Research article Special Issues

Investigation of SEIR model with vaccinated effects using sustainable fractional approach for low immune individuals

  • Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARS-COVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes.

    Citation: Huda Alsaud, Muhammad Owais Kulachi, Aqeel Ahmad, Mustafa Inc, Muhammad Taimoor. Investigation of SEIR model with vaccinated effects using sustainable fractional approach for low immune individuals[J]. AIMS Mathematics, 2024, 9(4): 10208-10234. doi: 10.3934/math.2024499

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  • Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARS-COVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes.



    Mathematics was initially utilized in biology in the 13th century by Fibonacci, who developed the famed Fibonacci series to explain an increasing population. Daniel Bernoulli utilized mathematics to describe the impact on tiny shapes. Johannes Reinke coined the phrase "bio maths" in 1901. Biomathematics involves the theoretical examination of mathematical models to understand the principles governing the formation and functioning of biological systems.

    Mathematical models are utilized to investigate specific questions related to the studied disease. For example, epidemiological models are important for predicting how infectious diseases spread, and help to control them by identifying important factors in the community. In this case, we want to analyze a particular model to understand how the COVID-19 virus behaves. This virus appeared in late 2019, and continues to be a global challenge. To gain a deeper insight into the underlying physical processes, we delve into the realm of fractional calculus. Previous literature has introduced various operators through the framework of fractional calculus [1,2]. In the field of Cη-Calculus, Golmankhaneh et al. [3] provided an explanation of the Sumudu transform and Laplace transform. Additionally, in 2019, Goyal [4] proposed a fractional model that demonstrated the potential to manage the Lassa hemorrhagic fever disease.

    Advancements in technology have led to significant progress in the field of epidemiology, enabling the examination of various infectious diseases for treatment, control, and cure [5]. It is essential to underscore that mathematical biology plays a pivotal role in investigating numerous diseases. Significant strides have been taken in the mathematical modeling of infectious diseases in recent decades, as evidenced by various studies [6,7]. Over the last thirty years, mathematical modeling has gained prominence in research, making substantial contributions to the development of effective public health strategies for disease control [8,9]. Mathematical models serve as invaluable tools for analyzing spatiotemporal patterns and the dynamic behavior of infections. Acknowledging their significance, researchers have approached the study of COVID-19 from diverse perspectives in the last three years [10,11].

    Various methodologies have been employed by researchers in this field to devise successful techniques for managing this condition, with recent studies offering additional insights [12,13]. For example, a recent investigation utilized a mathematical model to evaluate the impacts of immunization in nursing homes [14]. Furthermore, researchers have explored mathematical modeling and effective intervention strategies for controlling the COVID-19 outbreak [15]. Additionally, some studies have delved into COVID-19 mathematical models using stochastic differential equations and environmental white noise [16].

    In 2019, China experienced a notable outbreak of the coronavirus disease 2019 (COVID-19), prompting concerns about its potential to escalate into a worldwide pandemic [17]. Researchers from China, particularly Zhao et al., made significant contributions in addressing the challenges posed by COVID-19. This disease is attributed to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a viral infection. The initial verified case was documented in Wuhan, China, in December 2019 [18]. The infection rapidly spread worldwide, leading to the declaration of a COVID-19 pandemic.

    Transmission occurs through various means, including respiratory droplets from coughing, sneezing, close contact, and touching contaminated surfaces. Key preventive measures include consistent mask usage, frequent hand washing, and maintaining safe interpersonal distances [19]. Effective interventions and real-time data play a crucial role in managing the coronavirus outbreak [20]. Prior studies have employed real-time analysis to comprehend the transmission of the virus among individuals, the severity of the disease, and the early stages of the pathogen, particularly in the initial week of the outbreak [21].

    In December 2019, an outbreak of pneumonia cases was reported in Wuhan, initially with unidentified origins. Some cases were associated with exposure to wet markets and seafood. Chinese health authorities, in collaboration with the Chinese Center for Disease Control and Prevention (China CDC), initiated an investigation into the cause and spread of the disease on December 31, 2019 [22]. We conducted an analysis of temporal changes in the outbreak by examining the time interval between hospital admission dates and fatalities. Clinical studies on COVID-19 have indicated that symptoms typically manifest around 7 days after the onset of illness [23]. It is important to consider the duration between hospitalization and death in order to accurately assess the risk of mortality [24]. The information regarding the incubation period of COVID-19 and epidemiological data was sourced from publicly available records of confirmed cases [25].

    An established method for the fractional-order model is elaborated upon in [26]. Recent contributions include various fractional models related to COVID-19, such as the analysis by Atangana and Khan focusing on the pandemic's impact on China [27]. Additionally, the COVID-19 model's dynamical aspects were explored using fuzzy Caputo and ABC derivatives, as demonstrated in [28]. A similar type of approach using fractional operator techniques are given in [29,30,31]. Different author's have investigated the transmission of different infectious disease like COVID-19 with symptomatic and asymptomatic effects in the community by using the fractal-fractional definition [32,33].

    In light of the aforementioned significance, we aim is to address fundamental issues by concentrating on the distinctive challenges posed by the dynamics of COVID-19. To achieve this, we employ a model tailored to accurately capture the characteristics of COVID-19 dynamics and account for the limitations in our response to the pandemic. Initially, we examine the epidemic dynamics within a specific community characterized by a unique social pattern. For this analysis, we adopt a conventional SEIR design that accommodates prolonged incubation periods.

    Here, the previous model is given in [34] as follows:

    dSdt=aρ1SI1+γI(δ+ρ)S,dEdt=ρSδEρ2αEI,dIdt=ρ1SI1+γI+ρ2αEI(δ+μ0+ωb)I,dRdt=ωIδR. (1.1)

    Initial conditions corresponds to the aforementioned system:

    S0(t)=S0,E0(t)=E0,I0(t)=I0,R0(t)=R0.

