# New Variant of SARS-CoV-2 Dynamics with Imperfect Vaccine

Authors: Taye Samuel Faniran; Aatif Ali; Nawal E. Al-Hazmi; Joshua Kiddy K. Asamoah; Taher A. Nofal; Matthew O. Adewole
Journal: Complexity
 (2022)

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## Abstract

The occurrence of a new strain of SARS-CoV-2 cannot be ruled out. Therefore, this study seeks to investigate the possible effects of a hypothetical imperfect anti-COVID-19 vaccine on the control of not only the first variant of SARS-CoV-2 but also the second (new) variant of SARS-CoV-2. We further examine the ratesr and a, escape of quarantined infectious individuals from isolation centers. The control Rc and basic reproduction numbers R0 are computed which gives assess to obtain asymptotic stability of disease-free equilibrium point globally and the existence of a unique persistent equilibrium solution. Numerical results reveal that people infected with the second strain who are vaccinated with an imperfect vaccine are under control but the prevalence of the second variant enhances the prevalence of the first variant. Thus, discovering a vaccine that is effective (to a good extent) for the prevention of variant 2 (new variant) is necessary for the control of COVID-19. Numerical results also reveal that increase in the rate at which individuals infected with the first variant escape the isolation center gives rise to the population infected with the first variant and lowers the peak of the population infected with the second variant. This is probably because individuals infected with the second variant appear to be more careful with their lives and get vaccinated more than individuals infected with the first variant.


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## Body

## 1. Introduction

Since 2019, COVID-19 has wreaked havoc in several nations, resulting in too many infections and deaths. It has resulted in several economic failures. There were also millions of confirmed COVID-19 cases and deaths worldwide. As a result of its high transmission efficiency and catastrophic infection outcomes, the disease continues to represent a hazard to human health. Coughing, sneezing, encountering sick persons, or touching things or surfaces contaminated with fecal traces are the major transmission pathways for the SARS-CoV-2 virus, according to [1]. According to [2], an imperfect vaccination is one that is unable to protect all vaccinated vulnerable persons. Vaccines must elicit an immune response similar to that elicited by a typical illness without producing the true infectious disease [3]. According to [3], there are three types of defective vaccines: the first is known as a leaky vaccine, which reduces infection but does not eliminate the risk of illness following exposure to an infectious disease. The second vaccination is an all-or-nothing vaccine that confers lifetime protection to certain individuals but does not protect others. The third vaccination, which is only effective for a limited time, is diminishing. Through transmission dynamics, many researchers have attempted to investigate and comprehend the dynamical behavior of infectious diseases [4–10].COVID-19 articles may be found in [11–14]. In [15, 16], some optimal control problems are also established. Moshen et al. [17] also developed a mathematical model of media coverage impacts on COVID-19 dissemination. Moreover, the equilibrium solution stability requirements were discovered. Finally, they used numerical simulations to corroborate the theoretical conclusions and better understand the impact of changing factors on COVID-19 dissemination. Their findings suggest that media attention might be a useful tool for reducing illness transmission. Hattaf et al. [18] created a mathematical model to describe the dynamics of viral infections in vivo, including HIV infection. Three common incidence functions are used to represent the physical behavior of viral infection. They thoroughly studied the model basic features as well as its stability.Hattaf et al. [18] modeled a novel within-host between SARS-CoV-2 and host pulmonary epithelial cells. Also, they developed a new generalized fractional derivative and stability [19, 20]. They included the lytic and nonlytic immune responses, as well as both mechanisms of virus-to-cell infection in the model. They proved that the model was well-posed and that equilibria existed. They looked at the model dynamical behavior using two threshold parameters. Finally, the biological implications of the analytical data were provided. A mathematical model was used by Iboi et al. [21] to estimate the influence of an imperfect anti-COVID-19 vaccination on COVID-19 control in the United States. The model theoretical analysis, as well as model fitting and parameter estimates, was completed. They used baseline parameter values acquired by fitting the model using COVID-19 mortality data for the United States to run numerical simulations of the model. Their findings revealed that an anti-COVID-19 vaccine would need to be administered to at least 82 percent of the vulnerable US population, with an estimated protective effect of 80 percent. They came to the conclusion that combining the vaccination program with other treatments such as social distance, wearing face masks, and so on will considerably improve the flawed vaccine. Olaniyi et al. [22] developed an epidemic model that took into account transmission paths from sick, asymptomatic, and hospitalized people. Using the Least Squares approach, their model was fitted to the equivalent cumulative number of hospitalized persons provided by the Nigeria Center for Disease Control (NCDC). To test the model stability around a disease-free equilibrium point, they estimated the basic reproduction number and built the Lyapunov function. They also used Pontryagin’s Maximum Principle to determine the model’s best control. They used numerical simulations to back up their theoretical work, and the results showed that if the disease’s current effective transmission rate is decreased by 50%, the basic reproduction number in Nigeria may be reduced to less than one. Using early report case data, Nkamba and Martin Luther [23] constructed and assessed a mathematical model of the COVID-19 epidemic in Cameroon to anticipate the peak and examine the influence of containment efforts and undiagnosed sick persons on the epidemic pattern and features of COVID-19. Using appropriate Lyapunov functions, they calculated the fundamental reproduction number and established the global stability of both the disease-free and endemic equilibria. They used a sensitivity analysis to determine the most important parameters in influencing the dynamics of their model. Given the scenario in Cameroon, their findings revealed that a sluggish quarantine rate will raise the peak of infected persons, which will expand with further delays. Iboi et al. performed mathematical modeling and analysis of the COVID-19 epidemic in Nigeria [24]. They used a locally asymptotically stability analysis to provide a thorough investigation of their model. The COVID-19 Nigeria data are taken from the Nigeria Center for Disease Control (NCDC) to parameterize the model. This was used to evaluate the community-wide effect of different control and mitigation techniques over the whole Nigerian nation, as well as in two Nigerian federation states (Kano and Lagos) and the Federal Capital Territory (FCT) of Abuja. In areas where social distance, lock-down, and other community transmission reduction measures are not adopted, their findings indicated that Nigeria will have a devastatingly high COVID-19 death rate by April 2021 (in the hundreds of thousands). Parra et al. [25] developed a mathematical model of COVID-19 to investigate the effects of the emergence of a new, more transmissible SARS-CoV-2 stain in a specific location. In their model, they also included presymptomatic and asymptomatic infected people. Using the next-generation matrix approach, they were able to derive the model fundamental reproduction number, R0. Furthermore, they proceeded the equilibria’s local and global stability. Numerical simulations of the impact of a novel more infectious SARS-COV-2 strain in a population backed up their theoretical conclusions. According to their findings, a new, more infectious SARS-CoV-2 variant will arise, and the existing form prevalence will diminish.Our work varies from previous COVID-19 research in the following ways. Anggriani et al. [11] developed and assessed a COVID-19 mathematical model that included the transmission path of asymptomatic and symptomatic compartments, as well as decreasing immunity, without taking into account the effect of vaccination on the novel SARS-CoV-2 variant. To guarantee that COVID-19 vaccinations give maximum protection in the future, researchers must investigate the influence of an incomplete anti-COVID-19 vaccine on the new variation. To analyze and monitor the COVID-19 epidemic, Youssef et al. [26] suggested a modified version of the Susceptible-Exposed-Infectious-Quarantined-Recovered (SEIQR) essential disease dynamics model for the COVID-19 emergence. They only examined the standard Susceptible-Exposed-Infectious-Quarantined-Recovered of COVID-19, ignoring the possibility of a new SARS-CoV-2 variation and the impact of the present anti-COVID-19 vaccination on the new variety. Parra et al. [25] proposed an ordinary differential equation-based model to investigate a new more transmissible SARS-CoV-2 strain through the transmission route of exposed, presymptomatic, symptomatic, and asymptomatic classes, but they did not look into the impact of the COVID-19 vaccine on the new more transmissible SARS-CoV-2 strain. The absence of vaccinated persons in the presence of the new strain is costly since it is necessary to determine if the current COVID-19 vaccination protects against the new strain. As a result, the current study will fill in all of the gaps mentioned above.SARS-CoV-2 has recently been discovered in various countries with a novel strain that offers a greater hazard to humans. The current study investigates the impact of an imperfect vaccine on the dynamics of COVID-19 with two SARS-CoV-2 variants, which was inspired by a paper [25], in which they proposed two SARS-CoV-2 variants and presymptomatic infectious individuals on the dynamics of COVID-19. This distinguishes our work from that of [25]. The unique component of this work is modeling a new version of SARS-CoV-2 transmission dynamics in order to analyze the influence of a hypothetical imperfect anti-COVID-19 vaccination on the control of both the first and second variants of SARS-CoV-2. It is critical and necessary to determine how well the current antigen-specific COVID-19 vaccine protects individuals against the new variant so that scientists can determine whether an update to the current antigen-specific anti-COVID-19 vaccine is required to ensure that the COVID-19 vaccine continues to provide optimal protection as new antigenically distinct variants emerge in the future. Furthermore, we include the influence of the escape rate of quarantined infected persons from isolation facilities in Nigeria in our model owing to fears or beliefs that the government is utilizing the epidemic to enrich its pockets in Nigeria [27]. None of the most current and most recent publications on COVID-19, such as [11, 25, 26, 28–30], have taken these specific elements into account. Individuals should be aware that the COVID-19 vaccine does not provide 100 percent protection since immunization is one of the most effective advancements in disease prevention. On the other hand, it is crucial that everyone (with the exception of individuals with severe allergy diseases) obtains a vaccine. As a result, we develop a mathematical model that accounts for poor vaccination on both SARS-CoV-2 types. To support the analytical conclusions, a global study of the model is performed and numerical simulations are shown.The research aims and importance of the study are highlighted in the following sections based on the above.


### 1.1. Objectives

(1)
To explore the impact of the new variant of SARS-CoV-2 in a population, in the presence of an imperfect vaccine(2)
To examine the behavior of solution trajectories of the model when the new variant transmission rate of SARS-CoV-2 is higher than the pre-existing one [31](3)
To determine the impact of escape of quarantined infectious individuals from isolation centers


### 1.2. Significance

(1)
The study will help the health authorities to know the significance of imperfect vaccination on the transmission dynamics in the presence of a new variant of SARS-CoV-2.(2)
The study will help to know whether vaccinating the population against the new variant of SARS-CoV-2 plays a role in the control reproduction number.(3)
The evaluation of analytical results will provide useful information that will help to eradicate COVID-19 in the population.(4)
The study will help the policymakers on health to have a clear picture of the impact of imperfect vaccination against the new variant.(5)
The research will aid government officials in recognizing the risk of COVID-19 patients fleeing isolation facilities and developing strategies to educate them in their different local languages and dialects about COVID-19 and the repercussions of refusing the government free treatment.


## 1.1. Objectives

(1)
To explore the impact of the new variant of SARS-CoV-2 in a population, in the presence of an imperfect vaccine(2)
To examine the behavior of solution trajectories of the model when the new variant transmission rate of SARS-CoV-2 is higher than the pre-existing one [31](3)
To determine the impact of escape of quarantined infectious individuals from isolation centers


## 1.2. Significance

(1)
The study will help the health authorities to know the significance of imperfect vaccination on the transmission dynamics in the presence of a new variant of SARS-CoV-2.(2)
The study will help to know whether vaccinating the population against the new variant of SARS-CoV-2 plays a role in the control reproduction number.(3)
The evaluation of analytical results will provide useful information that will help to eradicate COVID-19 in the population.(4)
The study will help the policymakers on health to have a clear picture of the impact of imperfect vaccination against the new variant.(5)
The research will aid government officials in recognizing the risk of COVID-19 patients fleeing isolation facilities and developing strategies to educate them in their different local languages and dialects about COVID-19 and the repercussions of refusing the government free treatment.


## 2. Mathematical Formulation of COVID-19 Model

This section describes the transmission dynamics of COVID-19, and the formulated model is presented, respectively. The total individual population, denoted byN, is divided into nine categories: susceptible individuals S, vaccinated individuals V (individuals who get vaccinated), two classes of exposed individuals E1,2, two classes of quarantined infectious individuals Q1,2, two classes of unquarantined infectious individuals U1,2, and recovered individuals R, so that N=S+V+E1+E2+Q1+Q2+U1+U2+R.The susceptible individuals population is generated by the recruitment of individuals into the population, either by birth or immigration, at the rateΛ. We assume that individuals would belong to one of the classes described above, depending on the COVID-19 status [25]. We also assume that vaccine is not perfect, and therefore vaccinated individuals may contact the virus at the rates α1−ψ1 and η1−ψ2, where ψ1∈0,1 and ψ2∈0,1 represent the vaccine efficacy against variant 1 and variant 2. Furthermore, it is noteworthy that ψ1=1 and ψ2=1 imply that a vaccine provides 100% assurance against COVID-19, while ψ1=0 and ψ2=0 indicate a vaccine that does not secure individuals in any way. Furthermore, because the vaccine was designed to protect against variant 1, we assume that its efficacy against variant 2 is less than its efficacy against variant 1 (i.e., ψ2<ψ1). It is also assumed that someone who is infected with the first variant of SARS-CoV-2 cannot catch the second variant (the new variant) due to an acquired immune response [32, 33]. There is a progression rate from quarantined infectious individual compartments to unquarantined infectious individual compartments due to the probability of fleeing isolation centers. It is assumed that quarantined infectious individuals are not severely sick (i.e., they are moderately or mildly sick). By assumption, there is no loss of immunity [25].Applying the biological assumptions, nomenclature of parameters in Table1, the following system is formulated:(1)dSdt=Λ−αSU1−ηSU2−κ+μS,dVdt=κS−α1−ψ1VU1−η1−ψ2VU2−μV,dE1dt=αSU1+α1−ψ1VU1−β+μE1,dQ1dt=1−θβE1−r+τ1+μ+δ1Q1,dU1dt=θβE1+rQ1−τ2+δ2+μU1,dE2dt=ηSU2+η1−ψ2VU2−p+μE2,dQ2dt=1−bpE2−a+c1+μ+δ3Q2,dU2dt=bpE2+aQ2−c2+μ+δ4U2,dRdt=τ1Q1+τ2U1+c1Q2+c2U2−μR,with initial conditions such as(2)S0=S0>0,V0=V0>0,E10=E10>0,Q10=Q10>0,U10=U10>0,E20=E20>0,Q20=Q20>0,U20=U20>0,R0=R0>0,where the model parameters are nonnegative. For biological reasons, the model is analyzed in the feasible-region as follows:(3)D=S,V,E1,Q1,U1,E2,Q2,U2,R∈ℜ+9:0≤S≤Λμ+κ,0≤V≤κΛμμ+κ,N≤Λμ,that shows system (1) is to be positively invariant. Thus, the model is well-posed both mathematically and epidemiologically in D. The first eight equations of system (1) do not affect the compartments R, and therefore we have(4)dSdt=Λ−αSU1−ηSU2−κ+μS,dVdt=κS−α1−ψ1VU1−η1−ψ2VU2−μV,dE1dt=αSU1+α1−ψ1VU1−β+μE1,dQ1dt=1−θβE1−r+τ1+μ+δ1Q1,dU1dt=θβE1+rQ1−τ2+δ2+μU1,dE2dt=ηSU2+η1−ψ2VU2−p+μE2,dQ2dt=1−bpE2−a+c1+μ+δ3Q2,dU2dt=bpE2+aQ2−c2+μ+δ4U2.Table 1
Summary of the parameters.
ParameterMeaningValueReferenceN0Initial population size500,000AssumedαContact rate between S andU10.5/N0day−1[24]ηContact rate between S andU20.65/N0day−1[24]θFraction ofE1 who are unquarantined infectious0.3AssumedbFraction ofE2 who are unquarantined infectious0.3Assumedτ1Recovery rate of quarantined infectious individuals0.0815day−1[34]τ2Recovery rate of unquarantined infectious individuals1/14day−1[35]c1Recovery rate ofQ20.0815day−1[34]c2Recovery rate ofU21/14day−1[35]rProbability of fleeing isolation centers ofQ10.01day−1AssumedaProbability of fleeing isolation centers ofQ20.01day−1AssumedκVaccination rate1/42day−1AssumedΛRecruitment rateN0μday−1AssumedμNatural death rate0.01186year−1[36, 37]βIncubation rate of SARS-CoV-20.142day−1[23]pIncubation rate of new variant of SARS-CoV-20.0152day−1Assumedδ1Disease-induced death rate ofQ10.015[24]δ2Disease-induced death rate ofU10.015[24]δ3Disease-induced death rate ofQ20.02day−1Assumedδ4Disease-induced death rate ofU20.02day−1Assumedψ1Vaccine efficacy against variant 10.1−0.7Assumedψ2Vaccine efficacy against variant 20.1−0.65Assumed


