Predictive and Global Effect of Active Smoker in Asthma Dynamics with Caputo Fractional Derivative

1  Introduction

Asthma is a chronic disease that causes difficulty breathing by affecting the inner walls of the airways. Breathing difficulties and shortness of breath when exhaling are symptoms of the chronic illness asthma. Although there is no particular treatment, symptoms can be tracked. Patients should speak with medical professionals who can modify their treatment according to the patient’s condition. Chest tightness, discomfort, coughing, wheezing, respiratory system infections, dyspnea, and trouble sleeping are some of the symptoms. Children with asthma frequently exhibit wheezing during exhalation, which causes breathing problems. Asthmatic patients may have severe symptoms such as chest pain and loss of consciousness due to a shortage of oxygen.

According to reports, cigarette smoke can also cause asthma in addition to air pollutants such hexachloroplatinates and di-isocynates [1–3]. Smoking is harmful to the lungs and has numerous effects on our bodies. This leads to decreased lung function, which increases the lungs’ susceptibility to asthma triggers. Cigarette smoke contains high levels of irritants such as formaldehyde, ammonia, acrolein, and nitrogen oxides. The expert maintained that smoke which enter the host through a close smoker in the community, can harm the lungs. As a result, when someone inhales tobacco smoke, these disagreeable chemicals may trigger an asthma attack. Over 1,000,000 children with asthma worsen after being exposed to secondhand smoke, and over 15 million children are exposed to smoke every day [4,5]. Asthma is also caused by airborne pollutants, including dust from open-pit coal mining, coal-fired cement, and other operations, heavy metal cocktails from Solid Liquid Fuel (SLF) burning cement, and vanadium and nickel particles from power plants and oil refineries [6–8]. As a result, a susceptible population develops asthma as a result of ongoing exposure to air pollutants. To investigate the patterns of transmission of infectious diseases [9–12] and generate vital data regarding disease dynamics for control and prevention [13,14], mathematical models are crucial. Researchers have created a variety of numerical techniques to study disease dynamics. Researchers have recently examined the dynamics of asthma from a variety of angles in their study [15,16]. It is well known that ongoing smoking exposure is a major factor in the onset and progression of asthma.

Because fractional calculus is inherited and uses memory-based depictions, it is a flexible framework that provides accurate results in science and technology [17–19]. Using its derivative for a constant being zero, Caputo’s fractional derivative enables the solution of problems using conventional beginning and boundary conditions [20–22]. Nonlinear functional integral equations can now be solved with a variety of fractional operators (see [23,24]) thanks to new advances in fractional calculus. Dinku et al. [25] used various kernels of fractional differential equations to examine the effect of smoking on the development of cancer. To take into consideration the periodic impacts of smoking, they included a periodic function. For a variety of fractional-order derivatives, the study examined chaotic behavior, such as high smoking frequency and absence of immunological flux under low host cell starting circumstances. The goal of the study [26] was to compare the outcomes with conventional models by using fractional calculus to investigate adsorption and extraction processes. Equations were solved using the Caputo derivative, and processes such as the extraction of maize antioxidant compounds with varying solvent amounts and the adsorption of biodiesel glycerol at varying temperatures were examined. First-order and So and MacDonald’s models were used to compare the extraction of antioxidant chemicals. The diagnosis and therapy of lung cancer in patients with compromised immune systems utilizing cytokines and anti-PD-L1 inhibitors was investigated by Nisar et al. [27]. In addition to identifying early lung cancer management by cytokines and anti-PD-L1 inhibitors, which promote the generation of anti-cancer cells, they verified immune-mediated compartment linkages. IABNs were developed by Mukhtar et al. [28] to solve fractional order Parkinson’s disease scenarios. Through evaluations with regard numerical solutions and simulations, the IABN was examined. An excellent method for forecasting significant deformations of amorphous polymers is the variable order fractional parametric model, which was refined in [29]. It characterizes the mechanical behavior of polycarbonate at various temperatures. Using a fractional-order system, Farman et al. [30] demonstrate how decreased resources from forests affect toxin activity and fire caused by humans. These steps are intended to confirm that resource depletion has been resolved. In ecological modeling, fractional calculus is crucial for comprehending and overseeing maintaining forests in the face of human pressures, according to the study. Using GA, Adak et al. [31] optimized a SIR fractional order model that included medication, the media, and vaccine controls. The order of fractional derivatives was shown to affect the ideal management parameter values. The fractional operator in the Caputo sense was used by Bansal et al. [32] to create a mathematical model for studying the dynamics of transmitting monkeypox in both humans and other species.

