The most current and prevalent strain of SARS-CoV-2, the XEC Covid-19 variation, and multiple-dose vaccination approaches are examined in this section along with the coronavirus transmission dynamics model. To more precisely ascertain the disease dynamics scenario, the total population, N, is separated into seven compartments that do not overlap. To strengthen the efficiency of the coronavirus vaccine and health vaccination against the disease, a booster dosage of the vaccine is required.29 Individuals who are susceptible (S), vaccinated with the first, second, or booster doses (V1, V2, Vb), exposed (E), infected (I), and recovered (R) are the dynamic variables that are taken into consideration. These variables are mutually exclusive. The equation N = S + V1 + V2 + Vb + E + I + R represents the population's total number (N), which is thought to be constant and evenly distributed.

We take into account appropriate transmission rate parameters related to each compartment when designing the coronavirus's dynamic process. Each compartment includes the natural death rate, which is constant at μ per capita, and the rate of entry into the population (Π). In contrast, the infected compartment includes the mortality rate from the severe health crisis, which was θ. Individuals who get the first dose of the vaccination against the XEC Covid-19 variant at a constant rate ξ are moved to the V1 compartment. Additionally, individuals who receive the first dose of the vaccination have a reduced probability of contracting the infection again at the rate ψ. Those who received the first dosage of the vaccination receive the second dose at a steady pace τ.

A number of people shift to the R compartment at a rate of φ1 after receiving a second dose of the XEC Covid-19 vaccine. After receiving the booster dose at a steady pace φ2, the second-dose recipients proceed to the R compartment at a rate υ. The most sensitive metric for virus spread is the transmission rate β, which is used to infect people. A number of people transfer to the infected compartment and carry the virus at the rate ε from the exposed individuals. Some patients reach the recovery stage and recover from the disease at a rate of γ after getting the right medical care. A system of ordinary differential equations can be used to track the dynamics of disease transmission. A detailed description of the system of differential equations for the dynamic variables under consideration is given:

People transition to the R compartment through a process that occurs at a rate of φ1 after their completion of the XEC COVID-19 vaccine series. The individuals move from R to the R compartment through the rate φ2, when receiving another booster dose at a steady pace and transition to the R compartment through rate υ. The transmission rate β, contains the highest significance for spreading disease as it controls the rate at which new infections occur. The passage from exposure to the infected compartment happens through virus carrying at the rate υ. Medical care patients experience recovery at the rate γ Ordinary differential equations create models to describe disease transmission by representing the main variables and their behavioral trends over time, below are variables and parameters explanation in Table 1.

• 1.

  Invariant Region

Theorem: The analytical solutions obtained for Eq. (1) are viable within the parameter space Φ for t≥0.

Proof: Total population be defined as N=S+V1+V2+Vb+E+I+R.

Then, we have:

Simplified dNdt=Π−μN,:

further to get:

N≤Πμ−Π−μNμe−μt, as t→∞,we have N≤Πμ..

Thus, the proposed model can be analyzed within the feasible region. Φ=SV1V2VbEIR∈R7:N≤Πμ.

• 2.

  Positivity and boundedness of the Covid-19 model

Theorem: Given S>0,V1>0,V2>0,Vb>0,E>0,I>0,R>0,SV1V2VbEIR∈R7:N≤Πμ are positively invariant for t≥0..

Proof:

dSdt≥−βSI−ξS−μS⇒dSdt≥−βI+ξ+μS.

∫1SdS≥∫−βI+ξ+μdt,S≥S0e−βI+ξ+μt≥0.

Applying as follows:

• 3.

  Disease-free equilibrium (DFE)

The concept of DFE is essential in the analysis of epidemic models. It signifies a state in which the disease is absent from the community (E=I=0), meaning that no individuals are currently infected with the disease. Analyzing the DFE helps researchers to understand when a disease can be eliminated and the variables that inhibit an outbreak from spreading throughout the community.

Let dSdt=dV1dt=dV2dt=dVbdt=dEdt=dIdt=dRdt=0

and for S=V1=V2=Vb=E=I=R=0, DFE is given by:

• 4.

  Endemic equilibrium (EE)

The EE represents a steady state where the ailment continues in the population at a constant level. Unlike the DFE, the EE does not lead to the eradication of the disease; instead, it results in a stable level of infection that can be sustained over time. At endemic equilibrium E=I≠0, which shows the persistence of the disease. Let the unique EE points of the model be denoted by E1=S∗V1∗V2∗Vb∗E∗I∗R∗. The result for each can be obtained as follows:

S∗=γ+μ+θμ+εβε,V1∗=γ+μ+θμ+εβψ+τ+μ,V2∗=τγ+μ+θμ+εβψ+τ+μφ1+φ2+μ,Vb∗=φ2τγ+μ+θμ+εβψ+τ+μφ1+φ2+μμ+ν,

• 5.

  Basic reproductive numberR0

The basic reproduction number R0 represents an essential epidemiological value which predicts the amount of new cases that one infection would spread when it encounters a population that remains entirely vulnerable. The disease transmission possibility in communities becomes possible to estimate through this tool. The mathematical computation uses the next generation matrix approach to evaluate the disease classes E and R of the model. The disease remains active because of transmission factor R0 but ceases to exist when factor becomes zero.

