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

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 rates r and a , escape of quarantined infectious individuals from isolation centers. The control R c and basic reproduction numbers R 0 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.


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.
ere 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. e second vaccination is an all-ornothing vaccine that confers lifetime protection to certain individuals but does not protect others. e third vaccination, which is only effective for a limited time, is diminishing.
COVID-19 articles may be found in [11][12][13][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. eir 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. ree common incidence functions are used to represent the physical behavior of viral infection. ey 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]. ey included the lytic and nonlytic immune responses, as well as both mechanisms of virus-to-cell infection in the model. ey proved that the model was wellposed and that equilibria existed. ey 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. e model theoretical analysis, as well as model fitting and parameter estimates, was completed. ey 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. eir 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. ey 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. ey also used Pontryagin's Maximum Principle to determine the model's best control. ey 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. ey 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]. ey used a locally asymptotically stability analysis to provide a thorough investigation of their model. e COVID-19 Nigeria data are taken from the Nigeria Center for Disease Control (NCDC) to parameterize the model.
is 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, R 0 . 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. ey 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 equationbased 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. e absence of vaccinated persons in the presence of the new strain is costly since it is necessary to determine if the current 2 Complexity 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. e 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. is distinguishes our work from that of [25]. e 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][29][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. e research aims and importance of the study are highlighted in the following sections based on the above.

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

Significance.
(1) e 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) e 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) e evaluation of analytical results will provide useful information that will help to eradicate COVID-19 in the population.

Mathematical Formulation of COVID-19 Model
is section describes the transmission dynamics of COVID-19, and the formulated model is presented, respectively. e total individual population, denoted by N, is divided into nine categories: susceptible individuals S, vaccinated individuals V (individuals who get vaccinated), two classes of exposed individuals E 1,2 , two classes of quarantined infectious individuals Q 1,2 , two classes of unquarantined infectious individuals U 1,2 , and recovered individuals R, e susceptible individuals population is generated by the recruitment of individuals into the population, either by birth or immigration, at the rate Λ. 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 α(1 − ψ 1 ) and η(1 − ψ 2 ), where ψ 1 ∈ [0, 1] and ψ 2 ∈ [0, 1] represent the vaccine efficacy against variant 1 and variant 2. Furthermore, it is noteworthy that ψ 1 � 1 and ψ 2 � 1 imply that a vaccine provides 100% assurance against COVID-19, while ψ 1 � 0 and ψ 2 � 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., ψ 2 < ψ 1 ). 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]. ere 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 Table 1, the following system is formulated: with initial conditions such as where the model parameters are nonnegative. For biological reasons, the model is analyzed in the feasible-region as follows: that shows system (1) is to be positively invariant. us, the model is well-posed both mathematically and epidemiologically in D. e first eight equations of system (1) do not affect the compartments R, and therefore we have 4 Complexity

Basic Reproduction Number.
To determine the overall dynamical behavior of the equilibrium solution, the control reproduction number, R c , will be used. R c 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 where e disease-free equilibrium is , 0, 0, 0, 0, 0, 0 .
Consequently, R c for the first and second variants is given as where and 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 R 0 may be calculated from the value of R c , in the sense that ψ 1 � 0, ψ 2 � 0, and κ � 0. As a result, without any control mechanisms, the basic reproduction number is where and However, our analysis will be based on the control reproduction number, R c . Furthermore, it is assumed that the population is completely susceptible initially, and R c measures the average secondary cases at the beginning of the epidemic.

Stability Analysis
e 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.

Theorem 1.
e disease-free equilibrium is globally asymptotically stable if R c < 1.
Proof. Consider the Lyapunov function as follows: e time derivative of (10) along the COVID-19 model (4) is 6 Complexity us, the singleton set is the biggest compact invariant set in (S, ; V, ; E 1 , ; Q 1 , ; U 1 , ; E 2 , ; Q 2 , ; U 2 ) ∈ D : _ L � 0, and W0 is globally asymptotically stable in D according to LaSalle's Invariance Principle [40]. e 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 W 0 base of attraction. For R c < 1, this suggests that COVID-19 can be eradicated regardless of the original size of the infectious population. □

Existence and Uniqueness of Endemic Equilibrium
where S * , V * , E * 2 , and Q * 2 can be found from (16) to be e following equations for U * 1 and U * 2 are obtained after substituting (16) into (17), If we let so that _ f(x 1 , x 2 ) < 0, lim a⟶+∞ , f(x 1 , x 2 ) < 0, and In the sense that if R SV c > 1, then f(0) > 0. us, for the equation f(a) � 0, a unique positive solution exists only when R SV c > 1.

Global Stability of Endemic Equilibrium Solution (W 1 ).
In order to show that COVID-19 in the population persists and converges to W 1 , the following result is stated and proved.

Theorem 3.
e unique equilibrium W 1 is globally asymptotically stable (GAS) whenever R c > 1.
Proof. Given the following equations at the endemic equilibrium point W 1 : And consider the following Lyapunov function: where Differentiating V with respect to time gives After developing and using (21), we have the following: Substituting d 1 , d 2 , f 1 , and f 2 by their values and exploiting, we have Complexity 9 Define g � (1 − θ)r θμ + r , en, _ V(x) can be rewritten as By arithmetic-geometric means inequality, i.e., n − (m 1 + m 2 + · · · + m n ) ≤ 0, where m 1 .m 2 . . . m n � 1 and m 1 , m 2 , . . . , m n > 0, it follows that _ Hence, (W 1 ) is said to be GAS which follows LaSalle's Invariance Principle [12].  Figures 1 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 ψ 1 , μ, and τ 1 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 Figure 2(a), it is noticeable that the parameters ψ 2 , μ, and c 2 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

Numerical Simulations and Discussion
is 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 Table 1. Figure 3 shows the trajectories of the state variables in the proposed model when R c < 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. is confirms the result in eorem 1. In other words, the condition R 0 < 1 is sufficient for the disease control irrespective of the initial size of infection. In Figure 4, the number of susceptible individuals decreases and reaches a stable equilibrium. e 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 R c > 1. However, before approaching to endemic point, the infected individuals increase. us, after a certain percentage of the population has become vaccinated, infected, and recovered, the entire population will have certain level of indirect immunity. is is known as herd immunity. Figure 5 shows the impact of the new variant of SARS-CoV-2 in the presence of imperfect vaccine. R FV c � 0.7899 and R SV c � 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. us, discovering a vaccine which is effective (to a good extent) for the prevention against variant 2 is necessary for the control of COVID-19.

Impact of Intervention Strategies.
Simulation in Figure 6 is done to investigate the impact of escape of quarantined infectious individuals from isolation centers. e 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. is population later reduces as the susceptible population decreases (see subplot 1). It can also be seen from the subplots of 3-8 in  Table 1. With this choice of parameter values, R c � 0.6742, where S is the susceptible individuals, V is the vaccinated individuals, E 1 is the individuals exposed to variant 1, Q 1 is the quarantined individuals who are infected with variant 1, U 1 is the undetected individuals who are infected with variant 1, E 2 is the individuals exposed to variant 2, Q 2 is the quarantined individuals who are infected with variant 2, U 1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.    Table 1. With this choice of parameter values ψ 1 � ψ 2 � 0.1, α � 0.6/N 0 , η � 0.7/N 0 , and R c � 1.8883, where S is the susceptible individuals, V is the vaccinated individuals, E 1 is the individuals exposed to variant 1, Q 1 is the quarantined individuals who are infected with variant 1, U 1 is the undetected individuals who are infected with variant 1, E 2 is the individuals exposed to variant 2, Q 2 is the quarantined individuals who are infected with variant 2, U 1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.  Table 1.
With this choice of parameter values, R SV c � 0.8287, where S is the susceptible individuals, V is the vaccinated individuals, E 1 is the individuals exposed to variant 1, Q 1 is the quarantined individuals who are infected with variant 1, U 1 is the undetected individuals who are infected with variant 1, E 2 is the individuals exposed to variant 2, Q 2 is the quarantined individuals who are infected with variant 2, U 1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.

Conclusion
We   Figure 6: Investigation of the impact of escape of quarantined infectious individuals from isolation centers, where S is the susceptible individuals, V is the vaccinated individuals, E 1 is the individuals exposed to variant 1, Q 1 is the quarantined individuals who are infected with variant 1, U 1 is the undetected individuals who are infected with variant 1, E 2 is the individuals exposed to variant 2, Q 2 is the quarantined individuals who are infected with variant 2, U 1 is the undetected individuals who are infected with variant 2, and R is the recovered individuals.
Also, it gives an idea of the disease dynamics with the occurrence of a new variant of SARS-CoV-2. e control reproduction number, R c , is obtained using the next-generation matrix method. e basic reproduction number R 0 is obtained from the control reproduction number, R c , by setting the control measures to be zero, i.e., ψ 1 � ψ 2 � κ � 0. It is found that the control reproduction number, R c , is given by the maximum of the two threshold quantities, R FV c and R SV c . 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 R c < 1, and the other is endemic equilibrium which is shown to exist for R SV c > 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 R FV c and R SV c are less than one, i.e., R FV c , R SV c < 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. is 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. is 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.

Data Availability
No data were generated or analyzed during the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.