View of A STUDY TO KNOW THE IMPACT OF VACCINATION TO CONTROL COVID- 19 BY USING SVEI EPIDEMIC MODEL WITH MUTATE NON-MONOTONIC INCIDENCE RATE

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VOLUME: 08, Special Issue 08, (IC-RAAPAMAA-2021) Paper id-IJIERM-VIII-VIII, October 2021 25

A STUDY TO KNOW THE IMPACT OF VACCINATION TO CONTROL COVID- 19 BY USING SVEI EPIDEMIC MODEL WITH MUTATE NON-MONOTONIC INCIDENCE RATE

S. Vaidya

School of Studies in Mathematics Vikram University, Ujjain, 456010, M.P., India S. K. Tiwari

School of Studies in Mathematics Vikram University, Ujjain, 456010, M.P., India V. K. Gupta

Department of Mathematics, Govt. Madhav Science P.G. College, Ujjain, 456010, M.P., India Abstract: An approach of deterministic model is used to present the role of vaccination in controlling the pandemic infectious disease COVID-19. Four mutually exclusive compartments Susceptible, Vaccination, Exposed and Infective are used to show the horizontal transmission and its control with mutated non-monotonic incidence rate. By mathematical analysis it is observed that the presented model demonstrate no disease and endemic situation. Using epidemiological threshold parameter dynamic analysis of model is evaluated. Impact of vaccination in different situation is compared using epidemiological threshold parameter.

Different biological parametric values are considered to validate theoretical observation.

Keywords: Mathematical Model SVEI, Equilibrium points, Basic reproduction number, Lyapunov function, Routh-Herwitz criterion, Dulac‟s criterion, Stability.

1. INTRODUCTION

Today, the world is struggling to deal with COVID 19 virus, which has caused one of the worst pandemic in the modern times. Initial findings point the origin of COVID in China which soon spread over the world. It is an infectious & communicable disease, mostly spreads through respiratory droplets coming by cough, sneeze of an infected person. To curb and stop the disease, several preventive measures are taken as shutting of all public transports, lockdowns in cities, closure of schools & colleges, quarantine infected individuals and many more to reduce the incidence rate. To understand different real life situations with the pandemic, many mathematical models are used. These studies are going for last few years and vaccination is coming out as the most effective solution. It has been presented in past as well SEIV epidemic model [8] with a nonlinear incidence and a waning preventive vaccination has formulated and prove that there is always a backward bifurcation for increasing the rate at which infected individuals are treated. To include the impact of vaccination, in this paper, we will be analyzing the SVEI model where S (susceptible), V (vaccinated), E (exposed) & I (Infected) compartments include the whole population. Transmission of disease (λ is the transmission rate) is assumed to be based on muted non monotonic incidence rate λSI/(1+αsI+ αpI²)[3] , where αs and αp are social & psychological parameters respectively. These parameters add a totally new dimension of human effect with terms of social & psychological pressures like depression, due to loneliness because of long term social distancing, spending long hours at home, doubts &

confusion over vaccines etc. It has been observed that in many cases, people are getting infected after taking the first dose of vaccine and in few cases even after taking the second dose, creating the suspicion and the feeling of hopelessness in the human psyche.

The objective of the paper is to establish the model and find the equilibrium points of the presented model, to establish their existence with required conditions to justify the ultimate goal, which is to know the impact of vaccination to prevent COVID 19 virus, by comparing it with the help of different reproduction numbers produced in different real life situations.

2. FORMULATED MATHEMATICAL MODEL WITH ITS EXISTENCE AND BOUNDEDNESS

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VOLUME: 08, Special Issue 08, (IC-RAAPAMAA-2021) Paper id-IJIERM-VIII-VIII, October 2021 26 Fig.1 Transmission of diseases

Table 1 Parameters with presentation and description:

Symbols Description

b Recruitment rate of susceptible d Natural death rate

v Rate of susceptible getting vaccinated

l Rate of loosing immunity by vaccinated people over the time λ Transmission rate

am Immunity acquired by the body post infection by developing antibodies

ϒ Infection duration

σ Rate of getting infected after being exposed ξ Natural immunity rate

𝑺^′ 𝒕 = 𝒃 − 𝒅 + 𝒗 𝑺 + 𝒍𝑽 − 𝝀𝑺𝑰

𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐+ 𝟏 − 𝒂𝒎 𝜸𝑰 𝑽^′ 𝒕 = 𝒗𝑺 − 𝒍𝑽 − 𝒅𝑽 + 𝝃𝑬 + 𝒂_𝒎𝜸𝑰

𝑬^′ 𝒕 = ^𝝀𝑺𝑰

𝟏+𝜶_𝒔𝑰+𝜶_𝑷𝑰^𝟐− 𝒅𝑬 − 𝝃𝑬 − 𝝈𝑬 𝑰^′ 𝒕 = 𝝈𝑬 − 𝒅𝑰 − 𝜸𝑰

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As we are proposing this model to study human population so all the parameters are going to be non-negative which indicates S (t = 0), V (t = 0), E (t = 0), I (t = 0) all are strictly positive where their summation at any time indicates the whole population .The fractional gained immunity am, is also positive but negligible in size and sometimes almost zero i.e. 0

≤ am ≤ 1.

After adding all the four equations of (I) we get, 𝑁^′ 𝑡 = 𝑏 − 𝑑𝑁 which exhibits N (t) is moving towards b/d as the time becomes infinitely large by standard comparison theorem [5]

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So, this presents the set,ℵ = (S, V, E, I) ∈ R4+ : 0 ≤ S, V, E, I, S + V + E + I < 𝑏/d} with positively invariation by nature and if N (t = 0) > b/d, then two possibilities either the solution values will enter in the region ℵ at infinite time or the whole population converges to a constant b/d asymptotically. Finally the region ℵ will have all its solution values within it. As ℵ is positively invariant region is sufficient condition for considering existences, continuity and uniqueness of the dynamical results obtained by solving the presented model[2]This exhibits the following as a result

Lemma 2.1: The setℵ = (𝑆, 𝑉, 𝐸, 𝐼) ∈ 𝑅₊⁴: 0 ≤ 𝑆, 𝑉, 𝐸, 𝐼, 𝑆 + 𝑉 + 𝐸 + 𝐼 < 𝑏/𝑑 is positively invariant region for the proposed model and all solution of the model which start in ℵ remain in ℵ for all t

≥ 0.

3. QUALITATIVE STUDY:

The set ℵ , in terms of epidemiologically and mathematically is positively invariant indicates two type of equilibriums. Which are for the set ℵ and within it.

Disease free Equilibrium Point: Whenever the population is assumed to be free from the disease due to absence of the virus is popularly known as disease free equilibrium (𝜀0), which is possible by substituting zeros to all variables except the susceptible. Therefore by proposed model we have 𝜀0= 𝑆0, 𝑉0, 𝐸0, 𝐼0 = {_{𝑑+𝑙+𝑣 𝑑}^{𝑑+𝑙 𝑏} ,_{𝑑+𝑙+𝑣 𝑑}^𝑣𝑏 ,0, 0}

Basic Reproduction number: Well known with (𝑅₀), is most significant threshold parameter which supports in characterizing and analyzing the spread of of infection in class of susceptible mathematically. Its value indicates the expected mean value of new infective which are produced by an infective, when it mingled the population of susceptible. Next generation method is one of the most suitable procedures to calculate it mathematically [1].

Let X = [E, I]^T, then we receive dX/dt = F(X) –V(X) with F(X) = [ ^𝝀𝑺𝑰

𝟏+𝜶_𝒔𝑰+𝜶_𝑷𝑰^𝟐, 𝟎]^𝐓 and

V(X) = 𝑑 + 𝜉 + 𝜎 𝐸, −𝜎𝐸 + 𝑑 + 𝛾 𝐼 ^𝑇so by Jacobian matrix of F(X) and V(X) at the virus free position i.e. at no infection situation, (𝜀0), so the next generation matrix for the presented model will be 𝑓𝜗⁻¹ =