R0=σFV−1 such that σ is the spectral radius,

F=f1f2=βSI0 and V=v1v2=μ+εE−εE+γ+θ+μI.

The matrix of FandVcan be obtained by differentiating with respect to EandI. At E0,

F=0βS00=0βΠε+μ00 and V=ε+μ0−εγ+θ+μ..

Finding the inverse of V,FV−1=βΠεμ+ε2γ+θ+μβΠε+μγ+θ+μ00..

Hence, R0=βΠεμ+ε2γ+θ+μ.

• 6.

  Sensitivity analysis

We can obtain the sensitivity indices of all parameters in R0 by making use ϒR0X=∂R0∂X×XR0.

where X=βΠεγμθ and R0=βΠεμ+ε2γ+θ+μ..

Taking the parameters one after the other, we have the following by Maple 18 software package:

∂R0∂β×βR0=1,∂R0∂Π×ΠR0=1,∂R0∂ε×εR0=0.09090909127,

∂R0∂γ×γR0=−0.20000000, ∂R0∂μ×μR0=−1.690909091,∂R0∂θ×θR0=−0.20000000.

The ABC Covid-19 Model

Subject to:

According to Atangana and Owolabi's theory, we can use the ABC fractional integral to transform system1 into a voltera-type integral equation as follows:

Theorem: The kernels Ω1,Ω2,Ω3,Ω4,Ω5,Ω6,Ω7 in (2) satisfy the Lipschitz condition and contraction if the following inequality hold: 0≤η1,η2,η3,η4,η5,η6,η7<1..

Proof. Let Ω1tS=Π−βSI−ξS−μS+ψV1 and let kernels S1 and S2 be two functions, then we obtain the following.

The kernel of the Model, Eq. (1) can be expressed as:

Following recursive formula:

Next, we obtain the distinction between the expression's iterative terms.

Sn=∑n=0∞Θ1n,V1n=∑n=0∞Θ2n,V2n=∑n=0∞Θ3n,Vbn=∑n=0∞Θ4n,En=∑n=0∞Θ5n,In=∑n=0∞Θ6n and Rn=∑n=0∞Θ7n.

Applying triangular inequality and the norms on (7), we have:

These kernels satisfies the Lipschitz condition yielding the following:

Theorem: A solution to (2) exists if there is a tmax satisfying kiHω1−ω+tmaxωΓω such that i∈123…7..

Proof: Suppose S,V1,V2,Vb,E,I and R are bounded functions, we can apply the following relation to2:

Hence, Θkn, for k=1…7 in (8) exists. Also, to show that, Θkn, for k=1…7 are the solution of (2), we can construct:

Continuing this way recursively, we get:

Θ1n≤Bk1nk1Hω1−ω+tmaxωΓωn+1, where B=S−Sn−1. Taking the limit of both sides as n→∞, we get:

• 7.

  Stability analysis of the ABC Covid-19 model

We proved the stability of (2) here by using stability criteria of Ulam-Hyers and Generalized Ulam-Hyers.

Suppose ε>0, we can look into:

Definition 1. Let f:R→R, If there exist DΜ>0 and for all ε>0, a solution x¯∈R satisfying (10), then (2) is Ulam-Hyers stable if there exist x∈R of (2) with:

x¯−x≤DΜε,t∈ℕ, where DΜ=maxDΜiT.

Let x¯∈R satisfy:

Theorem 5. If x¯∈ℕ satisfy:

x¯ will then satisfies,

Proof.D0ABCtωx¯=Μtx¯+yt, and for:

We can use Mtx≤εt,∀tx∈ℕ×R7 and Eq. (11) to get:

Theorem 6. Let M:ℕ×R7→R be continuous for every x∈R, if there exist (14) with 1−ΨKΜ>0,(2) is then said to be Ulam-Hyers and afterward Ulam-Hyers stable.

Proof. Let x¯∈R satisfy (11) and x¯∈R is a unique solution of (2) using (10) and for all ε>0 we have:

This means that x¯−x≤DΜε such that DΜ=Ψ1−ΨKΜ. So, for ϕΜε=DΜε such that ϕΜ0=0. So, (2) is Ulam-Hyers stable.

• 8.

  Application of LADM

Here, we solved (2) using the Laplace Adomian Decomposition Method (LADM).

Simplifying,

Taking the inverse Laplace transform of (11), we obtain:

The model variables of12 can be represented as follows:

The nonlinear term can be decomposed as follows by Adomian:

By Eqs. (13) and (14), we have:

Let S0=m1,V10=m2,V20=m3,Vb0=m4,E0=m5,I0=m6,R0=m7, then Eq. (15) becomes:

The approximate solution is assumed as n→∞. So, S=limn→∞Sn,V1=limn→∞V1n,V2=limn→∞V2n,Vb=limn→∞Vbn,E=limn→∞En,I=limn→∞In,R=limn→∞Rn.

When n=0, using Mathematica 11 software package, Eq. (16) gives:

When n=1, Eq. (16) gives: