Skip to main content

Mathematical analysis of tuberculosis control model using nonsingular kernel type Caputo derivative

Abstract

This research work investigates some theoretical and semi-analytical results for the mathematical model of tuberculosis disease via derivative due to Caputo and Fabrizio. The concerned derivative involves exponential kernel and very recently it has been adapted for various applied problems. The required results are established by using some fixed point approach of Krasnoselskii and Banach. Further, by the use of iterative tools of Adomian decomposition and Laplace, the semi-analytical results are studied. Some graphical results are given with discussion.

Introduction

One of the most important diseases faced by human beings ever is tuberculosis (TB). TB is a spreadable, airborne bacterial infection which is caused by mycobacterium tuberculosis. This bacterium normally affects the lungs (pulmonary tuberculosis). This bacterium may also affect several other systems like kidneys, the brain, the lymphatic system, the central nervous system, spinal cord, etc. The presence of TB disease has been found since ancient times in various civilizations like Egypt, China, Roman, etc. (see [1]). One-third of the world population at the present time is infected with TB, and the number of infectious individuals increases at a rate of one per second [2]. The aforementioned disease was among the top ten death causes around the globe in the year 2015, where about 10.4 million individuals were infected from it. In the same year, about 1.8 million infectious individuals lost their lives from various diseases including 0.4 million with HIV and TB. 60% of the tuberculosis cases around the globe were concentrated in the six countries (Pakistan, India, Nigeria, China, Indonesia, and South Africa) [3]. Dye [4] gave some information that the major cause of death worldwide, in particularly in Sub-Saharan Africa, is due to TB and HIV. Further, HIV epidemic is a serious threat in many countries of the world. It is a clear evidence that worldwide children are protected from the early infection of the disease by vaccination like Bacillus Calmatte–Guerine(BCG) [5]. Therefore, detection and treatment of latent TB by modern therapy have been used recently to prevent the breakdown of rate of spread of the disease as only the members of the infectious class can transmit the disease to others.

Worldwide various procedures and methods have been used to understand the cause and control of these diseases in society. One of the powerful tools is the mathematical modeling through which we understand the dynamics of various disease transmission and suggest procedure how to control them in society. The mentioned area originated during 1927 for the first time. A variety of models have been developed and studied so far (refer to [610]). In this regard a five-compartment model for TB has also been constructed in [11] as follows:

$$ \textstyle\begin{cases} \dot{\mathscr{M}}(t)=\theta \rho -(\alpha +\mu ) \mathscr{M}, \\ \dot{\mathscr{S}}(t)=(1-\theta )\rho +\alpha \mathscr{M}-\beta { \mathscr{SI}}-\rho {\mathscr{S}}, \\ \dot{\mathscr{L}}(t)=\beta \mathscr{SI}-(\sigma +\tau +\mu ) \mathscr{L}, \\ \dot{\mathscr{I}}(t)=\tau \mathscr{L}- (\gamma +\mu +\delta )\mathscr{I}, \\ \dot{\mathscr{R}}(t)=\sigma \mathscr{L}+\gamma \mathscr{I}-\mu \mathscr{R}. \end{cases} $$
(1)

In the above model the whole populace is categorized into five classes: the immunized class \(\mathscr{M}\), the susceptible class \(\mathscr{S}\), the infected latently class \(\mathscr{L}\), the infectious class \(\mathscr{I}\), and the recovered class \(\mathscr{R}\). Parameters of the model under consideration are explained as follows: The constant of recruitment is represented by the symbol ρ, θ denotes the immunized portion at birth, α represents the rate of weaning off the vaccine, the natural death rate is denoted by the symbol μ, β represents the tuberculosis contraction rate, the successful treatment of infectious latent is denoted by the letter σ, the symbol τ is the rate of breakdown of latent TB into infectious TB, the successful cure of infectious TB individuals and the death resulting from the disease are respectively denoted by the symbols γ, δ.

Usually an integer order derivatives do not explore the dynamics of real world problems related to biology and physics well. In order to overcome this deficiency, fractional calculus has been given attention for the last few decades. Also we know that fractional calculus is increasingly used by mathematicians for mathematical modeling. Derivatives and integrals of noninteger order may be defined by a number of ways. Some well-known definitions are those given by Riemann and Liouville [12], Caputo, etc. (see [13]). The mentioned derivatives involve kernel of singular type. Frequently these two definitions have been increasingly used since fractional differential operator is in fact a definite integral operator for whom the definition of kernel is not unique or not regular. Further due to high degree of freedom in derivative of arbitrary order, researchers have given much attention to studying applied problems under these concepts. In this sense very recently some authors replaced the singular kernel by some nonlocal nonsingular and produced new definitions. Hence in 2015 Caputo and Fabrizio replaced the singular kernel by exponential kernel in the usual Caputo derivative and called it Caputo–Fabrizio fractional derivative (abbreviated as CFFD); for details, see [1416]. Therefore various researchers investigated different problems of applied nature under this concept. In various cases the mentioned derivative has produced significant results as compared to other forms of derivatives (see [1719]). The Caputo–Fabrizio derivative omits singular kernel by exponential kernel and hence makes the concerned differential operator nonlocal. Conventional fractional derivatives contain singular kernel which sometimes causes difficulty in explanation of some characteristics of various materials. To overcome this, Caputo and Fabrizio introduced a new definition of fractional integral and derivative which involves exponential kernel instead of singular one. Various studies can be found in the literature that have focused on the Caputo–Fabrizio fractional order derivatives, see, for instance, [2024]. Further, in various papers related to thermal sciences, the mentioned derivative has been proved powerful as compared to other type, we refer to [2530]. Keeping these points in mind and the nonsingular nature of the proposed derivative, we investigate the considered model for existence and analytical results.

Now the question is how to treat the problems involving derivative of fractional orders. For this need researchers have successfully updated the usual tools and methods to handle differential equations of fractional order (FODEs). Usual perturbation techniques and decomposition methods were greatly utilized to deal with ordinary FODEs. Also, for the mentioned problems, Adomian decomposition coupled with some integral transforms has been used very well (see [3134]). On the other hand, since the FODEs involving new type derivatives are very rarely used, very frequently authors [35] established some algorithms to handle such FODEs containing CFFD.

Hence we investigate the model given in (1) under CFFD as follows:

$$ \textstyle\begin{cases} {} _{0}^{CF}D_{t}^{\eta } \mathscr{M}(t)=\theta \rho -(\alpha +\mu ) \mathscr{M}, \\ {} _{0}^{CF}D_{t}^{\eta } \mathscr{S}(t)=(1-\theta )\rho +\alpha \mathscr{M}-\beta \mathscr{SI}-\rho \mathscr{S}, \\ {} _{0}^{CF}D_{t}^{\eta } \mathscr{L}(t)=\beta \mathscr{SI}-(\sigma + \tau +\mu )\mathscr{L}, \\ {} _{0}^{CF}D_{t}^{\eta } \mathscr{I}(t)=\tau \mathscr{L}-(\gamma +\mu + \delta )\mathscr{I}, \\ {} _{0}^{CF}D_{t}^{\eta } \mathscr{R}(t)=\sigma \mathscr{L}+\gamma \mathscr{I}-\mu \mathscr{R}. \end{cases} $$
(2)

We study model (2) subject to the biologically feasible initial conditions

$$\begin{aligned}& \mathscr{M}(0)=\mathbf{N}_{1}\geq 0,\qquad \mathscr{S}(0)= \mathbf{N}_{2} \geq 0, \\& \mathscr{L}(0)=\mathbf{N}_{3} \geq 0, \\& \mathscr{I}(0)=\mathbf{N}_{4}\geq 0, \qquad \mathscr{R}(0)= \mathbf{N}_{5} \geq 0. \end{aligned}$$

Initially we establish some conditions about the existence of solutions for model (2) by using some fixed point results like Banach and Krasnoselskii. After that, by using the considered tool of “Laplace Adomian decomposition method (LADM)” for \(\eta \in (0,1]\), we compute semi-analytical results. Finally, the approximate results are presented via graphs.

