Modeling and Optimal Control Analysis Applied to Real Cases of COVID-19 Pandemic with Double Dose Vaccination in Ethiopia

The novel coronavirus is a recently discovered member of one of the largest families of viruses with symptoms ranging from a simple cold to excruciating respiratory agony. In the present paper, a deterministic mathematical model is formulated to estimate the transmission dynamics of COVID-19 with the inclusion of control strategies like (i) double-dose vaccination, (ii) prevention, and (iii) treatment. In addition, instead of considering all infectious humans as one unit, we separate them into symptomatic and asymptomatic groups, and the impact is analyzed. This separation is meaningful because various reports indicate that the asymptomatic cases will spread the disease more than the symptomatic cases. The model is proved to be mathematically well-posed and biologically meaningful by showing positivity and boundedness of the solution using the appropriate initial conditions. For the reproduction number, a parametric formula is constructed, and also the associated numerical value is calculated from the reported real data in Ethiopia. Moreover, disease-free and endemic equilibria are determined, and their local and global stabilities are discussed using the Lyapunov function technique. These equilibria are found to be locally asymptotically stable if R 0 < 1 and R 0 > 1 , respectively. Following the model ﬁ tting and estimation of the parameter values, sensitivity analysis was performed in order to analyze the impact of each parameter on transmission dynamics. In other words, this study can be used to evaluate how major model parameters a ﬀ ect transmission dynamics and control. Utilizing Pontryagin ’ s maximal principle, the best control measures are implemented with the aim of lowering the burdens associated with infection, prevention, and treatment. To comprehend and visualize the impact of control techniques on the development of the disease and to illustrate the analytical ﬁ ndings generated in this study, numerical simulation studies are conducted. Finally, the output of the study illustrates that adhering to all the control strategies has a big impact on minimizing the transmission of the disease in society. Which means that if the control strategies are well managed by the concerned body, then the burden of the disease is reduced quickly in the Ethiopian population.