    The primary goal of this research is to employ novel fractional derivatives within mathematical analysis and simulation to enhance the COVID-19 model. COVID-19 is a highly dangerous disease that presents a significant risk to human life. Verification of the existence and distinct characteristics of the solution system is undertaken, coupled with a qualitative assessment of the system. So, we introduce vaccination measures for low immune individuals. We developed new mathematical model by taking vaccination measures which helps to control COVID-19 early which we shall observe on simulation easily. The research involves confirming the presence of a solution system with unique characteristics and conducting a qualitative evaluation of this system. Furthermore, the fractal-fractional derivative is utilized to investigate the real-world behavior of the newly developed mathematical model. Finally, numerical simulations are used to reinforce and authenticate the biological findings.

    Definition 1.1. If 0 <ξ 1 and 0 <λ 1, then the Riemann-Liouville operator for the fractal-fractional Operator (FFO) with a Mittag-Leffler (ML) kernel is defined as U(t) [35].

    FFM0Dξ,λtU(t)=AB(ξ)1ξt0EξdU(Ω)dtλ[ξ1ξ(tΩ)ξ]dΩ,

    involving 0<ξ,λ1, and AB(ξ)=1ξ+ξΓ(ξ).

    Therefore, the function U(t), which has an order of (ξ,λ) and a Mittag-Leffler (ML) kernel, is given as follows:

    FFM0Dξ,λtU(t)=λ(1ξ)tλ1U(t)AB(ξ)+ξλAB(ξ)t0Ωξ1(tΩ)U(Ω)dΩ.

    A newly developed model for SARS-COVID-19 includes the vaccinated effect, whereas the previous model used the SEIR framework. In this new model, we introduce a new variable called "Vaccinated." With the addition of Vaccinated, the new model is referred to as SEVIR, where "S" represents the Susceptible class, "E" represents the Exposed class, "V" represents the Vaccinated class, "I" represents the Infected class, and "R" represents the Recovered class.

    We define several parameters in this model: "a" represents the recruitment rate, "δ" represents the death rate due to natural causes, "ρ+ρ1" represents the contact rate from the Susceptible class to the Exposed class, "ρ2" represents the vaccination rate, "α" represents the rate at which the infection is reducing due to vaccination effects, "μ0" represents the infection death rate, "ω" represents the rate at which an individual recovers from vaccination and becomes recovered, "b" represents the recruitment rate to the Vaccinated class, "ϕ" represents the contact rate from the Vaccinated class to the Infected class, and "ψ" represents the recovery rate.

    We want to investigate spread of the SEIR model for SARS-COVID-19 with the vaccinated effect.

    So, the flow chart for newly developed model SEVIR is given as Figure 1.

    Figure 1.  The flow chart illustrates the newly developed model.

    The model that was developed based on the generalized hypothesis with the vaccinated effect is presented as follows:

    dSdt=a(δ+ρ+ρ1)S,dEdt=(ρ+ρ1)SδEρ2αEV,dVdt=ρ2αEV(δ+μ0+ωb+ϕ)V,dIdt=ϕV(μ0+δ+ψ)I,dRdt=ωV+ψIδR. (2.1)

    The following are initial conditions linked with the described system:

    S0(t)=S0,E0(t)=E0,V0(t)=V0,I0(t)=I0,R0(t)=R0.

    Using the fractal-fractional order (FFO) with a Mittag-Leffler (ML) definition, the above model becomes

    FFM0Dξ,λtS(t)=a(δ+ρ+ρ1)S,FFM0Dξ,λtE(t)=(ρ+ρ1)SδEρ2αEV,FFM0Dξ,λtV(t)=ρ2αEV(δ+μ0+ωb+ϕ)V,FFM0Dξ,λtI(t)=ϕV(μ0+δ+ψ)I,FFM0Dξ,λtR(t)=ωV+ψIδR. (2.2)

    Here, FFM0Dξ,λt is the fractal-fractional operator with Mittag-Leffler (ML), where 0<ξ1 and 0<λ1.

    The following are initial conditions linked with the described system:

    S0(t)=S0,E0(t)=E0,V0(t)=V0,I0(t)=I0,R0(t)=R0.

    Parameter descriptions are given in the following table.

    Parameters Representation Reference
    a contact rate from the susceptible to the exposed class [34]
    δ death rate due to natural causes [34]
    ρ+ρ1 contact rate from the susceptible to the exposed class [34]
    α rate at which the infection is reducing due to vaccination [34]
    ρ2 vaccination rate [34]
    μ0 death rate due to infection [34]
    ω rate at which an individuals becomes recovered [34]
    b recruitment rate to the vaccinated class [34]
    ϕ contact rate from the vaccinated to the infected class Assumed
    ψ recovery rate Assumed

    For this model, the point of equilibrium without disease (disease free) is

    D1(S,E,V,I,R)=(aδ+ρ+ρ1,a(ρ+ρ1)δ(δ+ρ+ρ1),0,0,0),

    as well as the endemic points of equilibrium D2(S,E,V,I,R), where

    S=aδ+ρ+ρ1,E=b+δ+ϕ+ω+μ0αρ2,V=(δ+ψ+μ0)Abδρ1+δ2ρ1+δϕρ1+δωρ1+δμ0ρ1aαρρ2aαρ1ρ2αϕ(bδϕωμ0)(δ+ρ+ρ1)ρ2,I=ϕ(Abδρ1+δ2ρ1+δϕρ1+δωρ1+δμ0ρ1aαρρ2aαρ1ρ2α(bδϕωμ0)(δ+ρ+ρ1)ρ2),R=(ϕψ+δω+ψω+ωμ0)×(Abδρ1+δ2ρ1+δϕρ1+δωρ1+δμ0ρ1aαρρ2aαρ1ρ2αδϕ(δ+ψ+μ0)(bδϕωμ0)(δ+ρ+ρ1)ρ2),

    where

    A=bδ2+δ3bδρ+δ2ρ+δ2ϕ+δρϕ+δ2ω+δρω+δ2μ0+δρμ0.

