Study of fractional order rabies transmission model via Atangana–Baleanu derivative

Abstract

In this work, we aim at disentangling the theoretical contribution through mathematical modeling approach to advance the understanding of rabies dynamics and control in livestock population. A fractional order model of rabies, using Atangana–Baleanu fractional operator is created. The analysis of suggested system and its application are both conducted. The capacity of proposed model to forecast the disease can help researchers and livestock health care agencies to take preventive actions.

Introduction

Rabies is a viral zoonosis, a disease that is transferrable between humans and animals. It is caused by infection with rabies virus of genus Lyssavirus¹. If left untreated, the disease is fetal. Humans and mammals both catch rabies through direct contact, food, bite or scratches given by an other infected animal. Commonly, the saliva of infected animals is contaminated by rabies virus and results in its transmission when they get in contact with other animals or humans².

Some of the primary reservoirs of rabies disease include animals like foxes, bats, skunks, rodents cats, dogs and raccoons. Rabies virus is transmitted in humans generally by dogs and livestock^3,4,5,6. Pets generally develop this disease when get bitten by rabid skunks. Basically, this virus gets into the nervous system, attacks it and leads to a critical neurological imbalance in the host body. Among other abnormalities, significant symptoms are hallucinations, anxiety, excess of saliva, paralysis and excitation⁷.

An annual report of WHO (World Health Organization) shows approximately 59,000 deaths across the globe due to rabies disease declaring it as a major health concern⁸. Though vaccination played a huge positive role in coping up with rabies in developed countries but most of the remote areas still suffer from harmful effects of rabies due to lack of access to vaccination.

Many factors such as existence of wildlife reservoirs, vaccine efficiency and its well timed implementation, exposure to infected ones, behavior and density of animal population makes the transmission dynamics of rabies a complex matter⁹.

Moreover, rabies is a major threat to livestock. There is always a danger of reintroducing rabies in domestic animals. Some species may go in extinction due to rabies. Regions that witness cases of rabies more often bear more loss. Its harmful effects can extend to their economy, tourism, agriculture, animal husbandry and much more. This threat to livestock and domestic animals puts more burden on veterinary doctors and welfare organization to cope up with rabies and overcome the loss⁸.

Mathematics has always been very efficient in problem solving and even for diseases, it finds a way to analyze them using mathematical modelling. By developing a mathematical model for different diseases and finding its solution, researchers provide a gateway to the public authorities to create beneficial and useful strategies to deal with diseases. Likewise, many mathematical models have been developed to get more clarity of rabies. An SEIR model was developed to study the dynamics of rabies in dogs by Tulu and his results showed that disease may end if the migration of dogs is restricted⁷. Also, same investigation was done by Renald et al. using an SEIV model and they suggested that vaccination can be beneficial to reduce the transmission of rabies¹⁰. Moreover, Liu et al. studied the two patch SEIRS model and found that vaccination rate should be improved and birth rate of dogs should be reduced to get rid of dog rabies in China¹¹. Furthermore, similar studies done by González-Roldán et al.¹², Kanda et al.¹³ and Lushasi et al.¹⁴ revealed that the improvisation of dog rabies vaccination program annually can help to control dog-to-dog and dog-to-human transmission.

Many mathematical models based on the fractional derivative have also been explored by researchers to study rabies transmission. Kumar et al. used Caputo fractional derivative to understand the dynamics of red fox rabies in Italy¹⁵. After this, Zarin et al. also incorporated Caputo fractional derivative in their research to explore the epidemic properties of rabies by considering convex incidence function¹⁶. Caputo Fabrizio derivative has been used by Aydogan et al. to look into the details of transmission pattern of the disease¹⁷. Recently, Garg and Chauhan used fixed-point approach to discuss the stability of solutions of fractional order rabies model¹⁸.

A new fractional differential operator having Mittag–Leffler function with a variety of applications was proposed by Atangana and Baleanu (AB)¹⁹. A considerable number of researchers took advantage of this differential operator to analyze the transmission mechanism of different infectious diseases. Bonyah et al. used this operator and explained the dynamics of co-infection of hepatitis and cancer. A study on tumors was held by Kumar et al. and Ghanbari et al. using this differential operator^20,21. In order to get improved comprehension of COVID-19 disease many authors benefited from this remarkable differential operator^22,23.

Both qualitative and quantitative research on many diseases like dengue fever, diabetes, tuberculosis and smoking have been made utilising Atangana–Baleuno derivative.

Considering dengue fever, Jajarmi et al.²⁴ presented a fractional model in which a sophisticated method has been provided to prove the stability. Baleanu et al. designed optimal control strategy to analyze the efficiency of chemotherapy on Tumor model²⁵. In order to better understanding of diabetes and tuberculosis co-existence, Jajarmi et al.²⁶ provided a very efficient numerical method. Ghanbari et al. contributed in image processing through the application of Atangana–Baleanu fractional operator²⁷. The impact of voluntary vaccination in COVID-19 has been studied mathematically by Ahmad et al.²⁸. Sensitive parameters involved in delayed pneumonia disease model has been studied by Naveed et al.²⁹. Atangana–Baleanu derivative in Caputo sense has been utilized by Din et al. to study Hepatitis B through a mathematical model³⁰. The treatment of polio disease has also been studied by Naveed et al. through delayed mathematical model³¹. Zafar et al. contributed in the study of public health awareness impact on COVID-19 through fractional modeling³². Fractional calculus has been used by Baleanu et al. to study the transmission pattern of Nipah virus³³. Investigation of Human liver has been carried out by Bhatter et al.³⁴ through the application of fractional derivative. Recently, Faiz et al. used Neural Network technique to study the fractional order dengue disease model³⁵. Using the idea of piece wise fractional order derivative in Caputo sense Shah et al. studied Nipah virus disease³⁶. Xue et al. performed stability analysis by considering linear delta operator system³⁷.

This study is based on the analysis of modified mathematical model discussed by Ndendya et al.³⁸. The incidence functio n has been modified as the harmonic mean incidence as suggested by Shah et al.³⁹. We organize our work as follows:

Some basic definitions are recalled in “Some important definitions” Section. “Characteristics of Atangana–Baleanu derivative” Section is devoted for the derivation of equivalent fractional integral equation to Antagana–Baleanu fractional differential equation in Reimaan sense by using the Laplace transform method. Mathematical model exploring the animal rabies disease is developed in “Mathematical model” Section. Existence of solutions and their uniqueness is proved in “Existence of solutions for the dissemination of the rabies diseasemodel” Section. In “Numerical scheme” Section, numerical scheme has been developed and graphical representation, for the illustration of analytical results, is given in “Numerical simulation” Section.

Some important definitions

First of all, we see the result of a function applying Caputo Fractional operator^40,41:

Definition 2.1

The Caputo fractional order derivative for function $\mathfrak {f}\in C^n$ of order $\alpha$ is defined as:

$$^{C}D_{t}^{\alpha }\mathfrak {f}(t)=I^n-^{\alpha }D^n \mathfrak {f} (t)=\frac{1}{\Gamma (n-\alpha )}\int _{0}^{t}\frac{\mathfrak {f}^n(\chi )}{(t-\chi )^{\alpha +n-1}}d\chi .$$

This derivative is well defined for absolutely continuous functions and $\alpha \in (n,n-1),$ where $n\in N.$ It should be noted that the value of the Caputo fractional derivative of the function $\mathfrak {f}$ at any point t has all the values of $\mathfrak {f}^n(\chi )$ for $0\ge \chi \le t.$ Clearly $^{C}D_{t}^{\alpha }\mathfrak {f}(t)$ approaches $\mathfrak {f}\prime (t)$ when $\alpha \rightarrow 1.$

Definition 2.2

The fractional integral of order $\alpha >0$ for a function $\mathfrak {f}:R^+\rightarrow R$ is defined as

$$\begin{aligned} I^\alpha _{t}\mathfrak {f}(t)=\frac{1}{\Gamma (\alpha )}=\int _{0}^{t}(t-\chi )^{\alpha -1} \mathfrak {f}(\chi )d\chi . \end{aligned}$$

Non-Local kernel without singularity help us to define fractional derivative with some characteristics^19,42.

Definition 2.3

Suppose $\mathfrak {f}\in H^{1}(m,n),n>m$ and $0\le \alpha \le 1.$ The Atangana–Baleanu derivative, in the sense of Caputo, is defined as

$$\begin{aligned} _{i}^{ABC}D_{t}^{\alpha }\mathfrak {f}(t)=\frac{\mathfrak {B}\left( \alpha \right) }{1-\alpha } \int _{i}^{t}\mathfrak {f}^{\prime }\left( \chi \right) E_{\alpha }\left[ -\alpha \frac{ \left( t-\chi \right) ^{\alpha }}{1-\alpha }\right] d\chi , \end{aligned}$$

(2.1)

where $\mathfrak {N}\left( \alpha \right)$ is the normalization function whose initial and final values are unity i.e. $\mathfrak {N}(0)=\mathfrak {N}(1)=1$ [16].