Traditional integer-order differential equations have been widely used to model the dynamics of asthma and its association with environmental pollutants, including tobacco smoke [33]. Incidents of asthma are closely related to environmental variables such as smoking, pollution in the air, and allergens. However, breathing difficulties can be reduced with medication-based therapy and limiting exposure to asthma triggers. Recent advancements have introduced fractional-order derivatives to better capture memory effects and hereditary properties inherent in biological systems [34,35]. Inspired by the debate above, we create a novel model in this work to examine the dynamics of different smoker classes that have the potential to spread asthma. As far as we are aware, this effort is novel and will investigate further perspectives on the problem. Furthermore, because fractional-order models better capture the intricate physical processes, we used the Caputo fractional operator rather than integer-order. The investigation of dynamical behavior throughout the entire time span and its precise mathematical characterization are currently the main challenges. The intricate dynamical processes found in smoking-induced asthma disease can be better described by fractional-order models.

The structure of this manuscript is as follows: A brief description of the fractional operator utilized in the suggested model is given in Section 2. The fractional-order-based smoking-induced asthma model is the main topic of Section 3, and a thorough examination of the generated model is included in Section 4. Section 5 explores the system’s stability using the Lyapunov stability approach. Inside the practicable spectrum, Section 6 examines the model’s performance under chaotic circumstances; Section 7 displays the numerical outcomes derived from the system; Section 8 supplies an in-depth analysis of the outcomes, and delivers a summary of the most significant results and the ultimate findings.

2  Formulation of Dynamical Model for Smoking-Induced Asthma

In this work, we developed a new model of smoking-induced asthma disease flow chart shown in Fig. 1. Smokers are divided into three classes as:

Figure 1: The flow chart of dynamical system

•   N(t); Non-smoker class;

•   A(t): Active-smoker class; and

•   L(t): Ex-smoker class.

Smoking-induced asthma population is categorized as follows:

•   B1(t): Asthma diagnose due to non-smoker class;

•   B2(t): Asthma diagnose due to active-smoker class; and

•   B3(t): Asthma diagnose due to ex-smoker class.

2.1 Assumptions

•   π represents the non-smokers N(t)′ recruiting rate, while δ1 represents the natural death rate. We assumed that the number of nonsmokers would decline when they begin to smoke at a rate of β and as a result of outside influences at a rate of η. Additionally, nonsmokers spread asthma at a rate of α1, and the number of nonsmokers who recover from asthma also contributes to the class N(t). Thus at time t, we have

N(t)=π−(β+η)N−α1N−δ1N+ρ1B1.(1)

•   Active smokers A(t) increase at the rate of β and η, and also with the temporary improvement rate of w due to ex-smokers. The number of active smokers decrease due to natural death rate of δ2, naturally smoking-quit rate of r, and as well as due to asthma at the rate of r1. Moreover, A(t) spread asthma at a rate of α2 and they decrease as they recover from asthma at a rate of ρ2. Therefore,

A(t)=(β+η)N+wL−(r+r1)A−(α2+δ2)A+ρ2B2.(2)

•   The class L(t) of ex-smokers rises at a smoking-quit rate of r and a quit rate of r1 because of asthma. Due to the recovered ex-smokers, the number of L(t) also rises at a rate of ρ3. The transmission rates of asthma and natural death, represented by α3 and δ3, respectively, cause a drop in the number of L(t).

L(t)=rA+r1B1−(α3+δ3−w)L+ρ3B3.(3)

•   The class B1(t) increases when ex-smokers (L(t)) come into contact with asthma and transmit it at a rate of α1. B1(t) class decreases as per natural death rate (μ1) and the recovery rate (ρ1) of active smokers having asthma.

B1(t)=α1N−(ρ1+μ1)B1.(4)

•   The class B2(t) increases when active smokers (A(t)) come into contact with asthma and transmit it at a rate of α2. B3(t) class reduces due to the natural death rate (μ2) and the recovery rate (ρ2) of active smokers having asthma.