    The reproductive number for the newly developed system by using the next generation method is

    R0=bδ(δ+ρ+ρ1)+aαρ2(ρ+ρ1)δ(δ+ρ+ρ1)(δ+ϕ+ω+μ0).

    In this section, we demonstrate the boundedness and positivity of the developed model.

    Theorem 3.1. The considered initial condition is

    {S0,E0,V0,I0,R0}Υ,

    and therefore the solutions {S,E,V,I,R} will be positive t0.

    Proof. We will begin the primary analysis to show the improved quality of the solutions. These solutions effectively address real-world issues and have positive outcomes. We will follow the methodology provided in references [36,37,38]. In this segment, we will examine the conditions required to ensure positive outcomes from the proposed model. To accomplish this, we will establish a standard.

    β=suptDββ(t),

    where "Dβ" represents the β domain. Now, we continue with S(t).

    FFM0Dξ,λtS(t)=a(δ+ρ+ρ1)S,t0,(δ+ρ+ρ1)S,t0.

    This yield

    S(t)S(0)Eξ[c1λξ(δ+ρ+ρ1)tξAB(ξ)(1ξ)(δ+ρ+ρ1)],t0,

    where "c" represents the time element. This demonstrates that the S(t) individuals must be positive t 0. Now, we have the E(t) individuals as follows:

    FFM0Dξ,λtE(t)=(ρ+ρ1)SδEρ2αEV,t0,(δ+ρ2αV)E,t0,(δ+ρ2αsuptDVV)E,t0,(δ+ρ2αV)E,t0.

    This yield

    E(t)E(0)Eξ[c1λξ(δ+ρ2αV)tξAB(ξ)(1ξ)(δ+ρ2αV)],t0,

    where "c" represents the time element. This demonstrates that the E(t) individuals must be positive t 0. Now, we have the V(t) individuals as follows:

    FFM0Dξ,λtV(t)=ρ2αEI(δ+μ0+ωb+ϕ)V,t0,(ρ2αE+μ0+δ+ωb+ϕ)V,t0,(ρ2αsuptDEE+μ0+δ+ωb+ϕ)V,t0,(ρ2αE+μ0+δ+ωb+ϕ)V,t0.

    This yield

    V(t)V(0)Eξ[c1λξ(ρ2αE+μ0+δ+ωb+ϕ)tξAB(ξ)(1ξ)(ρ2αE+μ0+δ+ωb+ϕ)],t0,

    where "c" represents the time element. This demonstrates that the V(t) individuals must be positive t 0. Now, we have the I(t) individuals as follows:

    FFM0Dξ,λtI(t)=ϕV(μ0+δ+ψ)I,t0,(μ0+δ+ψ)I(t),t0.

    This yield

    I(t)I(0)Eξ[c1λξ(μ0+δ+ψ)tξAB(ξ)(1ξ)(μ0+δ+ψ)],t0,

    where "c" represents the time element. This demonstrates that the I(t) individuals must be positive t 0. Now, we have the R(t) individuals as follows:

    FFM0Dξ,λtR(t)=ωV+ψIδR,t0,(δ)R(t),t0.

    This yield

    R(t)R(0)Eξ[c1λξ(δ)tξAB(ξ)(1ξ)(δ)],t0,

    where "c" represents the time element. This demonstrates that the R(t) individuals must be positive t 0.

    Theorem 3.2. Solutions of our developed model given in Eq (2.2) with positive initial values are all bounded.

    Proof. The above theorem demonstrates that the solutions of our developed model must be positive t0, and the strategies are described in [39]. Because X=S+E+V, then

    FFM0Dξ,λtX(t)=aδX(μ0+ωb+ϕ)V.

    We achieved as follows:

    Ψp={S,E,VR3+S+VX}t0.

    Further we have Xυ = I+R. So, we have

    FFM0Dξ,λtXυ(t)=(ϕ+ω)Vμ0IXυδ.

    Upon solving the above equation and taking t, we get

    Xυ(ϕ+ω)Vμ0Iδ.

    Thus,

    Ψυ={I,RR2+Xυ(ϕ+ω)Iμ0Qδ}t0.

    The model's mathematical solutions (2.2) are confined to the region Ψ.

    Ψ={S,E,V,I,RR5+S+VX,Xυ(ϕ+ω)Vμ0Iδ}t0.

    This demonstrates that for every t 0, all solutions remain positive, consistent with the provided initial conditions in the domain Ψ.

    Theorem 3.3. The proposed coronavirus model (2.2) in R5+ has positive invariant solutions, in addition to the initial conditions.

    Proof. In this particular scenario, we applied the procedure described in [40]. We have

    FFM0Dξ,λt(S(t))S=0=a0,FFM0Dξ,λt(E(t))E=0=(ρ+ρ1)S0,FFM0Dξ,λt(V(t))V=0=ρ2αEV+bV0,FFM0Dξ,λt(I(t))I=0=ϕV0,FFM0Dξ,λt(R(t))R=0=ωV+ψI0. (3.1)

    If (S0,E0,V0,I0,R0) R5+, then our obtained solution is unable to escape from the hyperplane, as stated in Eq (3.1). This proves that the R5+ domain is positive invariant.