Definition 2.4

Suppose $\mathfrak {f}(t)$ is a non differentiable function and $\mathfrak {f}\in H^{1}(m,n),n>m,$ and $\alpha \in \left[ 0,1\right] .$ The Atangana–Baleanu fractional derivative, in the sense of Riemann–Liouville, is defined as

$$\begin{aligned} _{i}^{ABR}D_{t}^{\alpha }\mathfrak {f}(t)=\frac{\mathfrak {N}\left( \alpha \right) }{1-\alpha } \frac{d}{dt }\int _{i}^{t}\mathfrak {f}\left( \chi \right) E_{\alpha }\left[ -\alpha \frac{\left( t-\chi \right) ^{\alpha }}{1-\alpha }\right] d\chi . \end{aligned}$$

(2.2)

Definition 2.5

The above fractional derivative has integral of order $\alpha$, in the form of fraction, as

$$\begin{aligned} _{i}^{AB}I_{t}^{\alpha }\mathfrak {f}(t)=\frac{1-\alpha }{\mathfrak {M}\left( \alpha \right) \Gamma \left( \alpha \right) }\int _{i}^{t}\mathfrak {f}\left( \chi \right) \left( t-\chi \right) ^{\alpha -1}d\chi . \end{aligned}$$

(2.3)

The initial function is obtained for $\alpha =0$ and for $\alpha =1,$ we get the ordinary integral.

Characteristics of Atangana–Baleanu derivative

Real world problems may be modeled by using the definitions given above. We establish the relationships among both definitions and Laplace transform, following are the relations for $n=1$ :

$$\begin{aligned} L\left\{ _{0}^{ABR}D_{t}^{\alpha }\mathfrak {f}(t)\right\} \left( \mathfrak {p}\right) =\frac{ \mathfrak {M}\left( \alpha \right) }{1-\alpha }\frac{\mathfrak {p}^{\alpha }L\left\{ \mathfrak {f}(t)\right\} \left( \mathfrak {p}\right) }{\mathfrak {p}^{\alpha }+\frac{\alpha }{1-\alpha }} \end{aligned}$$

(3.1)

and

$$\begin{aligned} L\left\{ _{0}^{ABC}D_{t}^{\alpha }\mathfrak {f}(t)\right\} \left( \mathfrak {p}\right) =\frac{ \mathfrak {M}\left( \alpha \right) }{1-\alpha }\frac{\mathfrak {p}^{\alpha }L\left\{ \mathfrak {f}(t)\right\} \left( \mathfrak {p}\right) -\mathfrak {p}^{\alpha -1}\mathfrak {f}(0)}{\mathfrak {p}^{\alpha }+\frac{\alpha }{1-\alpha }}. \end{aligned}$$

Theorem 3.1

Suppose a continuous function $\mathfrak {f}$ over a closed interval $\left[ m,n\right] .$ Then, we will get the following inequality on $\left[ m,n\right]$:

$$\begin{aligned} \left\| _{0}^{ABR}D_{t}^{\alpha }\mathfrak {f}(t)\right\| <\frac{\mathfrak {M}\left( \lambda \right) }{1-\alpha }\left\| \mathfrak {f}(x)\right\| . \end{aligned}$$

(3.2)

The values of $\left\| \mathfrak {f}(x)\right\|$ is taken as $\max _{m\le x\le n}\left| \mathfrak {f}(x)\right| .$

Theorem 3.2

Atangana–Baleanu derivative in Caputo and Riemann–Liouville sense satisfy following Lipschitz property:

$$\begin{aligned} \left\| _{0}^{ABR}D_{t}^{\alpha }\mathfrak {J}(t)-_{0}^{ABR}D_{t}^{\alpha }\mathfrak {L}(t)\right\| \le H\left\| \mathfrak {J}(t)-\mathfrak {L}(t)\right\| \end{aligned}$$

(3.3)

and

$$\begin{aligned} \left\| _{0}^{ABC}D_{t}^{\alpha }\mathfrak {J}(t)-_{0}^{ABC}D_{t}^{\alpha }\mathfrak {L}(t)\right\| \le H\left\| \mathfrak {J}(t)-\mathfrak {L}(t)\right\| . \end{aligned}$$

Theorem 3.3

A time fractional ODE is given below,

$$\begin{aligned} _{0}^{ABC}D_{t}^{\alpha }\mathfrak {f}(h)=\mathfrak {u}\left( t\right) , \end{aligned}$$

(3.4)

by making use of inverse Laplace transform and convolution theorem¹³, the equation given has one and only one solution, this solution is given below:

$$\begin{aligned} \mathfrak {f}\left( t\right) =\frac{1-\alpha }{\mathfrak {M}\left( \alpha \right) }\mathfrak {u}\left( t\right) + \frac{\alpha }{\mathfrak {M}\left( \alpha \right) \Gamma \left( \alpha \right) } \int _{c}^{t}\mathfrak {u}\left( y\right) \left( t-y\right) ^{\alpha -1}dy. \end{aligned}$$

(3.5)

Interested readers may find the proofs of theorems in detail in¹⁹.

Mathematical model

Animal rabies disease (ARB) is studied through the development of fractional model with the assumption of inclusion of infected immigrants. The whole rabies population is assumed to be composed of five mutually exclusive compartments. These compartments are labeled as Susceptible Rabies, Vaccinated, Exposed, Infectious and Recovered (Rabies). These classes are represented as $\mathcal {R}_s(t),\mathcal {R}_v(t),\mathcal {R}_e(t),\mathcal {R}_i(t),$ and $\mathcal {R}_s(t),$ respectively. The growth of rabies population is assumed via two routes: either by birth or infected immigrants, these are denoted by $\Theta$ and $\Omega ,$ respectively. The population in each class decays through natural death which is represented by the rate $\omega .$ However, infected rabies have additional removal rate represented by $\psi .$ It is assumed that vaccine is not fully effective and the rate of waning immunity is represented by $\phi .$ The incidence function for the transmission mechanism of rabies disease is considered as harmonic mean of $\mathcal {R}_s(t)$ and $\mathcal {R}_i(t)$ or $\mathcal {R}_s(t)$ and $\mathcal {R}_v(t)$ and are represented as $\frac{2\upsilon \mathcal {R}_s(t)\mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)}$ and $\frac{2\upsilon (1-\epsilon )\mathcal {R}_v(t)\mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)},$ with transmission rate $\upsilon ,$ respectively. The modification parameter which reduces the transmission of rabies among vaccinated individuals is represented as $\epsilon .$ The transition rates among susceptible to vaccinated, exposed to infectious and infected to recovered are represented as $\tau ,\sigma$ and $\rho ,$ respectively. We assume that recovered individuals don’t have permanent immunity and they join the susceptible class after $\frac{1}{\xi }$ time.

With these assumptions, now we are able to define the fractional model which is as follows:

$$\begin{aligned} \frac{d\mathcal {R}_s(t) }{dt}= & \Theta +\xi \mathcal {R}_r(t)-\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)}+\omega +\tau \right) \mathcal {R}_s(t)+\phi \mathcal {R}_v(t), \end{aligned}$$

(4.1)

$$\begin{aligned} \frac{d\mathcal {R}_v(t) }{dt}= & \tau \mathcal {R}_s(t)-\left( \left( 1-\epsilon \right) \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)} +\phi +\omega \right) \mathcal {R}_v(t), \end{aligned}$$

(4.2)

$$\begin{aligned} \frac{d\mathcal {R}_e(t) }{dt}= & \frac{2\upsilon \mathcal {R}_i(t)\mathcal {R}_s(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)} +\left( 1-\epsilon \right) \frac{2\upsilon \mathcal {R}_i(t)\mathcal {R}_v(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)} -\left( \sigma +\omega \right) \mathcal {R}_e(t),\nonumber \\ \frac{d\mathcal {R}_i(t) }{dt}= & \Omega +\sigma \mathcal {R}_e(t)-\left( \rho +\psi +\omega \right) \mathcal {R}_i(t), \end{aligned}$$

(4.3)

$$\begin{aligned} \frac{d\mathcal {R}_r(t) }{dt}= & \rho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t). \end{aligned}$$

(4.4)

Existence of solutions for the dissemination of the rabies disease model

In this paper, we have considered a non linear model of the rabies disease. We have no way to establish the analytical solution of this system of differential equations. But, one can give assurance, under some conditions, about the existence of exact solution. The interested readers may go through the work of Gu et al. for getting the approximate solutions of fractional differential equations^43,44,45. We will proceed here as follows to show the existence of solutions.

