-M_1(S_{n-1},t)\right\}\\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}\left\{M_1(S,y)-M_1(S_{n-1},y)\right\}\mathrm{d}y\Vert ,\end{array}$(30) (31) $\begin{array}{rl}\Vert G_n(t)\Vert & =\frac{1-\xi }{B(\xi )}\Vert M_1(S,t)-M_1(S_{n-1},t)\Vert \\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}\Vert {\int }_0^t{\left(t-y\right)}^{\xi -1}\{M_1(S,y)-M_1(S_{n-1},y)\}\mathrm{d}y\Vert ,\end{array}$(31) (32) $\begin{array}{rl}\Vert G_n(t)\Vert & \le \frac{1-\xi }{B(\xi )}{H}_1\Vert S-S_{n-1}\Vert +\frac{t^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}{H}_1\Vert S-S_{n-1}\Vert ,\end{array}$(32) (33) $\begin{array}{rl}\Vert G_n(t)\Vert & \le {\left(\frac{1-\xi }{B(\xi )}+\frac{t^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}\right)}^{n+1}{H_1}^{n+1}\Vert S-S_{n-1}\Vert .\end{array}$(33) At the point $t_0$, it becomes (34) $\Vert G_n(t)\Vert \le {\left(\frac{1-\xi }{B(\xi )}+\frac{t_0^{\xi }}{B(\xi )\mathrm{\Gamma \xi }}\right)}^{n+1}{H_1}^{n+1}\Vert S-S_{n-1}\Vert .$(34) Now as $n\to \mathrm{\infty }$, we get $\Vert G_n(t)\Vert \to 0$. Similarly, we also get $\Vert P_n(t)\Vert \to 0,$and $\Vert Q_n(t)\Vert \to 0$. Hence, the existence of the solution is verified.

5. Uniqueness of result

Let $S_1$ be another solution then, (35) $\begin{array}{rl}S(t)-S_1(t)& =\frac{1-\xi }{B(\xi )}{M}_1(S,t)+\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_1(S,y)\mathrm{d}y\\ & -\frac{1-\xi }{B(\xi )}{M}_1(S_1,t)-\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t{\left(t-y\right)}^{\xi -1}{M}_1(S_1,y)\mathrm{d}y.\end{array}$(35) Taking norms on both sides, we obtain (36) $\begin{array}{rl}\Vert S(t)-S_1(t)\Vert & =\frac{1-\xi }{B(\xi )}\Vert M_1(S,t)-M_1(S_1,t)\Vert \\ & +\frac{\xi }{B(\xi )\mathrm{\Gamma \xi }}{\int }_0^t\Vert {\left(t-y\right)}^{\xi -1}\left(M_1(S,y)-M_1(S_1,y)\right)\mathrm{d}y\Vert ,\end{array}$(36) Now, applying the Lipschitz constraint, we observe that the solution is bounded and it is possible only when $\begin{array}{rl}S(t)& =S_1(t),\\ I(t)& =I_1(t),\end{array}$and $N(t)=N_1(t),$It shows that the model has a unique solution.

