# Optimal Control Strategies of COVID-19 Dynamics Model

Authors: Temesgen Duressa Keno; Hana Tariku Etana
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this paper, we proposed the optimal control for the dynamics of COVID-19 with cost effectiveness strategies. First, we showed that model solution is positive, and bounded in a fixed domain. Besides, we used next-generation matrix to compute the basic reproduction number. If the basic reproduction number is less than one, then the disease-free equilibrium point is both locally and globally stable, respectively, via the help of Jacobian matrix and Lyapunov function, otherwise the endemic equilibrium occurs. The sensitivity analysis was determined with regard to all basic parameters. Then the model is fitted with COVID-19 infected cases reported from October 1, 2022 to October 30, 2022 in Ethiopia. The values of model parameters are then estimated from the data reported using the least-square method. Furthermore, using Pontryagin maximum principle, the model is extended to optimal control incorporating three control namely: personal protective, vaccination, and treatment of infected humans. Finally, based upon the numerical simulation of optimal controls and cost effectiveness analysis, the most optimal and less costly strategy to minimize the disease is combination of vaccination and treatment of infected.


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## Body

## 1. Introduction

The World Health Organization (WHO) declared COVID-19 to be a pandemic of international concern after the first human cases of the disease were reported by authorities in Wuhan City, China, in December 2019. As of June 1, 2022, More than 530 million infections and over 6.2 million people deaths had been caused worldwide by the virus, according to [1]. In India, the second COVID-19 wave has been extremely severe, with more than 30,000 new cases being confirmed every day during the final week of April 2021. The WHO reports that as of December 15, 2022, there have been 646,740,524 confirmed cases of COVID-19 and more than 6,637,512 deaths [2]. The first COVID-19 case to be reported in Africa occurred in Egypt on February 14, 2020. In Africa, there have been a total of 627,104,342 confirmed cases, 6,567,552 fatalities as of October 31, 2022 [2]. The first case of COVID-19 in Ethiopia was identified on March 13, 2020, in Addis Ababa, the country’s capital. Then on October 30, 2022, the Ethiopian Public Health Institute reported a total of 493,940 confirmed cases [1]. Sadly, this virus was able to spread very quickly, which increased the number of deaths each day. Therefore, it is crucial to stop the disease from spreading using non-drug methods like confinement. Due to COVID-19’s novelty, there is currently no known treatment. The disease is being treated, though, with the development of vaccines and antivirals. As the immune system of the infected person fights the virus, current case management focuses on reducing disease symptoms. It should be noted that COVID-19 complications are more likely to affect older people and people with underlying medical conditions like lung disease, asthma, cancer, diabetes, liver disease, and other immune compromised conditions [3]. The novel coronavirus disease can be prevented by adhering to measures, including regular hand washing or use of alcohol-based sanitizer, maintaining social distancing, avoiding crowded places, use of face masks, and use of hand gloves with personal protective clothing by healthcare workers while giving care to COVID-19 patients [3].A number of researchers have studied the modeling and an optimal control of COVID-19 transmission model to determines the role of control measures on disease transmission. For instance, Oke et al. [4] a mathematical model that has been proposed to investigate the COVID-19 outbreak in Africa. A system of differential equations makes up the model. The author’s findings lead to an increase in case detection and a decrease in the burden of the novel corona virus in Africa. Yang et al. [5] a mathematical model is presented to investigate the effect of vaccination on the spread of the COVID-19 virus. The authors recommended reducing contacts and vaccinating people to control COVID-19 diseases. Abriham et al. [6] the author proposed SEQISINHR compartmental model with the impact of prevention and control measures. The author comes to the conclusion that increasing isolation and quarantine rates will control COVID-19 disease. Noah and Kabir [7] presented a SEIHR compartmental non-linear deterministic epidemic model for COVID-19 transmission dynamics. The authors propose that social withdrawal, wearing a face mask in public, and isolating infected people are effective combinations for lowering COVID-19 diseases. Ahmed et al. [8] the author presented a mathematical model and method for controlling a novel coronavirus using fractional derivatives and ordinary differential equations. The authors recommended limiting contact between those who were exposed and those who were vulnerable. DarAssi et al. [9] studied a mathematical model for transmission dynamics of COCID-19. The authors concluded that the combination of vaccination, isolation and detection is the best strategies to mitigate