    The Riemann-Stieltjes integral has been widely recognized in the literature as the most commonly used integral. If

    Y(x)=y(x)dx,

    then the Riemann-Stieltjes integral is given as follows:

    Yw(x)=y(x)dw(x),

    where the y(x) global derivative with respect to w(x) is

    Dwy(x)=limh0y(x+h)y(x)w(x+h)w(x).

    If the above function's numerator and denominator are differentiated, we get

    Dwy(x)=y(x)w(x),

    assuming that w(x) 0, xDw. Now, we will test the impact on the coronavirus by using the global derivative instead of the classical derivative:

    DwS=a(δ+ρ+ρ1)S,DwE=(ρ+ρ1)SδEρ2αEV,DwV=ρ2αEV(δ+μ0+ωb+ϕ)V,DwI=ϕV(μ0+δ+ψ)I,DwR=ωV+ψIδR.

    For the sake of clean notation, we shall suppose that w is differentiable.

    S=w[a(δ+ρ+ρ1)S],E=w[(ρ+ρ1)SδEρ2αEV],V=w[ρ2αEV(δ+μ0+ωb+ϕ)V],I=w[ϕV(μ0+δ+ψ)I],R=w[ωV+ψIδR].

    An appropriate choice of the function w(t) will lead to a specific outcome. For instance, if w(t)=tα, where α is a real number, we will observe fractal movement. We had to take action due to the circumstances that

    w=suptDww(t)∣<N.

    The below example demonstrates the unique solution for the developed system:

    S=w[a(δ+ρ+ρ1)S]=Z1(t,S,G),E=w[(ρ+ρ1)SδEρ2αEV]=Z2(t,S,G),V=w[ρ2αEV(δ+μ0+ωb+ϕ)V]=Z3(t,S,G),I=w[ϕV(μ0+δ+ψ)I]=Z4(t,S,G),R=w[ωV+ψIδR]=Z5(t,S,G),

    where G=E,V,I,R.

    We need to confirm the first two requirements as follows:

    (1) Z(t,S,G)2<K(1+S2,

    (2) S1,S2, we have, Z(t,S1,G)Z(t,S2,G)2<ˉKS1S22.

    Initially,

    Z1(t,S,G)2=w[a(δ+ρ+ρ1)S]2,=w[a+(δρρ1)S]2,2w2(a2+(δρρ1)S2),2suptDww2a2+6suptDww2(ρ2+δ2+ρ1)S2,2w2a2+6w2(ρ2+δ2+ρ21)S2,2w2a2(1+3a2(ρ2+δ2+ρ21)S2),K1(1+S2),

    under the condition

    3a2(ρ2+δ2+ρ21)<1,

    involving

    K1=2w2a2.
    Z2(t,S,G)2=w[(ρ+ρ1)SδEρ2αEV]2,=w[(ρ+ρ1)S+(δρ2αV)E]2,2w2((ρ+ρ1)S2+(δρ2αV)E2),4suptDww2(ρ2+ρ21)suptDSS2+4suptDww2(δ2+ρ22α2suptDVV2)E2,4w2(ρ2+ρ21)S2+4w2(δ2+ρ22α2V2)E2,4w2(ρ2+ρ21)S2(1+(δ2+ρ22α2V2)E2(ρ2+ρ21)S2),K2(1+E2),

    under the condition

    (δ2+ρ22α2V2)(ρ2+ρ21)S2<1,

    where

    K2=2w2(ρ2+ρ21)S2.
    Z3(t,S,G)2=w[ρ2αEV(δ+μ0+ωb+ϕ)V]2,=w[ρ2αEV+bV+(δμ0ωϕ)V]2,2w2(ρ2αE+b2+(δμ0ωϕ)2)V2,4suptDww2[(ρ22α2suptDEE2+b2)+2(δ2+μ20+ω2+ϕ2)]V2,4∥∣w2[(ρ22α2E2+b2)+2(δ2+μ20+ω2+ϕ2)]V2,4∥∣w2(ρ22α2E2+b2)[1+2(δ2+μ20+ω2+ϕ2)(ρ22α2E2+b2)]V2,K3(2V2),

    under the condition

    2(δ2+μ20+ω2+ϕ2)(ρ22α2E2+b2)1,

    where

    K3=4∥∣w2(ρ22α2E2+b2).
    Z4(t,S,G)2=w[ϕV(μ0+δ+ψ)I]2,=w[ϕV+(μ0δψ)I]2,2w2(ϕV2+(μ0δψ)I2),2suptDww2ϕ2suptDVV2+6suptDww2(μ20+δ2+ψ2)I2,2w2ϕ2V2+6w2(μ20+δ2+ψ2)I2,2w2ϕ2I2(1+3(μ20+δ2+ψ2)I2ϕ2I2),K4(1+I2),

    under the condition

    3(μ20+δ2+ψ2)ϕ2V2<1,

    where

    K4=2w2ϕ2V2.
    Z5(t,S,G)2=w[ωV+ψIδR]2,=w[(ωV+ψI)+(δR)]2,2w2((ωV+ψI)2+δR2),4suptDww2(ω2suptDVV2+ψsuptDII2)+2suptDww2δ2R2,4w2(ω2V2+ψI2)+2w2δ2R2),4w2(ω2V2+ψ2I2)(1+δ2R22(ω2V2+ψ2I2)),K5(1+R2),

    under the condition

    δ22(ω2V2+ψ2I2)<1,

    where

    K5=4w2(ω2V2+ψ2I2).

    Hence the linear growth condition is satisfied.

    Further, we validate the Lipschitz condition.

    If

    Z1(t,S1,G)Z1(t,S2,G)2=w(δρρ1)(S1S2)2,Z1(t,S1,G)Z1(t,S2,G)2w2(3δ2+3ρ2+3ρ21)S1S22,suptDSZ1(t,S1,G)Z1(t,S2,G)2suptDww2(3δ2+3ρ2+3ρ21)suptDSS1S22,Z1(t,S1,G)Z1(t,S2,G)2w2(3δ2+3ρ2+3ρ21)S1S22,Z1(t,S1,G)Z1(t,S2,G)2¯K1S1S22,

    where

    ¯K1=∥w2(3δ2+3ρ2+3ρ21).