### 2.1. Basic Reproduction Number

To determine the overall dynamical behavior of the equilibrium solution, the control reproduction number,Rc, will be used. Rc is computed as the difference between the rate of new infection in each infected compartment F and the rate of transfer between each infected compartment G using the approach [38]. As a result, we have(5)dE1dtdQ1dtdU1dtdE2dtdQ2dtdU2dt=F−G=αSU1+α1−ψ1VU100ηSU2+η1−ψ2VU200−β+μE1−1−θβE1+r+m1Q1−θβE1−rQ1+m2U1p+μE2−1−bpE2+a+m3Q2−bpE2−aQ2+m4U2,where(6)m1=τ1+μ+δ1,m2=τ2+μ+δ2,m3=c1+μ+δ3,m4=c2+μ+δ4.The disease-free equilibrium is(7)W0=S0,V0,E10,Q10,U10,E20,Q20,U20=Λμ+κ,κΛμμ+κ,0,0,0,0,0,0.Consequently,Rc for the first and second variants is given as(8)Rc=maxRFVc,RSVc,where(9)RFVc=Λβαθm1+rμ+1−ψ1κm2μβ+μm1+rκ+μ,and(10)RSVc=Λpηbm3+aμ+1−ψ2κm4μp+μm3+aκ+μ.Rc is used as an invasion threshold for both predicting COVID-19 outbreaks and assessing control strategies that would reduce COVID-19 spread in the community by lowering Rc and parameters that would increase disease spread by increasing Rc. When control methods are ineffectual, the value of R0 may be calculated from the value of Rc, in the sense that ψ1=0, ψ2=0, and κ=0. As a result, without any control mechanisms, the basic reproduction number is(11)R0=maxRFV,RSV,where(12)RFV=Λβαθm1+rm2μβ+μm1+r,and(13)RSV=Λpηbm4+am5μp+μm4+a.However, our analysis will be based on the control reproduction number,Rc. Furthermore, it is assumed that the population is completely susceptible initially, and Rc measures the average secondary cases at the beginning of the epidemic.


## 2.1. Basic Reproduction Number

To determine the overall dynamical behavior of the equilibrium solution, the control reproduction number,Rc, will be used. Rc is computed as the difference between the rate of new infection in each infected compartment F and the rate of transfer between each infected compartment G using the approach [38]. As a result, we have(5)dE1dtdQ1dtdU1dtdE2dtdQ2dtdU2dt=F−G=αSU1+α1−ψ1VU100ηSU2+η1−ψ2VU200−β+μE1−1−θβE1+r+m1Q1−θβE1−rQ1+m2U1p+μE2−1−bpE2+a+m3Q2−bpE2−aQ2+m4U2,where(6)m1=τ1+μ+δ1,m2=τ2+μ+δ2,m3=c1+μ+δ3,m4=c2+μ+δ4.The disease-free equilibrium is(7)W0=S0,V0,E10,Q10,U10,E20,Q20,U20=Λμ+κ,κΛμμ+κ,0,0,0,0,0,0.Consequently,Rc for the first and second variants is given as(8)Rc=maxRFVc,RSVc,where(9)RFVc=Λβαθm1+rμ+1−ψ1κm2μβ+μm1+rκ+μ,and(10)RSVc=Λpηbm3+aμ+1−ψ2κm4μp+μm3+aκ+μ.Rc is used as an invasion threshold for both predicting COVID-19 outbreaks and assessing control strategies that would reduce COVID-19 spread in the community by lowering Rc and parameters that would increase disease spread by increasing Rc. When control methods are ineffectual, the value of R0 may be calculated from the value of Rc, in the sense that ψ1=0, ψ2=0, and κ=0. As a result, without any control mechanisms, the basic reproduction number is(11)R0=maxRFV,RSV,where(12)RFV=Λβαθm1+rm2μβ+μm1+r,and(13)RSV=Λpηbm4+am5μp+μm4+a.However, our analysis will be based on the control reproduction number,Rc. Furthermore, it is assumed that the population is completely susceptible initially, and Rc measures the average secondary cases at the beginning of the epidemic.


## 3. Stability Analysis

The global asymptotic stability for the disease-free equilibrium is examined by following a positive definite Lyapunov function. See [22, 23, 39] for further information on the creation of the Lyapunov function.