the disease. Peter et al. [10], formulated a mathematical model for controlling a novel coronavirus in Pakistan. The authors fitted the parameter values for a using the quantity of real cases from Pakistan. The authors demonstrated that a controlled rate of transmission can decrease the pandemic of the disease.Mbogo et al. [11] presented a mathematical model for transmission of COVID-19 epidemic. The author conclude that reduce the contact between susceptible and exposed populations can minimize the spread of the disease. Khajanchi et al. [12] developed COVID-19 SEIR compartment mathematical modeling. The author used social distances and isolation of effectiveness to control novel corona virus disease. Goswami and Shanmukha [13] a non-linear mathematical model was used to examine how the COVID-19 pandemic’s dynamics of transmission affected India. Additionally, they transformed the suggested model into an optimal control model. According to the authors, models with optimal control have more consistent results when it comes to lowering the number of infected people when compared to models without optimal control. Nana-Kyere and Sacrifice [14] studied the SEIRW COVID-19 compartmental model by extending to the optimal control model using three control measures. They come to the conclusion that a combination of personal protection, drug treatment and spraying considered by significantly reducing the number of exposed and infected people in the population. Seidu [15] presented optimal strategies for control of COVID-19 using deterministic ordinary differential equations. The authors conclude that controlling the spread of COVID-19 through physical distance and use face mask is the most effective strategy. Adesoye et al. [16] discussed dynamic COVID-19 and its optimal control by incorporating three control strategies. The authors suggested that mixed of three controls such as use of face mask and hand sanitizer with social distancing, treatment of COVID-19 patient and control against recurrence to reduce COVID-19 in Nigeria.Moore and Eric [17] has investigated four control mechanisms for preventing the spread of the corona virus: personal protection, hospitalization and treatment with early diagnosis, hospitalization and treatment with delayed diagnosis, environmental spraying, and cleaning of potentially infected surfaces. The authors come to the conclusion that quarantine and medical care for infected can minimize the transmission risk and mortality rates. Obsu and Balcha [18] non-linear ordinary differential equations were used to study the transmission risk of COVID-19 with the best possible control by applying three control measures. The authors come to the conclusion that medical care and intensive prevention are good control measures to reduce the number of populations that are exposed and infected. Asamoah et al. [19] proposed a COVID-19 model to study how to control the spread of the corona virus in Saudi Arabia using four control measures, including personal hygiene, proper safety precautions, practicing proper protocol, and fumigating schools. In the absence of vaccination, the author’s conclusion is that implementing physical or social distance protocols is the most effective and cost-efficient control intervention. Maryam [20] presented a compartment of the SEIR model with the optimal control problem using two control measures for COVID-19 transmission. In order to reduce COVID-19, the author suggests a combination of two control measures: vaccination and health education. Deressa and Duressa [21] investigated the SEIR model of COVID-19 transmission dynamics with ideal control to hospitalized and asymptomatic individuals. Using actual data from Ethiopia, the author estimated the parameter value. They come to the conclusion that a combination of three factors, including personal protection, public health education, and COVID-19 infection treatment, can reduce the number of coronavirus infections. Li et al. [22] developed a COVID-19 model with three control measures for optimal control. The authors concluded that the combination of vaccination, isolation and detection is the best strategies to mitigate the disease. Olaniyi et al. [23] the formulated optimal control of COVID-19 dynamics using Pontryagin maximum principle with different controls strategies. Finally, they conclude that these controls can reduce the prevalence of COVID-19 in the population.All of these studies, however, did not analyze the COVID-19 vaccine epidemic optimal control model with cost effectiveness analysis. In this paper, the model [24] is extended to the optimal control with cost effectiveness strategies. Moreover, the parameters are estimated using real data of confirmed cases from October 1, 2022 to October 30, 2022 in Ethiopia.This paper is arrange as follows: mathematical model is discussed in Section2. The model of analysis is describe in Section 3. Sensitivity analysis has been discussed in Section 4. By using Pontryagin maximum principle, the optimal control of the COVID-19 transmission model is analytically analyzed in Section 5. We show the numerical simulation in Section 6. The analysis of cost effectiveness is depicted in Section 7. Finally, in Section 8 we summary and discussion of the results.