Introduction
The novel coronavirus (COVID-19) is one of the largest fatal viruses which infected millions of humans throughout the world within a short period of time and has posed serious challenges to human life style including the economy and GDPs of the developed countries. The outbreak of the COVID-19 pandemic was first declared by the People's Republic of China on 31st December 2019, and it appeared in China, Wuhan City [1]. The infection caused by COVID-19 is identified by its signs and symptoms. Most of the signs and symptoms of COVID-19 are similar to the common cold illness, but it is more dangerous and causes severe respiratory problems. Despite efforts being made to curb the transmission of the COVID-19 disease, the disease is still persistent or endemic in many parts of the globe including Ethiopia [2]. Hence, globally, human beings are still facing a health problem due to the coronavirus. Numerous studies indicate that direct human contact or physical separation between susceptible individuals is the primary means of spreading this epidemic [3,4]. This is why COVID-19 becomes pandemic within a very short duration of time and is difficult to mitigate but reduce the transmission rate due the corresponding vaccine is obtained.
According to the nature of its mutation rate and the criticality of the virus, COVID-19 causes millions of deaths throughout the globe. Because of this, the World Health Organization (WHO) and scientists throughout the world face a lot of challenges and competition to get the corresponding vaccine. But now, starting from 7 March 2022, 10 vaccines exist and are granted by the WHO for emergency use in the case of coronavirus [5]. Up to March 07, 2022, around 10.9 billion COVID-19 vaccine doses were managed globally, and 63.4% of the population throughout the world has taken the minimum of one dose of a vaccine. This coverage almost represents developed countries due to the shortage of the vaccine in developing countries. Up to the date mentioned above, only 13.6% of people in developing countries have received the minimum of one dose of the vaccine [6]. In Ethiopia, until 5 March 2022, 26,178,996 amount of vaccine doses had been administered [7]. A deterministic epidemiological model can be used to evaluate the dynamics of any infectious disease's transmission and outbreak as of the middle of the 20th century. This model represents real-world situations and predicts the severity of those diseases using the concept of mathematics [2]. A lot of research work has been conducted on the transmission dynamics of novel coronavirus and different models are also presented. Additionally, a number of research works have been interested to identify the best vaccination strategy to control the transmission of coronavirus [8,9].
For instance, Matrajt et al. [10] presented a deterministic mathematical model depending on age to identify which group of age must be vaccinated first. They also verified that with low coverage, priority must be given to individuals whose age is 60 + years to reduce the number of mortality due to COVID-19. Semu et al. [11] developed a mathematical model to analyze the mitigation strategies for COVID-19 and take the following into account. The impact of asymptomatic patient individuals on the transmission rate of the disease in fully susceptible individuals, the effect of indirect spread of the disease through the population, and individuals behavioral changes in the community to use self-protective measures. Akuka et al. [12] formulated a SV 1 V 2 EIQR mathematical model with double-dose vaccination and quarantine to show that vaccination and treatment are very effective in controlling the spread of COVID-19 and that properly applying personal protective measures, that is, nonpharmaceutical public health interventions mechanisms such as proper use of face masks, regular hand washing, social distancing, and the like, should continue to be encouraged.
Since COVID-19 first arose in late 2019, numerous studies have been undertaken using mathematical models to begin studying the disease's transmission patterns and the pandemic's control mechanisms in various nations with the authors ( [13][14][15]). For example, Atifa et al. [16] developed and examined a mathematical compartmental model for the pandemic's transmission patterns while accounting for the impact of human reinfection after complete recovery utilizing Pakistan COVID-19 genuine cases recorded. By comparing the growth rates of COVID-19 patients in Pakistan and Malaysia before and after the imposition of lockdown, a mathematical model based on Lotka-Volterra equations was created to shed light on the impact of lockdown on COVID-19 transmission dynamics, as presented by Abro et al. [17]. Coronavirus and diabetes relationships are discussed by the authors Ozkose et al. [18] using actual data from Turkey. In order to explore COVID-19 transmission and its connection to diabetes, a pandemic fractional order model is first developed. In their approach, both types of diabetes those with and without complications are taken into account in connection to the quarantine strategy. Peter et al. [19] developed and analyzed an epidemic model of COVID-19 governed by eight compartmental models. In their study, they also consider double-dose vaccination in order to control the transmission dynamic of the pandemic. In particular, their study's findings show that when vaccination rates rise, the number of infected individuals in the population declines, lowering the disease burden in Malaysia.
In addition, Sharbayta et al. [20] presented a SV 1 V 2 EI a I s QHR deterministic mathematical model to conclude the transmission of COVID-19 in Ethiopia. In their study, they consider double-dose vaccination to mitigate the transmission of COVID-19 and use treatment strategy at quarantine and hospital classes to treat infected and critically ill individuals, respectively. By taking into account the combined effects of public health awareness, self-protective behavior, and treatment of hospitalized individuals, the authors [21][22][23] considered a deterministic mathematical model for the dynamics of COVID-19 transmission. This model is intended to reduce the spread of the disease. Additionally, by applying the combined effect least squares and Bayesian estimation mechanisms to the data gathered in Ethiopia and other nations, they calculate the unknown parameter values, identify the parameters that are most sensitive to the disease's transmission rate, and work to control its spread.
Therefore, the ongoing double-dose vaccination program and the two control techniques to slow the COVID-19 pandemic's spread among people are the driving forces behind this research. Here, we presented an optimal control analysis of the deterministic SV 1 V 2 EI a I s R compartmental model to investigate the effect of prevention, treatment, and double doses of vaccination against novel coronavirus mitigation. The newly developed model is parameterized using the least squares method on daily reported COVID-19 confirmed cases in Ethiopia from 01 July, 2022, to 31 August 2022. After receiving the first and second doses of the immunization, the virus' transmission and progression are also determined.
The remaining parts of our manuscript are organized as below. The developed model is formulated in Section 2, its analytical analysis is explained in Section 3, and the estimation of the parameters using real data is discussed in Section 4. Sensitivity analysis is discussed in Section 5, and the extension of the proposed model into optimal control is elaborated in Section 6. The numerical simulations performed to support the analytical results discussed in Section 3 are presented in Section 7. Finally, a conclusion is presented.