Let $\mathcal {L}\left( J\right)$ be the Banach space containing continuous real valued functions from $R\rightarrow R$ defined on interval J and $\mathcal {Q}=\mathcal {L}\left( J\right) *\mathcal {L}\left( J\right)$. We define the norm as

$$\begin{aligned} \left\| \mathcal {R}_s(t),\mathcal {R}_v(t),\mathcal {R}_e(t),\mathcal {R}_i(t),\mathcal {R}_r(t)\right\| =\left\| \mathcal {R}_s(t)\right\| +\left\| \mathcal {R}_v(t)\right\| +\left\| \mathcal {R}_e(t)\right\| +\left\| \mathcal {R}_i(t)\right\| +\left\| \mathcal {R}_r(t)\right\| \end{aligned}$$

(5.1)

Here, $\left\| \mathcal {R}_s(t)\right\| =\left\{ \sup \left\{ \left| \mathcal {R}_s(t) \right| \right\} :t\in J\right\}$, and all other norms for $\mathcal {R}_e(t),\mathcal {R}_v(t),\mathcal {R}_i(t)$ and $\mathcal {R}_r(t)$ are defined in the same way. By writing the above model using time fractional derivative of order $\mho \in [0,1]$, we have its following form

$$\begin{aligned} _0^{ABC}D_t^{\mho } \mathcal {R}_s(t)= & \Theta +\xi \mathcal {R}_r(t)-\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)}+\omega +\tau \right) \mathcal {R}_s(t)+\phi \mathcal {R}_v(t),\nonumber \\ _0^{ABC}D_t^{\mho } \mathcal {R}_v(t)= & \tau \mathcal {R}_s(t)-\left( \left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)}+\phi +\omega \right) \mathcal {R}_v(t), \nonumber \\ _0^{ABC}D_t^{\mho } \mathcal {R}_e(t)= & \frac{2\upsilon \mathcal {R}_i(t)\mathcal {R}_s(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)} +\left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(t)\mathcal {R}_v(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)} -\left( \sigma +\omega \right) \mathcal {R}_e(t),\\ _0^{ABC}D_t^{\mho } \mathcal {R}_i(t)= & \Omega +\sigma \mathcal {R}_e(t)-\left( \rho +\psi +\omega \right) \mathcal {R}_i(t),\nonumber \\ _0^{ABC}D_t^{\mho }\mathcal {R}_r(t)= & \rho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t),\nonumber \end{aligned}$$

(5.2)

having initial conditions

$$\begin{aligned} (\mathcal {R}_s(0),\mathcal {R}_v(0),\mathcal {R}_e(0),\mathcal {R}_i(0), \mathcal {R}_r(0))=(\mathcal {R}_{s0}.\mathcal {R}_{v0},\mathcal {R}_{e0}, \mathcal {R}_{i0},\mathcal {R}_{r0}). \end{aligned}$$

(5.3)

By making the use of Atangana–Baleanu fractional integral, the system of equations, stated above, may be transformed into Volterra type integral equations. The model in view of Theorem (3.3) may be expressed as

$$\begin{aligned} \mathcal {R}_s(t)-\mathcal {R}_s(0)= & \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \Theta +\xi \mathcal {R}_r(t)-\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+ \mathcal {R}_i(t)}+\omega +\tau \right) \mathcal {R}_s(t)+\phi \mathcal {R}_v(t) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \Theta +\xi \mathcal {R}_r(z)-\left( \frac{2\upsilon \mathcal {R}_i(z)}{\mathcal {R}_s(z)+\mathcal {R}_i(z)} +\omega +\tau \right) \mathcal {R}_s(z)+\phi \mathcal {R}_v(z) \right] dz, \nonumber \\ \mathcal {R}_v(t) -\mathcal {R}_v(0)= & \frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \tau \mathcal {R}_s(t)-\left( \left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)} +\phi +\omega \right) \mathcal {R}_v(t) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \tau \mathcal {R}_s(z)-\left( \left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(z)}{\mathcal {R}_v(z)+\mathcal {R}_i(z)}+\phi +\omega \right) \mathcal {R}_v(z) \end{array} \right] dz, \nonumber \\ \mathcal {R}_e(t) -\mathcal {R}_e(0)= & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\left\{ \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)} +\left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)} -\left( \sigma +\omega \right) \mathcal {R}_e(t) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \frac{2\upsilon \mathcal {R}_i(z)}{\mathcal {R}_s(z)+\mathcal {R}_i(z)} +\left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(z)}{\mathcal {R}_v(z)+\mathcal {R}_i(z)} -\left( \sigma +\omega \right) \mathcal {R}_e(z) \end{array} \right] dz, \nonumber \\ \mathcal {R}_i(t) -\mathcal {R}_i(0)= & \frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \Omega +\sigma \mathcal {R}_e(t)-\left( \mho +\psi +\omega \right) \mathcal {R}_i(t) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \Omega +\sigma \mathcal {R}_e(z)-\left( \mho +\psi +\omega \right) \mathcal {R}_i(z) \end{array} \right] dz, \nonumber \\ \mathcal {R}_r(t)-\mathcal {R}_r(0)= & \frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \mho \mathcal {R}_i(t)- \left( \xi +\omega \right) \mathcal {R}_r(t) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-\textsf{z}\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \mho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t) \end{array} \right] \textsf{dz},\nonumber \end{aligned}$$

(5.4)

For the sake of simplicity, we let

$$\begin{aligned} \mathbb {U}_{1}\left( t,\mathcal {R}_s\right)= & \Theta +\xi \mathcal {R}_r(t)-\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)}+\omega +\tau \right) \mathcal {R}_s(t)+\phi \mathcal {R}_v(t),\nonumber \\ \mathbb {U}_{2}\left( t,\mathcal {R}_v\right)= & \tau \mathcal {R}_s(t)-\left( \left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)}+\phi +\omega \right) \mathcal {R}_v(t), \nonumber \\ \mathbb {U}_{3}\left( t,\mathcal {R}_e\right)= & \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)} +\left( 1-\varepsilon \right) \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_v(t)+\mathcal {R}_i(t)} -\left( \sigma +\omega \right) \mathcal {R}_e(t),\nonumber \\ \mathbb {U}_{4}\left( t,\mathcal {R}_i\right)= & \Omega +\sigma \mathcal {R}_e(t)-\left( \rho +\psi +\omega \right) \mathcal {R}_i(t),\nonumber \\ \mathbb {U}_{5}\left( t,\mathcal {R}_r\right)= & \rho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t). \end{aligned}$$

(5.5)

Theorem 5.1

Lipschitz Condition and contraction will be satisfied by the kernels, defined above, $\mathbb {U}_{1},$ $\mathbb {U}_{2},$ $\mathbb {U}_{3},$ $\mathbb {U}_4$ and $\mathbb {U}_{5}$, if one has the following inequalities:

$$\begin{aligned} 0\le & \alpha _{1}\le 1, \nonumber \\ 0\le & \alpha _{2}\le 1, \nonumber \\ 0\le & \alpha _{3}\le 1, \nonumber \\ 0\le & \alpha _{4}\le 1, \nonumber \\ 0\le & \alpha _{5}\le 1. \end{aligned}$$

(5.6)

Proof

To show this, we consider the kernel $\mathbb {U}_{1}\left( t,\mathcal {R}_s\right) =\Theta +\xi \mathcal {R}_r(t)-\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)}+\omega +\tau \right) \mathcal {R}_s(t)+\phi \mathcal {R}_v(t).$ Consider two functions $\mathcal {R}_s$ and $\mathcal {R}_{s_1}$, then

$$\begin{aligned} \left\| \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_1}\right) \right\|= & \left\| -\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)} \right) \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) +\left( \omega +\tau \right) \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| \nonumber \\\le & \left\| -\left( \frac{2\upsilon \mathcal {R}_i(t)}{\mathcal {R}_s(t)+\mathcal {R}_i(t)} \right) \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| +\left( \omega +\tau \right) \left\| \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| \nonumber \\\le & 2\upsilon \left\| \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| +\left( \omega +\tau \right) \left\| \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| \nonumber \\\le & \left( 2\upsilon +\omega +\tau \right) \left\| \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| \nonumber \\ \end{aligned}$$

(5.7)

Taking, $\alpha _{1}=2\upsilon +\omega +\tau ,$ we arrive at the result

$$\begin{aligned} \left\| \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_1}\right) \right\| \le \alpha _{1}\left\| \left( \mathcal {R}_s(t)-\mathcal {R}_{s_1}(t)\right) \right\| . \end{aligned}$$

(5.8)

Consequently, we have the Lipschitz condition for the kernel $\mathbb {U}_{1}$, and furthermore if $0\le \alpha _{1}\le 1,$ so one has also the contraction for $\mathbb {U}_{1}$. In the similar way, we can show that all the remaining kernels fulfil the Lipschitz condition as below:

$$\begin{aligned} \left\| \mathbb {U}_{2}\left( t,\mathcal {R}_v\right) -\mathbb {U}_{2}\left( t,\mathcal {R}_{v_1}\right) \right\| \le \alpha _{2}\left\| \left( \mathcal {R}_v(t)-\mathcal {R}_{v_1}(t)\right) \right\| . \nonumber \\ \left\| \mathbb {U}_{3}\left( t,\mathcal {R}_e\right) -\mathbb {U}_{3}\left( t,\mathcal {R}_{e_1}\right) \right\| \le \alpha _{3}\left\| \left( \mathcal {R}_e(t)-\mathcal {R}_{e_1}(t)\right) \right\| . \nonumber \\ \left\| \mathbb {U}_{4}\left( t,\mathcal {R}_i\right) -\mathbb {U}_{4}\left( t,\mathcal {R}_{i_1}\right) \right\| \le \alpha _{4}\left\| \left( \mathcal {R}_i(t)-\mathcal {R}_{i_1}(t)\right) \right\| . \nonumber \\ \left\| \mathbb {U}_{5}\left( t,\mathcal {R}_r\right) -\mathbb {U}_{5}\left( t,\mathcal {R}_{r_1}\right) \right\| \le \alpha _{5}\left\| \left( \mathcal {R}_r(t)-\mathcal {R}_{r_1}(t)\right) \right\| . \end{aligned}$$