B2(t)=α2A−(ρ2+μ2)B2.(5)

•   The class B3(t) increases when ex-smokers (L(t)) come into contact with asthma and transmit it at a rate of α3. It also increases due to certain external impacts indicated by ϕ. B3(t) class falls due to the natural death rate (μ3) and the recovery rate (ρ3) of ex-smokers having asthma.

B3(t)=α3L+ϕB3−(ρ3+μ3)B3.(6)

2.2 Fractional-Order Model

A variable’s rate of change in classical modeling using ordinary differential equations is entirely dependent on its existing state. These models’ basic presumption that the system is memoryless restricts their capacity to represent biological processes impacted by prior actions. The Caputo fractional-order derivative presents a memory-dependent paradigm in which the system’s current state is impacted by every condition in the past via a weighted history. This characteristic is especially crucial for long-term respiratory disorders like asthma, where the effects of environmental exposures, like smoking, take time to manifest. Because the Caputo operator makes the switch from classical to fractional modeling more natural by allowing the integration of beginning conditions in the same form as classical derivatives, it is particularly well-suited for modeling such memory effects. A more realistic depiction of the disease’s course is achieved by incorporating the memory feature into the model, which successfully captures inflammatory accumulation, delayed immune responses, and residual smoking-related harm. Therefore, in comparison to conventional Ordinary Differential Equations (ODEs), the fractional-order model with the Caputo derivative offers a more thorough and biologically compatible approach.

There are numerous uses for fractional calculus. In contrast to other derivatives of non-integer orders, the Caputo fractional derivative is a fundamental idea in fractional calculus that finds practical tractability in problem modeling, providing rich and precise information. A possible physical manifestation of real-life phenomena, such the memory of smoking-induced asthma that explains a system’s behavior in the past, is the arbitrary fractional order. Here, we are discussing some fundamental findings that are essential to our work.

Definition 1. [36] For a continuous function Ψ(t), the Riemann-Liouville integral definition of order θ is:

Itθ0RΨ(t)=1Γ(θ)∫0t(t−τ)(θ−1)Ψ(τ)dτ.(7)

Definition 2. [36] The Caputo derivative operator of order θ can be written as:

Dtθ0CΨ(t)=1Γ(m−θ)[∫0t(t−τ)m−θ−1Ψ′(t)(τ)dτ],(8)

where m=[θ] is the smallest integer and m−1<θ<m.

Consequently, by improving the data fitting versatility, the fractional order parameter makes it possible to describe the asthma disease system using the Caputo derivative structure in an extra adaptable way.

DtθCN(t)=π−(β+η)N−α1N−δ1N+ρ1B1,DtθCA(t)=(β+η)N+wL−(r+r1)A−(α2+δ2)A+ρ2B2,DtθCL(t)=rA+r1B1−(α3+δ3−w)L+ρ3B3,DtθCB1(t)=α1N−(ρ1+μ1)B1,DtθCB2(t)=α2A−(ρ2+μ2)B2,DtθCB3(t)=α3L+ϕB3−(ρ3+μ3)B3,(9)

with initial condition N(0)≥0, A(0)≥0, L(0)≥0, B1(0)≥0, B2(0)≥0, B3(0)≥0. Here, θ is the order of Caputo fractional derivative (CDt) such that 0<θ≤1.

2.3 Positiveness and Boundedness of Solutions

In this part, we find the conditions ensuring the poistivity of the solutions, which are crucial for representing real-world scenarios. For N(t), we have

DtθCN(t)=π−(β+η)N−α1N−δ1N+ρ1B1,≥−(β+η+α1+δ1)N.(10)

This holds for all t≥0 when considering classical derivatives.

N(t)≥N(0)e−(β+η+α1+δ1)t.(11)

Notably, when dealing with classical derivatives, these properties hold for all t≥0.

A(t)≥A(0)e−(r+r1+α2+δ2)t,L(t)≥L(0)e−(α3+δ3−w)t,B1(t)≥B1(0)e−(ρ1+μ1)t,B2(t)≥B2(0)e−(ρ2+μ2)t,B3(t)≥B3(0)e−(ρ3+μ3−ϕ)t.(12)

While the positivity of system solutions in terms of Caputo derivative is defined by

N(t)≥N(0)Λθ[−(β+η+α1+δ1)tθ],A(t)≥A(0)Λθ[−(r+r1+α2+δ2)tθ],L(t)≥L(0)Λθ[−(α3+δ3−w)tθ],B1(t)≥B1(0)Λθ[−(ρ1+μ1)tθ],B2(t)≥B2(0)Λθ[−(ρ2+μ2)tθ],B3(t)≥B3(0)Λθ[−(ρ3+μ3−ϕ)tθ].(13)

2.4 Positively Invariant Region

Lemma 1. For the remainder of the time span, assuming the nonnegative starting conditions, the solutions of the system (9) remain in the confined positive region R+6.