### 3.1. Global Stability of Disease-Free Equilibrium

Theorem 1.
The disease-free equilibrium is globally asymptotically stable ifRc<1.Proof.
Consider the Lyapunov function as follows:(14)L=βθm1+rE1+rβ+μQ1+β+μm1+rU1+pbm3+aE2+ap+μQ2+p+μm3+aU2.
The time derivative of (10) along the COVID-19 model (4) is(15)L˙=βθm1+rαSU1+α1−ψ1VU1−β+μE1+rβ+μ1−θβE1−m1+rQ1+β+μm1+rθβE1+rQ1−m2U1+pbm3+aηSU2+η1−ψ2VU2−p+μE2+ap+μ1−bpE2−a+m3Q2+p+μm3+abpE2+aQ2−m4U2,=βθm1+rαSU1+α1−ψ1VU1−βθm1+rβ+μE1+rβ+μβE1−rβ+μθβE1−β+μm1+rm2U1+pbm3+aηSU2+η1−ψ2VU2−pbm4+ap+μE2+ap+μpE2−ap+μbpE2−p+μm3+am4U2,=βθm1+rαSU1+α1−ψ1VU1−βθm1+rβ+μE1+rβ+μ1−θβE1+β+μm1+rθβE1−β+μm1+rm2U1+pbm3+aηSU2+η1−ψ2VU2−pbm3+ap+μE2+ap+μ1−bpE2+p+μm4+abpE2−p+μm3+am4U2,=βθm1+rαSU1+α1−ψ1VU1−rβE1β+μ1−θ+rβ+μ1−θβE1−β+μm1+rm2U1+pbm3+a+ηSU2+η1−ψ2VU2−apE2p+μ1−b+ap+μ1−bpE2−p+μm3+am4U2,=β+μm1+rm2U1Λβαθm1+rμ+1−ψ1κm2μβ+μm1+rκ+μ−1+p+μm3+am4U2Λpηbm3+aμ+1−ψ2κm4μp+μm3+aκ+μ−1,=m2β+μm1+rRFVc−1U1+m5ρ+μm3+aRSVc−1U2,≤m2β+μm1+rU1+m4ρ+μm3+aU2Rc−1.
Furthermore,L˙=0↔Q1=0,U1=0 and Q2=0,U2=0 or S=S0, V=V0, and Rc=1.
Thus, the singleton set is the biggest compact invariant set inS,;V,;E1,;Q1,;U1,;E2,;Q2,;U2∈D:L˙=0, and W0 is globally asymptotically stable in D according to LaSalle’s Invariance Principle [40]. The aforementioned result has the epidemiological implication that the starting sizes of the model’s subpopulation (i.e., the initial number of sick persons) introduced into the community do not have to be within the model’s disease-free equilibrium W0 base of attraction. For Rc<1, this suggests that COVID-19 can be eradicated regardless of the original size of the infectious population.


### 3.2. Existence and Uniqueness of Endemic Equilibrium Solution

We will show that the formulated COVID-19 model contains an endemic equilibrium point,W1. The COVID-19 infection continues in the population at the endemic equilibrium point, which is a positive steady-state solution. The solution trajectories of the model are analyzed; when the transmission routes of the second variant of SARS-CoV-2 are higher than the first variant [25], we establish the existence of a single endemic equilibrium point if the condition ηU2∗>αU1∗ and then E1∗=U1∗=0 hold. The impact of the second SARS-CoV-2 variant on the dynamics of COVID-19 in light of defective vaccination in particular countries at the time of advance is determined analytically. The following are the equations:(16)Λ−αS∗U1∗−ηS∗U2∗−κ+μS∗=0,κS∗−α1−ψ1V∗U1∗−η1−ψ2V∗U2∗−μV∗=0,1−bpE2∗−a+c1+μ+δ3Q2∗=0,bpE2∗+aQ2∗−c2+μ+δ4U2∗=0.LetW1=S∗,V∗,E2∗,Q2∗ where S∗,V∗,E2∗, and Q2∗ can be found from (16) to be(17)S∗=ΛαU1∗+ηU2∗+κ+μ,V∗=ΛκαU1∗+ηU2∗+κ+μα1−ψ1U1∗+η1−ψ2U2∗+μ,E2∗=a+m3Q2∗1−bp,Q2∗=m41−bpm3+am3+apa+bm3U2∗.The following equations forU1∗ and U2∗ are obtained after substituting (16) into (17),(18)ηΛα1−ψ1U1∗+η1−ψ2U2∗+μ+κ1−ψ2αU1∗+ηU2∗+κ+μα1−ψ1U1∗+η1−ψ2U2∗+μ−p+μm4a+m3pbm3+a=0.If we let(19)fx1,x2=ηΛα1−ψ1x1+η1−ψ2x2+μ+κ1−ψ2αx1+ηx2+κ+μα1−ψ1x1+η1−ψ2x2+μ−p+μm4a+m3pbm3+a,so that f˙x1,x2<0, lima⟶+∞, fx1,x2<0, and(20)f0=m4p+μm3+apa+bm3RSVc−1.In the sense that ifRSVc>1, then f0>0. Thus, for the equation fa=0, a unique positive solution exists only when RSVc>1.Theorem 2.
Model (4) has a unique W1 whenever RSVc>maxRFVc,1.