𝜆𝜎 𝑆₀ 𝑑+𝜉+𝜎 (𝑑+𝛾)

𝜆𝑆₀ (𝑑+𝛾)

0 0 as the spectral radius of the next generation matrix is the reproduction number 𝑅₀= 𝑑+𝜉+𝜎 (𝑑+𝛾)^{𝜆𝜎 𝑆}⁰ =𝑑 𝑑+𝜉+𝜎 𝑑+𝛾 (𝑑+𝑙+𝑣)^{𝑏𝜎𝜆 (𝑑+𝑙)}

Endemic Equilibrium point: Whenever the disease stays as and spread with in human beings as in present situation of COVID -19 we find an interior equilibrium point known as endemic equilibrium (𝜀_𝑛). This can be calculated by equating equation of presented model to zero and we get (𝜀𝑛) = (𝑆𝑛, 𝑉𝑛, 𝐸𝑛, 𝐼𝑛) here,

𝐸_𝑛 = 𝑑 + 𝛾 𝐼_𝑛 𝜎 𝑆_𝑛= ^(𝟏+𝜶^𝒔^𝑰+𝜶_𝑅 ^𝑷^𝑰^𝟐^)𝑆⁰

0

𝑉𝑛= 𝑣𝑆₀

𝑅0(𝑙 + 𝑑)+ 𝑣𝑆₀𝜶_𝒔

𝑅0 +𝜉(𝑑 + 𝛾

𝜎 + 𝒂𝒎𝜸 𝐼_𝑛

(𝑙 + 𝑑)+ 𝑣𝑆0𝜶𝒑𝑰𝒏𝟐

𝑅0(𝑙 + 𝑑)

𝐼𝑛 = {−𝐴2+ ∆}/2𝐴1

𝑤ℎ𝑒𝑟𝑒 , 𝐴1= (𝑑 + 𝑙 + 𝑣)𝑆₀𝜶_𝒑 𝑅₀

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VOLUME: 08, Special Issue 08, (IC-RAAPAMAA-2021) Paper id-IJIERM-VIII-VIII, October 2021 28 𝐴₂= { 𝑑 + 𝑙 + 𝑣 𝑆₀𝜶𝒔

𝑅0 + 𝑑 + 𝛾 𝑙 + 𝑑 + 𝜉 + 𝜎

𝜎 + 𝑙 − 𝟏 − 𝒂_𝒎 𝜸 𝑎𝑛𝑑 𝐴₃= 𝑑 + 𝑙 + 𝑣 𝑆₀

𝑅0 (1 − 𝑅₀) 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 ,∆ = 𝐴₂²− 4𝐴₁𝐴₃ , ∆ > 0 𝑤ℎ𝑒𝑛 𝑅₀> 1

Local Stability: To show the local stability of system (I) at disease free equilibrium point. Let us consider

𝐹₁=𝑏

𝑑 𝑑 + 𝑙 − 𝑆 𝑑 + 𝑣 + 𝑙 − 𝝀𝑺𝑰

𝟏 + 𝜶_𝒔𝑰 + 𝜶_𝑷𝑰^𝟐− 𝒍𝑬 − 𝒍 − 𝟏 − 𝒂_𝒎 𝜸 𝑰 𝐹2= 𝝀𝑺𝑰

𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐− 𝒅 + 𝝃 + 𝝈 𝑬

𝐹₃= 𝜎𝐸 − 𝑑 + 𝛾 𝐼 (II)

With initial conditions S (t = 0), V (t = 0), E (t = 0), I (t = 0) all are strictly positive. The local stability for both equilibrium points is as follows:

The Variation matrix of 𝜀0 is presented as follows:

J[𝜀0]=

−(𝑑 + 𝑙 + 𝑣) −𝑙 −𝜆𝑆0− [𝑙 − 1 − 𝑎𝑚 𝛾]

0 −(𝑑 + 𝜉 + 𝜎) 𝜆𝑆0

0 𝜎 −(𝑑 + 𝛾)

Its characteristic equation is 𝐉 𝜀₀ − 𝛿 = 0

= [λ+ (d+ v+ l)] [𝜹^𝟐+ 𝟐𝒅 + 𝝃 + 𝝈 + 𝜸 𝜹 + 𝒅 + 𝝃 + 𝝈 𝒅 + 𝜸 (𝟏 − 𝑅0)] = 0

Here if 𝑅₀< 1 , then all the three Eigen roots of the characteristic equation will have negative real parts and if 𝑅0> 1 , then two of its Eigen values are with negative real parts and one with positive real part. This exhibits the following theorem.