    If

    Z2(t,S,E1,V,I,R)Z2(t,S,E2,V,I,R)2=w(δρ2αV)(E1E2)2,Z2(t,S,E1,V,I,R)Z2(t,S,E2,V,I,R)2w2(2δ2+2ρ22α2V2)E1E22,suptDEZ2(t,S,E1,V,I,R)Z2(t,S,E2,V,I,R)2suptDww2(2δ2+2ρ22α2suptDVV2)suptDE×E1E22,Z2(t,S,E1,V,I,R)Z2(t,S,E2,V,I,R)2w2(2δ2+2ρ22α2V2),E1E22,Z2(t,S,E1,V,I,R)Z2(t,S,E2,V,I,R)2¯K2E1E22,

    where

    ¯K2=∥w2(2δ2+2ρ22α2V2).

    If

    Z3(t,S,E,V1,I,R)Z3(t,S,E,V2,I,R)2=w(ρ2αE+b+(δμ0ωϕ))(V1V2)2,
    Z3(t,S,E,V1,I,R)Z3(t,S,E,V2,I,R)22w2(ρ2αE+b2+(δμ0ωϕ)2)×V1V22,
    suptDVZ3(t,S,E,V1,I,R)Z3(t,S,E,V2,I,R)24suptDww2(ρ22α2suptDEE2+b2+2(δ2+μ20+ω2+ϕ2))suptDVV1V22,Z3(t,S,E,V1,I,R)Z3(t,S,E,V2,I,R)24w2(ρ22α2E2+b2+2(δ2+μ20+ω2+ϕ2))V1V22,Z3(t,S,E,V1,I,R)Z3(t,S,E,V2,I,R)2¯K3V1V22,

    where

    ¯K3=4w2(ρ22α2E2+b2+2(δ2+μ20+ω2+ϕ2)).

    If

    Z4(t,S,E,V,I1,R)Z4(t,S,E,V,I2,R)2=w(μ0δψ)](Q1Q2)2,Z4(t,S,E,V,I1,R)Z4(t,S,E,V,I2,R)2=w2(3μ20+3δ2+3ψ2)I1I22,suptDIZ4(t,S,E,V,I1,R)Z4(t,S,E,V,I2,R)2=suptDww2(3μ20+3δ2+3ψ2)suptDII1I22,Z4(t,S,E,V,I1,R)Z4(t,S,E,V,I2,R)2w2(3μ20+3δ2+3ψ2)I1I22,Z4(t,S,E,V,I1,R)Z4(t,S,E,V,I2,R)2¯K4I1I22,

    where

    ¯K4=∥w2(3μ20+3δ2+3ψ2).

    If

    Z5(t,S,E,V,I,R1)Z5(t,S,E,V,I,R2)2=w(δ)(R1R2)2,Z5(t,S,E,V,I,R1)Z5(t,S,E,V,I,R2)2w2δ2(R1R2)2,suptDRZ5(t,S,E,V,I,R1)Z5(t,S,E,V,I,R2)2suptDww2δ2suptDRR1R22,Z5(t,S,E,V,I,R1)Z5(t,S,E,V,I,R2)2w2δ2R1R22,Z5(t,S,E,V,I,R1)Z5(t,S,E,V,I,R2)2¯K5R1R22,

    involving

    ¯K5=∥w2δ2.

    Then, given the condition, system (2.2) has a particular solution.

    max[3a2(ρ2+δ2+ρ21),(δ2+ρ22α2V2)(ρ2+ρ21)S2,2(δ2+μ20+ω2+ϕ2)(ρ22α2E2+b2),3(μ20+δ2+ψ2)ϕ2V2,δ22(ω2V2+ψ2I2)]<1.

    We use Lyapunov's approach and LaSalle's concept of invariance to analyze global stability and determine the conditions for eliminating diseases.

    Theorem 5.1. [41] When the reproductive number R0> 1, the endemic equilibrium points of the SEVIR model are globally asymptotically stable.

    Proof. The Lyapunov function can be expressed in the following manner:

    L(S,E,V,I,R)=(SSSlogSS)+(EEElogEE)+(VVVlogVV)+(IIIlogII)+(RRRlogRR).

    By applying a derivative on both sides,

    Dξ,λtL=˙L=(SSS)Dξ,λtS+(EEE)Dξ,λtE+(VVV)Dξ,λtV+(III)Dξ,λtI+(RRR)Dξ,λtR,

    we get

    Dξ,λtL=(SSS)(a(δ+ρ+ρ1)S)+(EEE)((ρ+ρ1)SδEρ2αEV)+(VVV)×(ρ2αEV(δ+μ0+ωb+ϕ)V)+(III)(ϕV(μ0+δ+ψ)I)+(RRR)×(ωV+ψIδR),

    and setting S=SS,E=EE,V=VV,I=II and R=RR results in

    Dξ,λtL=aaSS(δ+ρ+ρ1)(SS)2S+(ρ+ρ1)S(ρ+ρ1)S(ρ+ρ1)SEE+(ρ+ρ1)SEEδ(EE)2Eρ2αV(EE)2E+ρ2αV(EE)2E+ρ2αE(VV)2Vρ2αE(VV)2V+b(VV)2V(δ+μ0+ω+ϕ)(VV)2V+ϕVϕVϕVII+ϕVII(μ0+δ+ψ)(II)2I+ωVωVωVRR+ωVRR+ψIψIψIRR+ψIRRδ(RR)2R.

