A Theoretical Investigation of the SARS-CoV-2 Model via Fractional Order Epidemiological Model

1  Introduction

Corona-viruses family causes illnesses in humans, starting with the usual cold and leading to SARS. In the previous twenty years, two corona-virus epidemics have been reported [1–3]. One of them is SARS, which caused a large epidemic scale in various countries. This epidemic suffered approximately 8000 individuals with 800 deaths. Another type of this virus was the Middle East Respiratory Syndrome Coronavirus (MERS), initially reported in Saudi Arabia and then spreads to many countries, from which 2,500 cases were reported with 800 deaths and still the cause of sporadic cases [4]. A severe outbreak of respiratory illness was reported in Wuhan, China, in December 2019 [5]. The causative agent was identified and isolated from a single patient in early January 2020, the novel coronavirus (COVID-19). The scientific evidence indicated that the first source of transmission of the virus was an animal, while most cases rose due to the contact of infected humans with susceptible humans. The spread of this virus is a burning issue, which reached almost every nook of the World and therefore been reported in more than 200 countries. According to the current statistics, there are more than 494,504,712 confirmed cases, while 6,185,114 deaths occurred till 6th April 2022. It is a public health emergency declared by the World Health Organization (WHO). According to the severeness of this disease, the World Health Organization (WHO) announced it is a public health emergency for international concern. This virus seems to be very contagious, spreading very quickly to almost all over the world and therefore declaring it is a worldwide pandemic. It means that it is a severe public health risk, whose symptoms after infection include cough, fever, fatigue, breathing difficulties, etc.

Fractional computing is a growing field of applied mathematics and has attracted the attention of several researchers [6–12]. This analysis has widely been utilized to express the axioms of heritage and re-call different physical situations that occur in various fields of applied science. Many classical models are less accurate in predicting, while models with non-integer order are better for allocating and preserving the information that is missing [13–15]. Moreover, the derivative with classical order does not give the dynamics between two various points [16,17]. It could also be noted that the comparison of classical and fractional order epidemic models reveals that epidemiological models having non-integer order are the generalization of integer order. And so provide more accurate dynamics than classical order, (see for further detail [17–19]). A non-integer order model representing the complex dynamics of a biological system has been studied by Ali et al. [20]. Another study used a fractional-order model to explore toxoplasmosis dynamics in human and feline populations [21]. The stability analysis for the spread of pests in tea has been proposed using a fractional-order epidemic model [22]. Similarly, numerous authors studied various dynamical systems with fractional derivatives, e.g., Hadamard and Caputo, Rieman and Liouville [23–27]. For the solution of Caputo fractional-order epidemiological models, many iterative and numerical methods have been developed due to singular kernel complications arising. So, Caputo et al. presented an idea based on the non-singular kernel to overcome the limitation that arises in the above fractional-order derivatives [28].

Corona-virus disease (COVID-2019) has been recognized as a global threat and therefore got the attention of various researchers due to its novel nature. Modelling the dynamics of multiple infections disease has a rich literature [29–32]. A variety of mathematical models have been formulated to study the complicated dynamics of infectious disease and suggest optimal solutions for its future forecast [33–35]. The novel corona also has rich literature in which the formulation of models and forecasting its future dynamics are investigated [36–38]. For example, Wu et al. introduced a model to describe the transmission of the disease, based on reported data from 31.12.2019 to 28.01.2020 [39]. Imai et al. [40] studied the transmission of the disease with the help of computational modeling to estimate the disease outbreak in Wuhan, whose primary focus was on human-to-human transmission. Another study has been investigated by Zhu et al. [41] to analyze the infectivity of the novel coronavirus. The reported studies indicate that bats and minks may be two animal hosts of the novel coronavirus. Similarly, many more studies have been reported on the dynamics of a novel coronavirus, for instance, see [42,43]. Li et al. [44] organized a State-of-the-Art Survey using deep learning applications for COVID-19 analysis. Nevertheless, the work proposed by various researchers found in the literature is an excellent contribution; however, it could be improved by incorporating multiple essential and exciting factors related to the newly reported disease of coronavirus transmission.