## 2. Model Formulation and Description

The model formulate for the COVID-19 transmission is subdivided into five classes: SusceptibleSt, Exposed Et, Infected It, hospitalized Ht and Recovered Rt Classes. Thus, the size of total population is given as Nt=St+Et+It+Ht+Rt. The susceptible human population class is created by the recruitment rate π through birth and immigration, leaves by having active contact at the rate β, and also decreases by natural death at the rate μ. These people move up to exposed class Et. When an exposed person exhibits clinical symptoms, they are moved into the infected class It at a rate of δ, which causes the exposed class to decrease. Additionally, it decreases at rates of μ, σ, and τ1, respectively, through natural death, disease-related death, and recovery from early intervention (treatment). The population of infected people decreases due to hospitalization at a rate of θ, natural death at a rate of μ, and disease-related death at a rate of σ. The number of patients in hospitals increase when infected people move there at a rate of θ and decreases when people die naturally at a rate of μ. Additionally, it decreases through COVID-19-related deaths at a rate of σ and recovery at a rate of τ2. The population of the recovered class increase whenever the exposed class recovers from early treatment at rates of τ1, as well as when hospitalized patients recover from treatment after being hospitalized at rates of τ2, and it decreases due to natural death. The recovered individuals become again susceptible to the disease with at rate of ω. Figure 1 shows the flow diagram of COVID-19 and the description of all parameters are listed in Table 1.Figure 1
Flow diagram of COVID-19 transmission dynamics.Table 1
Parameters description used for the model (1).
ParametersParameters descriptionπHuman recruitment rateβProbability of contact rate transmissionτ1Early intervention recovery rateθHospitalization rate of infected peopleδRate at which those who have been exposed become infectedμNatural deaths rate of humanτ2The proportion of hospitalized patients who recoverσRate of COVID-19-related deathsωRate of recovered individuals to be again susceptible classBased on the above diagram, the description of the dynamics of the COVID-19 disease is depicted by following set of equations:(1)dSdt=π−βIS−μS+ωR,dEdt=βIS−δ+σ+μ+τ1E,dIdt=δE−θ+σ+μI,dHdt=θI−τ2+σ+μH,dRdt=τ1E+τ2H−μ+ωR,with initial conditions(2)S0=S0,E0=E0,I0=I0,H0=H0,R0=R0.


## 3. Mathematical Analysis of the Model

### 3.1. Invariant Region

Invariant region is used to determine in which the solution of the model is bounded.Theorem 1.
The closed regionΩ=S,E,I,H,R∈R+5:Nt≤π/μ is positive invariant set for the model (1).Proof.
Let the total populationN (t) at any time t is given by(3)Nt−=St+Et+It+Ht+Rt.
Differentiation of (3), with respect to t gives(4)dNdt=π−μN−E+I+Hσ.
This implies that(5)dNdt≤π−μN.
Integrating the (5), we obtain(6)π−μNt≥π−μN0e−μt.
By taking limit ast⟶∞ on the inequality (6), we get(7)Nt≤πμ.
Therefore, all the state variables with the initial values given in (1) for the human population are given by(8)Ω=S,E,I,H.R∈R+5:0≤Nt≤πμ,which is the set of feasible solutions for the model (1), all of which are contained within it, and we draw the conclusion that the model (1) is mathematically and epidemiologically well-posed within Ω [25].


### 3.2. Positivity and Boundedness

Theorem 2.
LetS0,E0,I0,H0 and R0 be non negative initial conditions, then the solutions (St,Et,It,Ht,Rt) of the proposed model in (1) are positive for all t>0.Proof.
Consider the following from the model’s first equation (1)(9)dsdt=π−βI+μS+ωR.
This is true that(10)dSdt≥−βI+μS.
Integrating (10) using the method of separating the variables by applying the initial condition when solving with respect to time, we obtain(11)St≥S0e−βI+μt>0.
Using the same justification, we can demonstrate that(12)Et≥E0e−τ1+δ+σ+μt>0,It≥I0e−θ+σ+μt>0,Ht≥H0e−τ2+σ+μt>0,Rt≥R0e−ω+μt>0.
This demonstrates that system (1) solution is positive for all t > 0. As a result, the suggested model (1) is epidemiologically significant and mathematically well posed in the domain Ω [25].


### 3.3. Disease Free Equilibrium (DFE)

In the absence of the COVID-19, the model has a COVID-19 free equilibrium that represents the system’s critical points (1). AtIt=0, we set the left side of model (1) equal to zero in order to determine the DFE. By using the symbol E0 to stand for the disease-free equilibrium point and solving for the non-infected state variable, the disease-free equilibrium point is obtained.(13)E0=S0,E0,I0,H0,R0=πμ,0,0,0,0.