    We can write Dξ,λtL=ΣΩ, where

    Σ=a+(ρ+ρ1)S+(ρ+ρ1)SEE+ρ2αV(EE)2E+ρ2αE(VV)2V+b(VV)2V+ϕV+ϕVII+ωV+ωVRR+ψI+ψIRR,

    and

    Ω=aSS+(δ+ρ+ρ1)(SS)2S+(ρ+ρ1)S+(ρ+ρ1)SEE+δ(EE)2E+ρ2αV(EE)2E+ρ2αE(VV)2V+(δ+μ0+ω+ϕ)(VV)2V+ϕV+ϕVII+(μ0+δ+ψ)(II)2I+ωV+ωVRR+ψV+ψIRR+δ(RR)2R.

    We conclude that if Σ<Ω, this yields Dξ,λtL<0, however when S=S,E=E,V=V,I=I and R=R, ΣΩ=0Dξ,λtL=0.

    We can observe that {(S,E,V,I,R)Γ:Dξ,λtL=0} represents the point D2 for the developed model.

    According to Lasalles' concept of invariance, D2 is globally uniformly stable in Γ if ΣΩ=0.

    Now, we will develop a solution using a numerical approach for our newly developed model given in Eq (2.2). We use the ML kernel in the current scenario instead of the classical derivative operator.

    Furthermore, we will use the variable order version.

    FFM0Dξ,λtS(t)=a(δ+ρ+ρ1)S,FFM0Dξ,λtE(t)=(ρ+ρ1)SδEρ2αEV,FFM0Dξ,λtV(t)=ρ2αEV(δ+μ0+ωb+ϕ)V,FFM0Dξ,λtI(t)=ϕV(μ0+δ+ψ)I,FFM0Dξ,λtR(t)=ωV+ψIδR.

    For clarity, we express the above equation as follows:

    FFM0Dξ,λtS(t)=S1(t,S,G),FFM0Dξ,λtE(t)=E1(t,S,G),FFM0Dξ,λtV(t)=V1(t,S,G),FFM0Dξ,λtI(t)=I1(t,S,G),FFM0Dξ,λtR(t)=R1(t,S,G).

    Where

    S1(t,S,G)=a(δ+ρ+ρ1)S,E1(t,S,G)=(ρ+ρ1)SδEρ2αEV,V1(t,S,G)=ρ2αEV(δ+μ0+ωb+ϕ)V,I1(t,S,G)=ϕV(μ0+δ+ψ)I,R1(t,S,G)=ωV+ψIδR.

    After using the fractal-fractional integral with the ML kernel, we obtain the following results:

    S(tη+1)=λ(1ξ)AB(ξ)tλ1ηS1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2tν+1tνS1(t,S,G)τξ1(tη+1τ)ξ1dτ,E(tη+1)=λ(1ξ)AB(ξ)tλ1ηE1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2tν+1tνE1(t,S,G)τξ1(tη+1τ)ξ1dτ,V(tη+1)=λ(1ξ)AB(ξ)tλ1ηV1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2tν+1tνV1(t,S,G)τξ1(tη+1τ)ξ1dτ,I(tη+1)=λ(1ξ)AB(ξ)tλ1ηI1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2tν+1tνI1(t,S,G)τξ1(tη+1τ)ξ1dτ,R(tη+1)=λ(1ξ)AB(ξ)tλ1ηR1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2tν+1tνR1(t,S,G)τξ1(tη+1τ)ξ1dτ, (6.1)

    where G(tη)=E(tη),V(tη),I(tη),R(tη).

    Remember that the Newton polynomial can be obtained by using the Newton interpolation formula.

    N(t,S,G)N(tη2,Sη2,Gη2)+1Δt[N(tη1,Sη1,Gη1)N(tη2,Sη2,Gη2)](τtη2)+12Δt2[N(tη,Sη,Eη,Iη,Qη,Rη)2N(tη1,Sη1,Gη1)N(tη2,Sη2,Gη2)](τtη2)(τtη1),

    where Gη2=Eη2,Vη2,Iη2,Rη2, Gη1=Eη1,Vη1,Iη1,Rη1.

    When we substitute the Newton polynomial into the system of Eqs (6.1), we obtain the following:

    Sη+1=λ(1ξ)AB(ξ)tλ1ηS1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2S1(tν2,Sν2,Gν2)×tλ1ν2tν+1tν(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=21Δt[tλ1ν1S1(tν1,Sν1,Gν1)tλ1ν2S1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=212Δt2[tλ1νS1(tν,Sν,Gν)2tλ1ν1S1(tν1,Sν1,Gν1)+tλ1ν2S1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(τtν1)(tη+1τ)ξ1dτ.Eη+1=λ(1ξ)AB(ξ)tλ1ηE1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2E1(tν2,Sν2,Gν2)×tλ1ν2tν+1tν(tη+1τ)ξ1dτ +ξλAB(ξ)Γ(ξ)ην=21Δt[tλ1ν1E1(tν1,Sν1,Gν1)tλ1ν2E1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=212Δt2[tλ1νE1(tν,Sν,Gν)2tλ1ν1E1(tν1,Sν1,Gν1)+tλ1ν2E1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(τtν1)(tη+1τ)ξ1dτ.Vη+1=λ(1ξ)AB(ξ)tλ1ηV1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2V1(tν2,Sν2,Gν2)×tλ1ν2tν+1tν(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=21Δt[tλ1ν1V1(tν1,Sν1,Gν1)tλ1ν2V1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=212Δt2[tλ1νV1(tν,Sν,Gν)2tλ1ν1V1(tν1,Sν1,Gν1)+tλ1ν2V1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(τtν1)(tη+1τ)ξ1dτ.Iη+1=λ(1ξ)AB(ξ)tλ1ηI1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2I1(tν2,Sν2,Gν2)×tλ1ν2tν+1tν(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=21Δt[tλ1ν1I1(tν1,Sν1,Gν1)tλ1ν2I1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=212Δt2[tλ1νI1(tν,Sν,Gν)2tλ1ν1Q1(tν1,Sν1,Gν1)+tλ1ν2I1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(τtν1)(tη+1τ)ξ1dτ.Rη+1=λ(1ξ)AB(ξ)tλ1ηR1(tη,S(tη),G(tη))+ξλAB(ξ)Γ(ξ)ην=2R1(tν2,Sν2,Gν2)×tλ1ν2tν+1tν(tη+1τ)ξ1dτ +ξλAB(ξ)Γ(ξ)ην=21Δt[tλ1ν1R1(tν1,Sν1,Gν1)tλ1ν2R1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(tη+1τ)ξ1dτ+ξλAB(ξ)Γ(ξ)ην=212Δt2[tλ1νR1(tν,Sν,Gν)2tλ1ν1R1(tν1,Sν1,Gν1)+tλ1ν2R1(tν2,Sν2,Gν2)]tν+1tν(τtν2)(τtν1)(tη+1τ)ξ1dτ, (6.2)