(5.9)

By using the kernels given in the Eqs. (5.8) and (5.9), system (5.4) can be written as

$$\begin{aligned} \mathcal {R}_s(t)-\mathcal {R}_s(0)= & \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{1}\left( z,\mathcal {R}_s\right) \right] dz, \nonumber \\ \mathcal {R}_v(t) -\mathcal {R}_v(0)= & \frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \mathbb {U}_{2}\left( t,\mathcal {R}_v\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{2}\left( z,\mathcal {R}_v\right) \right] dz, \nonumber \\ \mathcal {R}_e(t) -\mathcal {R}_e(0)= & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\left\{ \mathbb {U}_{3}\left( t,\mathcal {R}_e\right) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{3}\left( t,\mathcal {R}_e\right) \right] dz, \nonumber \\ \mathcal {R}_i(t) -\mathcal {R}_i(0)= & \frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \Omega +\sigma \mathcal {R}_e(t)-\left( \rho +\psi +\omega \right) \mathcal {R}_i(t) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \Omega +\sigma \mathcal {R}_e(z)-\left( \rho +\psi +\omega \right) \mathcal {R}_i(z) \end{array} \right] dz, \nonumber \\ \mathcal {R}_r(t)-\mathcal {R}_r(0)= & \frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \rho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-\textsf{z}\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \rho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t) \end{array} \right] \textsf{dz},\nonumber \end{aligned}$$

(5.10)

Now we can build the recursion relation as follows:

$$\begin{aligned} \mathcal {R}_{s_n}(t)= & \mathcal {R}_s(0)+\frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{1}\left( z,\mathcal {R}_s\right) \right] dz, \\ \mathcal {R}_{v_n}(t)= & \mathcal {R}_v(0) +\frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \mathbb {U}_{2}\left( t,\mathcal {R}_v\right) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{2}\left( z,\mathcal {R}_v\right) \right] dz, \\ \mathcal {R}_{e_n}(t)= & \mathcal {R}_e(0) +\frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\left\{ \mathbb {U}_{3}\left( t,\mathcal {R}_e\right) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{3}\left( t,\mathcal {R}_e\right) \right] dz, \\ \mathcal {R}_{i_n}(t)= & \mathcal {R}_i(0) +\frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \Omega +\sigma \mathcal {R}_e(t)-\left( \mho +\psi +\omega \right) \mathcal {R}_i(t) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \Omega +\sigma \mathcal {R}_e(z)-\left( \mho +\psi +\omega \right) \mathcal {R}_i(z) \end{array} \right] dz, \\ \mathcal {R}_{r_n}(t)= & \mathcal {R}_r(0)+\frac{\left( 1-\mho \right) }{ \mathfrak {M(\mho )} }\left\{ \mho \mathcal {R}_i(t)-\left( \mho +\omega \right) \mathcal {R}_r(t) \right\} \\ & +\frac{\mho }{\mathfrak {M(\mho )}\Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-\textsf{z}\right) ^{-\left( 1-\mho \right) }\left[ \begin{array}{c} \mho \mathcal {R}_i(t)-\left( \xi +\omega \right) \mathcal {R}_r(t) \end{array} \right] \textsf{dz}, \end{aligned}$$

The initial conditions are also given as

$$\begin{aligned} \mathcal {R}_{s_0}\left( t\right) =\mathcal {R}_{s_0},\mathcal {R}_{v_0}\left( t\right) =\mathcal {R}_{v_0},\mathcal {R}_{e_0}\left( t\right) =\mathcal {R}_{e_0}, \mathcal {R}_{i_0}\left( t\right) =\mathcal {R}_{i_0},\mathcal {R}_{r_0}\left( t\right) =\mathcal {R}_{r_0}. \end{aligned}$$

(5.11)

The difference among successive terms may be established as follows:

$$\begin{aligned} \Theta _{n}\left( t\right)= & \mathcal {R}_{s_n}\left( t\right) -\mathcal {R}_{s_{n-1}}\left( t\right) = \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{1}\left( t,\mathcal {R}_{s_n}\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_{n-1}}\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{1}\left( z,\mathcal {R}_{s_{n}}\right) -\mathbb {U}_{1}\left( z,\mathcal {R}_{s_{n-1}}\right) \right] dz, \nonumber \\ \xi _{n}\left( t\right)= & \mathcal {R}_{v_n}\left( t\right) -\mathcal {R}_{v_{n-1}}\left( t\right) = \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{2}\left( t,\mathcal {R}_{v_n}\right) -\mathbb {U}_{2}\left( t,\mathcal {R}_{v_{n-1}}\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{2}\left( z,\mathcal {R}_{v_{n}}\right) -\mathbb {U}_{2}\left( z,\mathcal {R}_{v_{n-1}}\right) \right] dz, \nonumber \\ \Pi _{n}\left( t\right)= & \mathcal {R}_{e_n}\left( t\right) -\mathcal {R}_{e_{n-1}}\left( t\right) = \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{3}\left( t,\mathcal {R}_{e_n}\right) -\mathbb {U}_{3}\left( t,\mathcal {R}_{e_{n-1}}\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{3}\left( z,\mathcal {R}_{e_{n}}\right) -\mathbb {U}_{3}\left( z,\mathcal {R}_{e_{n-1}}\right) \right] dz, \nonumber \\ \Upsilon _{n}\left( t\right)= & \mathcal {R}_{i_n}\left( t\right) -\mathcal {R}_{i_{n-1}}\left( t\right) = \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{4}\left( t,\mathcal {R}_{i_n}\right) -\mathbb {U}_{4}\left( t,\mathcal {R}_{i_{n-1}}\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{4}\left( z,\mathcal {R}_{i_{n}}\right) -\mathbb {U}_{4}\left( z,\mathcal {R}_{i_{n-1}}\right) \right] dz, \nonumber \\ \Phi _{n}\left( t\right)= & \mathcal {R}_{r_n}\left( t\right) -\mathcal {R}_{r_{n-1}}\left( t\right) = \frac{1-\mho }{\mathfrak {M(\mho )}} \left\{ \mathbb {U}_{5}\left( t,\mathcal {R}_{r_n}\right) -\mathbb {U}_{5}\left( t,\mathcal {R}_{r_{n-1}}\right) \right\} \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{5}\left( z,\mathcal {R}_{r_{n}}\right) -\mathbb {U}_{5}\left( z,\mathcal {R}_{r_{n-1}}\right) \right] dz, \end{aligned}$$

(5.12)

It is worth noticing that

$$\begin{aligned} \mathcal {R}_{s_n}\left( t\right) =\sum _{i=1}^{n}\Theta _{i}\left( t\right) , \mathcal {R}_{v_n}\left( t\right) =\sum _{i=1}^{n}\xi _{i}\left( t\right) , \mathcal {R}_{e_n}\left( t\right) =\sum _{i=1}^{n}\Pi _{i}\left( t\right) , \mathcal {R}_{i_n}\left( t\right) =\sum _{i=1}^{n}\Upsilon _{i}\left( t\right) , \mathcal {R}_{r_n}\left( t\right) =\sum _{i=1}^{n}\Phi _{i}\left( t\right) . \end{aligned}$$

(5.13)

Considering the system (5.12), applying the norm and triangular inequality, we get

$$\begin{aligned} \left\| \Theta _{n}\left( t\right) \right\|= & \left\| \mathcal {R}_{s_n}\left( t\right) -\mathcal {R}_{s_{n-1}}\left( t\right) \right\| \nonumber \\\le & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{1}\left\| \mathcal {R}_{s_{n-1}}\left( t\right) -\mathcal {R}_{s_{n-2}}\left( t\right) \right\| \\ & +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\left\| \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left[ \mathbb {U}_{1}\left( z,\mathcal {R}_{s_{n-1}}\right) -\mathbb {U}_{1}\left( z,\mathcal {R}_{s_{n-2}}\right) \right] dz\right\| . \nonumber \end{aligned}$$

(5.14)

As the Lipschitz condition are satisfied by the Kernal, we have

$$\begin{aligned} \left\| \mathcal {R}_{s_n}\left( t\right) -\mathcal {R}_{s_{n-1}}\left( t\right) \right\|\le & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{1}\left\| \mathcal {R}_{s_{n-1}}\left( t\right) -\mathcal {R}_{s_{n-2}}\left( t\right) \right\| \\ & +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\alpha _{1} \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left\| \mathcal {R}_{s_{n-1}}\left( t\right) -\mathcal {R}_{s_{n-2}}\left( t\right) \right\| dz, \nonumber \end{aligned}$$