It could be noted that the coronavirus disease spreading rises globally from human-to-human transmission, while the initial source of the disease was an animal/reservoir. It has also been confirmed from the characteristic of SARS-CoV-2 that various phases of the infections are very significant and influence the transmission of the disease. The latent individuals are notable because of having no symptoms while transmitting the infection. So a small number of latent individuals leads to a significant disaster. We formulate the model keeping in view the above aesthetic of the SARS-CoV-2 virus and study the temporal dynamics of the disease. Once to develop the model, we then fractionalize because of increased development that the epidemiological models having fractional order are more significant than integer-order. Therefore, the fractionalization of the model to its associated fractional-order version will be accorded with the application of fractional calculus. We prove the existence with uniqueness and discuss the feasibility of the developed epidemic problem with the help of the fixed point theory. We will also investigate whether the proposed model is bounded and possesses positive solutions. We perform the numerical visualization of the analytical results to verify the theocratical parts. We also show the difference between integer and non-integer order epidemiological cases.

2  Model Formulation

We formulate the proposed problem by considering the characteristic of the novel coronavirus disease. We classify the total human population Nh(t) in four various compartments of susceptible, latent, infected, and recovered individuals while assuming the reservoir class symbolized with M(t). We further assume the different transmission routes, i.e., from humans and reservoirs. We assume the following assumption before presenting the model:

a. The proposed model represents the dynamical population problem, so all the variables, parameters, and constants are positive.

b. Three different transmission routes transmit the disease, i.e., from a latent population, infected population to susceptible, and from the reservoir.

c. We assume that individuals with a strong immune system will recover in the latent period.

d. It is also assumed that there are two types of recovery from infection, i.e., naturally and due to treatment.

e. The death rate due to disease is assumed to be only in the infected compartment.

Moreover the prorogation of novel corona virus disease transmission is demonstrated by Fig. 1. Hence the combination of all these above assumptions with the disease propagation depicted in the flowchart lead to the system of non-linear differential equation represented by the following system:

{dSh(t)dt=Λ−(β1Lh(t)+γβ2Ih(t)+ψβ3M(t)+d)Sh(t),dLh(t)dt=(β1Lh(t)+γβ2Ih(t)+ψβ3M(t))Sh(t)−(γ1+γ2+d)Lh(t),dIh(t)dt=γ1Lh(t)−(γ3+d+d1)Ih(t),dRh(t)dt=γ2Lh(t)+γ3Ih(t)−dRh(t),dM(t)dt=η1Lh(t)+η2Ih(t)−αM(t),(1)

and the initial population sizes are assumed to be

Sh(0)>0,Lh(0)≥0,Ih(0)≥0,Rh(0)≥0,M(0)≥0.(2)

Figure 1: The graph describes the flowchart representing the transfer mechanism of the proposed epidemic problem

In the proposed problem, Λ is the ratio of the newborn. We also denote the disease transmission rates by β1, β2 and β2, which respectively represent the transmission from latent, infected, and reservoir, while γ and ψ are the transmission co-efficient. γ1 is the rate at which latent comes to the infected, while γ2 is the natural recovery rate. The recovery due to treatment is symbolized by γ3, and d is assumed to be the natural mortality rate while d1 is the disease-induced death rate. The parameters η1 and η2 are assumed to be the ratio contributing the virus into the seafood market M from latent and infected populations respectively, while the rate α is the removing rate of the virus from it. Let us assume that ρ is a fractional order parameter, i.e., 0<ρ<1. To extend the system (1) to corresponding fractional version, first we describe a few basic concepts related to fractional calculus. These concepts are helpful to retrieve our goals.

Definition 2.1. [17] Let T>0, and assume that ϕ∈H1(0,T), if n−1<ρ<n, and ρ>0 such that n∈N, then the derivative in the sense of Caputo as well as the Caputo-Fabrizio with ρ order are give by

CFD0,tρ{φ(t)}=K(ρ)(1−ρ)∫0tφ′(z)exp⁡((z−t)ρ1−ρ)dz,(3)

and

CD0,tρ{φ(t)}=1Γ(−ρ+n)∫0t(t−z)−1+n−ρφn(z)dz.(4)

In the above Eqs. (3) and (4), CF and C denote the Caputo-Fabrizio and Caputo. Moreover, t is positive, and K(ρ) symbolizes the normilization function, and K(0)=0=K(1).