### 3.4. Basic Reproduction NumberR0

In this section, the basic reproduction number, denoted byR0, is defined as the average amount of secondary infectious caused by primary infection in the given period [26]. We use the next-generation matrix method to compute reproduction number population classes. The first step is rewrite the model (1), by starting the infected human, exposed and hospitality population classes.(14)dEdt=βSI−δ+σ+μ+τ1E,dIdt=δE−θ+σ+μI,dHdt=θI−τ2+σ+μ.The right-hand side of system (11) for this model can be expressed as f-v, where(15)f=βSI00,v=τ1+δ+σ+μEθ+σ+μI−δEτ2+σ+μ−θI.The Jacobian matrices atF and Vdisease-free equilibrium point are obtained by taking a partial derivative with respect to E, I and H as follows:(16)F=0βπμ0000000,V=τ1+δ+σ+μ00−δθ+σ+μ00−θτ2+σ+μ,V−1=1τ1+δ+σ+μ00δτ1+δ+σ+μθ+σ+μ1θ+σ+μ0θσ−θ+σ+μτ1+δ+σ+μθ+σ+μτ2+σ+μ01τ2+σ+μ.Moreover, the product ofFV−1 is obtain as:(17)FV−1=βπδμτ1+δ+σ+μθ+σ+μβπμθ+σ+μ0000000.Given thatF is non-negative and V is non-singular, V−1 and FV−1 are both non-negative. The matrix FV−1 is referred to as the model’s next-generation matrix [27]. It is possible to calculate the eigenvalues of the matrix FV−1 as det (FV−1−λ) = FV−1−λ = 0. This implies.(18)FV−1−λ=βπδμτ1+δ+σ+μθ+σ+μ−λβπμθ+σ+μ000−λ0000−λ=0.Final from (18), the basic reproduction number, R0 = ρ (FV−1) is the largest eigenvalue of the product FV−1 and given as:(19)R0=δβπμτ1+μ+δ+σθ+σ+μ.


### 3.5. Local Stability of Disease Free-Equilibrium

Theorem 3.
The disease free-equilibrium point of system (1) is locally asymptotically stable in Ω if R0<1.Proof.
The system’s Jacobian matrix of model (1) is calculated as(20)JE0=−βI+μ0−βS0ωβI−τ1+δ+σ+μβS000δ−θ+σ+μ0000θ−τ2+σ+μ00τ10τ2−μ+ω.
The Jacobian matrix at DFEE0=π/μ,0,0,0,0 of system (1) is given as(21)JE0=−μ0−βπμ0ω0−τ1+δ+σ+μβπμ000δ−θ+σ+μ0000θ−τ2+σ+μ00τ10τ2−μ+ω.
From (21), the corresponding characteristic equation has the following form:(22)−μ−λ−λ−μ+ω−τ2+σ+μ−λλ2+b1λ+b2=0,where(23)b1=τ1+δ+σ+μ+θ+σ+μ,b2=τ1+δ+σ+μθ+σ+μ−δπβμ.
From (22), we obtain that(24)λ1=−μ<0,λ2=−μ+ω<0,λ3=−τ2+σ+μ<0,and from the last characteristic equation, we have(25)λ2+b1λ+b2=0.
By applying the Routh-Hurwitz criteria [28], equation (25) has a real root that is strictly negative if b1>0 and b2>0. Due to the fact that b1 is the sum of positive parameters, we can see that b1>0, we have(26)b2=τ1+δ+σ+μθ+σ+μ−δπβμ=1−R0.
However,1−R0 must be positive for b2> 0 to hold true, which implies that R0 < 1. Since R0<1, the disease-free equilibrium of model (1) is locally asymptotically stable in Ω.


### 3.6. Globally Stability of Disease Free-Equilibrium

Theorem 4.
IfR0<1, then the disease free-equilibrium point of system (1) is globally asymptotically stable in Ω.Proof.
Take the Lyapunov function, as described by:(27)L=δδ+σ+μ+τ1θ+σ+μE+1θ+σ+μI,where the DFE point E0 is in the open neighborhood defined by Ω. For equations Et and It, the Lyapunov function L is continuously differentiable, and L>0 for all E,I∈Ω as well as L=0 at DFE.(28)dLdt=δδ+σ+μ+τ1θ+σ+μdEdt+1θ+σ+μdIdt=δδ+σ+μ+τ1θ+σ+μβSI−δ+σ+μ+τ1E+1θ+σ+μδE−θ+σ+μI,+−δδ+σ+μ+τ1θ+σ+μδ+σ+μ+τ1+1θ+σ+μδE,=δβπδ+σ+μ+τ1θ+σ+μμ−1I+−δθ+σ+μ+δθ+σ+μE,=δβπδ+σ+μ+τ1θ+σ+μμ−1I.
This implies that(29)dLdt=R0−1I.
Hence,dL/dt≤0, we must have R0−1≤0. Thus, dL/dt<0 if R0<1 and dL/dt=0 if and only if I=0. Therefore, the singleton set E0=π/μ,0,0,0,0 is the largest compact invariant set in E,I∈Ω. As a result, every solution equations of model (1) with initial conditions in the Ω range approaches the disease-free equilibrium point as time (t) tends to infinity whenever R0<1, according to Lasalle’s invariant principle [29]. As a result, if R0<1, the DFE pont is globally asymptotically stable in Ω.