6. Results of model by applying laplace transform

In this part, we found the solution to the Rabies model using the Laplace transform. For this, we have adopted the iterative technique. The reduced system is as follows: (37) $\begin{array}{l}{}^{\mathrm{ABC}}{D}^{\xi }S(t)=\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI},\\ {}^{\mathrm{ABC}}{D}^{\xi }I(t)=\mathrm{LI}+\mathrm{XSI}-D_{N}I-\mathrm{EI},\\ {}^{\mathrm{ABC}}{D}^{\xi }N(t)=\mathrm{LN}-D_{N}N-\mathrm{EI}.\end{array}\}$(37) Applying Laplace changes both sides in the first equation of the system (37(37) $\begin{array}{l}{}^{\mathrm{ABC}}{D}^{\xi }S(t)=\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI},\\ {}^{\mathrm{ABC}}{D}^{\xi }I(t)=\mathrm{LI}+\mathrm{XSI}-D_{N}I-\mathrm{EI},\\ {}^{\mathrm{ABC}}{D}^{\xi }N(t)=\mathrm{LN}-D_{N}N-\mathrm{EI}.\end{array}\}$(37) ). We have $\begin{array}{rl}& \mathrm{LT}\left\{{}^{\mathrm{ABC}}{D}^{\xi }S(t)\right\}=\mathrm{LT}\left\{\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right\},\\ & \frac{M(\xi )}{1-\xi }.\frac{s^{\xi }\mathrm{LT}\left\{S(t)\right\}-s^{\xi -1}S(0)}{s^{\xi }+\frac{\xi }{1-\xi }}=\mathrm{LT}\left\{\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right\},\\ & \frac{s^{\xi }\mathrm{LT}\left\{S(t)\right\}-s^{\xi -1}S(0)}{s^{\xi }+\frac{\xi }{1-\xi }}=\frac{1-\xi }{M(\xi )}.\mathrm{LT}\left\{\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right\},\\ & s^{\xi }\mathrm{LT}\left\{S(t)\right\}-s^{\xi -1}S(0)\\ & =\left(s^{\xi }+\frac{\xi }{1-\xi }\right).\frac{1-\xi }{M(\xi )}.\mathrm{LT}\left\{\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right\},\\ & s^{\xi }\mathrm{LT}\left\{S(t)\right\}\\ & =s^{\xi -1}S(0)+\left(s^{\xi }+\frac{\xi }{1-\xi }\right).\frac{1-\xi }{M(\xi )}\\ & .\mathrm{LT}\left\{\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right\},\\ & \mathrm{LT}\left\{S(t)\right\}\\ & =s^{-1}S(0)+\left(1+\frac{\xi s^{-\xi }}{1-\xi }\right).\frac{1-\xi }{M(\xi )}\\ & .\mathrm{LT}\left\{\mathrm{LN}+M(N-S-I)-D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\right\}.\end{array}$Using inverse Laplace transform on both sides, we have $\begin{array}{rl}S(t)& =S(0)+L{T}^{-1}[\frac{\left(1-\xi +\xi s^{-\xi }\right)}{M(\xi )}.\mathrm{LT}\{\mathrm{LN}+M(N-S-I)\\ & -D_{N}S-\mathrm{QS}-\mathrm{LI}-\mathrm{XSI}\}\phantom{\frac{\left(1-\xi +\xi s^{-\xi }\right)}{M(\xi )}}],\end{array}$or, repeating the method, we get the recurrence relation as follows: (38) $\begin{array}{rl}{S}_{n+1}(t)& =S(0)\\ & +L{T}^{-1}[\frac{\left(1-\xi +\xi s^{-\xi }\right)}{M(\xi )}.\mathrm{LT}\{L{N}_n+M(N_n-S_n-I_n)\\ & -D_N{S}_n-Q{S}_n-L{I}_n-X{S}_n{I}_n.\}\phantom{\frac{\left(1-\xi +\xi s^{-\xi }\right)}{M(\xi )}}].\end{array}$(38) In the same way, we get the other expressions from the second and third equations of the system, (39) $I_{n+1}(t)=I(0)+L{T}^{-1}\left[\frac{\left(1-\xi +\xi s^{-\xi }\right)}{M(\xi )}.\mathrm{LT}\left\{L{I}_n+X{S}_n{I}_n-D_N{I}_n-E{I}_n\right\}\right],$(39) and (40) $N_{n+1}(t)=N(0)+L{T}^{-1}\left[\frac{\left(1-\xi +\xi s^{-\xi }\right)}{M(\xi )}.\mathrm{LT}\left\{L{N}_n-D_N{N}_n-E{I}_n\right\}\right].$(40) The result is found by $\begin{array}{rl}S(t)& =\underset{n\to \mathrm{\infty }}{lim}{S}_n(t),\\ I(t)& =\underset{n\to \mathrm{\infty }}{lim}{I}_n(t),\end{array}$and $N(t)=\underset{n\to \mathrm{\infty }}{lim}{N}_n(t).$

7. Numerical and graphical representations

For numerical solutions, we have used the values of parameters [4], which are explained and given in the below table:

Table

S.N.	Variable	Symbol	Value
1	No. of susceptible at t=0	$S_0$	9
2	No. of infected at t=0	$I_0$	1
3	Total population at t=0	$N_0$	10
4	Rate of per capita birth	l	0.083
5	Rabies transmission rate	x	0.1
6	Mortality rate in disease	ε	0.05
7	Vaccination rate	q	0
8	Waning immunity	m	0
9	Rate function of per capita death depending upon N	$d_N$	0.05

After using the above parameter values, we have found the graphs which are shown in the later part of article. Here, one thing is to be noted that we have taken the fractional values of the order of equations as well as we have also found the results at $\xi =1$. These variations shows that fractional-order system is better than integer order system since at integer value, susceptible are decreasing rapidly than those of fractional values. In the same way, the number of infected are increasing more rapidly for integer value as compared to fractional order. So, here we can say that fractional ordered model gives better results as compared to integer order system. Using these values in our analysis, we have found the following graphs of various factors below: If we compare the results obtained here and of [4], then we can clearly see the differences of taking the non-singular and non-local derivative over the Caputo derivative. In [4], the graphs are in something like wave form but in our representation, it give the clarity of thoughts about the various factors with respect to time. They used stability analysis approach while we obtain our significant results by using Laplace transform approach.

8. Conclusion

We have studied a mathematical model of the deadly disease rabies in this work. We have used an iterative technique to apply the Laplace transform method to the Atangana–Baleanu fractional operator in order to obtain both numerical and graphical results. To confirm that our new approach is one, we have also employed a fixed-point approach. Since our primary goal in using this approach was to examine the effects of vaccination, the graphical results indicate that the number of susceptible people is declining over time (which is not surprising given that the number of infected people is rising over time, indicating that susceptible people are becoming infected). The graphs of susceptibility and infection indicate that, as a result of the vaccination process's absence, the population's daily rate of infection is rising. Therefore, based on the study, we can conclude that immunization is crucial for preventing rabies. For the future work, we can change the parameter's values and kernel of the system too to compare the existing results as well as for some new findings.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Conflict of interest statement

Researchers state that there aren't any conflicts of interest with the article that is being discussed.

Data availability statement

The availability of statistics is already cited in the article.

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University, Saudi Arabia, for funding this research work through the project number ‘IMSIU-RP23002’