Theorem 3.4.1: The Disease free equilibrium is locally asymptotically stable if 𝑅0< 1 and unstable if 𝑅0> 1.

To know the stability behavior at local level at 𝜀_𝑛 again we will generate variation matrix for it

J[𝜀_𝑛]=

− 𝑑 + 𝑙 + 𝑣 −_𝟏+𝜶_𝒔^𝝀𝑰𝒏_𝑰+𝜶_𝑷_𝑰𝟐 −𝑙 −_(𝟏+𝜶^{𝝀𝑺(𝟏−𝜶}^𝑷^𝑰^𝟐

𝒔𝑰+𝜶_𝑷𝑰^𝟐)^𝟐− [𝑙 − 1 − 𝑎𝑚 𝛾]

𝝀𝑰𝒏

𝟏+𝜶_𝒔𝑰+𝜶_𝑷𝑰^𝟐 −(𝑑 + 𝜉 + 𝜎) _(𝟏+𝜶^{𝝀𝑺(𝟏−𝜶}^𝑷^𝑰^𝟐

𝒔𝑰+𝜶_𝑷𝑰^𝟐)^𝟐

0 𝜎 −(𝑑 + 𝛾)

and for characteristic

equation

− 𝑑 + 𝑙 + 𝑣 −_𝟏+𝜶^𝝀𝑰𝒏

𝒔𝑰+𝜶_𝑷𝑰^𝟐− 𝝎 −𝑙 −_(𝟏+𝜶^{𝝀𝑺(𝟏−𝜶}^𝑷^𝑰^𝟐

𝒔𝑰+𝜶_𝑷𝑰^𝟐)^𝟐− [𝑙 − 1 − 𝑎𝑚 𝛾]

𝝀𝑰𝒏

𝟏+𝜶_𝒔𝑰+𝜶_𝑷𝑰^𝟐 − 𝑑 + 𝜉 + 𝜎 − 𝜔 _(𝟏+𝜶^{𝝀𝑺(𝟏−𝜶}^𝑷^𝑰^𝟐

𝒔𝑰+𝜶_𝑷𝑰^𝟐)^𝟐

0 𝜎 − 𝑑 + 𝛾 − 𝜔

= 0

=> P (𝝎)= 𝝎^𝟑+ 𝑩𝟏𝝎^𝟐+ 𝑩𝟐𝝎 + 𝑩𝟑 = 0, where

𝑩𝟏= 𝟑𝒅 + 𝒗 + 𝒍 + 𝝃 + 𝝈 + 𝜸 + 𝝀𝑰𝒏 𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐 𝑩𝟐=𝜶_𝒔 𝒅 + 𝝃 + 𝝈 (𝒅 + 𝜸)𝐼_𝑛

𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐 + 𝟐𝒅 + 𝝃 + 𝝈 +ϒ { 𝒅 + 𝒗 + 𝒍 + 𝝀𝐼_𝑛

𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐} + 𝒍𝝀𝐼_𝑛 𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐

𝑩𝟑= 𝐼𝑛

𝟏 + 𝜶𝒔𝑰 + 𝜶𝑷𝑰^𝟐 𝜶_𝒔 𝒅 + 𝒗 + 𝒍 𝒅 + 𝝃 + 𝝈 𝒅 +ϒ + 𝝀 𝒅 +ϒ 𝒅 + 𝝃 + 𝒍 + 𝝀𝒅𝝈 + 𝝀𝝈(𝒍 + 𝑎𝑚𝛾)]

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As all the three coefficients of characteristic equation at endemic equilibrium is strictly positive so we get 𝐵1𝐵2− 𝐵3> 0 so by Routh - Hurwitz criterion the 𝜀𝑛 is locally asymptotically stable which exhibits the following theorem:

Theorem 3.4.2: If 𝑅0> 1 , then the 𝜀𝑛 is locally asymptotically stable.

Global Stability

To analyze the global stability of the equilibrium point. Firstly we will work for 𝜀0Of the set (II)

By considering the Lyapunov Function [7]

L = σ E+ 𝒅 + 𝝃 + 𝝈 𝑰

 L‟ = σ E‟+ 𝒅 + 𝝃 + 𝝈 𝑰′

 L‟ = σ { ^𝝀𝑺𝑰

𝟏+𝜶_𝒔𝑰+𝜶_𝑷𝑰^𝟐− 𝒅𝑬 − 𝝃𝑬 − 𝝈𝑬 }+ 𝒅 + 𝝃 + 𝝈 {𝝈𝑬 − 𝒅𝑰 − 𝜸𝑰}

 L‟= 𝒅 + 𝝃 + 𝝈 𝒅 + 𝜸 𝑰 _𝟏+𝜶^𝑅⁰

𝒔𝑰+𝜶_𝑷𝑰^𝟐− 𝟏

L‟=0 if and only if I =0 and L‟ < 0 if R0 < 1 for all different values of incidence rate. Hence 𝜀0 is the largest invariant set in {(S, V, E, I) L‟=0}.So by Lyapunov – Lasalle invariance principle [4]

The disease free equilibrium E⁰ is globally asymptotically stable. This justifies the following theorem.

Theorem 3.5.1: If R0 < 1, then 𝜀0 is globally asymptotically stable.

Now, for global stability at endemic equilibrium we use Dulac plus Poincare Bendixson theorem [4] as,

𝐻₁ 𝑆, 𝑉, 𝐸, 𝐼 =¹_𝑆. 𝑉. 𝐸. 𝐼Where 𝑆 > 0, 𝐸 > 0, 𝐼 > 0, 𝑉 > 0 𝜑1= 𝒃 − 𝒅 + 𝒗 𝑺 + 𝒍𝑽 − 𝝀𝑺𝑰

𝟏 + 𝜶_𝒔𝑰 + 𝜶_𝑷𝑰^𝟐+ 𝟏 − 𝒂𝒎 𝜸𝑰 𝜑₂= 𝒗𝑺 − 𝒍𝑽 − 𝒅𝑽 + 𝝃𝑬 + 𝒂_𝒎𝜸𝑰

𝜑3=_𝟏+𝜶^𝝀𝑺𝑰

𝒔𝑰+𝜶_𝑷𝑰^𝟐− 𝒅𝑬 − 𝝃𝑬 − 𝝈𝑬 𝜑₄= 𝝈𝑬 − 𝒅𝑰 − 𝜸𝑰 𝛻 𝐻1. 𝐹 = 1

𝑆. 𝑉. 𝐸. 𝐼

𝜕 𝐻. 𝜑1

𝜕𝑆 +𝜕 𝐻. 𝜑2

𝜕𝑉 +𝜕 𝐻. 𝜑3

𝜕𝐸 +𝜕 𝐻. 𝜑4

𝜕𝑅 = − (𝑑 + 𝑣)

𝐸𝐼 +(𝑙 + 𝑑) 𝑆𝐸𝐼

(𝑑 + 𝜉 + 𝜎)

𝑆𝐼 +(𝑑 + 𝛾) SE < 0

Hence by Dulac‟ Criterion, there is no closed orbit in the first quadrant. Therefore, the endemic equilibrium is globally asymptotically stable.