### 3.7. Endemic Equilibrium Point

The steady state solution where the disease continues to affect the population is the endemic equilibrium point, denoted by the symbolE1=S∗,E∗,I∗,H∗,R∗. If we assume that the model’s equilibrium point is typically S∗,E∗,I∗,H∗,R∗, then model (1) gives;(30)S∗=π+ωRR0−1+μ,E∗=πR0−1R0−1+μτ1+δ+σ+μ,I∗=R0−1β,H∗=θR0−1τ2+σ+μ,R∗=τ1βπIτ2+σ+μ+τ2θIμτ1+δ+σ+μμτ2+σ+μτ1+δ+σ+μR0.It follows logically from the the above that wheneverR0>1, a unique positive endemic equilibrium point exists.


### 3.8. Globally Stability of Endemic Equilibrium

Theorem 5.
IfR0>1, then the endemic equilibrium point of the model equation blue1 is globally asymptotically stable.Proof.
Consider the lyapunov functionL defined by:(31)L=12S−S∗+E−E∗+I−I∗+H−H∗+R−R∗2.
The partial derivative ofblue24 with respect to time t corresponding to system (1) is obtain as,(32)dLdt=S−S∗+E−E∗+I−I∗+H−H∗+R−R∗ddtS+E+I+H+R=S−S∗+E−E∗+I−I∗+H−H∗+R−R∗dNdt.
Then, from the equations (5) we have,(33)dNdt=ddtS+E+I+H+R≤π−μN.
By substituting equations of model (33) into the equation (32), and the result is(34)dLdt=S−S∗+E−E∗+I−I∗+H−H∗+R−R∗ddtS+E+I+H+R,≤S−S∗+E−E∗+I−I∗+H−H∗+R−R∗π−μN,≤N−πμπ−μN.
We get the following result by rearranging and simplifying the equation (34).(35)dLdt≤−1μπ−μN2.
Hence,dL/dtS,E,I,H,R≤0 and dL/dt=0, if and only if S∗=S, E∗=E, I∗=I, H∗=H, and R∗=R. Therefore, the largest positive invariant set in S∗,E∗,I∗,H∗,R∗∈Ω:dL/dt=0 is a singleton E1. Thus, E1 is globally asymptotically stable in the set Ω in accordance to Lasalle’s invariant principle [29].


## 3.1. Invariant Region

Invariant region is used to determine in which the solution of the model is bounded.Theorem 1.
The closed regionΩ=S,E,I,H,R∈R+5:Nt≤π/μ is positive invariant set for the model (1).Proof.
Let the total populationN (t) at any time t is given by(3)Nt−=St+Et+It+Ht+Rt.
Differentiation of (3), with respect to t gives(4)dNdt=π−μN−E+I+Hσ.
This implies that(5)dNdt≤π−μN.
Integrating the (5), we obtain(6)π−μNt≥π−μN0e−μt.
By taking limit ast⟶∞ on the inequality (6), we get(7)Nt≤πμ.
Therefore, all the state variables with the initial values given in (1) for the human population are given by(8)Ω=S,E,I,H.R∈R+5:0≤Nt≤πμ,which is the set of feasible solutions for the model (1), all of which are contained within it, and we draw the conclusion that the model (1) is mathematically and epidemiologically well-posed within Ω [25].


## 3.2. Positivity and Boundedness

Theorem 2.
LetS0,E0,I0,H0 and R0 be non negative initial conditions, then the solutions (St,Et,It,Ht,Rt) of the proposed model in (1) are positive for all t>0.Proof.
Consider the following from the model’s first equation (1)(9)dsdt=π−βI+μS+ωR.
This is true that(10)dSdt≥−βI+μS.
Integrating (10) using the method of separating the variables by applying the initial condition when solving with respect to time, we obtain(11)St≥S0e−βI+μt>0.
Using the same justification, we can demonstrate that(12)Et≥E0e−τ1+δ+σ+μt>0,It≥I0e−θ+σ+μt>0,Ht≥H0e−τ2+σ+μt>0,Rt≥R0e−ω+μt>0.
This demonstrates that system (1) solution is positive for all t > 0. As a result, the suggested model (1) is epidemiologically significant and mathematically well posed in the domain Ω [25].


## 3.3. Disease Free Equilibrium (DFE)

In the absence of the COVID-19, the model has a COVID-19 free equilibrium that represents the system’s critical points (1). AtIt=0, we set the left side of model (1) equal to zero in order to determine the DFE. By using the symbol E0 to stand for the disease-free equilibrium point and solving for the non-infected state variable, the disease-free equilibrium point is obtained.(13)E0=S0,E0,I0,H0,R0=πμ,0,0,0,0.