Preliminaries

In this section of the manuscript, we present some fundamental definitions as follows.

Definition 1

([15])

Let \(\varphi \in \mathcal{H}^{1}(a,b)\), \(b>a\), \(\eta \in (0,1)\), then the CFFD is given as

$$ {}_{0}^{CF}D_{t}^{\eta } \varphi (t)=\frac{\mathcal{K(\eta )}}{1-\eta } \int _{\alpha }^{t} \varphi ^{\prime }(\theta ) \exp \biggl[- \frac{t-\theta }{1-\theta } \biggr]\,d\theta , $$
(3)

where \(\mathcal{K(\eta )}\) in (3) is the normalization function with \(\mathcal{K}(1)=\mathcal{K}(0)=1\). If the function failed to exist in \(\mathcal{H}^{1}(a,b)\), then the above derivative can be reformulated as

$$ {}_{0}^{CF}D_{t}^{\eta }\varphi (t)= \frac{\mathcal{K(\eta )}}{1-\eta } \int _{\alpha }^{t} \varphi (t)- \varphi (\theta ) \exp \biggl[- \frac{t-\theta }{1-\theta } \biggr]\,d\theta . $$

Definition 2

([36])

Let \(\eta \in (0,1]\), then the integral of fractional order η of the function φ is

$$\begin{aligned} {}_{0}^{CF}I_{t}^{\eta }\varphi (t)= \frac{(1-\eta )}{\mathcal{K}(\eta )} \varphi (t) +\frac{\eta }{\mathcal{K}(\eta )}\varphi (t) \int _{0}^{t} \varphi (\theta )\,d\theta . \end{aligned}$$

Definition 3

([35, 37])

The Laplace transform of CFFD \({}_{0}^{CF}D_{t}^{\eta }\) of \(M(t) \) is given as

$$ \mathscr{L} \bigl[{}_{0}^{CF}D_{t}^{\eta } M(t) \bigr]= \frac{s \mathscr{L}[M(t)]-M(0)}{s+\eta (1-s)}, \quad s\geq 0, \eta \in (0,1]. $$

Equilibrium points and the basic reproduction number

Before proceeding further, we consider it advantageous to find the equilibrium points and the basic reproduction number of the model under consideration. There are two types of possible equilibrium points of the model. The first one is the point where there is no disease in the community, i.e., the disease-free equilibrium point. This is obtained by setting the right-hand side of each equation in the model to zero along with \(\mathscr{L}=\mathscr{I}=\mathscr{R}=0\). Solving the system then gives \(\mathscr{M}^{0}=\frac{\theta \rho }{\alpha +\mu }\) and \(\mathscr{S}^{0}=\frac{\alpha +\mu -\mu \theta }{\alpha +\mu }\). Thus the disease-free equilibrium point of the model under investigation is given by \(\mathscr{E}^{0}= (\frac{\theta \rho }{\alpha +\mu }, \frac{\alpha +\mu -\mu \theta }{\alpha +\mu },0,0,0 )\).

To find the basic reproduction number, we consider only the infectious classes of the model. Let \(\mathbf{{\mathit{{V}}}= (\mathscr{L},\mathscr{I} )^{T}}\), by the help of the given model one can write d V d t =FV= [ β S I 0 ] [ ( σ + τ + μ ) L τ L + ( γ + μ + σ ) I ] . The Jacobian matrices of \(\mathscr{F}\) and \(\mathscr{V}\) are given by F= [ 0 β S 0 0 0 ] and V= [ σ + τ + μ 0 τ γ + μ + δ ] . The inverse matrix of \(\mathcal{V}\) is given by V 1 = [ 1 σ + τ + μ 0 τ 1 γ + μ + δ ] . Hence the next generation matrix \(\mathcal{F}\mathcal{V}^{-1}\) is calculated as

F V 1 = [ β S 0 τ ( γ + μ + δ ) ( σ + τ + δ ) β S 0 γ + μ + δ 0 0 ] .
(4)

The spectral radius of the next generation matrix (4) gives the threshold quantity \(R_{0}\) [38]. Thus

$$ R_{0}= \frac{\beta \tau (\alpha +\mu -\mu \theta )\tau }{(\alpha +\mu )((\gamma +\mu +\delta )(\sigma +\tau +\delta )}. $$

Three-dimensional plots of the basic reproduction number \(R_{0}\) versus different parameters in the model under consideration are depicted in Fig. 1. This quantity plays the key role in stability analysis and in finding conditions for the said purpose.

Figure 1
figure1

Three-dimensional plot of the basic reproduction number versus different parameters involved in the model under consideration. The parametric values are given in Table 1 at the end

Existence and uniqueness results for tuberculosis disease model of fractional order

In the following we derive existence results related to our model (2) exploiting the so-called fixed point theorem due to Banach. To proceed further, we first of all define the functions given below:

$$ \textstyle\begin{cases} f_{1}(t,M,S,L,I,R)=\theta \rho -(\alpha +\mu )M, \\ f_{2}(t,M,S,L,I,R)=(1-\theta )+\alpha {M}-\beta {SI}-\rho {S}, \\ f_{3}(t,M,S,L,I,R)=\beta {SI}-(\sigma +\tau +\mu ){L}, \\ f_{4}(t,M,S,L,I,R)=\tau {L}-(\gamma +\mu +\delta ){I}, \\ f_{5}(t,M,S,L,I,R)=\sigma {L}+\gamma {I}-\mu {R}, \end{cases} $$
(5)

where

$$ M(0)=\mathbf{N}_{1},\qquad S(0)=\mathbf{N}_{2},\qquad L(0)= \mathbf{N}_{3},\qquad I(0)= \mathbf{N}_{4},\qquad R(0)= \mathbf{N}_{5}. $$

So our problem becomes

$$ \textstyle\begin{cases} {} _{0}^{CF}D_{t}^{\eta }M(t)= f_{1}(t,M,S,L,I,R)=\theta \rho -(\alpha + \mu )M, \\ {} _{0}^{CF}D_{t}^{\eta } S(t)= f_{2}(t,M,S,L,I,R)=(1-\theta )+\alpha {M}- \beta {SI}-\rho {S}, \\ {} _{0}^{CF}D_{t}^{\eta } L(t)= f_{3}(t,M,S,L,I,R)=\beta {SI}-(\sigma + \tau +\mu ){L}, \\ {} _{0}^{CF}D_{t}^{\eta } I(t)= f_{4}(t,M,S,L,I,R)=\tau {L}-(\gamma + \mu +\delta ){I}, \\ {} _{0}^{CF}D_{t}^{\eta }R(t) = f_{5}(t,M,S,L,I,R)=\sigma {L}+\gamma {I}- \mu {R}, \end{cases} $$
(6)

where \(M(0)=\mathbf{N}_{1} \), \(S(0)=S_{0} \), \(L(0)=L_{0} \), \(I(0)=I_{0} \), \(R(0)=R_{0}\). Application of \({}_{0}^{CF}I^{\eta }\) on both sides of (2) gives the following system of integral equations:

$$ \textstyle\begin{cases} M(t)=M(0)+G [f_{1} (t,M,S,L,I,R)-f_{01} ] \\ \hphantom{M(t)=}{} +\overline{G} \int _{0}^{t}f_{1} (\xi ,M,S,L,I,R)\,d\xi , \\ S(t)=S(0)+G [f_{2} (t,M,S,L,I,R)-f_{02} ] \\ \hphantom{S(t)=}{} +\overline{G} \int _{0}^{t}f_{2} (\xi ,M,S,L,I,R)\,d\xi , \\ L(t)=L(0)+G [f_{3} (t,M,S,L,I,R)-f_{03} ] \\ \hphantom{L(t)=}{}+\overline{G} \int _{0}^{t}f_{3} (\xi ,M,S,L,I,R)\,d\xi , \\ I(t)=I(0)+G [f_{4} (t,M,S,L,I,R)-f_{04} ] \\ \hphantom{I(t)=}{}+\overline{G} \int _{0}^{t}f_{4} (\xi ,M,S,L,I,R)\,d\xi , \\ R(t)=R(0)+G [f_{5} (t,M,S,L,I,R)-f_{05} ] \\ \hphantom{R(t)=}{}+\overline{G} \int _{0}^{t}f_{5} (\xi ,M,S,L,I,R)\,d\xi , \end{cases} $$
(7)

where \(G=\frac{(1-\eta )}{ \mathcal{K}(\eta )}\), \(\overline{G}= \frac{\eta }{\mathcal{K}(\eta )} \). Further, we will use \(f_{0i}=f_{i}(0, M(0), S(0), L(0), I(0), R(0))\), \(i=1,2,3,4,5\). Using the initial conditions, we have

$$ \textstyle\begin{cases} M(t)=\mathbf{N}_{1}+G [f_{1} (t,M,S,L,I,R)-f_{01} ] \\ \hphantom{M(t)=}{} +\overline{G} \int _{0}^{t}f_{1} (\xi ,M,S,L,I,R)\,d\xi , \\ S(t)=\mathbf{N}_{2}+G [f_{2} (t,M,S,L,I,R)-f_{02} ] \\ \hphantom{S(t)=}{}+\overline{G} \int _{0}^{t}f_{2} (\xi ,M,S,L,I,R)\,d\xi , \\ L(t)=\mathbf{N}_{3}+G [f_{3} (t,M,S,L,I,R)-f_{03} ] \\ \hphantom{L(t)=}{}+\overline{G} \int _{0}^{t}f_{3} (\xi ,M,S,L,I,R)\,d\xi , \\ I(t)=\mathbf{N}_{4}+G [f_{4} (t,M,S,L,I,R)-f_{04} ] \\ \hphantom{I(t)=}{} +\overline{G} \int _{0}^{t}f_{4} (\xi ,M,S,L,I,R)\,d\xi , \\ R(t)=\mathbf{N}_{5}+G [f_{5} (t,M,S,L,I,R)-f_{05} ] \\ \hphantom{R(t)=}{}+\overline{G} \int _{0}^{t}f_{5} (\xi ,M,S,L,I,R)\,d\xi . \end{cases} $$
(8)

Here, we denote Banach space by \(X=C([0, T]\times \mathscr{R}^{5}, \mathscr{R})\) under the norm

$$ \Vert W \Vert =\max_{t\in [0,T]} \bigl\{ \bigl\vert W(t) \bigr\vert :W=(M,S,L,I,R) \bigr\} , $$

where \(T>0\) such that \(0\leq t\leq T<\infty \).

Theorem 1

(Krasnoselskii fixed point theorem)

Let X be a Banach space and D be a closed and convex subset of X, then there exist two operators A, B for which the following hold:

  1. 1.

    The sum \(Ax+By\) belongs to D;

  2. 2.

    The operator A is a contraction, while the operator B is continuous and compact;

  3. 3.

    at least one solution z in a way that \(Az+Bz=z\) holds.

Let us assume

W(t)= [ M S L I R ] ,W(0)(t)= W 0 = [ N 1 N 2 N 3 N 4 N 5 ] ,F= [ f 1 f 2 f 3 f 4 f 5 ] .

Therefore, system (8) reduces to

$$\begin{aligned}& W(t)=W_{0} +G F(t,M,S,L,I,R)-G F_{0}+\overline{G} \int _{0}^{t} F(\xi ,M,S,L,I,R)\,d\xi, \\& W(t)=W_{0}+G \bigl[F(t,W)-F_{0} \bigr]+ \overline{G} \int _{0}^{t} F(\xi ,W)\,d\xi . \end{aligned}$$
(9)

For further analysis, we suppose that the following assumptions hold:

(\(\mathcal{H}_{1}\)):

There exists \(\mathcal{K}_{F}>0\) for which

$$ \bigl\vert F(t,W)-F(t,\overline{W}) \bigr\vert \leq \mathcal{K}_{F} \vert W-\overline{W} \vert . $$
(\(\mathcal{H}_{2}\)):

There exist \(C_{F}>0\) and \(M_{F}>0\) such that

$$ \bigl\vert F(t,W) \bigr\vert \leq C_{F} \vert W \vert +M_{F}. $$

Theorem 2

In the view of Theorem 1, problem (9) has at least one solution provided \(G \mathcal{K}_{F}\) is less than unity.

Proof

To prove the theorem, we define a compact and closed set D such that \(D={W \in X: \|W\|\leq r}\). Next, we define operators A and B as follows:

$$ \begin{aligned} & AW(t)=W_{0} +G F(t,M,S,L,I,R)-G F_{0}, \\ & BW(t)=\overline{G} \int _{0}^{t} F(\xi ,M,S,L,I,R)\,d\xi . \end{aligned} $$
(10)

To verify that the operator A in (10) is a contraction, we assume \(W,\overline{W} \in X\), so that

$$\begin{aligned} \Vert AW-A\overline{W} \Vert =&\max \bigl\vert AW(t)-A\overline{W} (t) \bigr\vert \\ =&\max \bigl\vert G \bigl[F(t,W)-F(t,\overline{W}) \bigr] \bigr\vert , \end{aligned}$$

from which we have

$$ \Vert AW-A\overline{W} \Vert \leq G K_{F} \bigl[ \Vert W- \overline{W} \Vert \bigr], $$

which clearly indicates that the operator A is a contraction.

Now we show that the operator B is compact and continuous. Consider

$$\begin{aligned} \bigl\vert BW(t) \bigr\vert =& \biggl\vert \overline{G} \int _{0}^{t} F(\xi ,M,S,L,I,R)\,d\xi \biggr\vert \\ \leq &\overline{G} \int _{0}^{t} \bigl\vert F \bigl(\xi ,W(\xi ) \bigr) \bigr\vert \,d\xi . \end{aligned}$$
(11)

Taking max of (11), we have

$$\begin{aligned} \Vert BW \Vert \leq & \overline{G}\max _{t \in [0,T]} \int _{0}^{t} \bigl\vert F \bigl(\xi ,W( \xi ) \bigr) \bigr\vert \,d\xi \\ \leq & \overline{G}\max_{t \in [0,T]} \int _{0}^{t} \bigl[C_{F} \Vert W \Vert +M_{F} \bigr]\,d\xi \\ \leq & \overline{G} T(C_{F} r+M_{F}). \end{aligned}$$
(12)

This implies that B is bounded in (12). Let us assume that in the domain of t we have \(t_{1}< t_{2}\). One may write

$$\begin{aligned} \bigl\vert BW(t_{2} )-BW(t_{1}) \bigr\vert =& \biggl\vert \overline{G} \int _{0}^{t_{2}}F(\xi ,W)\,d\xi -\overline{G} \int _{0}^{t_{1}}F(\xi ,W)\,d\xi \biggr\vert \\ \leq & \overline{G}(C_{F} r+M_{F}) (t_{2}-t_{1}). \end{aligned}$$
(13)

If \(t_{2}\) approaches \(t_{1}\), then the right-hand side of (13) goes to zero. Consequently, \(t_{2} \rightarrow t_{1}\), which leads to

$$ \bigl\vert BW(t_{2} )-BW(t_{1}) \bigr\vert \rightarrow 0. $$

It follows that B is equicontinuous and, consequently, B is compact continuous. This implies that B is a completely continuous operator. Hence, all the conditions of Theorem 1 are satisfied. One may immediately conclude that model (2) has at least one solution. □

Theorem 3

There is a unique solution of the model under consideration (2) if the functions \(f_{1}\), \(i=1,2,3,4\), are continuous and \(\overline{G} K_{F}(1+T)<1\).