Definition 2.2. [17] If 0<ρ<1 and φ(t), varies with time t, then the integral is described by

RLJ0,tρ{φ(t)}=1Γ(ρ)∫0t(t−z)ρ−1φ(z)dz,(5)

is known as the Riemann-Liouville integral and

CFJ0,tρ{φ(t)}=2(2−ρ)K(ρ){(1−ρ)φ(t)+ρ∫0tφ(z)dz}.(6)

is said to be the Caputo-Fabrizio-Caputo (CF) integral.

Using the theory of fractional calculus to take the associated fractional version of the considered model. Since ρ is the fractional order parameter, and we assume ρ1=γ1ρ+γ2ρ+dρ and ϱ2=γ3ρ+dρ+d1ρ for the shake of simplicity, therefore the application of the above definitions to the proposed systems gives

{CFD0,tρSh(t)=Λρ−(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t)+dρ)Sh(t),CFD0,tρLh(t)=(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t))Sh(t)−ρ1Lh(t),CFD0,tρIh(t)=γ1ρLh(t)−ρ2Ih(t),CFD0,tρRh(t)=γ2ρLh(t)+γ3ρIh(t)−dρRh(t),CFD0,tρM(t)=η1ρLh(t)+η2ρIh(t)−αρM(t).(7)

To discuss the feasibility of the above epidemiological system (7) of fractional order, we discuss the existence as well as uniqueness analysis in the subsequent sections.

2.1 Existence and Uniqueness

We exploit fixed point theory to show the model's existence and uniqueness under-considered, as Equation reported (7). We transform the reported system into its associated integral equations as

{Sh(t)=Sh(0)+CFJ0,tρ{Λρ−(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t)+dρ)Sh(t)},Lh(t)=Lh(0)+CFJ0,tρ{(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t))Sh(t)−ρ1Lh(t)},Ih(t)=Ih(0)+CFJ0,tρ{γ1ρLh(t)−ρ2Ih(t)},Rh(t)=Rh(0)+CFJ0,tρ{γ2ρLh(t)+γ3ρIh(t)−dρRh(t)},M(t)=M(0)+CFJ0,tρ{η1ρLh(t)+η2ρIh(t)−αρM(t)}.

We apply the definition of CF integral, which ultimately implies that

Sh(t)=Sh(0)+2(1−ρ)K(ρ)(2−ρ){Λρ−(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t)+dρ)Sh(t)}+2ρK(ρ)(2−ρ)∫0t{Λρ−(β1ρLh(z)+γρβ2ρIh(z)+ψρβ3ρM(z)+dρ)Sh(z)}dz,Lh(t)=Lh(0)+2(1−ρ)(2−ρ)K(ρ){(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t))Sh(t)−ρ1Lh(t)}+2ρK(ρ)(2−ρ)∫0t{(β1ρLh(z)+γρβ2ρIh(z)+ψρβ3ρM(z))Sh(z)−ρ1Lh(z)}dz,Ih(t)=Ih(0)+2(1−ρ)K(ρ)(2−ρ){γ1ρLh(t)−ρ2Ih(t)}+2ρ(2−ρ)K(ρ)∫0t{γ1ρLh(z)−ϱ2Ih(z)}dz,Rh(t)=Rh(0)+2(1−ρ)(2−ρ){γ2ρLh(t)+γ3ρIh(t)−dρRh(t)}1K(ρ)+2ρK(ρ)(2−ρ)∫0t{γ2ρLh(z)+γ3ρIh(z)−dρRh(z)}dz,M(t)=M(0)+2(1−ρ)(2−ρ){η1ρLh(t)+η2ρIh(t)−αρM(t)}1K(ρ)+2ρK(ρ)(2−ρ)∫0t{η1ρLh(z)+η2ρIh(z)−αρM(z)}dz.