(5.15)

for which we have,

$$\begin{aligned} \left\| \Theta _{n}\left( t\right) \right\| \le \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{1}\left\| \Theta _{n-1}\left( t\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\alpha _{1} \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left\| \Theta _{n-1}\left( t\right) \right\| dz, \end{aligned}$$

(5.16)

In the similar manner, the following results are obtained:

$$\begin{aligned} \left\| \xi _{n}\left( t\right) \right\|\le & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{2}\left\| \xi _{n-1}\left( t\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\alpha _{2} \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left\| \xi _{n-1}\left( t\right) \right\| dz,\nonumber \\ \left\| \Pi _{n}\left( t\right) \right\|\le & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{3}\left\| \Pi _{n-1}\left( t\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\alpha _{3} \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left\| \Pi _{n-1}\left( t\right) \right\| dz,\nonumber \\ \left\| \Upsilon _{n}\left( t\right) \right\|\le & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{4}\left\| \Upsilon _{n-1}\left( t\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\alpha _{4} \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left\| \Upsilon _{n-1}\left( t\right) \right\| dz,\nonumber \\ \left\| \Phi _{n}\left( t\right) \right\|\le & \frac{\left( 1-\mho \right) }{\mathfrak {M(\mho )} }\alpha _{5}\left\| \Phi _{n-1}\left( t\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} +\Gamma \left( \mho \right) }\alpha _{5} \int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\left\| \Phi _{n-1}\left( t\right) \right\| dz. \end{aligned}$$

(5.17)

$\square$

After establishing the above results, it is very easy for us to state and prove the following new Theorem.

Theorem 5.2

The Rabies disease model, represented by (5.2), has a solution with the condition

$$\begin{aligned} \frac{1-\mho }{\mathfrak {M(\mho )} }\alpha _{i}+\frac{t_{\max }^{\mho }}{ \mathfrak {M(\mho )} \Gamma \left( \mho \right) }\alpha _{i}<1, for~i=1,2,3,4,5. \end{aligned}$$

(5.18)

Proof

Since all the functions taken in the model (5.2) are bounded having kernels satisfying the Lipschitz condition, therefore from (5.16), we get the succeeding terms as :

$$\begin{aligned} \left\| \Theta _{n}\left( t\right) \right\|\le & \left\| \mathcal {R}_s(0) \right\| \left[ \frac{1-\mho }{\mathfrak {M(\mho )} }\alpha _{1}+ \frac{t_{\max }^{\mho }}{\Gamma \left( \mho \right) \mathfrak {M(\mho )} } \alpha _{1}\right] ^{n}, \nonumber \\ \left\| \xi _{n}\left( t\right) \right\|\le & \left\| \mathcal {R}_v(0) \right\| \left[ \frac{1-\mho }{\mathfrak {M(\mho )} }\alpha _{2}+ \frac{t_{\max }^{\mho }}{\Gamma \left( \mho \right) \mathfrak {M(\mho )} } \alpha _{2}\right] ^{n}, \nonumber \\ \left\| \Pi _{n}\left( t\right) \right\|\le & \left\| \mathcal {R}_e(0) \right\| \left[ \frac{1-\mho }{\mathfrak {M(\mho )} }\alpha _{3}+ \frac{t_{\max }^{\mho }}{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \alpha _{3}\right] ^{n}, \nonumber \\ \left\| \Upsilon _{n}\left( t\right) \right\|\le & \left\| \mathcal {R}_i(0) \right\| \left[ \frac{1-\mho }{\mathfrak {M(\mho )} }\alpha _{4}+ \frac{t_{\max }^{\mho }}{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \alpha _{4}\right] ^{n}, \nonumber \\ \left\| \Phi _{n}\left( t\right) \right\|\le & \left\| \mathcal {R}_r(0) \right\| \left[ \frac{1-\mho }{\mathfrak {M(\mho )} } \alpha _{5}+\frac{t_{\max }^{\mho }}{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\alpha _{5}\right] ^{n}. \end{aligned}$$

(5.19)

Thus equalities given in (5.13) exist. The given functions are smooth. Now we prove the existence of solution of model (5.2) and these solutions are expressed in the form of above functions. For this, we assume

$$\begin{aligned} \mathcal {R}_s\left( t\right)= & \mathcal {R}_s(0)+\mathcal {R}_{s_{n}}\left( t\right) -a_{n}\left( t\right) , \nonumber \\ \mathcal {R}_{v}\left( t\right)= & \mathcal {R}_{v}\left( 0\right) +\mathcal {R}_{v_{n}}\left( t\right) -b_{n}\left( t\right) , \nonumber \\ \mathcal {R}_{e}\left( t\right)= & \mathcal {R}_{e}\left( 0\right) +\mathcal {R}_{e_{n}}\left( t\right) -c_{n}\left( t\right) , \nonumber \\ \mathcal {R}_{i}\left( t\right)= & \mathcal {R}_{i}\left( 0\right) +\mathcal {R}_{i_{n}}\left( t\right) -d_{n}\left( t\right) , \nonumber \\ \mathcal {R}_{r}\left( t\right)= & \mathcal {R}_{r}\left( 0\right) +\mathcal {R}_{r_{n}}\left( t\right) -e_{n}\left( t\right) . \end{aligned}$$

(5.20)

Our purpose is to show that $\left\| a_{\infty }\left( t\right) \right\| \rightarrow 0$ ultimately. For this, we get

$$\begin{aligned} \left\| a_{n}\left( t\right) \right\|\le & \left\| \frac{1-\mho }{ \mathfrak {M(\mho )} }\mathbb {U}_{1}\left( t,\mathcal {R}_s\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_{n-1}}\right) + \frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{\mho -1}\left( \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_{n-1}}\right) \right) dz\right\| ,\nonumber \\ \left\| a_{n}\left( t\right) \right\|\le & \frac{1-\mho }{N\left( \mho \right) }\left\| \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_{n-1}}\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{\mho -1}\left\| \left( \mathbb {U}_{1}\left( t,\mathcal {R}_s\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s_{n-1}}\right) \right) \right\| dz \nonumber \\\le & \frac{1-\mho }{\mathfrak {M(\mho )} }\tau _{1}\left\| \mathcal {R}_s-\mathcal {R}_{s_{n-1}}\right\| +\frac{t^{\mho }}{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\tau _{1}\left\| \mathcal {R}_s-\mathcal {R}_{s_{n-1}}\right\| . \end{aligned}$$

(5.21)

Repeating this process recursively, we obtain

$$\begin{aligned} \left\| a_{n}\left( t\right) \right\| \le \left[ \frac{1-\mho }{ \mathfrak {M(\mho )} }+\frac{t^{\mho }}{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\right] ^{n+1}\tau _{1}^{n}L. \end{aligned}$$

(5.22)

Then at $t_{\max },$ we have

$$\begin{aligned} \left\| a_{n}\left( t\right) \right\| \le \left[ \frac{1-\mho }{ \mathfrak {M(\mho )} }+\frac{t_{\max }^{\mho }}{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\right] ^{n+1}\tau _{1}^{n}L. \end{aligned}$$

(5.23)

Applying the limit $n->\infty$, on both sides, we get $\left\| a_{\infty }\left( t\right) \right\| \rightarrow 0$. Hence, the assertion is completed. $\square$

Uniqueness of the solution

Now, in this section, we prove the uniqueness of solutions for the system (5.2). To achieve this goal, we assume that another solution say $(\mathcal {R}_{s1}, \mathcal {R}_{v1}, \mathcal {R}_{e1}, \mathcal {R}_{i1}, \mathcal {R}_{r1})$, exists. We will proceed as follows,

$$\begin{aligned} \left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| \le \frac{ 1-\mho }{\mathfrak {M(\mho )} }[\mathbb {U}_{1}\left( t,\mathcal {R}_{s}\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s1}\right) ] +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \int \limits _{0}^{t}\left( t-z\right) ^{\mho -1}\left\| \left( \mathbb {U}_{1}\left( t,\mathcal {R}_{s}\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s1}\right) \right) \right\| dz. \end{aligned}$$

(5.24)

We get the following result, using the property of norm:

$$\begin{aligned} \left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| \le \frac{ 1-\mho }{\mathfrak {M(\mho )} }\left\| \mathbb {U}_{1}\left( t,\mathcal {R}_{s}\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s1}\right) \right\| +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\int \limits _{0}^{t}\left( t-z\right) ^{\mho -1}\left\| \mathbb {U}_{1}\left( t,\mathcal {R}_{s}\right) -\mathbb {U}_{1}\left( t,\mathcal {R}_{s1}\right) \right\| dz. \end{aligned}$$

(5.25)

By the application of Lipschitz condition, one has

$$\begin{aligned} \left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| \le \frac{ 1-\mho }{\mathfrak {M(\mho )} }\alpha _{1}\left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| +\frac{\alpha _{1}t^{\mho }}{N\left( \mho \right) \Gamma \left( \mho \right) }\left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| . \end{aligned}$$

(5.26)

This gives

$$\begin{aligned} \left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| \left( 1-\frac{ 1-\mho }{\mathfrak {M(\mho )} }\alpha _{1}-\frac{t^{\mho }\alpha _{1}}{N\left( \mho \right) \Gamma \left( \mho \right) }\right) \le 0. \end{aligned}$$

(5.27)

According to Theorem (5.2), $\left( 1-\frac{ 1-\mho }{\mathfrak {M(\mho )} }\alpha _{1}+\frac{t^{\mho }\alpha _{1}}{N\left( \mho \right) \Gamma \left( \mho \right) }\right) >0,$ for $\mho \in [0,1]$ and $\alpha _{1} \in [0,1]$. Thus

$$\begin{aligned} \left\| \mathcal {R}_{s}\left( t\right) -\mathcal {R}_{s1}\left( t\right) \right\| =0. \end{aligned}$$

Thus we have

$$\begin{aligned} \mathcal {R}_{s}\left( t\right) =\mathcal {R}_{s1}\left( t\right) . \end{aligned}$$

(5.28)

Applying the same procedure, it can be shown that

$\mathcal {R}_{v}\left( t\right) =\mathcal {R}_{v1}\left( t\right) , \mathcal {R}_{e}\left( t\right) =\mathcal {R}_{e1}\left( t\right) , \mathcal {R}_{i}\left( t\right) =\mathcal {R}_{i1}\left( t\right) , \mathcal {R}_{r}\left( t\right) =\mathcal {R}_{r1}\left( t\right)$. In this way, solution of system (5.2) is unique.

Numerical scheme

Some special problems involving rate of change having non-local kernel without singularity, has been solved numerically by applying a very novel technique recently developed by Toufik and Atangana [19]. In their work, two salient features can be easily seen, one the quick convergence of the problem to the solution and other is high accuracy. Their applied method can be comprehend easily by considering an ordinary differential equation in non linear form as:

$$\begin{aligned} _0^{ABC}D_t^{\mho } \mathfrak {u}\left( t\right)= & \mathfrak {h}\left( t,\mathfrak {u}\left( t\right) \right) \\ \mathfrak {u}\left( 0\right)= & \mathfrak {u}_{0}. \nonumber \end{aligned}$$

(6.1)

The fractional integral equivalent to initial value problem (6.1) is given as

$$\begin{aligned} \mathfrak {u}\left( t\right) -\mathfrak {u}\left( 0\right) \le \frac{1-\mho }{\mathfrak {M(\mho )} } \mathfrak {h}\left( t,\mathfrak {u}\left( t\right) \right) +\frac{\mho }{\Gamma \left( \mho \right) \mathfrak {M(\mho )} }\int \limits _{0}^{t}\mathfrak {f}\left( z,\mathfrak {u}\left( z\right) \right) \left( t-z\right) ^{\mho -1}dz. \end{aligned}$$

(6.2)

The integral Eq. (6.2), for $t=t_{n+1}$, $n=0,$ 1,  $2 \ldots ,$ can be written as

$$\begin{aligned} \mathfrak {u}\left( t_{n+1}\right) -\mathfrak {u}\left( 0\right) \le \frac{1-\mho }{M\left( \mho \right) }\mathfrak {h}\left( t_{n},\mathfrak {u}\left( t_{n}\right) \right) +\frac{\mho }{M\left( \mho \right) \Gamma \left( \mho \right) }\int \limits _{0}^{t_{n+1}}\mathfrak {h}\left( z,\mathfrak {u}\left( z\right) \right) \left( t_{n+1}-z\right) ^{\mho -1}dz. \end{aligned}$$

(6.3)

The application of Lagrange polynomial interpolation on the function $\mathfrak {h}\left( z,\mathfrak {u}\left( z\right) \right)$ in two steps for the interval $\left[ t_{k},t_{k+1}\right]$ gives the result as

$$\begin{aligned} p_{k}\left( z\right)= & \mathfrak {h}\left( z,\mathfrak {u}\left( z\right) \right) \nonumber \\= & \frac{z-t_{k-1}}{t_{k}-t_{k-1}}\mathfrak {h}\left( t_{k},\mathfrak {u}\left( t_{k}\right) \right) -\frac{z-t_{k}}{t_{k}-t_{k-1}}\mathfrak {h}\left( t_{k-1},\mathfrak {u}\left( t_{k-1}\right) \right) \nonumber \\= & \frac{\mathfrak {h}\left( t_{k},\mathfrak {u}\left( t_{k}\right) \right) }{h}\left( z-t_{k-1}\right) -\frac{\mathfrak {h}\left( t_{k-1},\mathfrak {u}\left( t_{k-1}\right) \right) }{h} \left( z-t_{k}\right) \nonumber \\\simeq & \frac{\mathfrak {h}\left( t_{k},\mathfrak {u}_{k}\right) }{h}\left( z-t_{k-1}\right) -\frac{ \mathfrak {h}\left( t_{k-1},\mathfrak {u}_{k-1}\right) }{h}\left( z-t_{k}\right) \end{aligned}$$

(6.4)

If we again consider Eq. (6.3) for $\mathfrak {f}\left( z,u\left( z\right) \right)$ together with the Lagrange polynomial interpolation, we have:

$$\begin{aligned} \mathfrak {u}_{n+1}=\mathfrak {u}\left( 0\right) +\frac{1-\mho }{\mathfrak {M(\mho )} }\mathfrak {h}\left( t_{n},\mathfrak {u}\left( t_{n}\right) \right) +\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\sum \limits _{k=0}^{n}\left( \begin{array}{c} \frac{\mathfrak {h}\left( t_{k},\mathfrak {u}_{k}\right) }{h}\int \limits _{t_{k}}^{t_{k+1}}\left( z-t_{k-1}\right) \left( t_{n+1}-z\right) ^{\mho -1}dz \\ -\frac{\mathfrak {h}\left( t_{k-1},\mathfrak {u}_{k-1}\right) }{h}\int \limits _{t_{k}}^{t_{k+1}} \left( z-t_{k}\right) \left( t_{n+1}-z\right) ^{\mho -1}dz \end{array} \right) , \end{aligned}$$

(6.5)

We obtain the following equality, after calculating the integral expression given in the above sum,

$$\begin{aligned} \mathfrak {u}_{n+1}= & \mathfrak {u}\left( 0\right) +\frac{1-\mho }{\mathfrak {M(\mho )} }\mathfrak {h}\left( t_{n},\mathfrak {u}\left( t_{n}\right) \right) +\frac{\mho }{\mathfrak {M(\mho )} } \sum \limits _{k=0}^{n}\frac{h^{\mho }\mathfrak {h}\left( t_{k},\mathfrak {u}_{k}\right) }{\Gamma \left( \mho +2\right) }\left( n+1-k\right) ^{\mho +1}\left( n-k+2+\mho \right) \nonumber \\ & -\frac{\mho }{\mathfrak {M(\mho )} }\sum \limits _{k=0}^{n}\frac{h^{\mho }\mathfrak {h}\left( t_{k},\mathfrak {u}_{k}\right) }{\Gamma \left( \mho +2\right) }\left( n-k\right) ^{\mho }\left( n-k+2+2\mho \right) \nonumber \\ & \ -\frac{h^{\mho }\mathfrak {h}\left( t_{k-1},\mathfrak {u}_{k-1}\right) }{\Gamma \left( \mho +2\right) }\left( \left( n+1-k\right) ^{\mho +1}-\left( n-k\right) ^{\mho }\left( n-k+1+\mho \right) \right) +R_{n}^{\mho }, \end{aligned}$$

(6.6)

where $R_{n}^{\mho }$ is the remainder term which is expressed as

$$\begin{aligned} R_{n}^{\mho }=\frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) } \sum \limits _{k=0}^{n}\int \limits _{t_{k}}^{t_{k-1}}\frac{\left( z-t_{k}\right) \left( z-t_{k-1}\right) }{2}.\frac{\partial ^{2}}{\partial z^{2}}\left[ \mathfrak {h}\left( z,y\left( z\right) \right) \right] _{z=\varepsilon z}\left( t_{n+1}-z\right) ^{\mho -1}dz. \end{aligned}$$

(6.7)