    where Gν2=Eν2,Vν2,Iν2,Rν2, Gν1=Eν1,Vν1,Iν1,Rν1, Gν=Eν,Vν,Iν,Rν, and G(tη)=E(tη),V(tη),I(tη),R(tη).

    We can perform the following calculations for the integral in Eq (6.2):

    tν+1tν(tη+1τ)ξ1dτ=(Δt)ξξ[(ην+1)ξ(ην)ξ],tν+1tν(τtν2)(tη+1τ)ξ1dτ=(Δt)ξ+1ξ(ξ+1)[(ην+1)ξ×(ην+3+2ξ)(ην)ξ(ην+3+3ξ)].tν+1tν(τtν2)(τtν1)(tη+1τ)ξ1dτ=(Δt)ξ+2ξ(ξ+1)(ξ+2)[(ην+1)ξ{2(ην)2+(3ξ+10)×(ην)+2ξ2+9ξ+12}(ην)ξ{2(ην)2+(5ξ+10)(ην)+6ξ2+18ξ+12}], (6.3)

    substituting integral calculation values into Eq (6.2).

    We acquire the numerical solutions S(t),E(t),V(t),I(t)andR(t):

    Sη+1=λ(1ξ)AB(ξ)tλ1ηS1(tη,S(tη),G(tη))+ξλ(Δt)ξAB(ξ)Γ(ξ+1)ην=2S1(tν2,Sν2,Gν2)×tλ1ν2[(ην+1)ξ(ην)ξ]+ξλ(Δt)ξAB(ξ)Γ(ξ+2)ην=2[tλ1ν1S1(tν1,Sν1,Gν1)tλ1ν2S1(tν2,Sν2,Gν2)][(ην+1)ξ(ην+3+2ξ)(ην)ξ(ην+3+3ξ)]+ξλ(Δt)ξ2AB(ξ)Γ(ξ+3)ην=2[tλ1νS1(tν,Sν,Gν)2tλ1ν1S1(tν1,Sν1,Gν1)+tλ1ν2S1(tν2,Sν2,Gν2)][(ην+1)ξ{2(ην)2+(3ξ+10)(ην)+2ξ2+9ξ+12}(ην)ξ×{2(ην)2+(5ξ+10)(ην)+6ξ2+18ξ+12}],Eη+1=λ(1ξ)AB(ξ)tλ1ηE1(tη,S(tη),G(tη))+ξλ(Δt)ξAB(ξ)Γ(ξ+1)ην=2E1(tν2,Sν2,Gν2)×tλ1ν2[(ην+1)ξ(ην)ξ]+ξλ(Δt)ξAB(ξ)Γ(ξ+2)ην=2[tλ1ν1E1(tν1,Sν1,Gν1)tλ1ν2E1(tν2,Sν2,Gν2)][(ην+1)ξ(ην+3+2ξ)(ην)ξ(ην+3+3ξ)]+ξλ(Δt)ξ2AB(ξ)Γ(ξ+3)ην=2[tλ1νE1(tν,Sν,Gν)2tλ1ν1E1(tν1,Sν1,Gν1)+tλ1ν2E1(tν2,Sν2,Gν2)][(ην+1)ξ{2(ην)2+(3ξ+10)(ην)+2ξ2+9ξ+12}(ην)ξ×{2(ην)2+(5ξ+10)(ην)+6ξ2+18ξ+12}],Vη+1=λ(1ξ)AB(ξ)tλ1ηV1(tη,S(tη),G(tη))+ξλ(Δt)ξAB(ξ)Γ(ξ+1)ην=2V1(tν2,Sν2,Gν2)×tλ1ν2[(ην+1)ξ(ην)ξ]+ξλ(Δt)ξAB(ξ)Γ(ξ+2)ην=2[tλ1ν1V1(tν1,Sν1,Gν1)tλ1ν2V1(tν2,Sν2,Gν2)][(ην+1)ξ(ην+3+2ξ)(ην)ξ(ην+3+3ξ)]+ξλ(Δt)ξ2AB(ξ)Γ(ξ+3)ην=2[tλ1νV1(tν,Sν,Gν)2tλ1ν1V1(tν1,Sν1,Gν1)+tλ1ν2V1(tν2,Sν2,Gν2)][(ην+1)ξ{2(ην)2+(3ξ+10)(ην)+2ξ2+9ξ+12}(ην)ξ×{2(ην)2+(5ξ+10)(ην)+6ξ2+18ξ+12}],Iη+1=λ(1ξ)AB(ξ)tλ1ηI1(tη,S(tη),G(tη))+ξλ(Δt)ξAB(ξ)Γ(ξ+1)ην=2I1(tν2,Sν2,Gν2)×tλ1ν2[(ην+1)ξ(ην)ξ]+ξλ(Δt)ξAB(ξ)Γ(ξ+2)ην=2[tλ1ν1I1(tν1,Sν1,Gν1)tλ1ν2I1(tν2,Sν2,Gν2)][(ην+1)ξ(ην+3+2ξ)(ην)ξ(ην+3+3ξ)]+ξλ(Δt)ξ2AB(ξ)Γ(ξ+3)ην=2[tλ1νI1(tν,Sν,Gν)2tλ1ν1I1(tν1,Sν1,Gν1)+tλ1ν2I1(tν2,Sν2,Gν2)][(ην+1)ξ{2(ην)2+(3ξ+10)(ην)+2ξ2+9ξ+12}(ην)ξ×{2(ην)2+(5ξ+10)(ην)+6ξ2+18ξ+12}],Rη+1=λ(1ξ)AB(ξ)tλ1ηR1(tη,S(tη),G(tη))+ξλ(Δt)ξAB(ξ)Γ(ξ+1)ην=2R1(tν2,Sν2,Gν2)×tλ1ν2[(ην+1)ξ(ην)ξ]+ξλ(Δt)ξAB(ξ)Γ(ξ+2)ην=2[tλ1ν1R1(tν1,Sν1,Gν1)tλ1ν2R1(tν2,Sν2,Gν2)][(ην+1)ξ(ην+3+2ξ)(ην)ξ(ην+3+3ξ)]+ξλ(Δt)ξ2AB(ξ)Γ(ξ+3)ην=2[tλ1νR1(tν,Sν,Gν)2tλ1ν1R1(tν1,Sν1,Gν1)+tλ1ν2R1(tν2,Sν2,Gν2)][(ην+1)ξ{2(ην)2+(3ξ+10)(ην)+2ξ2+9ξ+12}(ην)ξ×{2(ην)2+(5ξ+10)(ην)+6ξ2+18ξ+12}].