Proof

To prove the theorem, we define an operator \(P:X\rightarrow X\) such that

$$ PW(t)=W_{0} +G F(t,W)-GF_{0}+\overline{G} \int _{0}^{t_{1}} F(\xi ,W)\,d\xi . $$

Let \(W, \overline{W} \in X\), we may write

$$\begin{aligned} \bigl\Vert P(W)-P(\overline{W}) \bigr\Vert =& \max \bigl\vert {P(W) (t)-P(\overline{W}) (t)} \bigr\vert \\ \leq & \max \bigl\vert G (F(t, W)-F(t,\overline{W}) \bigr\vert + \overline{G} \max | \int _{0}^{t} \bigl\vert (F(t,W)-F(t,P( \overline{W}) \bigr\vert d \xi \\ \leq & \overline{G} K_{F} \Vert W-\overline{W} \Vert +G K_{F} T \Vert W- \overline{W} \Vert . \end{aligned}$$

It follows that

$$\begin{aligned} \Vert W-\overline{W} \Vert \leq \overline{G} K_{F}(1+T) \Vert W-\overline{W} \Vert . \end{aligned}$$
(14)

Consequently, model (2) under investigation has a unique solution. □

Construction of general algorithm for the required solution of the considered model

To derive the series type solution for the considered problem, we take \(\mathcal{K}(\eta )=1\) and apply the Laplace transform on both sides of (2). We construct the following algorithm:

$$ \textstyle\begin{cases} \frac{s \mathscr{L}[M(t)]-M(0)}{s+\eta (1-s)}= \mathscr{L} [\theta \rho -(\alpha +\mu ) M ], \\ \frac{s \mathscr{L}[S(t)]-S(0)}{s+\eta (1-s)}= \mathscr{L} [(1- \theta )\rho +\alpha M-\beta SI- \mu S ], \\ \frac{s \mathscr{L}[L(t)]-L(0)}{s+\eta (1-s)}= \mathscr{L} [\beta SI-( \sigma +\tau +\mu )L ], \\ \frac{s \mathscr{L}[I(t)]-I(0)}{s+\eta (1-s)}= \mathscr{L} [\tau L-( \gamma +\mu +\delta )I ], \\ \frac{s \mathscr{L}[R(t)]-R(0)}{s+\eta (1-s)}= \mathscr{L} [\sigma L+ \gamma I-\mu R ]. \end{cases} $$
(15)

After rearranging terms in (15), we have

$$ \textstyle\begin{cases} \mathscr{L} [M(t) ]= \frac{M(0)}{s}+ \frac{s+\eta (1-s)}{s}\mathscr{L} [\theta \rho -(\alpha +\mu ) M ], \\ \mathscr{L} [S(t) ]= \frac{S(0)}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [(1-\theta )\rho +\alpha M-\beta SI- \mu S ], \\ \mathscr{L} [L(t) ]= \frac{L(0)}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [ \beta SI-(\sigma +\tau +\mu )L ], \\ \mathscr{L} [I(t) ]= \frac{I(0)}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [ \tau L-(\gamma +\mu +\delta )I ], \\ \mathscr{L} [R(t) ]= \frac{R(0)}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [ \sigma L+\gamma I-\mu R ]. \end{cases} $$
(16)

Using the initial conditions of system (2), one has

$$ \textstyle\begin{cases} \mathscr{L} [M(t) ]= \frac{\mathbf{N}_{1}}{s}+ \frac{s+\eta (1-s)}{s}\mathscr{L} [\theta \rho -(\alpha +\mu ) M ], \\ \mathscr{L} [S(t) ]= \frac{\mathbf{N}_{2}}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [(1-\theta )\rho +\alpha M-\beta SI- \mu S ], \\ \mathscr{L} [L(t) ]= \frac{\mathbf{N}_{3}}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [\beta SI-(\sigma +\tau +\mu )L ], \\ \mathscr{L} [I(t) ]= \frac{\mathbf{N}_{4}}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [\tau L-(\gamma +\mu +\delta )I ], \\ \mathscr{L} [R(t) ]= \frac{\mathbf{N}_{5}}{s}+\frac{s+\eta (1-s)}{s} \mathscr{L} [\sigma L+\gamma I-\mu R ]. \end{cases} $$
(17)

Let the solution we compute be in the form of an infinite series as follows:

$$\begin{aligned}& M(t)=\sum_{n=0}^{\infty }M_{n} (t),\qquad S(t)=\sum_{n=0}^{\infty }S_{n} (t), \qquad L(t)=\sum_{n=0}^{\infty }L_{n} (t), \\& I(t)=\sum_{n=0}^{\infty }I_{n} (t),\qquad R(t)=\sum_{n=0}^{\infty }R_{n} (t), \end{aligned}$$

and decompose the nonlinear term SI in terms of Adomian polynomial as follows:

$$ S(t)I(t)=\sum_{n=0}^{\infty }A_{n}(t), $$

where \(A_{n}=\frac{1}{\Gamma {(n+1)}}\frac{d^{n}}{d\lambda ^{n}} [ ( \sum_{k=0}^{n} \lambda ^{k} S_{k} ) (\sum_{k=0}^{n} \lambda ^{k} I_{k} ) ]|_{\lambda =0}\)

$$\begin{aligned}& n=0:\quad A_{0}=S_{0}(t)I_{0}(t), \\& n=1:\quad A_{1}=S_{0}(t) I_{1}(t)+S_{1}(t) I_{0}(t), \\& n=2:\quad A_{2}=S_{0}(t) I_{2}(t)+S_{1}(t) I_{1}(t)+S_{2}(t) I_{0}(t), \\& n=3:\quad A_{3}= S_{0}(t) I_{3}(t)+S_{1}(t) I_{2}(t) +S_{2}(t) I_{1}(t)+S_{3}(t) I_{0}(t), \\& n=4:\quad A_{4}=S_{0}(t) I_{4}(t)+S_{1}(t) I_{3}(t)+S_{2}(t) I_{2}(t)+S_{3}(t) I_{1}(t) +S_{4}(t) I_{0}(t), \\& \cdots \\& n=n:\quad A_{n}=S_{0}(t) I_{n}(t)+S_{1}(t) I_{(n-1)}(t)+\cdots +S_{(n-1)}(t) I_{1}(t)+S_{n}(t) I_{0}(t). \end{aligned}$$