### 3.3. Global Stability of Endemic Equilibrium SolutionW1

In order to show that COVID-19 in the population persists and converges toW1, the following result is stated and proved.Theorem 3.
The unique equilibriumW1 is globally asymptotically stable (GAS) whenever Rc>1.Proof.
Given the following equations at the endemic equilibrium pointW1:(21)Λ=μS∗+μV∗+αS∗U1∗+ηS∗U2∗+αU1∗1−ψ1V∗+ηU2∗1−ψ2V∗,κS∗=α1−ψ1V∗U1∗+η1−ψ2V∗U2∗+μV∗,β+μE1∗=αS∗U1∗+α1−ψ1V∗U1∗,p+μE2∗=ηS∗U2∗+η1−ψ2V∗U2∗,Q1∗=1−θμU1∗θμ+r,Q2∗=1−bμU2∗bμ+a.
Let(22)X=S∗,V∗,E1∗,E2∗,Q1∗,Q2∗,U1∗,U2∗T∈ℜ+8.
And consider the following Lyapunov function:(23)Vx=S−S∗−S∗lnSS∗+V−V∗−V∗lnVV∗+E1−E1∗−E1∗lnE1E1∗+d1Q1−Q1∗−Q1∗lnQ1Q1∗+d2Q2−Q2∗−Q2∗lnQ2Q2∗+f1U1−U1∗−U1∗lnU1U1∗+f2U2−U2∗−U2∗lnU2U2∗,where(24)d1=αrS∗+1−ψ1V∗μμ+r,d2=ηaS∗+1−ψ2V∗μ,f1=αS∗+1−ψ1V∗μ,f2=ηS∗+1−ψ2V∗μ.
DifferentiatingV with respect to time gives(25)V˙x=1−S∗SS˙+1−V∗VV˙+1−E1∗E1E˙1+1−E2∗E2E˙2+d11−Q1∗Q1Q˙1+d21−Q2∗Q2Q˙2+f11−U1∗U1U˙1+f21−U2∗U2U˙2.V˙x=1−S∗SΛ−αSU1−ηSU2−κ+μS+1−V∗VκS−α1−ψ1VU1−η1−ψ2VU2−μV+1−E1∗E1αSU1+α1−ψ1VU1−β+μE1+1−E2∗E2ηSU2+η1−ψ2VU2−p+μE2+d11−Q1∗Q11−θβE1−r+m1Q1+d21−Q2∗Q21−bpE2−a+m3Q2+f11−U1∗U1θβE1+rQ1−m2U1+f21−U2∗U2bpE2+aQ2−m4U2.
After developing and using (21), we have the following:(26)V˙x=−μSS−S∗2+αU1∗S∗+1−ψ1V∗+ηU2∗S∗+1−ψ2V∗+κS∗+μV∗−μV−κSV∗V+d11−θβ+f1θβE1−μ−βE1+d21−bp+f2bpE2−μ−pE2−αSU1E1∗E1−1−ψ1αVU1E1∗E1−1−ψ2ηVU2E2∗E2+μ+βE1∗−p+μE2∗+f1r−d1μ+rQ1+f2a−d2μ+aQ2−d11−θβE1Q1∗Q1−d21−bpE2Q2∗Q2+d1μ+rQ1∗+d2μ+aQ2∗+−f1μ+αU1∗+1−ψ1αV∗U1+−f2μ+ηU2∗+1−ψ2ηV∗U2−f1θβE1U1∗U1−f2pbE2U∗U2−f1rQ1U1∗U1−f2aQ2U2∗U2+f1μU1∗+f2μU2∗.
Substitutingd1, d2, f1, and f2 by their values and exploiting, we have(27)V˙x=−μSS−S∗2+μV∗3−S∗S−VV∗−SV∗S∗V+αU1∗S∗3+r1−θθμ+r+ηU2∗S∗3+a1−bbμ+a+1−ψ1αU1∗V∗4+r1−θθμ+r+1−ψ2ηU2∗V∗4+a1−bbμ+a−ηSU2E2∗E2−αSU1E1∗E1−1−ψ1αVU1E1∗E1−1−ψ2ηVU2E2∗E2−1−ψ1αSU1∗V∗2S∗V∗−1−ψ2ηSU2∗V∗2S∗V∗−αr1−θS∗+1−ψ1V∗U1∗θμ+r−ηa1−bS∗+1−ψ2V∗U2∗bμ+a−ηaS∗+1−ψ2V∗Q2U2∗μU2−αθβS∗+1−ψ1V∗E1U1∗μU1−ηbpS∗+1−ψ2V∗E2U2∗μU2−αrS∗+1−ψ1V∗Q1U1∗μU1.
Define(28)g=1−θrθμ+r,k=θμ+rθμ+r,h=1−babμ+a,s=bμ+abμ+a.
Then,(29)g+h=1,k+s=1,3+g=3g+3k+g,=3k+4g,3+h=3h+3s+h,=3s+4h,4+g=4k+5g4+h=4s+5h.
V˙x can be rewritten as(30)V˙x=−μSS−S∗2+μV∗3−S∗S−VV∗−SV∗S∗V+3h+4gαS∗U1∗+3s+4kηS∗U2∗+4h+5g1−ψ1αV∗U1∗+4s+5k1−ψ2ηV∗U2∗−αSU1E1∗E1−ηSU2E2∗E2−1−ψ1αVU1E1∗E1−1−ψ2ηVU2E2∗E2−1−ψ1αSU1∗V∗2S∗V∗−1−ψ2ηSU2∗V∗2S∗V∗−αr1−θS∗+1−ψ1V∗U1∗θμ+r−ηa1−bS∗+1−ψ2V∗U2∗bμ+a−αθβS∗+1−ψ1V∗E1U1∗μU1−ηbpS∗+1−ψ2V∗E2U2∗μU2−αrS∗+1−ψ1V∗Q1U1∗μU1−ηaS∗+1−ψ2V∗Q2U2∗μU2,V˙x=−μSS−S∗2+μV∗3−S∗S−VV∗−SV∗S∗V+kαS∗U1∗3−S∗S−SU1E1∗S∗U2∗E1−θβE1kμU1+sηS∗U2∗3−S∗S−SU2E2∗S∗U2∗E2−bpE2sμU2+gαS∗U1∗4−S∗S−SU1E1∗S∗U1∗E1−rQ1gμU1−r1−θβE1Q1∗gμμ+rU1∗Q1+hηS∗U2∗4−S∗S−SU2E2∗S∗U2∗E2−aQ2hμU2−a1−bpE2Q2∗hμμ+aU2∗Q2+k1−ψ1αV∗U1∗4−S∗S−VU1E1∗V∗U1∗E1−θβE1kμQ1−SV∗S∗V+s1−ψ2αV∗U2∗4−S∗S−VU2E2∗V∗U2∗E2−bpE2sμQ2−SV∗S∗V+g1−ψ1αV∗U1∗5−S∗S−VU1E1∗V∗U1∗E1−SV∗S∗V−r1−θβQ1∗E1gμμ+rU1∗Q1−rQ1gμU1+h1−ψ2ηV∗U2∗5−S∗S−VU2E2∗V∗U2∗E2−SV∗S∗V−a1−bpQ2∗E2hμμ+aU2∗Q2−aQ2hμU2.
By arithmetic-geometric means inequality, i.e.,n−m1+m2+⋯+mn≤0, where m1.m2…mn=1 and m1,m2,…,mn>0, it follows that V˙≤0 with V=0 if and only if S=S∗, V=V∗, E1=E1∗, E2=E2∗, Q1=Q1∗, Q2=Q2∗, U1=U1∗, and U2=U2∗. Hence, W1 is said to be GAS which follows LaSalle’s Invariance Principle [12].


### 3.4. Sensitivity and Uncertainty Analysis

Figures1 and 2 show the sensitivity and uncertainty analysis SA/UA of the obtained reproduction numbers for both variants with 2000 samples (up to 5 simulations for Figure 1 and a single simulation for Figure 2), getting a reliability indicator of 100%. From the SA in Figure 1(a), it is noticeable that the parameters ψ1,μ, and τ1 contribute most to the disease reduction while the parameters Λ,θ, and α contribute to the spread of the disease. On the other hand, the UA in Figure 1(b) shows that the range of the RFVc is approximately 0.4−1.8, though most of the outputs are concentrated in low values 0.4−1.1. From the SA in Figure 2(a), it is noticeable that the parameters ψ2,μ, and c2 contribute most to the disease reduction while the parameters b,a,η,p, and Λ contribute to the spread of the disease. On the other hand, the UA in Figure 2(b) shows that the range ofRSVc is approximately 0.2−2, and most of the outputs are concentrated in low values 0.2−1.1.Figure 1
Global sensitivity and uncertainty analysis of variant 1. (a) Partial rank correlation coefficients (PRCC) for variant 1. (b) The box plot of variant 1.
(a)(b)Figure 2
Global sensitivity and uncertainty analysis of variant 2. (a) Partial rank correlation coefficients (PRCC) for variant 2. (b) The box plot of variant 2.
(a)(b)


## 3.1. Global Stability of Disease-Free Equilibrium

Theorem 1.
The disease-free equilibrium is globally asymptotically stable ifRc<1.Proof.
Consider the Lyapunov function as follows:(14)L=βθm1+rE1+rβ+μQ1+β+μm1+rU1+pbm3+aE2+ap+μQ2+p+μm3+aU2.
The time derivative of (10) along the COVID-19 model (4) is(15)L˙=βθm1+rαSU1+α1−ψ1VU1−β+μE1+rβ+μ1−θβE1−m1+rQ1+β+μm1+rθβE1+rQ1−m2U1+pbm3+aηSU2+η1−ψ2VU2−p+μE2+ap+μ1−bpE2−a+m3Q2+p+μm3+abpE2+aQ2−m4U2,=βθm1+rαSU1+α1−ψ1VU1−βθm1+rβ+μE1+rβ+μβE1−rβ+μθβE1−β+μm1+rm2U1+pbm3+aηSU2+η1−ψ2VU2−pbm4+ap+μE2+ap+μpE2−ap+μbpE2−p+μm3+am4U2,=βθm1+rαSU1+α1−ψ1VU1−βθm1+rβ+μE1+rβ+μ1−θβE1+β+μm1+rθβE1−β+μm1+rm2U1+pbm3+aηSU2+η1−ψ2VU2−pbm3+ap+μE2+ap+μ1−bpE2+p+μm4+abpE2−p+μm3+am4U2,=βθm1+rαSU1+α1−ψ1VU1−rβE1β+μ1−θ+rβ+μ1−θβE1−β+μm1+rm2U1+pbm3+a+ηSU2+η1−ψ2VU2−apE2p+μ1−b+ap+μ1−bpE2−p+μm3+am4U2,=β+μm1+rm2U1Λβαθm1+rμ+1−ψ1κm2μβ+μm1+rκ+μ−1+p+μm3+am4U2Λpηbm3+aμ+1−ψ2κm4μp+μm3+aκ+μ−1,=m2β+μm1+rRFVc−1U1+m5ρ+μm3+aRSVc−1U2,≤m2β+μm1+rU1+m4ρ+μm3+aU2Rc−1.
Furthermore,L˙=0↔Q1=0,U1=0 and Q2=0,U2=0 or S=S0, V=V0, and Rc=1.
Thus, the singleton set is the biggest compact invariant set inS,;V,;E1,;Q1,;U1,;E2,;Q2,;U2∈D:L˙=0, and W0 is globally asymptotically stable in D according to LaSalle’s Invariance Principle [40]. The aforementioned result has the epidemiological implication that the starting sizes of the model’s subpopulation (i.e., the initial number of sick persons) introduced into the community do not have to be within the model’s disease-free equilibrium W0 base of attraction. For Rc<1, this suggests that COVID-19 can be eradicated regardless of the original size of the infectious population.


## 3.2. Existence and Uniqueness of Endemic Equilibrium Solution

We will show that the formulated COVID-19 model contains an endemic equilibrium point,W1. The COVID-19 infection continues in the population at the endemic equilibrium point, which is a positive steady-state solution. The solution trajectories of the model are analyzed; when the transmission routes of the second variant of SARS-CoV-2 are higher than the first variant [25], we establish the existence of a single endemic equilibrium point if the condition ηU2∗>αU1∗ and then E1∗=U1∗=0 hold. The impact of the second SARS-CoV-2 variant on the dynamics of COVID-19 in light of defective vaccination in particular countries at the time of advance is determined analytically. The following are the equations:(16)Λ−αS∗U1∗−ηS∗U2∗−κ+μS∗=0,κS∗−α1−ψ1V∗U1∗−η1−ψ2V∗U2∗−μV∗=0,1−bpE2∗−a+c1+μ+δ3Q2∗=0,bpE2∗+aQ2∗−c2+μ+δ4U2∗=0.LetW1=S∗,V∗,E2∗,Q2∗ where S∗,V∗,E2∗, and Q2∗ can be found from (16) to be(17)S∗=ΛαU1∗+ηU2∗+κ+μ,V∗=ΛκαU1∗+ηU2∗+κ+μα1−ψ1U1∗+η1−ψ2U2∗+μ,E2∗=a+m3Q2∗1−bp,Q2∗=m41−bpm3+am3+apa+bm3U2∗.The following equations forU1∗ and U2∗ are obtained after substituting (16) into (17),(18)ηΛα1−ψ1U1∗+η1−ψ2U2∗+μ+κ1−ψ2αU1∗+ηU2∗+κ+μα1−ψ1U1∗+η1−ψ2U2∗+μ−p+μm4a+m3pbm3+a=0.If we let(19)fx1,x2=ηΛα1−ψ1x1+η1−ψ2x2+μ+κ1−ψ2αx1+ηx2+κ+μα1−ψ1x1+η1−ψ2x2+μ−p+μm4a+m3pbm3+a,so that f˙x1,x2<0, lima⟶+∞, fx1,x2<0, and(20)f0=m4p+μm3+apa+bm3RSVc−1.In the sense that ifRSVc>1, then f0>0. Thus, for the equation fa=0, a unique positive solution exists only when RSVc>1.Theorem 2.
Model (4) has a unique W1 whenever RSVc>maxRFVc,1.