Proof. We obtain the following:

DtθCN(t)∣N=0=π+ρ1B1≥0,DtθCA(t)∣A=0=(β+η)N+wL+ρ2B2≥0,DtθCL(t)∣L=0=rA+r1B1+ρ3B3≥0,DtθCB1(t)∣B1=0=α1N≥0,DtθCB2(t)∣B2=0=α2A≥0,DtθCB3(t)∣B3=0=α3L≥0.(14)

Now, we sum up the all equations in (9). Then we obtain

M(t)=π−δ1N+(w−r1−δ2)A+(2w−δ3)L+(r1−μ1)B1−μ2B2+(ϕ−μ3)B3,≥π−κM.(15)

This yields

limt→∞SupM(t)≤πκ.(16)

Thus, we can conclude that

Υ=((N,A,L,B1,B2,B3)∈R+6:0≤M(t)≤πκ),(17)

is a positively invariant region.□

2.5 Equilibrium Points

By setting the system’s left side to zero, this part offers a thorough examination of equilibrium locations. Let there are no active smokers and in the absence of asthma we get the points free of disease as

E0=(N0, A0, L0, B10, B20, B30),=(π(β+η+α1+δ1),0,0,0,0,0).(18)

The asthma-persistent equilibrium is given by

E∗=(N∗,A∗,L∗,B1∗,B2∗,B3∗),(19)

where

N∗=π+ρ1B1∗β+η+α1+δ1,B1∗=α1N∗ρ1+μ1,A∗=(β+η)N∗+wL∗+ρ2B2∗r+r1+α2+δ2,B2∗=α2A∗ρ2+μ2,L∗=rA∗+r1B1∗+ρ3B3∗α3+δ3−w,B3∗=α3L∗ρ3+μ3−ϕ.(20)

2.6 Reproductive Number

In order to find the reproductive number, we consider the system:

DtθCB1(t)=α1N−(ρ1+μ1)B1,DtθCB2(t)=α2A−(ρ2+μ2)B2,DtθCB3(t)=α3L+ϕB3−(ρ3+μ3)B3.(21)

The matrices F and V−1 are calculated using the next generation matrix technique as follows:

F=[α1000α2000α3],V−1=[1(ρ1+μ1)0001(ρ2+μ2)0001(ρ3+μ3−ϕ)].(22)

Then, the reproduction number is obtained as

R0=max(α1(ρ1+μ1),α2(ρ2+μ2),α3(ρ3+μ3−ϕ)).(23)

2.7 Sensitivity of R0

In the parameters model, the reproduction number R0, which measures the disease distribution, has sensitivity indices that are found by sensitivity analysis. By calculating a parameter’s uncertainty from its distribution, a frequency distribution for the fundamental reproduction number can be produced. It is possible to investigate the sensitivity of R0 by taking into account important factors and computing partial derivatives of the reproductive number. Let

R0N=α1(ρ1+μ1),   R0A=α2(ρ2+μ2),   R0L=α3(ρ3+μ3−ϕ),(24)

then, we calculate the sensitivity of R0N, R0A, and R0L as follows:

∂R0N∂α1=1(ρ1+μ1)>0,∂R0N∂ρ1=∂R0N∂μ1=−α1(ρ1+μ1)2<0,(25)

∂R0A∂α2=1(ρ2+μ2)>0,∂R0A∂ρ2=∂R0A∂μ2=−α2(ρ2+μ2)2<0,(26)

∂R0L∂α3=1(ρ3+μ3−ϕ)>0,∂R0L∂ρ3=∂R0L∂μ3=−α3(ρ3+μ3−ϕ)2<0,∂R0L∂ϕ=α3(ρ3+μ3−ϕ)2>0.(27)

An increase in that parameter’s value will result in an increase in the related reproductive number since a positive value is directly proportional to related reproductive. Relative to related reproductive, a negative value is inversely proportional, meaning that if we raise the value of that parameter, the related reproductive number will fall. Sensitivity of these reproductive numbers with respect to these parameters is depicted in Figs. 2 and 3.