Let us assume that ℓi, i=1,2,…,5 describes the kernels such that

{ℓ1(Sh(t),t)=Λρ−(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t)+dρ)Sh(t),ℓ2(Lh(t),t)=(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t))Sh(t)−ϱ1Lh(t),ℓ3(Ih(t),t)=γ1ρLh(t)−ϱ2Ih(t),ℓ4(Rh(t),t)=γ2ρLh(t)+γ3ρIh(t)−dρRh(t),ℓ5(Mi(t),t)=η1ρLh(t)+η2ρIh(t)−αρM(t).(8)

Theorem 2.1. The kernels ℓi, satisfies axioms of Lipschitz conditions.

Proof 1. We assume that Sh and S1h, Lh and Lh1, Ih and Ih1, Rh and Rh1, and M and M1 are the two functions for the above kernels ℓ1, ℓ2, ℓ3, ℓ4 and ℓ5 respectively, then establishing the system is given by

{ℓ1(Sh(t),t)−ℓ1(Sh1(t),t)={Λρ−(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t)+dρ)Sh(t)},ℓ2(Lh(t),t)−ℓ2(Lh1(t),t)={(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t))Sh(t)−ϱ1Lh(t)},ℓ3(Ih(t),t)−ℓ3(Ih1(t),t)={γ1ρLh(t)−ϱ2Ih(t)},ℓ4(Rh(t),t)−ℓ4(Rh1(t),t)=γ2ρLh(t)+γ3ρIh(t)−dρRh(t),ℓ5(M(t),t)−ℓ5(M(t),t)=η1ρLh(t)+η2ρIh(t)−αρM(t),

Upon, the application of Cauchy's inequality leads to

{‖ℓ1(Sh(t),t)−ℓ1(Sh1(t),t)‖≤‖{Λρ−(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t)+dρ)Sh(t)}‖,‖ℓ2(Lh(t),t)−ℓ2(Lh1(t),t)‖≤‖{(β1ρLh(t)+γρβ2ρIh(t)+ψρβ3ρM(t))Sh(t)−ϱ1Lh(t)}‖,‖ℓ3(Ih(t),t)−ℓ3(Ih1(t),t)‖≤‖γ1ρLh(t)−ϱ2Ih(t)‖,‖ℓ4(Rh(t),t)−ℓ4(Rh1(t),t)‖≤‖γ2ρLh(t)+γ3ρIh(t)−dρRh(t)‖,‖ℓ5(M(t),t)−ℓ5(M(t),t)‖≤‖η1ρLh(t)+η2ρIh(t)−αρM(t)‖.

We then obtain recursively the following relations:

Sh(t)=2(2−ρ){(1−ρ)ℓ1(Sh(n−1)(t),t)+ρ∫0tℓ1(Sh(n−1)(z),z)dz}1K(ρ),Lh(t)=2(2−ρ){(1−ρ)ℓ2(Lh(n−1)(t),t)+ρ∫0tℓ2(Lh(n−1)(z),z)dz}1K(ρ),Ih(t)=2(2−ρ){(1−ρ)ℓ3(Ih(n−1)(t),t)+ρ∫0tℓ3(Ih(n−1)(z),z)dz}1K(ρ),Rh(t)=2(2−ρ)K(ρ){(1−ρ)ℓ5(Rn−1(t),t)+ρ∫0tℓ5(Rh(n−1)(z),z)dz},M(t)=2(2−ρ)K(ρ){(1−ρ)ℓ6(Mn−1(t),t)+ρ∫0tℓ6(Mh(n−1)(z),z)dz}.(9)

The difference of two successive terms with the application of norm and majorizing, one may obtain