Model Formulation
Mathematical modeling is one of the best mechanisms which is used to analyze the epidemiology of a disease. In 2 Journal of Applied Mathematics order to properly examine the transmission dynamics and control of the pandemic in the community, we design and analyze a deterministic mathematical model with the inclusion of control strategies and estimation of the parameters. According to their disease status at any moment t, the population size NðtÞ, which represents the suggested model, is divided into seven mutually exclusive individuals. These different classes include the number of susceptible humans SðtÞ , who are uninfected individuals with the disease but has a full chance to be infected with the diseases; the vaccinated individuals with the first dose V 1 ðtÞ, who still have a chance to be susceptible again and infected by COVID-19; the second dose vaccinated individuals V 2 ðtÞ, who are people those who have finished both dose vaccination within a time but still have a chance to be susceptible again due to waning vaccination during double dose vaccination time. As most of the researchers estimated, the vaccine will protect against 85 percent of infections with the most optimistic assumption. Hence, due to this, it is assumed that the population in the second dose of vaccinated humans goes to the recovered class at a rate Φ and the other moves to the susceptible class at a rate ϕ 2 .The force of infection λ = βððωI a + I s Þ/NÞ, where β is the effective transmission rate and ω is the reduction in disease transmission in asymptomatic people. Since the considered population is assumed to be constant, the recruitment into the susceptible class is only through birth at the rate of Λ. Due to the first dose of the vaccine's inadequacy to prevent against COVID-19, susceptible humans enter the vaccinated compartment at a rate of θ after receiving the first dose, first-dose vaccinated humans enter the susceptible class at a rate of ϕ 1 , and the remaining individuals enter the vaccinated compartment at a rate of ρ after receiving the second dose. The number of Exposed individuals EðtÞ get infected without symptoms (asymptomatic) I a ðtÞ or with symptoms (symptomatic) I s ðtÞ at the rate of pν and ð1 − pÞ ν, respectively. Where ν is the progression rate from the incubation period (exposed class) to the infected classes, and p is a fraction of exposed individuals becoming infected asymptomatically while the remaining (1-p) fractions are infected symptomatically. γ a and γ s are the rates at which I a ðtÞ and I s ðtÞ are recovered from COVID-19 patient due to natural immunity and treatment of infected classes, respectively. RðtÞ stands for the recovered compartment number; this class consists of either tested persons who have already had a COVID-19 negative result and are no longer in contact with contaminated areas or diseased individuals, or dead cases. In general, it is assumed that the size of the entire human population, indicated by NðtÞ, is constant and evenly distributed, as shown by: The proposed COVID-19 model is governed by the following assumptions: (i) All parameters are nonnegative (ii) To keep the size of the population constant, consider all dead humans as newborns which are replaced in the susceptible class (iii) The susceptible individuals are recruited by birth, and it is denoted by Λ through this manuscript (iv) The death rate due to COVID-19 is considered in both symptomatic and asymptomatic classes as α s and α a , respectively.
(v) Full-dose vaccinated individuals become susceptible again due to waning vaccination The proposed model considers first and second-dose vaccination as compartment with prevention and treatment as a control strategy to eradicate the transmission dynamics of coronavirus in Ethiopia. The proposed model parameters are described in Table 1 with their source. The model used in studying the transmission dynamics of COVID-19 in this study is governed by the next system of ordinary differential equation, i.e., Equation (2), while the corresponding flow diagram is depicted in Figure 1.

Qualitative Analysis of the Model
In this section, some basic properties of the formulated model defined in Equation (2) including the feasible region, the positivity of the solution, equilibria, and their stability, are discussed.    Proof. Choosing any one of equation from Equation (2) above, let us take the first equation i.e.,

Positivity and Boundedness of Solution
Integrating Equation (5) by separation of variable method, we obtain Using initial condition Sð0Þ = S 0 and Equation (6), we obtain In a similar way, the remaining equations of Equation (2) can be proved and gives the following outputs: Proof. From the total population size over a time t, we have Since NðtÞ is assumed to be constant over a time t, then ðdN/dtÞ = 0.
Hence, Ω is a positively invariant set, and on this set, the proposed model is well defined epidemiologically and mathematically. Therefore, it is sufficient to investigate the dynamics of the presented model in Ω.

Disease-Free Equilibrium Point (DFE).
To calculate the DFE of Equation (2), we equate the right-hand side of Equation (2) to zero and make the disease state variables E = I a = I s = 0 and DF state variables S, V 1 , V 2 , and R nonzero. But at infection-free, the variables E, I a , I s , and R are zero. After that, solve the remaining equations of Equation (2) will give us When there is no vaccination, then Equation (15) is reduced to a fully susceptible, and it is given by If θ = 1, then we get a DFE in which every susceptible human is vaccinated with the first dose, and 3.2.2. Basic Reproduction Number ðR 0 Þ. The threshold quantity ðR 0 Þ is the average number of secondary cases produced by one primary infection during the infectious period in a fully susceptible population, and the control threshold quantity (in our case denoted by R c ) is used to represent the same quantity for a system incorporating vaccine as intervention strategies [24]. To calculate the threshold quantity, follow the work of Watmough and Driessche method. From Equation (2) and the notation X = ðE, I a , I s Þ, we have represent the transfer of individuals into and out of the infected classes. The Jacobian matrices of F(X) and V(X) at DFE are given as blow, respectively.
Now, it is simple to determine the inverse of V and given by Using the next-generation matrix, and the corresponding eigenvalues are λ 1 = 0 which is a double root and From λ 1 and λ 2 , we conclude that λ 2 is the spectral radius of the next-generation matrix which equal with R c and Setting θ = ρ = ϕ 1 = ϕ 2 = 0 in Equation (24) will give the basic reproduction number, R 0 as Now, rewriting Equation (24) in terms of R 0 as , then all populations are vaccinated with the first dose of vaccine. Hence, the corresponding effective (control) reproduction number is given by In general, if R c > 1, then COVID-19 infections will persist in the community, and it will eventually disappear from society if R c < 1.

Existence of Endemic Equilibrium (EE) Point of the Model.
In this section, we study the disease EE of Equation (2). The main interesting idea behind the EE is that it is used to determine the existence of the disease in the human being. To find such conditions for the existence of an equilibrium for which COVID-19 is endemic in the community, at least one of E * ≠ 0, I * a ≠ 0, or I * s ≠ 0. Let us denote the steady-state solution of Equation (2) To find E e , equate Equation (2) to zero, and the reduced part of Equation (2) becomes Hence, the existence of E e depends on R c , that is, E e of Equation (2) exists if R c > 1.

6
Journal of Applied Mathematics

Stability Analysis of Equilibrium Points.
To determine the local and global asymptotic stability of the equilibria of Equation (2), we use the Jacobian matrices at DFE and EE for local stability and construct the Lyapunov function for global stability.

Local Stability Analyses of DFE
Theorem 3. The disease-free-equilibrium point, E dfe of Equation (2) is locally asymptotically stable (LAS) if R c < 1 and unstable otherwise.
Proof. To proof the above theorem, first calculate the Jacobian matrix of Equation (2) at E dfe . That is We now need to demonstrate that all of JðE dfe Þ's eigenvalues are negative. Only the diagonal element −μ appears in the seventh column, indicating that −μ is one of the necessary eigenvalues. The remaining eigenvalues can be found in the reduced matrix, J 1 ðE dfe Þ.
The characteristic equation of M 1 is where Clearly, we see that b 1 > 0 and b 2 > 0 because they are the sum of positive parameters. But from the third equation, i.e., We also say that b 3 > 0 if and only if R c < 1. In the same fashion, the characteristic equation of M 2 is The proof is as straight-forward as [25].

Global Stability Analyses of DFE
Theorem 5. The disease-free equilibrium point of the dynamical system on Equation (2) is globally asymptotically stable (GAS) if R c < 1 and unstable otherwise.
Proof. To proof global stability of E def , we first construct the Lyapunov function [25]. Let the Lyapunov function be V : R 7 + ⟶ R + and defined as To determine constants, equate the partial derivatives of Equation (37) with respect to S, E, I a , and I s to zero. Hence, to determine a 1 , equate the partial derivatives of Equation (37) in terms of S to zero, and we will obtain Partial derivative in terms of E gives us −a 2 k 4 + a 3 pν + a 4 ðð1 − pÞνÞ = 0, and from this we obtain Partial derivative in terms of I a gives us −a 1 βωðS 0 /NÞ + a 2 βωðS 0 /NÞ − a 3 k 5 = 0, and this gives us Partial derivative in terms of I s gives us −a 1 βðS 0 /NÞ + a 2 βðS 0 /NÞ − a 4 k 6 = 0, and from this gives Since S ≤ S 0 , and substituting a 1 , a 2 , a 3 , and a 4 into Equation (37) and after some simplification Since a 2 k 4 ≥ 0, then dV/dt = a 2 k 4 ðR c − 1Þ ≤ 0 where R c ≤ 1 and dV/dt = 0 if and only if S = E = I a = I s = 0. This implies that the only trajectory of Equation (2)

Global Stability Analyses of EEP
Proof. Since diseases EEP exist for R c > 1, then to prove the GAS behavior of E e , we apply [26,27] approach as where x i is a state variable, i = 1,…7 and x * i is the endemic equilibrium point. From Equation (45), we have Having derivative of Equation (45) in the direction of the solution of Equation (2) Expanding the above equation will give us Further simplification result to Now, we can write dL/dt as dL/dt = Ψ 1 − Ψ 2 , where + γ a I a + γ s I s + ΦV 2 + μR * , From the fact that all the parameter values considered in Equation (2)

Estimation of the parameter's Value
In this manuscript, we solve a dynamics of parameter estimation problem via a daily-wise reported number of coronavirus confirmed cases in Ethiopia from July 1, 2022, to August 31, 2022, see Table 2. The best fit corresponding to the daily-wise reported confirmed individuals via our model is shown in Figure 2 using the MATLAB routine. Here, we fitted some number of the model parameter's value such as p, ω, θ, ρ, ϕ 1 , ϕ 2 , and Φ, whereas the remaining parameter's value was taken from well-published literature or estimated, see Table 1. Furthermore, the initial conditions of the presented model are estimated according to the demographic data of Ethiopia and the real reported COVID-19 patient by the Ethiopian Ministry of Health 2022 and Ethiopian World in Data (Our World in Data, COVID-19 2022) between July 01, 2022, and August 31, 2022 [28]. The calculated parameters from the reported data in Ethiopia are estimated as follows: The total population size of Ethiopia in 2022 is approximately N = 110, 000, 000 during the specified days, and the life expectancy of Ethiopians for the year 2022 is 67.8 [23]. And the natural death rate (μ) is calculated as μ = 1/ð365 * 67:8Þ, while Λ = μ * Nð0Þ. According to the Ethiopian Federal Ministry of Health report, there have been 281 active COVID-19 cases, i.e., I s ð0Þ = 281, and from the WHO report, 80% of infected humans become asymptomatic; hence, we estimated I a as I a ð0Þ = 281/0:8. We let Eð0Þ = 650, which is approximately equivalent to the sum of I a and I s [20]. Consequently, assuming V 1 ð0Þ=500, V 2 ð0Þ=300, Rð0Þ = 200, and Nð0Þ = 3002282 then Sð0Þ is estimated as Sð0Þ = Nð0Þ − ðV 1 ð0Þ + V 2 ð0Þ + Eð0Þ + I a ð0Þ + I s ð0Þ + Rð0ÞÞ.
The fitted parameter's values are demonstrated using the least-squares method to minimize the summation of squared errors defined on Equation (53). The system of Equation (2) is formulated as a standard dynamic system in the following form: y t 0 ð Þ = y 0 , ð52Þ 9 Journal of Applied Mathematics where y represent vector of the dependent variables (compartments) and p is the vector of unknown parameters. The associated sum-of-square error is given by where y i stands for the real data and y i = yðt i , pÞ is the solution of Equation (51) with initial condition Equation (52) according to the number of infected individuals at time t i for a given set of estimated parameters p. The main goal is to minimize the objective function subject to Equation (53) to obtain the required fitted parameter values [29].

Estimation of Basic Reproduction Number (R c ). The basic
reproduction number of any disease shows how fast the disease can spread throughout susceptible populations. In this case, we have a control reproduction parameter, R c , in the presence of a vaccine and a simple reproduction number R 0 in the absence of a vaccine in our model. Particularly, if R c > 1, then the disease can spread into the susceptible population, and if R c < 1, then the disease dies out from the community. So in this section, we estimate the numerical value of R c for Equation (2). Substituting the corresponding parameter's value in Equation (25) and Equation (26) from Table 1, we get the following estimated results. R 0 = 2:105062463, R c = 0:0007495074407, and R 1 = 0:00008555327934. Therefore, the value of R 0 is approximately 2.11 which indicates 2 or 3 susceptible individuals can be infected by a single infected human because of no vaccination and any other control mechanisms in the model. Similarly, R c is approximately 0.001 indicates that the diseases cannot spread into the population due to the presence of control strategies in the model like vaccination.

Sensitivity Analysis
In this section, we carried out a sensitivity analysis of the parameters in the basic reproduction number to identify their impact on the transition dynamics of the COVID-19 pandemic. To go through this section, we used the normalized sensitivity index definition as stated in [30]. The Normalized forward sensitivity index of a variable, R c , that    Infected  Days  Infected   1  281  17  55  33  38  48  26  2  250  18  0  34  59  49  20  3  86  19  199  35  37  50  41  4  161  20  124  36  49  51  25  5  154  21  99  37  15  52  8  6  190  22  134  38  15  53  42  7  185  23  87  39  58  54  22  8  151  24  30  40  41  55 For φ represents all the parameters in R c The same method can be used to determine the index values of the other parameters. And as depicted in Figure 3, the parameters (β, ω, ν, ϕ 1 , and ϕ 2 ) have positive indices. This shows us they have a great impact on the spreading of the disease in the community; as their values increase the disease expands through the population, and as their values decrease, the diseases become eliminated from the population. In addition, the parameters in which their sensitivity indices are negative (θ, α a , α s , γ 1 , γ 1 , Φ, and ρ) have an effect of reducing the burden of the disease in the population as their values increase. Therefore, research advice for stakeholders is to work on minimizing the positive indices and maximizing the negative indices of the parameters.

Optimal Control Analysis
In this section, we examine the following two suggested control methods, which are crucial in reducing the dynamics of the new coronavirus epidemic transmission. Thus, c 1 ðtÞ and c 2 ðtÞ, which stand for the following terms, are used to signify time-dependent control variables: (i) The preventive measures focused on inhibiting the transmission dynamics of the virus from symptomatic and asymptomatic humans. This can be obeyed through public health advocacy for wearing face masks in public places, regular hand washing, social distancing, quarantine, and isolation that help reduce the contact rate. Noting that c 1 ðtÞ = 1 implies the strategy is effective for protection against infection, while c 1 ðtÞ = 0 implies failure of the strategy (ii) The treatment measure includes medical care for all the confirmed cases to increase the recovery rate of infected individuals and to decrease disease-induced mortality rate. This can be achieved by rapidly giving patients who are hospitalized with COVID-19 a critical case of additional oxygen or mechanical breathing Consequently, by including these control variables in Equation (2), we obtain  β, ν, α a , α s , γ a , γ s , ω, θ, ρ, Φ, ϕ 1 , and ϕ 2 .

Journal of Applied Mathematics
The primary task at hand is to identify the best possible control for preventive measures c 1 and c 2 , given the relatively low costs of these techniques. The Pontryagin maximum principle [31] is used to establish the necessary and sufficient conditions for the existence of optimal control. The function that minimizes the number of exposed cases and symptomatically and asymptomatically infected cases over a time interval of ½0, t f can be defined as subject to Equation (57) with associated initial conditions defined on Equation (3). The positive constants b 1 , b 2 , and b 3 are used to balance the units of the integrand to reduce the dominance of any one of the terms in the integral, and w 1 and w 2 measure the relative cost of intervention strategies over the time interval ½0, t f . Minimizing Equation (58) provides an optimal control c * i = 1, 2 such that subject to the control-induced state Equation (57) with prescribed initial data and where C = ðc 1 , c 2 Þ: each c i is measurable with 0 ≤ c i < 1, i = 1, 2 for t ∈ ½0, t f . The solution of Equation (57) or the optimal control problem is the vector function ðS * , V * 1 , V * 2 , E, I * a , I * s , R * Þ ∈ R 7 + associated with an optimal control pair ðc * 1 , c * 2 Þ ∈ C on the time interval ½0, t f that minimizes the cost functional Equation (58).
(2) The optimal condition of the system Moreover, from Equation (61) the corresponding optimal controls c * 1 and c * 2 are given by c * 2 = max 0, min 1, (3) Hamiltonian system which is the combination of the control system and the adjoint system, respectively, defined in Journal of Applied Mathematics (4) The transversality condition of the model is also holds true. That is Next, we discuss the characterization of optimal controls and adjoint variables.
Theorem 7 (Necessary optimality conditions). Suppose ðc * 1 , c * 2 Þ is the optimal control and (S * , V * 1 , V * 2 , E * , I * a , I * s , and R * ) is the corresponding unique optimal solution of the optimal control problem given in Equation (57) with the initial condition defined in Equation (3) and Equations (60) and (61) with a fixed final time t f . Then there exists an adjoint function λ * i ðtÞ, i = 1, 2, 3, 4, 5, 6, 7 satisfying the following adjoint system:

Numerical Simulations and Its Discussion
In order to validate the analytical findings mentioned in Section 3, numerical simulations of the model utilizing the MATLAB programming language's ordinary differential equation solvers are presented in this section. Furthermore, we may statistically observe how control measures affect the coronavirus outbreak's spread in the model outlined in Section 6. To illustrate the numerical results, use the initial conditions defined in Section 4 and the parameters in Table 1.
To determine the optimal solution of the optimality system, an iterative scheme is used. Since our aim is to control the diseases, the numerical simulations carried out on the control system Equation (57), adjoint equations defined on Equation (64) with respect to Equation (65) and characterizations of the control defined on Equation (62) and Equation (63) are run in MATLAB using the Runge-Kutta forward-backward sweep method to support the analytical results discussed in Sections 3 and 6.
An iterative scheme is used to find the optimal solution to the optimality system. Since the state Equation (2) has initial conditions and the adjoint systems (66) have final conditions, we solve the state system using a forward fourth-order Runge-Kutta method and solve the adjoint system using a backward fourth-order Runge-Kutta method. The iterative solution scheme involves making a guess of the controls and using that guess to solve the state system. The initial guess of the control together with the solution  Journal of Applied Mathematics of the state systems is used to solve the adjoint systems. The controls are then updated using a convex combination of the previous controls and the values obtained using the characterizations. The solution of the state and adjoint systems is then repeated using the updated controls. This process is repeated until the values in the current iteration are close enough to the previous iteration values.
The impact of firstand second-dose vaccines, prevention, and treatment control strategies on susceptible, exposed, infected, and recovered humans is discussed qualitatively below. Figure 4 illustrates the impact of full-dose vaccination on the susceptible population. In Figure 4 14 Journal of Applied Mathematics influence of the first dose vaccination rate ðθÞ on susceptible individuals. Also, we observed that the susceptibility of individuals to COVID-19 can be decreased significantly when we increase the first-shot vaccination rate sufficiently. It is visible from Figure 4 that taking full-dose vaccination in parallel with prevention and treatment control strategies decreases the susceptibility of individuals to the COVID-19 virus. From a biological view, when the second dose vaccination rate Φ increases, it provides humans with herd immunity against the virus.

Impact Control Strategies on E(T).
The blue solid line in Figure 5(a) depicts the proportion of exposed people who received vaccinations but did not use any additional preventative measures. It rises initially before progressively falling off after 15 days. But when we use vaccinations and control strategies simultaneously, as shown in Figure 5(a), with a red solid line from the beginning, the exposed cases decrease rapidly until there are no exposed humans as time increases. This depicts how the use of both doses of vaccination, prevention, and treatments at the same time can eliminate the risk of an outbreak of the diseases. From Figure 5(b), we can conclude that if there are no control strategies, then the exposed cases increase with time. From a biological explanation, when there is no vaccination and other control strategies, anybody can be easily exposed to the disease. 7.3. Impact Control Strategies on I s ðtÞ. The effect of vaccination, preventative, and treatment control techniques on people with symptoms of infection is shown in Figure 6. As clearly depicted in Figure 6(a), the red line represents the controlled symptomatically infected humans with full dose vaccine and two control strategies, while the green line represents the controlled symptomatically infected individuals with full-dose vaccine only. This clearly indicates that the COVID-19 pandemic can be controlled fast if effective prevention, proper treatment, and vaccinations are carried out simultaneously. Moreover, if no control strategies are implemented, then symptomatically infested humans increases, as shown in Figure 6(b). Hence, we recommended that both dose vaccine, prevention, and treatment are essential to obtaining maximum protection from coronavirus. Figure 7 illustrates the impact of effective treatment, proper prevention, and vaccination on asymptotically infected humans with COVID-19. In Figure 7(a), the red line represents the controlled asymptomatically infected humans with full dose vaccine only, while the blue one shows the controlled asymptomatically infected humans with the use of full dose vaccine and the two control strategies simultaneously. The green solid line in Figure 7(b) shows that if no control strategies are used, then the rate of asymptotically infected individuals rises with time. This clearly indicates the COVID-19 pandemic can be controlled fast if effective prevention, proper treatment, and both shot vaccinations are carried out simultaneously. If the rate of the first dose of vaccination θ escalated from 0.11411 to 0.34110, then we observed that both symptomatically and asymptomatically infected cases decreased effectively, as shown in Figures 8(a) and 8(b).

Impact Control Strategies on I a ðtÞ.
7.5. Impact Control Strategies on I a ðtÞ and I s ðtÞ. Figure 9 shows that asymptomatic infected individuals can spread the disease more than symptomatically infected individuals, and it is dangerous to control them relative to the symptomatic ones using control strategies like prevention and treatment. The firstand second-shot vaccine, i.e., θ, and ρ rate has a negative correlation to controlling the pandemic (see Figure 8 and Table 3). This led us to the conclusion that if we sufficiently enhance the vaccination rate for both doses and implement efficient preventive and appropriate treatment, the infection case in both symptomatic and asymptomatic instances may be significantly reduced. Generally, Journal of Applied Mathematics from Figures 9(a) and 9(b), we conclude that asymptomatic individuals have a greater effect on the transmission of COVID-19 than the symptomatic one. In this stage, we can see that the number of cases in infected subgraphs dropped sharply after a few days, which also indicates that the combination of vaccination, personal protection, and treatment can effectively and quickly inhibit the distribution of the virus.
7.6. The Influence of Control Strategies on RðtÞ. Figure 10 shows that the recovered rate increases when full-dose vaccination, prevention, and treatment strategies are implemented simultaneously. But without thus controls, the number of recovered individuals is almost null. This reveals that the number of infectious averted humans from the COVID-19 pandemic due to the intervention of these controls observe that the biological interpretation of the model states that practicing the full dose vaccination and the two control strategies concurrently at the highest level from the beginning of the pandemic significantly minimized the number of cases in the populations to the extent that there are almost no cases, as shown in Figures 5-9, but Figure 10 shows the recovery rate increaseing.