Figure 2: The sensitivity analysis of R0’s parameters

Figure 3: R0’s parameters Sensitivity Indices

3  Existence and Uniqueness Analysis

This study highlights the distinctiveness of the suggested Caputo model, which is founded on the Banach contraction principle and guarantees its existence and dependability in forecasting the dynamics of smoking-induced asthma. To enhance clarity and simplicity, we first assume the following four kernals:

DtθCN(t)=ξ1(t,N),DtθCA(t)=ξ2(t,A),DtθCL(t)=ξ3(t,L),DtθCB1(t)=ξ4(t,B1),DtθCB2(t)=ξ5(t,B2),DtθCB3(t)=ξ6(t,B3).(28)

Applying the fractional-order Caputo operator to Eq. (16), we obtain

N(t)−N(0)=φ(θ)ξ1(t,N)+ψ(θ)∫0tξ1(χ,N)dχ,A(t)−A(0)=φ(θ)ξ2(t,A)+ψ(θ)∫0tξ2(χ,A)dχ,L(t)−L(0)=φ(θ)ξ3(t,L)+ψ(θ)∫0tξ3(χ,L)dχ,B1(t)−B1(0)=φ(θ)ξ4(t,B1)+ψ(θ)∫0tξ4(χ,B1)dχ,B2(t)−B2(0)=φ(θ)ξ5(t,B2)+ψ(θ)∫0tξ5(χ,B2)dχ,B3(t)−B3(0)=φ(θ)ξ6(t,B3)+ψ(θ)∫0tξ6(χ,B3)dχ,(29)

where φ(θ) and ψ(θ) are the positive real constants. We will now establish the Lipschitz condition for the Caputo system (9).

Theorem 1. The aforementioned kernals system ξ1(t,N), ξ2(t,A), ξ3(t,L), ξ4(t,B1), ξ5(t,B2), and ξ6(t,B3) satisfy the Lipschitz condition.

Proof. First, we justify the Lipschitz condition for the kernel ξ1. Considering two function, N and N^, the corresponding norm is given by:

∥ξ1(t,N)−ξ1(t,N^)∥≤∥[π−(β+η)N−α1N−δ1N+ρ1B1]−[π−(β+η)N^−α1N^−δ1N^+ρ1B1^]∥,≤(β+η+α1+δ1)∥N−N^∥+ρ1∥B1−B1^∥.(30)

where β, η, α1, δ1, and ρ1 are bounded functions. In the same way, the norms for the other model equations can be determined.

The subsequent recursive formula for (29) is the result of the Caputo model having at least one solution.

Nm(t)=φ(θ)ξ1+ψ(θ)∫0tξ1(χ,Nm+1)dχ,Am(t)=φ(θ)ξ2+ψ(θ)∫0tξ2(χ,Am+1)dχ,Lm(t)=φ(θ)ξ3+ψ(θ)∫0tξ3(χ,Lm+1)dχ,B1m(t)=φ(θ)ξ4+ψ(θ)∫0tξ4(χ,B1(m+1))dχ,B2m(t)=φ(θ)ξ5+ψ(θ)∫0tξ5(χ,B2(m+1))dχ,B3m(t)=φ(θ)ξ6+ψ(θ)∫0tξ6(χ,B3(m+1))dχ.(31)

The positive initial conditions are the first iterative values. Then

Π1m=Nm(t)−Nm−1(t),=φ(θ)[ξ1(t,Nm−1)−ξ1(t,Nm−2)]+ψ(θ)∫0t[ξ1(χ,Nm−1)−ξ1(χ,Nm−2)]dχ,(32)

Π2m=Am(t)−Am−1(t),=φ(θ)[ξ2(t,Am−1)−ξ2(t,Am−2)]+ψ(θ)∫0t[ξ2(χ,Am−1)−ξ2(χ,Am−2)]dχ,(33)

Π3m=Lm(t)−Lm−1(t),=φ(θ)[ξ3(t,Lm−1)−ξ3(t,Lm−2)]+ψ(θ)∫0t[ξ3(χ,Lm−1)−ξ3(χ,Lm−2)]dχ,(34)