## 3.3. Global Stability of Endemic Equilibrium SolutionW1

In order to show that COVID-19 in the population persists and converges toW1, the following result is stated and proved.Theorem 3.
The unique equilibriumW1 is globally asymptotically stable (GAS) whenever Rc>1.Proof.
Given the following equations at the endemic equilibrium pointW1:(21)Λ=μS∗+μV∗+αS∗U1∗+ηS∗U2∗+αU1∗1−ψ1V∗+ηU2∗1−ψ2V∗,κS∗=α1−ψ1V∗U1∗+η1−ψ2V∗U2∗+μV∗,β+μE1∗=αS∗U1∗+α1−ψ1V∗U1∗,p+μE2∗=ηS∗U2∗+η1−ψ2V∗U2∗,Q1∗=1−θμU1∗θμ+r,Q2∗=1−bμU2∗bμ+a.
Let(22)X=S∗,V∗,E1∗,E2∗,Q1∗,Q2∗,U1∗,U2∗T∈ℜ+8.
And consider the following Lyapunov function:(23)Vx=S−S∗−S∗lnSS∗+V−V∗−V∗lnVV∗+E1−E1∗−E1∗lnE1E1∗+d1Q1−Q1∗−Q1∗lnQ1Q1∗+d2Q2−Q2∗−Q2∗lnQ2Q2∗+f1U1−U1∗−U1∗lnU1U1∗+f2U2−U2∗−U2∗lnU2U2∗,where(24)d1=αrS∗+1−ψ1V∗μμ+r,d2=ηaS∗+1−ψ2V∗μ,f1=αS∗+1−ψ1V∗μ,f2=ηS∗+1−ψ2V∗μ.
DifferentiatingV with respect to time gives(25)V˙x=1−S∗SS˙+1−V∗VV˙+1−E1∗E1E˙1+1−E2∗E2E˙2+d11−Q1∗Q1Q˙1+d21−Q2∗Q2Q˙2+f11−U1∗U1U˙1+f21−U2∗U2U˙2.V˙x=1−S∗SΛ−αSU1−ηSU2−κ+μS+1−V∗VκS−α1−ψ1VU1−η1−ψ2VU2−μV+1−E1∗E1αSU1+α1−ψ1VU1−β+μE1+1−E2∗E2ηSU2+η1−ψ2VU2−p+μE2+d11−Q1∗Q11−θβE1−r+m1Q1+d21−Q2∗Q21−bpE2−a+m3Q2+f11−U1∗U1θβE1+rQ1−m2U1+f21−U2∗U2bpE2+aQ2−m4U2.
After developing and using (21), we have the following:(26)V˙x=−μSS−S∗2+αU1∗S∗+1−ψ1V∗+ηU2∗S∗+1−ψ2V∗+κS∗+μV∗−μV−κSV∗V+d11−θβ+f1θβE1−μ−βE1+d21−bp+f2bpE2−μ−pE2−αSU1E1∗E1−1−ψ1αVU1E1∗E1−1−ψ2ηVU2E2∗E2+μ+βE1∗−p+μE2∗+f1r−d1μ+rQ1+f2a−d2μ+aQ2−d11−θβE1Q1∗Q1−d21−bpE2Q2∗Q2+d1μ+rQ1∗+d2μ+aQ2∗+−f1μ+αU1∗+1−ψ1αV∗U1+−f2μ+ηU2∗+1−ψ2ηV∗U2−f1θβE1U1∗U1−f2pbE2U∗U2−f1rQ1U1∗U1−f2aQ2U2∗U2+f1μU1∗+f2μU2∗.
Substitutingd1, d2, f1, and f2 by their values and exploiting, we have(27)V˙x=−μSS−S∗2+μV∗3−S∗S−VV∗−SV∗S∗V+αU1∗S∗3+r1−θθμ+r+ηU2∗S∗3+a1−bbμ+a+1−ψ1αU1∗V∗4+r1−θθμ+r+1−ψ2ηU2∗V∗4+a1−bbμ+a−ηSU2E2∗E2−αSU1E1∗E1−1−ψ1αVU1E1∗E1−1−ψ2ηVU2E2∗E2−1−ψ1αSU1∗V∗2S∗V∗−1−ψ2ηSU2∗V∗2S∗V∗−αr1−θS∗+1−ψ1V∗U1∗θμ+r−ηa1−bS∗+1−ψ2V∗U2∗bμ+a−ηaS∗+1−ψ2V∗Q2U2∗μU2−αθβS∗+1−ψ1V∗E1U1∗μU1−ηbpS∗+1−ψ2V∗E2U2∗μU2−αrS∗+1−ψ1V∗Q1U1∗μU1.
Define(28)g=1−θrθμ+r,k=θμ+rθμ+r,h=1−babμ+a,s=bμ+abμ+a.
Then,(29)g+h=1,k+s=1,3+g=3g+3k+g,=3k+4g,3+h=3h+3s+h,=3s+4h,4+g=4k+5g4+h=4s+5h.
V˙x can be rewritten as(30)V˙x=−μSS−S∗2+μV∗3−S∗S−VV∗−SV∗S∗V+3h+4gαS∗U1∗+3s+4kηS∗U2∗+4h+5g1−ψ1αV∗U1∗+4s+5k1−ψ2ηV∗U2∗−αSU1E1∗E1−ηSU2E2∗E2−1−ψ1αVU1E1∗E1−1−ψ2ηVU2E2∗E2−1−ψ1αSU1∗V∗2S∗V∗−1−ψ2ηSU2∗V∗2S∗V∗−αr1−θS∗+1−ψ1V∗U1∗θμ+r−ηa1−bS∗+1−ψ2V∗U2∗bμ+a−αθβS∗+1−ψ1V∗E1U1∗μU1−ηbpS∗+1−ψ2V∗E2U2∗μU2−αrS∗+1−ψ1V∗Q1U1∗μU1−ηaS∗+1−ψ2V∗Q2U2∗μU2,V˙x=−μSS−S∗2+μV∗3−S∗S−VV∗−SV∗S∗V+kαS∗U1∗3−S∗S−SU1E1∗S∗U2∗E1−θβE1kμU1+sηS∗U2∗3−S∗S−SU2E2∗S∗U2∗E2−bpE2sμU2+gαS∗U1∗4−S∗S−SU1E1∗S∗U1∗E1−rQ1gμU1−r1−θβE1Q1∗gμμ+rU1∗Q1+hηS∗U2∗4−S∗S−SU2E2∗S∗U2∗E2−aQ2hμU2−a1−bpE2Q2∗hμμ+aU2∗Q2+k1−ψ1αV∗U1∗4−S∗S−VU1E1∗V∗U1∗E1−θβE1kμQ1−SV∗S∗V+s1−ψ2αV∗U2∗4−S∗S−VU2E2∗V∗U2∗E2−bpE2sμQ2−SV∗S∗V+g1−ψ1αV∗U1∗5−S∗S−VU1E1∗V∗U1∗E1−SV∗S∗V−r1−θβQ1∗E1gμμ+rU1∗Q1−rQ1gμU1+h1−ψ2ηV∗U2∗5−S∗S−VU2E2∗V∗U2∗E2−SV∗S∗V−a1−bpQ2∗E2hμμ+aU2∗Q2−aQ2hμU2.
By arithmetic-geometric means inequality, i.e.,n−m1+m2+⋯+mn≤0, where m1.m2…mn=1 and m1,m2,…,mn>0, it follows that V˙≤0 with V=0 if and only if S=S∗, V=V∗, E1=E1∗, E2=E2∗, Q1=Q1∗, Q2=Q2∗, U1=U1∗, and U2=U2∗. Hence, W1 is said to be GAS which follows LaSalle’s Invariance Principle [12].


## 3.4. Sensitivity and Uncertainty Analysis

Figures1 and 2 show the sensitivity and uncertainty analysis SA/UA of the obtained reproduction numbers for both variants with 2000 samples (up to 5 simulations for Figure 1 and a single simulation for Figure 2), getting a reliability indicator of 100%. From the SA in Figure 1(a), it is noticeable that the parameters ψ1,μ, and τ1 contribute most to the disease reduction while the parameters Λ,θ, and α contribute to the spread of the disease. On the other hand, the UA in Figure 1(b) shows that the range of the RFVc is approximately 0.4−1.8, though most of the outputs are concentrated in low values 0.4−1.1. From the SA in Figure 2(a), it is noticeable that the parameters ψ2,μ, and c2 contribute most to the disease reduction while the parameters b,a,η,p, and Λ contribute to the spread of the disease. On the other hand, the UA in Figure 2(b) shows that the range ofRSVc is approximately 0.2−2, and most of the outputs are concentrated in low values 0.2−1.1.Figure 1
Global sensitivity and uncertainty analysis of variant 1. (a) Partial rank correlation coefficients (PRCC) for variant 1. (b) The box plot of variant 1.
(a)(b)Figure 2
Global sensitivity and uncertainty analysis of variant 2. (a) Partial rank correlation coefficients (PRCC) for variant 2. (b) The box plot of variant 2.
(a)(b)


## 4. Numerical Simulations and Discussion

This section presents numerical solutions of the proposed model and discussion of our findings. All simulations are done on MATLAB platform while parameter values used are in Table1.


### 4.1. Investigation of Stability

Figure3 shows the trajectories of the state variables in the proposed model when Rc<1. Different initial values are used as shown in Figure 3. Although the number of individuals in the infected classes (i.e., exposed individuals, quarantined individuals, and undetected individuals) first increases, it later decreases and attains a disease-free state. This confirms the result in Theorem 1. In other words, the condition R0<1 is sufficient for the disease control irrespective of the initial size of infection.Figure 3
These graphs are obtained by numerically solving the proposed model usingψ1=0.7andψ2=0.65 and other parameter values as contained in Table 1. With this choice of parameter values, Rc=0.6742, where S is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.In Figure4, the number of susceptible individuals decreases and reaches a stable equilibrium. The populations of the vaccinated individuals exhibit ups and downs until a stable equilibrium is reached. Infected classes (i.e., exposed individuals, quarantined individuals, and undetected individuals) approach the equilibrium position after early rises. It very well may be found in Figure 4 that the infectious population approaches the endemic point with Rc>1. However, before approaching to endemic point, the infected individuals increase. Thus, after a certain percentage of the population has become vaccinated, infected, and recovered, the entire population will have certain level of indirect immunity. This is known as herd immunity.Figure 4
These graphs are obtained by numerically solving model 1 using parameter values as contained in Table1. With this choice of parameter values ψ1=ψ2=0.1, α=0.6/N0, η=0.7/N0, and Rc=1.8883, where S is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.