## 3.4. Basic Reproduction NumberR0

In this section, the basic reproduction number, denoted byR0, is defined as the average amount of secondary infectious caused by primary infection in the given period [26]. We use the next-generation matrix method to compute reproduction number population classes. The first step is rewrite the model (1), by starting the infected human, exposed and hospitality population classes.(14)dEdt=βSI−δ+σ+μ+τ1E,dIdt=δE−θ+σ+μI,dHdt=θI−τ2+σ+μ.The right-hand side of system (11) for this model can be expressed as f-v, where(15)f=βSI00,v=τ1+δ+σ+μEθ+σ+μI−δEτ2+σ+μ−θI.The Jacobian matrices atF and Vdisease-free equilibrium point are obtained by taking a partial derivative with respect to E, I and H as follows:(16)F=0βπμ0000000,V=τ1+δ+σ+μ00−δθ+σ+μ00−θτ2+σ+μ,V−1=1τ1+δ+σ+μ00δτ1+δ+σ+μθ+σ+μ1θ+σ+μ0θσ−θ+σ+μτ1+δ+σ+μθ+σ+μτ2+σ+μ01τ2+σ+μ.Moreover, the product ofFV−1 is obtain as:(17)FV−1=βπδμτ1+δ+σ+μθ+σ+μβπμθ+σ+μ0000000.Given thatF is non-negative and V is non-singular, V−1 and FV−1 are both non-negative. The matrix FV−1 is referred to as the model’s next-generation matrix [27]. It is possible to calculate the eigenvalues of the matrix FV−1 as det (FV−1−λ) = FV−1−λ = 0. This implies.(18)FV−1−λ=βπδμτ1+δ+σ+μθ+σ+μ−λβπμθ+σ+μ000−λ0000−λ=0.Final from (18), the basic reproduction number, R0 = ρ (FV−1) is the largest eigenvalue of the product FV−1 and given as:(19)R0=δβπμτ1+μ+δ+σθ+σ+μ.


## 3.5. Local Stability of Disease Free-Equilibrium

Theorem 3.
The disease free-equilibrium point of system (1) is locally asymptotically stable in Ω if R0<1.Proof.
The system’s Jacobian matrix of model (1) is calculated as(20)JE0=−βI+μ0−βS0ωβI−τ1+δ+σ+μβS000δ−θ+σ+μ0000θ−τ2+σ+μ00τ10τ2−μ+ω.
The Jacobian matrix at DFEE0=π/μ,0,0,0,0 of system (1) is given as(21)JE0=−μ0−βπμ0ω0−τ1+δ+σ+μβπμ000δ−θ+σ+μ0000θ−τ2+σ+μ00τ10τ2−μ+ω.
From (21), the corresponding characteristic equation has the following form:(22)−μ−λ−λ−μ+ω−τ2+σ+μ−λλ2+b1λ+b2=0,where(23)b1=τ1+δ+σ+μ+θ+σ+μ,b2=τ1+δ+σ+μθ+σ+μ−δπβμ.
From (22), we obtain that(24)λ1=−μ<0,λ2=−μ+ω<0,λ3=−τ2+σ+μ<0,and from the last characteristic equation, we have(25)λ2+b1λ+b2=0.
By applying the Routh-Hurwitz criteria [28], equation (25) has a real root that is strictly negative if b1>0 and b2>0. Due to the fact that b1 is the sum of positive parameters, we can see that b1>0, we have(26)b2=τ1+δ+σ+μθ+σ+μ−δπβμ=1−R0.
However,1−R0 must be positive for b2> 0 to hold true, which implies that R0 < 1. Since R0<1, the disease-free equilibrium of model (1) is locally asymptotically stable in Ω.