Π4m=B1(m)(t)−B1(m−1)(t),=φ(θ)[ξ4(t,B1(m−1))−ξ4(t,B1(m−2))]+ψ(θ)∫0t[ξ4(χ,B1(m−1))−ξ4(χ,B1(m−2))]dχ,(35)

Π5m=B2(m)(t)−B2(m−1)(t),=φ(θ)[ξ5(t,B2(m−1))−ξ5(t,B2(m−2))]+ψ(θ)∫0t[ξ5(χ,B2(m−1))−ξ5(χ,B2(m−2))]dχ,(36)

Π6m=B3(m)(t)−B3(m−1)(t),=φ(θ)[ξ6(t,B3(m−1))−ξ6(t,B3(m−2))]+ψ(θ)∫0t[ξ6(χ,B3(m−1))−ξ6(χ,B3(m−2))]dχ.(37)

It is important to note that

∑j=0mΠ1j=Nm(t),    ∑j=0mΠ2j=Am(t),    ∑j=0mΠ3j=Lm(t),∑j=0mΠ4j=B1(m)(t),    ∑j=0mΠ5j=B2(m)(t),    ∑j=0mΠ6j=B3(m)(t).(38)

Then, we have

Π1m=∥Nm(t)−Nm−1(t)∥,=∥φ(θ)[ξ1(t,Nm−1)−ξ1(t,Nm−2)]+ψ(θ)∫0t[ξ1(χ,Nm−1)−ξ1(χ,Nm−2)]dχ∥.(39)

By applying the triangle inequality, the above equation can be reduced

∥Nm(t)−Nm−1(t)∥≤φ(θ)∥[ξ1(t,Nm−1)−ξ1(t,Nm−2)]∥+ψ(θ)∥∫0t[ξ1(χ,Nm−1)−ξ1(χ,Nm−2)]dχ∥.(40)

Following from Eq. (29), the kernel ξ1(t,N) satisfies the Lipschitz condition. Consequently, it can be expressed as:

∥Nm(t)−Nm−1(t)∥≤φ(θ)ζ1∥Nm−1−Nm−2∥+ψ(θ)ζ1∫0t∥Nm−1−Nm−2∥dχ.(41)

We can simplify the above inequality as follows:

∥Π1m(t)∥≤φ(θ)ζ1∥Π(m−1)(t)∥+ψ(θ)ζ1∫0t∥Π(m−1)(χ)∥dχ.(42)

∥Π2m(t)∥≤φ(θ)ζ2∥Π2(m−1)(t)∥+ψ(θ)ζ2∫0t∥Π2(m−1)(χ)∥dχ.(43)

∥Π3m(t)∥≤φ(θ)ζ3∥Π3(m−1)(t)∥+ψ(θ)ζ3∫0t∥Π3(m−1)(χ)∥dχ.(44)

∥Π4m(t)∥≤φ(θ)ζ4∥Π4(m−1)(t)∥+ψ(θ)ζ4∫0t∥Π4(m−1)(χ)∥dχ.(45)

∥Π5m(t)∥≤φ(θ)ζ5∥Π5(m−1)(t)∥+ψ(θ)ζ5∫0t∥Π5(m−1)(χ)∥dχ.(46)

∥Π6m(t)∥≤φ(θ)ζ6∥Π6(m−1)(t)∥+ψ(θ)ζ6∫0t∥Π6(m−1)(χ)∥dχ.(47)

□

Theorem 2. Given the condition:

φ(θ)ζj+ψ(θ)ζjΠ1,0<1,     j=1,2,...,6.

The fractional-order smoking-induces asthma system has an analytical solution at Π0.

Proof. Following from the recursive relation, we observe that the all state variables are bounded functions, confirming that the Lipschitz criteria.

∥Π1m∥≤∥N(0)∥[φ(θ)ζj+ψ(θ)ζjt]m(48)

Although the solutions listed are legitimate, we take into consideration the following to make sure the functions listed accurately reflect the suggested model:

N(t)−N(0)=Nm(t)−ℸ1m(t).

This implies that

∥ℸ1m(t)∥≤∥φ(θ)[ξ1(t,Nm−1)−ξ1(t,Nm−2)]+ψ(θ)∫0t[ξ1(χ,Nm−1)−ξ1(χ,Nm−2)]dχ∥.