‖Un(t)‖=‖Shn(t)−Sh1,h(n−1)(t)‖≤2(1−ρ)K(ρ)(2−ρ)‖ℓ1(Sh(n−1)(t),t)−ℓ1(Sh1,h(n−2)(t),t)‖+2ρK(ρ)(2−ρ)||∫0t[ℓ1(Sh(n−1)(z),z)−ℓ1(Sh1,h(n−2)(z),z)]dz||,‖Wn(t)‖=‖Lh(t)−Lh1,h(n−1)(t)‖≤2(1−ρ)(2−ρ)K(ρ)‖ℓ2(Lh(n−1)(t),t)−ℓ2(Lh1,h(n−2)(t),t)‖+2ρ(2−ρ)K(ρ)||∫0t[ℓ2(Lh(n−1)(z),z)−ℓ2(Lh1,a(n−2)(z),z)]dz||,‖Xn(t)‖=‖Ihn(t)−Ih1,h(n−1)(t)‖≤2(1−ρ)K(ρ)(3−ρ)‖ℓ3(Ih(n−1)(t),t)−ℓ3(Ih1,h(n−2)(t),t)‖+2ρ(2−ρ)||∫0t[ℓ3(Ih(n−1)(z),z)−ℓ3(Ih1,h(n−2)(z),z)]dz||1K(ρ),‖Yn(t)‖=‖Rhn(t)−Rh1,h(n−1)(t)‖≤2(1−ρ)K(ρ)(2−ρ)‖ℓ4(Rh(n−1)(t),t)−ℓ4(Rh1,h(n−2)(t),t)‖+2ρ(2−ρ)K(ρ)||∫0t[ℓ4(Rh(n−1)(z),z)−ℓ4(Rh1,h(n−2)(z),z)]dz||,‖Zn(t)‖=‖Mn(t)−M1,h(n−1)(t)‖≤2(1−ρ)K(ρ)(2−ρ)‖ℓ5(M(n−1)(t),t)−ℓ5(M1,h(n−2)(t),t)‖+2ρ(2−ρ)K(ρ)||∫0t[ℓ4(M(n−1)(z),z)−ℓ5(M1,h(n−2)(z),z)]dz||,(10)

with

{∑i=0∞Ui(t)=Shi(t),∑i=0∞Wi(t)=Lhi(t),∑i=0∞Xi(t)=Ihi(t),∑i=0∞Yi(t)=Rhi(t),∑i=0∞Zi(t)=Mi(t)(11)

It could be noted that the kernels ℓi satisfy Lipschitz conditions, therefore we may obtain

‖Un(t)‖=‖Shn(t)−Sh1,h(n−1)(t)‖≤2(1−ρ)K(ρ)(2−ρ)τ1‖Sh(n−1)(t)−Sh1,h(n−2)(t)‖+2ρ(2−ρ)K(ρ)τ2∫0t||Sh(n−1)(z)−Sh1,h(n−2)(z)||dz,‖Wn(t)‖=‖Lhn(t)−Lh1,h(n−1)(t)‖≤2(1−ρ)(2−ρ)K(ρ)τ5‖Lh(n−1)(t)−Lh1,h(n−2)(t)‖+2ρK(ρ)(2−ρ)τ6∫0t||Lh(n−1)(z)−Lh1,h(n−2)(z)||dz,‖Xn(t)‖=‖Ihn(t)−Ih1,h(n−1)(t)‖≤2(1−ρ)(2−ρ)K(ρ)τ5‖Ih(n−1)(t)−Ih1,h(n−2)(t)‖+2ρK(ρ)(2−ρ)τ6∫0t||Ih(n−1)(z)−Ih1,h(n−2)(z)||dz,‖Yn(t)‖=‖Rhn(t)−Rh1,h(n−1)(t)‖≤2(1−ρ)K(ρ)(2−ρ)τ9‖Rh(n−1)(t)−Rh1,h(n−2)(t)‖+2ρ(2−ρ)K(ρ)τ10∫0t||Rh(n−1)(z)−Rh1,h(n−2)(z)||dz,‖Zn(t)‖=‖Mn(t)−M1,(n−1)(t)‖≤2(1−ρ)K(ρ)(2−ρ)τ9‖Mn−1(t)−M1,(n−2)(t)‖+2ρ(2−ρ)K(ρ)τ10∫0t||Mn−1(z)−M1,(n−2)(z)||dz.(12)

Theorem 2.2. The epidemiological model of fractional order (7) possesses a solution.