4. FINDINGS AND NUMERICAL SIMULATION

The main concern of the paper is to study the impact of vaccination in controlling the effect of COVID-19 in different situations for this calculated R0 of the presented model is 𝑅0=

𝑏𝜎𝜆 (𝑑+𝑙)

𝑑 𝑑+𝜉+𝜎 𝑑+𝛾 (𝑑+𝑙+𝑣)used as so by taking „𝑣‟ as variable parameter and after working along it we get is 𝑅𝑣=𝑑 𝑑+𝜉+𝜎 𝑑+𝛾 (𝑑+𝑙+𝑣)^{−𝑏𝜎𝜆 (𝑑+𝑙)} ² Which indicates the intake of vaccination is helping in reducing

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the effect of diseases definitely many parameters are involved around it which creates different situations. This can be discussed as follows:

4.1 When there is no Vaccination: to execute this situation as per presented parameters we consider 𝑣 = 0. This converts the model to SEI model with 𝑅_𝑣0=𝑑 𝑑+𝜉+𝜎 𝑑+𝛾 ^𝑏𝜎𝜆

If we compare these two reproduction numbers we get 𝑅0< 𝑅𝑣0 i.e. the average number of new infective produced by an infective after interacting with susceptible class is more when there is no vaccinations are placed among population.

4.2 Whenever the vaccination is optimal and long-lasting recovery: to represent this situation if 𝑙 = 𝛾 = 0 will modifies the values of 𝑅0 𝑡𝑜 𝑅0 this gives much lesser value. So we get 𝑅𝑙0< 𝑅0< 𝑅𝑣0 it means 𝑅𝑙0 is to present the perfefect vaacination and permanent recovery of individuals.

4.3:Complete vaccination : if we desire to have complete vaccination among population , mathematically can be presented by𝑣 = 1, so the 𝑅₀ in this way will be 𝑅_𝑣1=𝑑 𝑑+𝜉+𝜎 𝑑+𝛾 𝑑+𝑙+1 ^{𝑏𝜎𝜆 𝑑+𝑙}

and by comparing , we get 𝑅0> 𝑅𝑣1 and if the disease free equilibrium point is considered which is provide as asymptotically stable with 𝑅₀< 1 so by combining we get 𝑅_𝑣1< 𝑅₀< 1.

4.4: Perfect Vaccination with sound proof vaccination strategy: assuming 𝑣 = 1, 𝛾 = 𝑙 = 𝜎 = 0with sufficient‟ 𝜉‟with permanent level of recruitment. This specific situation leads for the best eradication of the system .after substituting all parametric level we get, 𝑅𝑝 =𝑑 𝑑+𝜉 𝑑+1 ^𝑏𝜎𝜆 which is supports the assumption.

These different situations can clearly be visualized by graph below where different parametric values are kept which are taken from published papers and previous data available and by simulating this using MATLAB we receive it which supports to our results. The obtained graph express clearly that the reproduction number quantitatively is maximum when there is no vaccination and the best results are obtained at the position of absolute vaccination which means when everyone is vaccinated and earning permanent immunity against the infection.

Definitely it will reduce the impact of disease with time. Just cent percent vaccination to the crowd is not enough boosting at psychological, sociology, medical level is equally required to maintain the immunity among individuals.

Graph 1: Impact of vaccination to control the diseases.

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VOLUME: 08, Special Issue 08, (IC-RAAPAMAA-2021) Paper id-IJIERM-VIII-VIII, October 2021 31 5. CONCLUSION

We presented the SVEI model with a new dimension of social and psychological parameters by the muted non monotonic incidence rate. The mathematical analysis of the model shows that in case of no vaccination, the model will behave as a SEI model where the reproduction number of the model will always be greater than R0.If we go with perfect vaccination, which means the ultimate effectiveness of vaccine, though not all population has received the vaccine yet, we see the positive results start showing and the reproduction number of the model is observed as lesser than R0 . If we see that everyone is vaccinated, then ideally the reproduction number of the model should be equal to R0, but we see some shortcomings due to social and psychological parameters which effects people with a negative impact on their thought process & confidence which indirectly affects their immunity to fight the pandemic. So, to decrease the incidence rate, we should work on improving and increasing the social and psychological strength of people which will help in achieving the perfect vaccination with optimized values of other parameters that will result in limiting the value of R0 to less than 1. This situation will finally eradicate the pandemic.

REFERENCES

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