## 3.6. Globally Stability of Disease Free-Equilibrium

Theorem 4.
IfR0<1, then the disease free-equilibrium point of system (1) is globally asymptotically stable in Ω.Proof.
Take the Lyapunov function, as described by:(27)L=δδ+σ+μ+τ1θ+σ+μE+1θ+σ+μI,where the DFE point E0 is in the open neighborhood defined by Ω. For equations Et and It, the Lyapunov function L is continuously differentiable, and L>0 for all E,I∈Ω as well as L=0 at DFE.(28)dLdt=δδ+σ+μ+τ1θ+σ+μdEdt+1θ+σ+μdIdt=δδ+σ+μ+τ1θ+σ+μβSI−δ+σ+μ+τ1E+1θ+σ+μδE−θ+σ+μI,+−δδ+σ+μ+τ1θ+σ+μδ+σ+μ+τ1+1θ+σ+μδE,=δβπδ+σ+μ+τ1θ+σ+μμ−1I+−δθ+σ+μ+δθ+σ+μE,=δβπδ+σ+μ+τ1θ+σ+μμ−1I.
This implies that(29)dLdt=R0−1I.
Hence,dL/dt≤0, we must have R0−1≤0. Thus, dL/dt<0 if R0<1 and dL/dt=0 if and only if I=0. Therefore, the singleton set E0=π/μ,0,0,0,0 is the largest compact invariant set in E,I∈Ω. As a result, every solution equations of model (1) with initial conditions in the Ω range approaches the disease-free equilibrium point as time (t) tends to infinity whenever R0<1, according to Lasalle’s invariant principle [29]. As a result, if R0<1, the DFE pont is globally asymptotically stable in Ω.


## 3.7. Endemic Equilibrium Point

The steady state solution where the disease continues to affect the population is the endemic equilibrium point, denoted by the symbolE1=S∗,E∗,I∗,H∗,R∗. If we assume that the model’s equilibrium point is typically S∗,E∗,I∗,H∗,R∗, then model (1) gives;(30)S∗=π+ωRR0−1+μ,E∗=πR0−1R0−1+μτ1+δ+σ+μ,I∗=R0−1β,H∗=θR0−1τ2+σ+μ,R∗=τ1βπIτ2+σ+μ+τ2θIμτ1+δ+σ+μμτ2+σ+μτ1+δ+σ+μR0.It follows logically from the the above that wheneverR0>1, a unique positive endemic equilibrium point exists.


## 3.8. Globally Stability of Endemic Equilibrium

Theorem 5.
IfR0>1, then the endemic equilibrium point of the model equation blue1 is globally asymptotically stable.Proof.
Consider the lyapunov functionL defined by:(31)L=12S−S∗+E−E∗+I−I∗+H−H∗+R−R∗2.
The partial derivative ofblue24 with respect to time t corresponding to system (1) is obtain as,(32)dLdt=S−S∗+E−E∗+I−I∗+H−H∗+R−R∗ddtS+E+I+H+R=S−S∗+E−E∗+I−I∗+H−H∗+R−R∗dNdt.
Then, from the equations (5) we have,(33)dNdt=ddtS+E+I+H+R≤π−μN.
By substituting equations of model (33) into the equation (32), and the result is(34)dLdt=S−S∗+E−E∗+I−I∗+H−H∗+R−R∗ddtS+E+I+H+R,≤S−S∗+E−E∗+I−I∗+H−H∗+R−R∗π−μN,≤N−πμπ−μN.
We get the following result by rearranging and simplifying the equation (34).(35)dLdt≤−1μπ−μN2.
Hence,dL/dtS,E,I,H,R≤0 and dL/dt=0, if and only if S∗=S, E∗=E, I∗=I, H∗=H, and R∗=R. Therefore, the largest positive invariant set in S∗,E∗,I∗,H∗,R∗∈Ω:dL/dt=0 is a singleton E1. Thus, E1 is globally asymptotically stable in the set Ω in accordance to Lasalle’s invariant principle [29].


## 4. Sensitivity Analysis

In this section, we have calculated the sensitivity analysis of parameters in the model (1), which affected the basic reproduction number in relation to parameter values in the Covid-19 model. Sensitivity analysis, as described in [30], for the basic reproductive number mainly helps to discover parameters that have a high impact on the values of R0 and hence should be targeted for designing intervention strategy. These parameters can increase or decrease the basic reproduction number if their value increase or decrease and vice versa. Determine the normalized forward sensitivity index of the reproduction number regarding various model parameters in order to find the parameters that have a significant impact on the basic reproduction numbers R0.Definition 1.
The normalized forward sensitivity index ofR0 that can be differentiable with respect to a particular parameter M is defined as [31, 32](36)kMR0=∂R0∂M×MR0.
The sensitivity index ofR0 with respect to parameter β, for instance, is defined as follows:(37)kβR0=∂R0∂β×βR0=0.5>0.
Using the same technique with respect to the parameters,kπR0, kδR0, kτ1R0, kσR0, kμR0, kθR0 are computed and the sensitivity index are given in Table 2.Table 2
Sensitivity indices of parameters.
Parameters symbolSensitivity indexπ0.5δ0.5β0.5σ−0.152158μ−0.152158τ1−0.0315θ−0.4092


### 4.1. Interpretation of the Sensitivity Indices

The sensitivity indices ofR0 with respect to basic parameters are described in Table 2. This finding demonstrates how the parameters π,δ, and β have positive sensitivity indices that increase the value of R0 as their values increase while the other parameters remain constant. And given that the μ,σ,τ1 and θ parameters all have negative indices, increasing their values while holding the other parameters constant will decrease the value of R0.