Using the Lipschitz condition,

∥ℸ1m(t)∥≤φ(θ)ζ1∥N−Nm−1∥+ψ(θ)ζ1∥N−Nm−1∥t.(49)

This yields,

∥ℸ1m(t)∥≤[φ(θ)+ψ(θ)t]m+1ζ1m+1ε.(50)

Next, at t0, we obtain,

∥ℸ1m(t)∥≤[φ(θ)+ψ(θ)t0]m+1ζ1m+1ε.(51)

When m approaches ∞, we reach at

∥ℸ1m(t)∥→0.(52)

In the same direction, we may deduce

∥ℸ2m(t)∥→0,∥ℸ3m(t)∥→0,∥ℸ4m(t)∥→0,∥ℸ5m(t)∥→0,∥ℸ6m(t)∥→0.(53)

We must examine an other solution N(t) to the suggested model in order to demonstrate the answer’s uniqueness.

N(t)−N^(t)=φ(θ)[ξ1(t,N)−ξ1(t,N^)]+ψ(θ)∫0t[ξ1(χ,N)−ξ1(χ,N^)]dχ.(54)

Utilizing the norm in Eq. (54), we obtain

∥N(t)−N^(t)∥(1−φ(θ)ζ1+ψ(θ)ζ1t)≤0.(55)

□

Theorem 3. In the following circumstance, the analytical solution for the Caputo fractional model is unique, which is:

(1−φ(θ)ζ1+ψ(θ)ζ1t)>0.(56)

Proof. Note that (56) is equivalent to (55) so that

∥N(t)−N^(t)∥(1−φ(θ)ζ1+ψ(θ)ζ1t)≤0.(57)

Thus,

∥N(t)−N^(t)∥=0.

This suggests that the solution is distinctive and is therefore

N(t)=N^(t).

Also, we have

A(t)=A^(t),     L(t)=L^(t),     B1(t)=B1^(t),     B2(t)=B2^(t),     B3(t)=B3^(t).(58)

□

4  Stability Analysis

Theorem 4. The endemic equilibrium points E∗ are globally asymptotically stable if the reproductive number mentioned in (23) is greater than 1.

Proof. Let us represent the Lyapunov function:

F(N∗,A∗,L∗,B1∗,B2∗,B3∗)≤(N−N∗−N∗lnN∗N)+(A−A∗−A∗lnA∗A)+(L−L∗−L∗lnL∗L)+(B1−B1∗−B1∗lnB1∗B1)+(B2−B2∗−B2∗lnB2∗B2)+(B3−B3∗−B3∗lnB3∗B3).(59)

Then, we have

DtθCF≤(N−N∗N)DtθCN+(A−A∗A)DtθCA+(L−L∗L)DtθCL+(B1−B1∗B1)DtθCB1+(B2−B2∗B2)DtθCB2+(B3−B3∗B3)DtθCB3.(60)

From (9), we can write as

DtθCF≤(N−N∗N)(π−(β+η)N−α1N−δ1N+ρ1B1)+(A−A∗A)((β+η)N+wL−(r+r1)A−(α2+δ2)A+ρ2B2)+(L−L∗L)(rA+r1B1−(α3+δ3−w)L+ρ3B3)+(B1−B1∗B1)(α1N−(ρ1+μ1)B1)+(B2−B2∗B2)(α2A−(ρ2+μ2)B2)+(B3−B3∗B3)(α3L+ϕB3−(ρ3+μ3)B3).(61)

Replacing N=N−N∗, A=A−A∗, L=L−L∗, B1=B1−B1∗, B2=B2−B2∗, B3=B3−B3∗, we can have the following

DtθCF≤(N−N∗N)(π−(β+η)(N−N∗)−α1(N−N∗)−δ1(N−N∗)+ρ1(B1−B1∗))+(A−A∗A)((β+η)(N−N∗)+w(L−L∗)−(r+r1)(A−A∗)−(α2+δ2)(A−A∗)+ρ2(B2−B2∗))+(L−L∗L)(r(A−A∗)+r1(B1−B1∗)−(α3+δ3−w)(L−L∗)+ρ3(B3−B3∗))+(B1−B1∗B1)(α1(N−N∗)−(ρ1+μ1)(B1−B1∗))+(B2−B2∗B2)(α2(A−A∗)−(ρ2+μ2)(B2−B2∗))+(B3−B3∗B3)(α3(L−L∗)+ϕ(B3−B3∗)−(ρ3+μ3)(B3−B3∗)).(62))