Conclusion
With double dose vaccination and control strategies as prevention and treatment classes for the susceptible and infected populations, respectively, the newly emerged coronavirus transmission dynamics have been modeled and analyzed in this manuscript using non-linear order differential equations. Due to its human-to-human transmission method, COVID-19 encounters numerous difficulties worldwide and quickly spreads to become a pandemic. Therefore, in order to prevent this pandemic with limited resources, public health must be protected through appropriate vaccination as well as by enforcing any other control measures. In order to analyze the COVID-19 transmission dynamics using data that is readily available from Ethiopia, the SV 1 V 2 EI a I s R compartmental disease model will be developed and mathematically examined. A mathematical model fitting to estimate the unknown parameters in our proposed model is also performed using real COVID-19 confirmed case data in Ethiopia from July 01, 2022, to August 31, 2022. We ran analytical and numerical simulations of the model provided using these parameters and identified two different equilibrium points: a disease-endemic equilibrium point and a disease-free equilibrium point. We also calculated the model's reproduction number in both the vaccine's presence and absence. We called this reproduction number the control reproduction number in the vaccine's presence, and we denoted it by R c . The two equilibrium points of Equation (2) are locally and globally asymptotically stable are also shown. From the epidemiological perspective, we can draw the conclusion that the COVID-19 pandemic will approximately fade out of the population when R c < 1; otherwise, it persists in society if R c > 1. Therefore, the estimated R c for the suggested model utilizing the pandemic's reported cases can accurately ascertain the disease's true dynamics and identify the disease's current state in the Ethiopian population.
A sensitivity analysis has been performed to explore the effect of the model parameters on COVID-19 transmission dynamics. For instance, minimizing β and ω will highly help in reducing the infection burden in society. While boosting θ has a significant influence on decreasing the infection, employing θ and ρ jointly has a greater impact on minimizing the infection. The parameter β has 100% responsible for spreading the COVID-19 virus. Therefore, the effective transmission rate needs to be reduced in order to lessen the spread and burden of COVID-19 in the population. This can be accomplished by using the right preventative measures, such as health procedures like good personal hygiene, social seclusion, and the appropriate use of face masks.
This study draws special attention to controlling the transmission of the virus in society with the significance of firstand second-dose vaccination with other control strategies like perversion and treatment using real reported data in Ethiopia. Moreover, the impact of the first and second vaccine dose rate was demonstrated. Increasing vaccination rates has the predicted effect of reducing the number of infected people in the contaminated compartments. Increases in the second dose vaccination rate, in particular, have been shown to slow the spread of COVID-19 and control the endemic disease. However, COVID-19 continues to pose a hazard to the human population even after the first and second dose rates of the vaccine were implemented. Therefore, additional control measures, such as prevention and treatment for those who are most at risk of developing a significant COVID-19 illness, are necessary to further eradicate the disease in the population and stop the spread of another pandemic. The numerical simulation has demonstrated that both doses of vaccination have a negative impact on both symptomatic and asymptomatic cases. Therefore, from the numerical simulation and analytical analysis, we conclude that the use of both dose vaccination and control strategies such as treatment and perversion will highly eliminate the COVID-19 pandemic from the Ethiopian population. Therefore, the output of this study can be used as a policy input for different countries with COVID-19 pandemic. In order to contain the disease, the World Health Organization (WHO) and nations around the world must implement a policy that requires that not only do full-dose vaccinations prevent pandemic outbreaks, but also that preventive measures and treatment are made necessary for all members of the population. The spread of COVID-19 can be greatly slowed down with a full-dose vaccine in conjunction with personal protection measures and aggressive treatment of affected patients. This will protect everyone on the planet from the deadly coronavirus of our generation.

Data Availability
The data used to support the findings of this study are in this paper.

Conflicts of Interest
It is stated by the authors that they have no competing interests.