In view of these values, model (8) becomes

$$ \textstyle\begin{cases} \mathscr{L} [\sum_{k=0}^{\infty }M_{k} (t) ]= \frac{\mathbf{N}_{1}}{s} +\frac{s+\eta (1-s)}{s}\mathscr{L} [\theta \rho -(\alpha +\mu ) \sum_{k=0}^{\infty }M_{k} (t) ], \\ \mathscr{L} [\sum_{k=0}^{\infty }S_{k} (t) ] \\ \quad = \frac{\mathbf{N}_{2}}{s} +\frac{s+\eta (1-s)}{s}\mathscr{L} [(1-\theta )\rho +\alpha \sum_{k=0}^{\infty }M_{k} (t) -\beta \sum_{k=0}^{\infty }A_{k}- \mu \sum_{k=0}^{\infty }S_{k} (t) ], \\ \mathscr{L} [\sum_{k=0}^{\infty }L_{k} (t) ]= \frac{\mathbf{N}_{3}}{s} +\frac{s+\eta (1-s)}{s}\mathscr{L} [\beta \sum_{k=0}^{\infty }A_{k}-( \sigma +\tau +\mu )\sum_{k=0}^{\infty }L_{k} (t) ], \\ x \mathscr{L} [\sum_{k=0}^{\infty }I_{k} (t) ]= \frac{\mathbf{N}_{4}}{s} +\frac{s+\eta (1-s)}{s}\mathscr{L} [\tau \sum_{k=0}^{\infty }L_{k} (t)-(\gamma +\mu + \delta )\sum_{k=0}^{\infty }I_{k} (t) ], \\ \mathscr{L} [\sum_{k=0}^{\infty }R_{k} (t) ] \\ \quad = \frac{\mathbf{N}_{5}}{s} +\frac{s+\eta (1-s)}{s}\mathscr{L} [\sigma \sum_{k=0}^{\infty }L_{k} (t)+\gamma \sum_{k=0}^{\infty }I_{k} (t)-\mu \sum_{k=0}^{\infty }R_{k} (t) ]. \end{cases} $$
(18)

Now, comparing terms on both sides of (18), we get the following series of problems.

Case 1. When \(n=0 \), we have

$$ \textstyle\begin{cases} \mathscr{L} [M_{0}(t) ]= \frac{\mathbf{N}_{1}}{s}+ \frac{s+\eta (1-s)}{s}\mathscr{L}[\theta \rho ], \\ \mathscr{L} [S_{0}(t) ]=\frac{\mathbf{N}_{2}}{s}+ \frac{s+\eta (1-s)}{s}\mathscr{L} [(1-\theta )\rho ], \\ \mathscr{L} [L_{0}(t) ]=\frac{\mathbf{N}_{3}}{s}, \\ \mathscr{L} [I_{0}(t) ]=\frac{\mathbf{N}_{4}}{s}, \\ \mathscr{L} [R_{0}(t) ]=\frac{\mathbf{N}_{5}}{s}. \end{cases} $$
(19)

Evaluating the inverse Laplace transform, we get

$$ \textstyle\begin{cases} M_{0}(t)=\mathbf{N}_{1}+ \theta \rho [1+\eta (t-1) ], \\ S_{0}(t)= \mathbf{N}_{2}+(1-\theta )\rho [1+\eta (t-1) ], \\ L_{0}(t)=\mathbf{N}_{3}, \\ I_{0}(t)=\mathbf{N}_{4}, \\ R_{0}(t)=\mathbf{N}_{5}. \end{cases} $$
(20)

Case 2. When \(n=1 \), we have

$$ \textstyle\begin{cases} \mathscr{L} [M_{1}(t) ]= \frac{s+\eta (1-s)}{s} \mathscr{L} [-(\alpha +\mu ) M_{0}(t) ], \\ \mathscr{L} [S_{1}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [(\alpha M_{0}(t)- \beta A_{0}(t) - \mu S_{0}(t) ], \\ \mathscr{L} [L_{1}(t) ]=\frac{s+\eta (1-s)}{s}\mathscr{L} [\beta A_{0}(t)-( \sigma +\tau +\mu )L_{0}(t) ], \\ \mathscr{L} [I_{1}(t) ]=\frac{s+\eta (1-s)}{s}\mathscr{L} [\tau L_{0}(t) -(\gamma +\mu +\delta )I_{0}(t) ], \\ \mathscr{L} [R_{1}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [\sigma L_{0}(t)+ \gamma I_{0}(t)-\mu R_{0}(t) ]. \end{cases} $$
(21)

Evaluating the inverse Laplace transform, we get

$$ \textstyle\begin{cases} M_{1}(t)= [-(\alpha +\mu ) M_{0}(t) ] [1+\eta (t-1) ], \\ S_{1}(t)= (\alpha M_{0}(t)-\beta S_{0}(t) I_{0}(t)-\mu S_{0}(t) ) [1+ \eta (t-1) ], \\ L_{1}(t)= (\beta S_{0}(t) I_{0}(t)-( \sigma +\tau +\mu )L_{0}(t) ) [1+ \eta (t-1) ], \\ I_{1}(t)= (\tau L_{0}(t) -(\gamma +\mu +\delta )I_{0}(t) ) [1+\eta (t-1) ], \\ R_{1}(t)= (\sigma L_{0}(t) +\gamma I_{0}(t) )-\mu R_{0}(t)) [1+\eta (t-1) ]. \end{cases} $$
(22)

Case 3. When \(n=2 \), we have

$$ \textstyle\begin{cases} \mathscr{L} [M_{2}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [-(\alpha +\mu ) M_{1}(t) ], \\ \mathscr{L} [S_{2}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [(\alpha M_{1}(t)- \beta S_{1}(t) - \mu S_{1}(t) ], \\ \mathscr{L} [L_{2}(t) ]=\frac{s+\eta (1-s)}{s}\mathscr{L} [\beta S_{1}(t)-( \sigma +\tau +\mu )L_{1}(t) ], \\ \mathscr{L} [I_{2}(t) ]=\frac{s+\eta (1-s)}{s}\mathscr{L} [\tau L_{1}(t) -(\gamma +\mu +\delta )I_{1}(t) ], \\ \mathscr{L} [R_{2}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [\sigma L_{1}(t)+ \gamma I_{1}(t)-\mu R_{1}(t) ]. \end{cases} $$
(23)

Evaluating the inverse Laplace transform, we get

$$ \textstyle\begin{cases} M_{2}(t)=(\alpha +\mu )^{2} M_{0} [1+2 \eta (t-1)+{ \eta }^{2} (\frac{t^{2}}{2!}-2t+1 ) ], \\ S_{2}(t)= [-\alpha (\alpha +\mu )M_{0}-\beta \{S_{0} (\tau L_{0}-( \gamma +\mu +\delta )I_{0} ) \\ \hphantom{S_{2}(t)=}{}+I_{0}(\alpha M_{0} -\beta S_{0} I_{0}-\mu S_{0}) \} -\mu (\alpha M_{0} - \beta S_{0} I_{0}-\mu S_{0}) ] \\ \hphantom{S_{2}(t)=}{}\times [1+2 \eta (t-1)+{\eta }^{2} (\frac{t^{2}}{2!}-2t+1 ) ], \\ L_{2}(t)= [\beta \{S_{0} (\tau L_{0}-(\gamma +\mu +\delta )I_{0} )+I_{0}( \alpha M_{0} -\beta S_{0} I_{0}-\mu S_{0}) \} \\ \hphantom{L_{2}(t)=}{}- \{(\sigma +\tau +\mu ) ( \{\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \} \} ] \\ \hphantom{L_{2}(t)=}{}\times [1+2 \eta (t-1) +{\eta }^{2} (\frac{t^{2}}{2!}-2t+1 ) ], \\ I_{2}(t)= [\tau \{\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \}- \{(\gamma + \mu +\delta ) (\tau L_{0}-(\gamma +\mu +\delta )I_{0} \} ] \\ \hphantom{I_{2}(t)=}{}\times [1+2 \eta (t-1)+{\eta }^{2} (\frac{t^{2}}{2!}-2t+1 ) ], \\ R_{2}(t)= [\tau \{\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \} \\ \hphantom{R_{2}(t)=}{}- \{(\gamma +\mu +\delta ) (\tau L_{0}-\mu (\sigma L_{0}+\gamma I_{0}- \mu R_{0}) \} ] \\ \hphantom{R_{2}(t)=}{}\times [1+2 \eta (t-1)+{\eta }^{2} ( \frac{t^{2}}{2!}-2t+1 ) ]. \end{cases} $$
(24)

Case 4. When \(n=3 \), we have

$$ \textstyle\begin{cases} \mathscr{L} [M_{3}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [-(\alpha +\mu ) M_{2} ], \\ \mathscr{L} [S_{3}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L} [(\alpha M_{2}- \beta P_{2} - \mu S_{2} ], \\ \mathscr{L} [L_{3}(t) ]=\frac{s+\eta (1-s)}{s}\mathscr{L} [\beta P_{2}-( \sigma +\tau +\mu )L_{2} ], \\ \mathscr{L} [I_{3}(t) ]=\frac{s+\eta (1-s)}{s}\mathscr{L} [\tau L_{2} -( \gamma +\mu +\delta )I_{2} ], \\ \mathscr{L} [R_{3}(t) ]=\frac{s+\eta (1-s)}{s} \mathscr{L}[\sigma L_{2}+ \gamma I_{1}-\mu R_{2} ] \end{cases} $$
(25)

and so on. The other terms may similarly be computed.

Evaluating the inverse Laplace transform, we get

$$ \textstyle\begin{cases} M_{3}(t)=-(\alpha +\mu )^{3} M_{0} [1+3 \eta (t-1)+ \eta ^{2} (\frac{t^{2}}{2!}-2t+1 ) \\ \hphantom{M_{3}(t)=}{} +\eta ^{3} ( \frac{t^{3}}{3!}-3\frac{t^{2}}{2!}+3t-1 ) ] \\ S_{3}(t)= [ \alpha (\alpha +\mu )^{2}M_{0}- \beta [ \{S_{0}\tau \{\beta S_{0} I_{0}- (\sigma +\tau +\mu )L_{0} \} \\ \hphantom{S_{3}(t)=}{} - \{(\gamma + \mu + \delta ) (\tau L_{0}-(\gamma +\mu +\delta )I_{0} ) \} ] \\ \hphantom{S_{3}(t)=}{} - [\alpha (\alpha +\mu )M_{0}-\beta \{S_{0} (\tau L_{0}-(\gamma +\mu + \delta )I_{0} ) \\ \hphantom{S_{3}(t)=}{}+I_{0}(\alpha M_{0} -\beta S_{0} I_{0}- \mu S_{0}) \} -\mu (\alpha M_{0}-\beta S_{0} I_{0}-\mu S_{0}) ] I_{0} \\ \hphantom{S_{3}(t)=}{} -\mu [ - \alpha (\alpha +\mu )M_{0}-\beta \{S_{0} (\tau L_{0}-( \gamma +\mu + \delta )I_{0} ) \\ \hphantom{S_{3}(t)=}{} +I_{0}(\alpha M_{0} -\beta S_{0} I_{0}- \mu S_{0}) \}-\mu (\alpha M_{0} - \beta S_{0} I_{0}-\mu S_{0}) ] \\ \hphantom{S_{3}(t)=}{} \times [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3 \frac{t^{2}}{2!}+3t-1 ) ] \\ \hphantom{S_{3}(t)=}{} + [ (\alpha M_{0}-\beta S_{0} I_{0}- \mu S_{0}) (\tau L_{0} -( \gamma +\mu +\delta )I_{0} ) ] \\ \hphantom{S_{3}(t)=}{} \times [1+3 \eta (t-1)+\eta ^{2} (2{t^{2}}-2t+1 )+\eta ^{3} (\frac{t^{3}}{{3}}-2{t^{2}}+3t-1 ) ], \\ L_{3}(t)=\beta [ \{S_{0}\tau \{\beta S_{0} I_{0}- (\sigma +\tau + \mu )L_{0} \} - \{(\gamma +\mu +\delta ) (\tau L_{0}-(\gamma +\mu + \delta )I_{0} ) \} ] \\ \hphantom{L_{3}(t)=}{} + [-\alpha (\alpha +\mu )M_{0}-\beta \{S_{0} (\tau L_{0}-(\gamma +\mu + \delta )I_{0} ) \\ \hphantom{L_{3}(t)=}{}+I_{0}(\alpha M_{0} -\beta S_{0} I_{0}- \mu S_{0}) \} \\ \hphantom{L_{3}(t)=}{} -\mu (\alpha M_{0} -\beta S_{0} I_{0}-\mu S_{0}) I_{0} ]- [(\sigma + \tau +\mu ) [\beta \{S_{0} (\tau L_{0}-(\gamma +\mu +\delta )I_{0} ) \\ \hphantom{L_{3}(t)=}{} +(\alpha M_{0} -\beta S_{0} I_{0}-\mu S_{0}) I_{0} ]-(\sigma +\tau + \mu ) \{\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \} ] \\ \hphantom{L_{3}(t)=}{} \times [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3 \frac{t^{2}}{2!}+3t-1 ) ] \\ \hphantom{L_{3}(t)=}{} +[ (\alpha M_{0}-\beta S_{0} I_{0}-\mu S_{0}) (\tau L_{0} -(\gamma + \mu +\delta )I_{0} ) \\ \hphantom{L_{3}(t)=}{} \times [1+3 \eta (t-1)+\eta ^{2} (2{t^{2}}-2t+1 )+\eta ^{3} ( \frac{t^{3}}{{3}}-2{t^{2}}+3t-1 ) ], \\ I_{3}(t)=[\tau \{\beta S_{0} (\tau L_{0}-(\gamma +\mu +\delta )I_{0} )+ I_{0}(\alpha M_{0} -\beta S_{0} I_{0}- \mu S_{0}) \}\\ \hphantom{I_{3}(t)=}{}-\{(\sigma +\tau + \mu ) \{ \beta S_{0} I_{0} -(\sigma +\tau +\mu )L_{0} \} \\ \hphantom{I_{3}(t)=}{}- [(\gamma +\mu +\delta ) \{\tau (\beta S_{0}I_{0}-( \sigma +\tau +\mu )L_{0} \}-(\gamma +\mu +\delta ) \\ \hphantom{I_{3}(t)=}{}\times (\tau L_{0} -(\gamma +\mu +\delta )I_{0} ) ] \\ \hphantom{I_{3}(t)=}{}\times [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3 \frac{t^{2}}{2!}+3t-1 ) ], \\ R_{3}(t)= \sigma \{\beta S_{0} (\tau L_{0}-(\gamma +\mu +\delta )I_{0} )+ I_{0}(\alpha M_{0} -\beta S_{0} I_{0}- \mu S_{0}) \} \\ \hphantom{R_{3}(t)=}{} -(\sigma +\tau +\mu ) \{\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \} \\ \hphantom{R_{3}(t)=}{}-[( \gamma +\mu +\delta ) \{\tau (\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \} \\ \hphantom{R_{3}(t)=}{} +\gamma [\tau \{\beta S_{0} I_{0}-(\sigma +\tau + \mu )L_{0} \}- \{( \gamma +\mu +\delta ) (\tau L_{0}-(\gamma +\mu +\delta )I_{0} \} \\ \hphantom{R_{3}(t)=}{} -\mu [\tau \{\beta S_{0} I_{0}-(\sigma +\tau +\mu )L_{0} \} \\ \hphantom{R_{3}(t)=}{}- \{(\gamma + \mu +\delta ) (\tau L_{0}-\mu (\sigma L_{0}+\gamma I_{0}- \mu R_{0}) \} ] \\ \hphantom{R_{3}(t)=}{} \times [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3 \frac{t^{2}}{2!}+3t-1 ) ], \end{cases} $$
(26)