The error upper bound has been given in $\left[ 19 \right] .$

Numerical scheme for system of equations representing rabies model

Now, we consider Rabies model given in (5.2). We have seen that the model can be expressed in the form of kernels through the application of Atangana–Baleanu fractional integral. New form will be as follows:

$$\begin{aligned} \mathcal {R}_{s}\left( t\right)= & \mathcal {R}_{s}\left( 0\right) +\frac{\left( 1-\mho \right) }{M\left( \mho \right) }\mathbb {U}_{1}\left( t,\mathcal {R}_{s}\left( t\right) \right) +\frac{\mho }{M\left( \mho \right) +\Gamma \left( \mho \right) }\int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\mathbb {U}_{1}\left( z,\mathcal {R}_{s}\left( z\right) \right) dz, \nonumber \\ \mathcal {R}_{v}\left( t\right)= & \mathcal {R}_{v}\left( 0\right) +\frac{\left( 1-\mho \right) }{M\left( \mho \right) }\mathbb {U}_{2}\left( t,V\left( t\right) \right) +\frac{\mho }{M\left( \mho \right) +\Gamma \left( \mho \right) }\int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\mathbb {U}_{2}\left( z,\mathcal {R}_{v}\left( z\right) \right) dz, \nonumber \\ \mathcal {R}_{e}\left( t\right)= & \mathcal {R}_{e}\left( 0\right) +\frac{\left( 1-\mho \right) }{M\left( \mho \right) }\mathbb {U}_{2}\left( t,\mathcal {R}_{e}\left( t\right) \right) +\frac{\mho }{M\left( \mho \right) +\Gamma \left( \mho \right) }\int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\mathbb {U}_{2}\left( z,\mathcal {R}_{e}\left( z\right) \right) dz, \nonumber \\ \mathcal {R}_{i}\left( t\right)= & \mathcal {R}_{i}\left( 0\right) +\frac{\left( 1-\mho \right) }{M\left( \mho \right) }\mathbb {U}_{3}\left( t,\mathcal {R}_{i}\left( t\right) \right) +\frac{\mho }{M\left( \mho \right) +\Gamma \left( \mho \right) }\int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\mathbb {U}_{3}\left( z,\mathcal {R}_{i}\left( z\right) \right) dz, \nonumber \\ \mathcal {R}_{r}\left( t\right)= & \mathcal {R}_{r}\left( 0\right) +\frac{\left( 1-\mho \right) }{M\left( \mho \right) }\mathbb {U}_{4}\left( t,\mathcal {R}_{r}\left( t\right) \right) +\frac{\mho }{M\left( \mho \right) +\Gamma \left( \mho \right) }\int \limits _{0}^{t}\left( t-z\right) ^{-\left( 1-\mho \right) }\mathbb {U}_{4}\left( z,\mathcal {R}_{r}\left( z\right) \right) dz. \end{aligned}$$

(6.8)

Also the initial conditions

$$\begin{aligned} \mathcal {R}_{s}\left( 0\right)= & \mathcal {R}_{v}\left( 0\right) =\mathcal {R}_{e}\left( 0\right) =\mathcal {R}_{i}\left( 0\right) =\mathcal {R}_{r}\left( 0\right) =0. \end{aligned}$$

(6.9)

$$\begin{aligned} \mathcal {R}_{s_{n+1}}= & \mathcal {R}_{s_{0}}+\frac{1-\mho }{\mathfrak {M(\mho )} }\mathbb {U}_{1}\left( t_{n},\mathcal {R}_{s}\left( t_{n}\right) \right) \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} }\sum \limits _{k=0}^{n} \left( \frac{h^{\mho }\mathbb {U}_{1}\left( t_{k},\mathcal {R}_{s_{k}}\right) }{\Gamma \left( \mho +2\right) }\left( \begin{array}{c} \left( n+1-k\right) ^{\mho }\left( n-k+2+\mho \right) -\left( n-k\right) ^{\mho }\left( n-k+2+2\mho \right) \end{array} \right) \right) \nonumber \\ & -\frac{h^{\mho }\mathbb {U}_{1}\left( t_{k-1},\mathcal {R}_{s_{k-1}}\right) }{ \Gamma \left( \mho +2\right) }\left( \left( n+1-k\right) ^{\mho +1}-\left( n-k\right) ^{\mho }\left( n-k+1+\mho \right) \right) +^{1}R_{n}^{\mho },\end{aligned}$$

(6.10)

$$\begin{aligned} \mathcal {R}_{v_{n+1}}= & \mathcal {R}_{v_{0}}+\frac{1-\mho }{\mathfrak {M(\mho )} }\mathbb {U}_{2}\left( t_{n},\mathcal {R}_{v}\left( t_{n}\right) \right) \nonumber \\ & +\frac{\mho }{\mathfrak {M(\mho )} }\sum \limits _{k=0}^{n} \left( \frac{h^{\mho }\mathbb {U}_{2}\left( t_{k},\mathcal {R}_{v_{k}}\right) }{\Gamma \left( \mho +2\right) }\left( \begin{array}{c} \left( n+1-k\right) ^{\mho }\left( n-k+2+\mho \right) -\left( n-k\right) ^{\mho }\left( n-k+2+2\mho \right) \end{array} \right) \right) \nonumber \\ & -\frac{h^{\mho }\mathbb {U}_{1}\left( t_{k-1},\mathcal {R}_{v_{k-1}}\right) }{ \Gamma \left( \mho +2\right) }\left( \left( n+1-k\right) ^{\mho +1}-\left( n-k\right) ^{\mho }\left( n-k+1+\mho \right) \right) +^{1}R_{n}^{\mho },\end{aligned}$$

(6.11)

$$\begin{aligned} \mathcal {R}_{e_{n+1}}= & \mathcal {R}_{e_{0}}+\frac{1-\mho }{\mathfrak {M(\mho )} }\mathbb {U}_{3}\left( t_{n},\mathcal {R}_{e}\left( t_{n}\right) \right) +\frac{\mho }{\mathfrak {M(\mho )} }\sum \limits _{k=0}^{n} \left( \frac{h^{\mho }\mathbb {U}_{3}\left( t_{k},\mathcal {R}_{e_{k}}\right) }{\Gamma \left( \mho +2\right) }\left( \begin{array}{c} \left( n+1-k\right) ^{\mho }\left( n-k+2+\mho \right) \\ -\left( n-k\right) ^{\mho }\left( n-k+2+2\mho \right) \end{array} \right) \right) \nonumber \\ & -\frac{h^{\mho }\mathbb {U}_{3}\left( t_{k-1},\mathcal {R}_{e_{k-1}}\right) }{ \Gamma \left( \mho +2\right) }\left( \left( n+1-k\right) ^{\mho +1}-\left( n-k\right) ^{\mho }\left( n-k+1+\mho \right) \right) +^{2}R_{n}^{\mho }, \end{aligned}$$

(6.12)

$$\begin{aligned} \mathcal {R}_{i_{n+1}}= & \mathcal {R}_{i_{0}}+\frac{1-\mho }{\mathfrak {M(\mho )} } \mathbb {U}_{4}\left( t_{n},\mathcal {R}_{i}\left( t_{n}\right) \right) +\frac{\mho }{\mathfrak {M(\mho )} }\sum \limits _{k=0}^{n} \left( \frac{h^{\mho }\mathbb {U}_{4}\left( t_{k},\mathcal {R}_{i_{k}}\right) }{\Gamma \left( \mho +2\right) }\left( \begin{array}{c} \left( n+1-k\right) ^{\mho }\left( n-k+2+\mho \right) \\ -\left( n-k\right) ^{\mho }\left( n-k+2+2\mho \right) \end{array} \right) \right) \nonumber \\ & -\frac{h^{\mho }\mathbb {U}_{4}\left( t_{k-1},\mathcal {R}_{i_{k-1}}\right) }{ \Gamma \left( \mho +2\right) }\left( \left( n+1-k\right) ^{\mho +1}-\left( n-k\right) ^{\mho }\left( n-k+1+\mho \right) \right) +^{3}R_{n}^{\mho }, \end{aligned}$$

(6.13)

$$\begin{aligned} \mathcal {R}_{r_{n+1}}= & \mathcal {R}_{r_{0}}+\frac{1-\mho }{\mathfrak {M(\mho )} } \mathbb {U}_{5}\left( t_{n},\mathcal {R}_{r}\left( t_{n}\right) \right) +\frac{\mho }{\mathfrak {M(\mho )} }\sum \limits _{k=0}^{n} \left( \frac{h^{\mho }\mathbb {U}_{5}\left( t_{k},\mathcal {R}_{r_{k}}\right) }{\Gamma \left( \mho +2\right) }\left( \begin{array}{c} \left( n+1-k\right) ^{\mho }\left( n-k+2+\mho \right) \\ -\left( n-k\right) ^{\mho }\left( n-k+2+2\mho \right) \end{array} \right) \right) \nonumber \\ & -\frac{h^{\mho }\mathbb {U}_{5}\left( t_{k-1},\mathcal {R}_{r_{k-1}}\right) }{ \Gamma \left( \mho +2\right) }\left( \left( n+1-k\right) ^{\mho +1}-\left( n-k\right) ^{\mho }\left( n-k+1+\mho \right) \right) +^{4}R_{n}^{\mho }, \end{aligned}$$