### 4.2. Impact of Intervention Strategies

Figure5 shows the impact of the new variant of SARS-CoV-2 in the presence of imperfect vaccine. RFVc=0.7899andRSVc=0.8287 imply that the spread of variant 1 of SARS-CoV-2 is relatively under control than the spread of variant 2. Furthermore, Figure 5 shows that prevalence of variant 2 enhances the prevalence of variant 1. Thus, discovering a vaccine which is effective (to a good extent) for the prevention against variant 2 is necessary for the control of COVID-19.Figure 5
These graphs are obtained by numerically solving the model usingψ1=0.62 and other parameter values as contained in Table 1. With this choice of parameter values, RSVc=0.8287, where S is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.Simulation in Figure6 is done to investigate the impact of escape of quarantined infectious individuals from isolation centers. The parameters responsible for this are r and a. When r=a=0.01, a good number of susceptible individuals become vaccinated leading to an increase in vaccinated population. This population later reduces as the susceptible population decreases (see subplot 1). It can also be seen from the subplots of 3–8 in Figure 6 that, with increase in the rate at which individuals infected with variant 1 escape the isolation center, vaccination of susceptible individuals lowers the peak of the population infected with variant 2 while there is an increase in the population infected with variant 1. This is probably because individuals infected with variant 2 appear to be more careful with their lives and get vaccinated more than individuals infected with variant 1.Figure 6
Investigation of the impact of escape of quarantined infectious individuals from isolation centers, whereS is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.


## 4.1. Investigation of Stability

Figure3 shows the trajectories of the state variables in the proposed model when Rc<1. Different initial values are used as shown in Figure 3. Although the number of individuals in the infected classes (i.e., exposed individuals, quarantined individuals, and undetected individuals) first increases, it later decreases and attains a disease-free state. This confirms the result in Theorem 1. In other words, the condition R0<1 is sufficient for the disease control irrespective of the initial size of infection.Figure 3
These graphs are obtained by numerically solving the proposed model usingψ1=0.7andψ2=0.65 and other parameter values as contained in Table 1. With this choice of parameter values, Rc=0.6742, where S is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.In Figure4, the number of susceptible individuals decreases and reaches a stable equilibrium. The populations of the vaccinated individuals exhibit ups and downs until a stable equilibrium is reached. Infected classes (i.e., exposed individuals, quarantined individuals, and undetected individuals) approach the equilibrium position after early rises. It very well may be found in Figure 4 that the infectious population approaches the endemic point with Rc>1. However, before approaching to endemic point, the infected individuals increase. Thus, after a certain percentage of the population has become vaccinated, infected, and recovered, the entire population will have certain level of indirect immunity. This is known as herd immunity.Figure 4
These graphs are obtained by numerically solving model 1 using parameter values as contained in Table1. With this choice of parameter values ψ1=ψ2=0.1, α=0.6/N0, η=0.7/N0, and Rc=1.8883, where S is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.


## 4.2. Impact of Intervention Strategies

Figure5 shows the impact of the new variant of SARS-CoV-2 in the presence of imperfect vaccine. RFVc=0.7899andRSVc=0.8287 imply that the spread of variant 1 of SARS-CoV-2 is relatively under control than the spread of variant 2. Furthermore, Figure 5 shows that prevalence of variant 2 enhances the prevalence of variant 1. Thus, discovering a vaccine which is effective (to a good extent) for the prevention against variant 2 is necessary for the control of COVID-19.Figure 5
These graphs are obtained by numerically solving the model usingψ1=0.62 and other parameter values as contained in Table 1. With this choice of parameter values, RSVc=0.8287, where S is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.Simulation in Figure6 is done to investigate the impact of escape of quarantined infectious individuals from isolation centers. The parameters responsible for this are r and a. When r=a=0.01, a good number of susceptible individuals become vaccinated leading to an increase in vaccinated population. This population later reduces as the susceptible population decreases (see subplot 1). It can also be seen from the subplots of 3–8 in Figure 6 that, with increase in the rate at which individuals infected with variant 1 escape the isolation center, vaccination of susceptible individuals lowers the peak of the population infected with variant 2 while there is an increase in the population infected with variant 1. This is probably because individuals infected with variant 2 appear to be more careful with their lives and get vaccinated more than individuals infected with variant 1.Figure 6
Investigation of the impact of escape of quarantined infectious individuals from isolation centers, whereS is the susceptible individuals, V is the vaccinated individuals, E1 is the individuals exposed to variant 1, Q1 is the quarantined individuals who are infected with variant 1, U1 is the undetected individuals who are infected with variant 1, E2 is the individuals exposed to variant 2, Q2 is the quarantined individuals who are infected with variant 2, U1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.


## 5. Conclusion

We presented a COVID-19 model consisting of nine mutually exclusive compartments representing COVID-19 dynamics. The model considers a vaccinated population and a new variant of SARS-CoV-2 in which the vaccine is not perfect and the transmission rate of the new variant (second variant) of SARS-CoV-2 is higher than the transmission rate of the first variant. The model also considers the escape rate of quarantined infectious individuals. This study is necessary and important as it gives a clear understanding of the impact of the imperfect vaccine on the new variant (second variant). Also, it gives an idea of the disease dynamics with the occurrence of a new variant of SARS-CoV-2. The control reproduction number,Rc, is obtained using the next-generation matrix method. The basic reproduction number R0 is obtained from the control reproduction number, Rc, by setting the control measures to be zero, i.e., ψ1=ψ2=κ=0. It is found that the control reproduction number, Rc, is given by the maximum of the two threshold quantities, RFVc and RSVc. It is shown that there are two possible equilibria of the model; one is a disease-free equilibrium that exists and is globally asymptotically stable if Rc<1, and the other is endemic equilibrium which is shown to exist for RSVc>1. Analytically, results reveal that the second variant (new variant) of SARS-CoV-2 dominates the first variant. Hence, the first variant of SARS-CoV-2 clears out of the population over time, even with the control reproduction number greater than one. Moreover, the first and second variant can only be cleared out of the population if RFVc and RSVc are less than one, i.e., RFVc,RSVc<1. Further, the impact of the imperfect vaccine on the new variant is explored through numerical simulations. Numerical results reveal that individuals infected with the new variant (second variant) of SARS-CoV-2 who are vaccinated with an imperfect vaccine are under control but the prevalence of the second variant enhances the prevalence of the first variant. Numerical results also reveal that increase in the rate at which individuals infected with the first variant escape the isolation center gives rise to the population infected with the first variant and lowers the peak of the population infected with the second variant. This is probably because individuals infected with the second variant appear to be more careful with their lives and get vaccinated more than individuals infected with the first variant. As a result, current antiviral methods such as frequent hand washing, use of mask, physical separation, excellent ventilation, and avoiding crowded locations or enclosed settings continue to function against the first and second types by limiting viral transmission. This study can be extended by introducing fractional-order into the formulated model. In this case, the model will be given fractional differential equations. All these directions need more investigation, and therefore they shall be left for future works.



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