and so on. In this way, next terms of the series solution may be computed. Therefore, we get the required solution as follows:

$$ \textstyle\begin{cases} M(t)=M_{0} (t)+M_{1} (t)+M_{2} (t)+M_{3} (t)+\cdots, \\ S(t)=S_{0} (t)+S_{1} (t)+S_{2} (t)+S_{3} (t)+\cdots, \\ L(t)=L_{0} (t)+L_{1} (t)+L_{2} (t)+L_{3} (t)+\cdots, \\ I(t)=I_{0} (t)+I_{1} (t)+I_{2} (t)+I_{3} (t)+\cdots, \\ R(t)=R_{0} (t)+R_{1} (t)+R_{2} (t)+R_{3} (t)+\cdots . \end{cases} $$
(27)

Theorem 4

Let \(\mathscr{X}\) be the Banach space and \(\mathbf{T} : \mathscr{X}\rightarrow \mathscr{X}\) be a contractive nonlinear operator such that, for all \(W, {\bar{W}} \in \mathscr{X}\), \(\|\mathbf{T}(W)-\mathbf{T}({\bar{W}}) \|_{\mathscr{X}}\leq \kappa \|W-{\bar{w}}\|_{\mathscr{X}}\), \(0<\kappa <1\). Using the Banach contraction principle, T has a unique point W such that \(\mathbf{T}{W}={W}\), where \(W=(x,y,z)\). By applying LADM, the series given in (26) can be written as

$$ W_{n}=\mathbf{T}W_{n-1},\qquad W_{n-1}=\sum _{j=1}^{n-1}W_{j},\quad j=1,2,3, \ldots, $$

and let \(W_{0}=W_{0}\in B_{\varepsilon }(W)\), where \(B_{\varepsilon }(W)={\bar{\mathbf{w}}\in \mathscr{X}:\| \bar{\mathbf{w}}-W\|_{\mathscr{X}}<\varepsilon }\), then one has

  1. (i)

    \(W_{n} \in B_{r}(W)\);

  2. (ii)

    \(\lim_{n\rightarrow \infty }W_{n}=W\).

Proof

The proof of the above theorem can be derived in a way similar to that in [39]. □

Numerical results and discussion

In this part of the paper, we present numerical results along with illustration regarding the approximate solution of the model under discussion. We take the approximate values for the parameters as given in Table 1. In light of these values, we get the series solution as follows:

$$\begin{aligned} \textstyle\begin{cases} M(t)=90-4.194[1+\eta (t-1)]+0.192411[1+2 \eta (t-1)+{\eta }^{2}( \frac{t^{2}}{2!}-2t+1)] \\ \hphantom{M(t)=}{} -0.008966 [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3\frac{t^{2}}{2!}+3t-1 ) ], \\ S(t)=400-1823.361[1+\eta (t-1)]+8408.2894[1+2 \eta (t-1)+{\eta }^{2}( \frac{t^{2}}{2!}-2t+1)] \\ \hphantom{S(t)=}{} -39200.2682 [1+3 \eta (t-1)+ \eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3\frac{t^{2}}{2!}+3t-1 ) ] \\ \hphantom{S(t)=}{} -355.6365 [1+3 \eta (t-1)+{\eta }^{2}(2{t^{2}}-6t+3)+{\eta }^{3} (\frac{t^{3}}{{3}}-2{t^{2}}+3t-1 ) ], \\ L(t)=100+1811.44[1+\eta (t-1)]-8488.5579[1+2 \eta (t-1)+{\eta }^{2}( \frac{t^{2}}{2!}-2t+1)] \\ \hphantom{L(t)=}{} +39597.60959 [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3\frac{t^{2}}{2!}+3t-1 ) ] \\ \hphantom{L(t)=}{} +355.6365 [1+3 \eta (t-1)+{\eta }^{2}(2{t^{2}}-6t+3) +{\eta }^{3} (\frac{t^{3}}{{3}}-2{t^{2}}+3t-1 ) ], \\ I(t)=50-2.14545[1+\eta (t-1)]+22.6071 [1+2 \eta (t-1)+{\eta }^{2} (\frac{t^{2}}{2!}-2t+1 ) ] \\ \hphantom{I(t)=}{} -106.7888 [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3\frac{t^{2}}{2!}+3t-1 ) ], \\ R(t)=10+4.0454[1+\eta (t-1)]+61.8304 [1+2 \eta (t-1)+{\eta }^{2} (\frac{t^{2}}{2!}-2t+1 ) ] \\ \hphantom{R(t)=}{} +289.93398 [1+3 \eta (t-1)+\eta ^{2} (\frac{t^{2}}{2!}-2t+1 )+\eta ^{3} (\frac{t^{3}}{3!}-3\frac{t^{2}}{2!}+3t-1 ) ]. \end{cases}\displaystyle \end{aligned}$$
(28)

Now we plot the solution up to five terms as given in (28) in Figures 15, corresponding to different fractional orders.

Table 1 Table of parametric values used for simulations of the problem under consideration

It can be observed from Fig. 1 that the immunized population decreases with different fractional orders at different ratio. In the same way the susceptible population is increasing as shown in Fig. 2. The infected and the latently infected population is also increasing as shown in Figs. 3 and 4, respectively. Because the susceptible population is converted to infected or latently infected. Proper cure is applied, then the recovered population will increase as shown in Fig. 5. The process of increase or decrease is initially fastest at lower fractional order and some time after the process is reversed, and the greater is the fractional order the faster is the increasing or decreasing process of population of the respective compartments. It means that fractional order derivatives can express the behavior more globally. The recovered population gradually increases and converges to equilibrium state as time progresses, as shown in Fig. 6.

Figure 2
figure2

Graphical representation of approximate solutions up to five terms of immunized population \(M(t)\) at different fractional order of the considered model (2)

Figure 3
figure3

Graphical representation of approximate solutions up to five terms of susceptible population \(S(t)\) at different fractional order of the considered model (2)

Figure 4
figure4

Graphical representation of approximate solutions up to five terms of latently infected population \(L(t)\) at different fractional order of the considered model (2)

Figure 5
figure5

Graphical representation of approximate solutions up to five terms of infected population \(I(t)\) at different fractional order of the considered model (2)

Figure 6
figure6

Graphical representation of approximate solutions up to five terms of recovered population \(R(t)\) at different fractional order of the considered model (2)

Conclusion

We have investigated a biological model of TB under Caputo–Fabrizio fractional derivative. We have also established some sufficient results regarding the existence as well as the uniqueness of solution for the considered problem with the help of fixed point theorems. Further we have used a hybrid type method to compute series type solutions for the proposed model. To the best of our knowledge, the said techniques were very rarely used to handle the analytical solutions of FODEs involving nonsingular derivative of Caputo–Fabrizio type in the past. Further the numerical results have been displayed via graphs indicating that the established technique can be used to handle solution of those FODEs involving CFFD efficiently. Further, the mentioned method can be utilized to investigate more nonlinear problems of FODEs involving CFFD.

Availability of data and materials

Data sharing is not applicable to this paper.

References

  1. 1.