Proof 2. From the assertions derived in Eq. (11) with the utilization of recursive formulas we obtain

‖Un(t)‖≤‖Sh(0)‖+{2(2−ρ)K(ρ)}n{(tρτ2)n+((1−ρ)τ1)n},‖Wn(t)‖≤‖Lh(0)‖+{2(2−ρ)K(ρ)}n{(τ3(1−ρ))n}+{(tρτ4)n},‖Xn(t)‖≤‖Ih(0)‖+{2(2−ρ)K(ρ)}n{(tρτ6)n}+{((1−ρ)τ5)n},‖Yn(t)‖≤‖Rh(0)‖+{2(2−ρ)K(ρ)}n{((1−ρ)τ7)n}+{(ρτ8t)n},‖Zn(t)‖≤‖M(0)‖+{2(2−ρ)K(ρ)}n{((1−ρ)τ9)n}+{(ρτ10t)n}.(13)

So, the relations as described by the above equation are smooth and exists, however to investigate that the functions in these relations are the solutions for system (7), we making the substitutions

{Sh(t)=Shn(t)−Υ1,n(t),Lh(t)=Lhn(t)−Υ2,n(t),Ih(t)=Ihn(t)−Υ3,n(t),Rh(t)=Rhn(t)−Υ5,n(t),M(t)=Mn(t)−Υ6,n(t),(14)

where Υi,n(t), i=1,2,…,5 denote remainder terms of the series, thus

Sh(t)−Sh(n−1)(t)=2(1−ρ)ℓ1(Sh(t)−Υ1,n(t))(2−ρ)K(ρ)+2ρK(ρ)(2−ρ)∫0tℓ1(Sh(z)−Υ1,n(z))dz,Lh(t)−Lh(n−1)(t)=2ℓ2(Lh(t)−Υ2,n(t))(1−ρ)(2−ρ)K(ρ)+2ρK(ρ)(2−ρ)∫0tℓ2(Lh(z)−Υ2,n(z))dz,Ih(t)−Ih(n−1)(t)=2(1−ρ)ℓ3(Ih(t)−Υ3,n(t))(2−ρ)K(ρ)+2ρK(ρ)(2−ρ)∫0tℓ3(Ih(z)−Υ3,n(z))dz,Rh(t)−Rh(n−1)(t)=(1−ρ)2ℓ4(Rh(t)−Υ4,n(t))K(ρ)(2−α)+2α(2−ρ)K(ρ)∫0tℓ4(Rh(z)−Υ4,n(z))dz,M(t)−M(n−1)(t)=(1−ρ)2ℓ5(M(t)−Υ5,n(t))K(ρ)(2−ρ)+2ρ(2−ρ)K(ρ)∫0tℓ5(M(z)−Υ5,n(z))dz.(15)

The application of norm on both sides of the above system with utilization of the Lipschitz axiom gives that

||Sh(t)−2ρK(ρ)(2−ρ)∫0tℓ1(Sh(z),z)dz−2(1−ρ)ℓ1(Sh(t),t)(2−ρ)K(ρ)−Sh(0)||≤‖Υ1,n(t)‖{2ρτ2t(2−ρ)K(ρ)+(τ12(1−ρ)K(ρ)(2−ρ))+1},||Lh(t)−2ρ(2−ρ)K(ρ)∫0tℓ2(Lh(z),z)dz−2ℓ2(Lh(t),t)(1−ρ)K(ρ)(2−ρ)−Lh(0)||≤‖Υ2,n(t)‖{(2τ3(1−ρ)(2−ρ)K(ρ)+2tτ4ρk(ρ)(2−ρ))+1},||Ih(t)−2ℓ3(Ih(t),t)(1−ρ)(2−ρ)K(ρ)−Ih(0)−2ρ(2−ρ)K(ρ)∫0tℓ3(Ih(z),z)dz||≤‖Υ3,n(t)‖{(2tρτ6(2−ρ)K(ρ)+(1−ρ)2τ5(2−ρ)K(ρ))+1},||Rh(t)−2ℓ4(Rh(t),t)(1−ρ)(2−ρ)K(ρ)−Rh(0)−2ρK(ρ)(2−ρ)∫0tℓ4(Rh(z),z)dz||≤‖Υ4,n(t)‖{(2τ7(1−ρ)(2−ρ)K(ρ)+2tτ8ρ(2−ρ)K(ρ))+1},||M(t)−ℓ5(M(t),t)2(1−ρ)(2−ρ)K(ρ)−M(0)−2ρK(ρ)(2−ρ)∫0tℓ5(M(z),z)dz||≤‖Υ5,n(t)‖{(2ρtτ12K(ρ)(2−ρ)+(1−ρ)2τ11(2−ρ)K(ρ))+1}.(16)