    In this section, we utilized an advanced technique to obtain theoretical outcomes and assess their effectiveness. The newly developed SEVIR system was analyzed through simulation. By applying non-integer parametric values in the SARS-COVID-19 model, we obtained interesting findings. Figures 26 display the solutions for S(t), E(t), V(t), I(t), and R(t) by reducing the fractional values to the desired level. To validate the efficiency of the theoretical outcomes, we provide the following examples. Numerical simulations for the SARS-COVID-19 model were performed using MATLAB. The initial conditions used in the newly developed model are S(0)=217.342565,E(0)=100,V(0)=1.386348,I(0)=1.1 and R(0)=1.271087. The parameter values used in the developed system are as follows: a=1.43, δ=0.000065, ρ=0.45, ρ1=0.10, ρ2=0.020, α=0.0008601, μ0=0.19, ω=0.98, b=0.135, ϕ=0.0001, and ψ=0.0001. Figures 2, 4, and 5 illustrate the changes in susceptible, vaccinated, and infected individuals respectively, showing a sharp decrease before reaching a stable position. Meanwhile, Figures 3 and 6 demonstrate the dynamics of exposed and recovered individuals respectively at different fractional orders in which both individuals increases and after certain time the number of individuals approaches stable state using different dimensions. The research predicts future infection rates and suggests ways to decrease the spread of infection units more effectively. By utilizing a fractal-fractional approach, the study yields reliable and accurate results for all compartments at non-integer order derivatives, which are more trustworthy when fractional values are reduced as well a by reducing its dimensions. The findings suggest that the number of infected individuals decreases significantly due to vaccination measures, while the number of recovered individuals increases due to a decline in infected individuals and the effect of vaccination.

    Figure 2.  The value of S(t) using the fractal-fractional operator with various fractional values at different dimensions.
    Figure 3.  The value of E(t) using the fractal-fractional operator with various fractional values at different dimensions.
    Figure 4.  The value of V(t) using the fractal-fractional operator with various fractional values at different dimensions.
    Figure 5.  The value of I(t) using the fractal-fractional operator with various fractional values at different dimensions.
    Figure 6.  The value of R(t) using the fractal-fractional operator with various fractional values at different dimensions.

    This article employs a fractional order SEVIR model for SARS-COVID-19 with vaccinated effects using an FFO to find reliable solutions. We provide advice on controlling this virus to help our community overcome the pandemic by implementing vaccinated measures for low immune individuals. We analyze the dangerous coronavirus disease with the effect of vaccination to understand its real impact on the community. Qualitative and quantitative analyses are conducted to verify its stable position in a continuous dynamical system. We also verify that the fractional order coronavirus disease model has bounded and unique solutions. We examine the impact of global measures to control the spread of the coronavirus disease. Also, analyses are performed to see how the rate of infection changes after the implementation of vaccination measures. We ensure that our findings are reliable and realistic. FFO is used for continuously monitoring the spread as well as control of the disease in society after vaccination measures. It was observed that infected individuals recover quickly due to the vaccinated strategy. The fractal-fractional operator (FFO) is used for continuously monitoring for the spread of the diseases using different fractional values as well as reliable solutions. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator which provides the real behavior of the spread as well as control of COVID-19 with different dimensions and continuous monitoring, which can be observed in simulation. We conduct numerical simulations to observe how the disease controlled in the community after the implementation of vaccination measures using different fractional values with different dimensions. Additionally, future estimates are provided based on our findings, which can help in mitigating the risk of the disease spreading in the environment.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to extend their sincere appreciation to Researchers Supporting Project number (RSP2024R472), King Saud University, Riyadh, Saudi Arabia.

    All authors declare no conflicts of interest in this paper.



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