    Morse, D., Brothwell, D.R., Ucko, P.J.: Tuberculosis in ancient Egypt. Am. Rev. Respir. Dis. 90(4), 524–541 (1964)

    Google Scholar 

  2. 2.

    Aparicio, J.P., Capurro, A.F., Castillo-Chavez, C.: Transmission and dynamics of tuberculosis on generalized households. J. Theor. Biol. 206, 327–341 (2000)

    Article  Google Scholar 

  3. 3.

    Floyd, K., Glaziou, P., Zumla, A., Raviglione, M.: The global tuberculosis epidemic and progress in care, prevention, and research: an overview in year 3 of the end TB era. Lancet Respir. Med. 6(4), 299–314 (2018)

    Article  Google Scholar 

  4. 4.

    Dye, C.: Global epidemiology of tuberculosis. Lancet 367(9514), 938–940 (2006)

    Article  Google Scholar 

  5. 5.

    Colditz, G.A., Brewer, T.F., Berkey, C.S., Wilson, M.E., Burdick, E., Fineberg, H.V., Mosteller, F.: Efficacy of BCG vaccine in the prevention of tuberculosis: meta-analysis of the published literature. JAMA 271(9), 698–702 (1994)

    Article  Google Scholar 

  6. 6.

    Arqub, O.A., El-Ajou, A.: Solution of the fractional epidemic model by homotopy analysis method. J. King Saud Univ., Sci. 25(1), 73–81 (2013)

    Article  Google Scholar 

  7. 7.

    Rafei, M., Ganji, D.D., Daniali, H.: Solution of the epidemic model by homotopy perturbation method. Appl. Math. Comput. 187(2), 1056–1062 (2007)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Zhao, S., Xu, Z., Lu, Y.: A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China. Int. J. Epidemiol. 29(4), 744–752 (2000)

    Article  Google Scholar 

  9. 9.

    Haq, F., Shah, K., Khan, A., Shahzad, M., Rahman, G.: Numerical solution of fractional order epidemic model of a vector born disease by Laplace Adomian decomposition method. Punjab Univ. J. Math. 49(2), 13–22 (2017)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Ullah, I., Ahmad, S., Al-Mdallal, Q., Khan, Z.A., Khan, H., Khan, A.: Stability analysis of a dynamical model of tuberculosis with incomplete treatment. Adv. Differ. Equ. 2020(1), 499 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Enagi, A.I., Ibrahim, M.O., Akinwande, N.I., Bawa, M., Wachin, A.A.: A mathematical model of tuberculosis control incorporating vaccination, latency and infectious treatments (case study of Nigeria). Int. J. Math. Comput. Sci. 12(2), 97 (2017)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Podlubny, I.: Fractional Differential Equations: Mathematics in Science and Engineering. Academic Press, New York (1999)

    Google Scholar 

  13. 13.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland, Amsterdam (2006)

    Google Scholar 

  14. 14.

    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)

    Google Scholar 

  15. 15.

    Caputo, M., Fabrizio, M.: Application of new time and spatial fractional derivatives with exponential kernels. Prog. Fract. Differ. Appl. 2, 1–11 (2016)

    Article  Google Scholar 

  16. 16.

    Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    El-Saka, H.A.A.: The fractional-order SIS epidemic model with variable population size. J. Egypt. Math. Soc. 22(1), 50–54 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Toledo-Hernandez, R., Rico-Ramirez, V., Iglesias-Silva, G.A., Diwekar, U.M.: A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: fractional models for biological reactions. Chem. Eng. Sci. 117, 217–228 (2014)

    Article  Google Scholar 

  19. 19.

    Wang, Z., Yang, D., Ma, T., Sun, N.: Stability analysis for nonlinear fractional-order systems based on comparison principle. Nonlinear Dyn. 75(1–2), 387–402 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integro-differential inclusions via the extended fractional Caputo–Fabrizio derivation. Bound. Value Probl. 2019(1), 79 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020(1), 1 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Aydogan, M.S., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integro-differential equations including the Caputo–Fabrizio derivative. Bound. Value Probl. 2018(1), 90 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential equations. Bound. Value Probl. 2017(1), 1 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Ullah, I., Ahmad, S., Rahman, M., Arfan, M.: Investigation of fractional order Tuberculosis (TB) model via caputo derivative. Chaos Solitons Fractalss (2020, in press)

  25. 25.

    Algahtani, O.J.J.: Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fractals 89, 552–559 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Saad, K.M.: Comparing the Caputo, Caputo–Fabrizio and Atangana–Baleanu derivative with fractional order: fractional cubic isothermal auto-catalytic chemical system. Eur. Phys. J. Plus 133(3), 1–12 (2018)

    Article  Google Scholar 

  27. 27.

    Goufo, E.F.D.: Application of the Caputo–Fabrizio fractional derivative without singular kernel to Korteweg–de Vries–Burgers equation. Math. Model. Anal. 21(2), 188–198 (2016)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Ahmad, B., Alsaedi, A., Nazemi, S.Z., Rezapour, S.: Some existence theorems for fractional integro-differential equations and inclusions with initial and non-separated boundary conditions. Bound. Value Probl. 2014(1), 249 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Rezapour, S., Samei, M.E.: On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation. Bound. Value Probl. 2020(1), 1 (2020)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Baleanu, D., Rezapour, S., Mohammadi, H.: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 371(1990), 20120144 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Haq, F., Shah, K., Rahman, G., Shahzad, M.: Numerical solution of fractional order smoking model via Laplace Adomian decomposition method. Alex. Eng. J. 57(2), 1061–1069 (2018)

    Article  Google Scholar 

  32. 32.

    Ali, A., Shah, K., Khan, R.A.: Numerical treatment for traveling wave solutions of fractional Whitham–Broer–Kaup equations. Alex. Eng. J. 57(3), 1991–1998 (2018)

    Article  Google Scholar 

  33. 33.

    Kiymaz, O.: An algorithm for solving initial value problems using Laplace Adomian decomposition method. Appl. Math. Sci. 3(29–32), 1453–1459 (2009)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Khuri, S.A.: A Laplace decomposition algorithm applied to a class of nonlinear differential equations. J. Appl. Math. 1(4), 141–155 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Shaikh, A., Tassaddiq, A., Nisar, K.S., Baleanu, D.: Analysis of differential equations involving Caputo–Fabrizio fractional operator and its applications to reaction-diffusion equations. Adv. Differ. Equ. 2019(1), 178 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 87–92 (2015)

    Google Scholar 

  37. 37.

    Khan, S.A., et al.: Existence theory and numerical solutions to smoking model under Caputo–Fabrizio fractional derivative. Chaos, Interdiscip. J. Nonlinear Sci. 29(1), 013128 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Zhao, X.Q.: The theory of basic reproduction ratios. In: Dynamical Systems in Population Biology, pp. 285–315. Springer, Cham (2017)

    Google Scholar 

  39. 39.

    Shah, K., Khalil, H., Khan, R.A.: Analytical solutions of fractional order diffusion equations by natural transform method. Iran. J. Sci. Technol., Trans. A, Sci. 42(3), 1479–1490 (2018)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

All authors contributed equally in writing of this manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Rafi Ullah.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ahmad, S., Ullah, R. & Baleanu, D. Mathematical analysis of tuberculosis control model using nonsingular kernel type Caputo derivative. Adv Differ Equ 2021, 26 (2021). https://doi.org/10.1186/s13662-020-03191-x

Download citation

Keywords

  • Tuberculosis-(TB) mathematical model
  • Natural transforms
  • Approximate solutions
  • Caputo–Fabrizio fractional order derivative
  • Numerical simulations
\