(6.14)

where $^{i}R_{n}^{\mho },$ $i=1,2,3,4,5$ are remainders which has representation as

$$\begin{aligned} ^{1}R_{n}^{\mho }= & \frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\sum \limits _{k=0}^{n}\int \limits _{t_{k}}^{t_{k-1}}\frac{\left( z-t_{k}\right) \left( z-t_{k-1}\right) }{2}.\frac{\partial ^{2}}{\partial z^{2}}\left[ \mathbb {U}_{1}\left( z,\mathcal {R}_{s}\left( z\right) \right) \right] _{z=\varepsilon z}\left( t_{n+1}-z\right) ^{\mho -1}dz, \nonumber \\ ^{2}R_{n}^{\mho }= & \frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\sum \limits _{k=0}^{n}\int \limits _{t_{k}}^{t_{k-1}}\frac{\left( z-t_{k}\right) \left( z-t_{k-1}\right) }{2}.\frac{\partial ^{2}}{\partial z^{2}}\left[ \mathbb {U}_{2}\left( z,\mathcal {R}_{v}\left( z\right) \right) \right] _{z=\varepsilon z}\left( t_{n+1}-z\right) ^{\mho -1}dz, \nonumber \\ ^{3}R_{n}^{\mho }= & \frac{\mho }{\mathfrak {M(\mho )} \Gamma \left( \mho \right) }\sum \limits _{k=0}^{n}\int \limits _{t_{k}}^{t_{k-1}}\frac{\left( z-t_{k}\right) \left( z-t_{k-1}\right) }{2}.\frac{\partial ^{2}}{\partial z^{2}}\left[ \mathbb {U}_{3}\left( z,\mathcal {R}_{e}\left( z\right) \right) \right] _{z=\varepsilon z}\left( t_{n+1}-z\right) ^{\mho -1}dz, \nonumber \\ ^{4}R_{n}^{\mho }= & \frac{\mho }{\mathfrak {M}\left( \mho \right) \Gamma \left( \xi \right) }\sum \limits _{k=0}^{n}\int \limits _{t_{k}}^{t_{k-1}}\frac{\left( z-t_{k}\right) \left( z-t_{k-1}\right) }{2}.\frac{\partial ^{2}}{\partial z^{2}}\left[ \mathbb {U}_{4}\left( z,\mathcal {R}_{i}\left( z\right) \right) \right] _{z=\varepsilon z}\left( t_{n+1}-z\right) ^{\mho -1}dz,\nonumber \\ ^{5}R_{n}^{\mho }= & \frac{\mho }{\mathfrak {M}\left( \mho \right) \Gamma \left( \xi \right) }\sum \limits _{k=0}^{n}\int \limits _{t_{k}}^{t_{k-1}}\frac{\left( z-t_{k}\right) \left( z-t_{k-1}\right) }{2}.\frac{\partial ^{2}}{\partial z^{2}}\left[ \mathbb {U}_{5}\left( z,\mathcal {R}_{r}\left( z\right) \right) \right] _{z=\varepsilon z}\left( t_{n+1}-z\right) ^{\mho -1}dz, \end{aligned}$$

(6.15)

Numerical simulation

In this section, we give the graphical illustration of the numerical scheme developed earlier. We simulate the model by assuming the initial values of the variables and suitable values of the parameters. The initial values of susceptible, vaccinated, exposed, infectious and recovered are taken as 400, 200, 100, 20 and 10,  respectively. Parameter values are taken as $\Theta =125,\xi =0.1,2\upsilon =0.878,\omega =0.081,\tau =0.054,\phi =0.1,\varepsilon =0.8,\sigma =0.0078,\phi =0.8378,\psi =0.8,\Omega =15.$ We simulate the model up to 1000 days for different values of $\mho$ and get the solution for each class. Figure 1 shows the solutions of $\mathcal {R}_{s},\mathcal {R}_{v},\mathcal {R}_{e},\mathcal {R}_{i}$ and $\mathcal {R}_{r}$ for the variation of $\mho$ from 0.8 to 1. It can be observed that all the classes of animal population approach the same endemic level and it is independent of the fractional value of $\mho .$ Figure 1a shows the endemic level of susceptible animals corresponding to the values of $\mho$ from 0.8 to 1. There is a slight change in endemic level when the value of $\mho$ approach 1. Figure 1b express the constant level of vaccinated animals. This level changes from 1019 to 1031 as $\mho$ changes from 0.8 to 1. Figure 1c indicates the reduction in exposed population and it can be seen that the animals belonging to the exposed class approach the same number even if the values of $\mho$ is varied from 0.8 to 1. The behavior of infectious animals having rabies disease is expressed in Fig. 1d. We can see that infectious animals approach the unique endemic level. Figure 1e shows that the recovered population approach the unique constant level. Figure 2 shows the endemic level of susceptible animals. Figure 2a,b shows that susceptible animals approach the unique endemic level for the fractional values of $\mho .$ This unique endemic level also occurs for $\mho =1.$ This phenomena is shown in Fig. 2c. Taking the time period of 1000 days vaccinated class is plotted in Fig. 3. Figure 3a shows that vaccinated animals increase initially and then approach the constant level with the passage of time but with the slight difference. Figure 3b indicates that small difference in the endemic level disappear when we take $\mho =1.$ Solutions of exposed class is expressed in Fig. 4. It can be observed, from Fig. 4a that the solutions of this class differ slightly when the value of $\mho =0.8.$ When fractional derivative is taken 0.9 the endemic level tends to uniqueness and unique value can be observed for $\mho =1.$ and it can be seen in Fig. 4b,c. However Fig. 4b shows the minor change in the endemic level for $\mho =0.9,$ but there is no change in exposed cattle for integral value of $\mho$ as shown in Fig. 4c. Infected animals having rabies disease is shown in Fig. 5. We can see, from Fig. 5a–c, that animals with rabies disease approach the unique constant level for $\mho =0.8,\mho =0.9$ and $\mho =1.$ Figure 5a shows that the number of infected animals approach the endemic level after passing some days. The constant is approached earlier by increasing the value of $\mho$ as shown in Fig. 5b. Moreover, after very short period the number of infected animals become constant when we assume the integral value as expressed in Fig. 5c. Recovered class also show the unique endemic level for fractional and integral values of $\mho .$ Figure 6a–c show this behavior. It can be seen, from Fig. 6a, significant change occurs in the early days when we take different initial values for $\mho =0.8.$ This difference reduce when we approach fractional to integral value as shown in Fig. 6b. There is no change in the recovery class occurs for the integral value of $\mho ,$ even though we change the initial values of recovered animals. It is shown in Fig. 6c. A very interesting phenomena of all classes can be seen in Fig. 7. When the time period is taken 50 days, we can see, from Fig. 7a, that oscillatory behavior occurs in each class of population. This pattern is also appear in Fig. 7b when time period is 100 days. However, the population approach constant level when we take $t=500$ and it can be seen in Fig. 7c. The behavior of population, for $\mho =1$, is shown in Fig. 8. We can see that the oscillator behavior of each class of population disappear when the value of $\mho$ is taken 1 irrespective of the time period. Figure 8a–c show that for oscillatory behavior disappear for $t=50,100$ and 500 for the integral value of $\mho .$ Figure 8a shows that the cattle population approach the constant level when the time period is taken as 500 days. When we short the time interval up to 100 days, the population approach same constant level as shown in Fig. 8b. It can be confirmed from, Fig. 8c, that the number of cattle belonging to each compartment remain same after 50 days. Different behavior of vaccinated class is shown in Fig. 9. It can be seen, from graph shown in Fig. 9a, that vaccinated individuals increase without showing the oscillatory behavior when the time period is take as 50 days. Similar pattern of vaccination class is shown in Fig. 9b–c for $t=100$ and 500. The value of $\mho$ is taken as 0.8.

Conclusion

In this work, the rabies model with harmonic incidence function has been solved through the application of fractional derivative. The non-local kernal in the absence of singularity has been assumed while obtaining the outcomes of mathematical model recently suggested by Atangana and Baleanu. We have proved the existence of solutions of nonlinear ordinary differential equations through the fixed-point theorem. The uniqueness of solutions is also proved analytically. Borrowing the idea from Atangana and Toufik, numerical scheme has been developed for the fractional order $\mho .$ The solutions have been presented graphically which shows that the uniqueness of solutions retained either the fractional value of the derivative is changed or the initial conditions are varied. However, it is shown graphically that, for small time interval, the fractional order present the extra features of all the classes, except vaccinated class, which is unable to present in the case of ordinary derivative.We have studied the impact of different values of fractional operator by varying initial values. We assumed the parameter values and it has been shown that for different initial values of classes, the solutions ultimately approach the same constant level. One can apply the model on actual available data and estimate the values of parameters. By utilizing these and taking the actual values as initial one can check that whether the population approach unique endemic level or not. If cattle attain the same values then identifying the significant parameters through sensitivity analysis, effective control measures could be taken to reduce the number of infected cattle. However, the complete eradication of the disease is not possible because the efficacy of vaccine to certain level which exhibits the phenomena of backward bifurcation.

Fig. 1

Solution of the system (5.2) for different values of $\mho$.

Fig. 2

Endemic level of Susceptible Animals for different values of $\mho .$.

Fig. 3

Endemic level of Vaccinated Animals for different values of $\mho .$.

Fig. 4

Endemic level of Exposed Animals for different values of $\mho .$.

Fig. 5

Endemic level of Infectious Animals for different values of $\mho .$.

Fig. 6

Endemic level of Recovered Animals for different values of $\mho .$.

Fig. 7

Fluctuation in the solutions for different time periods.

Fig. 8

Behavior of solutions for integral value of $\mho .$.

Fig. 9

Graphs of Vaccinated class for different time periods.