### 4.2. Parameter Estimations

In this section, we apply the suggested model to the total confirmed COVID-19 infection cases in Ethiopia and estimate the unknown model parameters using monthly cumulative data. Table3 shows monthly data of all COVID-19 infection cases in Ethiopia from October 1, 2022, to October 30, 2022, taken from WHO situation reports (WHO, 2022).Table 3
Monthlye cases of COVID-19 in Ethiopia from October 1, 2022 to October 30, 2022.
NoMonthCOVID-19 confirmed cases11 October493,58722 October493,58833 October493,61644 October493,62755 October493,64066 October493,65477 October493,67188 October493,68299 October493,6841010 October493,6981111 October493,7081212 October493,7231313 October493,7381414 October493,7671510 October493,7811616 October493,7831717 October493,7831818 October493,8031919 October493,8182020 October493,8372121 October493,8372222 October493,8642323 October493,8752424 October493,8852525 October493,8942626 October493,9052727 October493,9122828 October493,9372929 October493,9393030 October493,940We formulate system (1) in the following method to solve the dynamic parameter estimation problem:(38)z′=ft,z,θ,zt0=z0,where θ is the vector of unknown parameters and z is the vector of dependent variables. The sum of squares error (SSE) used to measure the error is represented by(39)SSEθ=argmin∑i=1zit−z¯it2,where n is the number of real data points that are available and . represents the Euclidean norm in the variable Rn. The expression zit is the corresponding model solution at time ti, and the z¯it represents the actual total confirmed cases of COVID-19. During least-squares fitting, we seek a value for the model parameter θ such that the squared sum of errors is at its lowest. This value is θ¯. Given that the dependence of a solution zt,θ on the parameter θ is through a highly non-linear system of differential equations, it is clear that this problem is a non-linearleast-squares problem.As shown in Figure2, the model parameters of system (1) are estimated using least-square fitting methods, which results in a better fit for the model solution to the real data. The corresponding parameter values are shown in Table 4.Figure 2
SEIHR model fit with real data on the number of COVID-19 cases in Ethiopia.Table 4
Parameters from literature value and estimated values used for model (1).
ParametersEstimated valueReferenceμ0.0000301Estimatedπ4995Estimatedδ0.000011618[16]σ0.0218Estimatedω1.2981Estimatedβ0.38974Estimatedτ10.03232Assumedτ20.6450Estimatedθ0.06813/day[33]As shown in Figure2, the model solution fits the real data better when the model parameters of the model system (1) are estimated using least-square fitting methods. The corresponding parameter values are shown in the color blue table Table 4 below. The model’s basic reproduction number estimate is given by the expression R0=2.64196>1. The prevalence of COVID-19 will cause an epidemic because R0>1.


## 4.1. Interpretation of the Sensitivity Indices

The sensitivity indices ofR0 with respect to basic parameters are described in Table 2. This finding demonstrates how the parameters π,δ, and β have positive sensitivity indices that increase the value of R0 as their values increase while the other parameters remain constant. And given that the μ,σ,τ1 and θ parameters all have negative indices, increasing their values while holding the other parameters constant will decrease the value of R0.


## 4.2. Parameter Estimations

In this section, we apply the suggested model to the total confirmed COVID-19 infection cases in Ethiopia and estimate the unknown model parameters using monthly cumulative data. Table3 shows monthly data of all COVID-19 infection cases in Ethiopia from October 1, 2022, to October 30, 2022, taken from WHO situation reports (WHO, 2022).Table 3
Monthlye cases of COVID-19 in Ethiopia from October 1, 2022 to October 30, 2022.
NoMonthCOVID-19 confirmed cases11 October493,58722 October493,58833 October493,61644 October493,62755 October493,64066 October493,65477 October493,67188 October493,68299 October493,6841010 October493,6981111 October493,7081212 October493,7231313 October493,7381414 October493,7671510 October493,7811616 October493,7831717 October493,7831818 October493,8031919 October493,8182020 October493,8372121 October493,8372222 October493,8642323 October493,8752424 October493,8852525 October493,8942626 October493,9052727 October493,9122828 October493,9372929 October493,9393030 October493,940We formulate system (1) in the following method to solve the dynamic parameter estimation problem:(38)z′=ft,z,θ,zt0=z0,where θ is the vector of unknown parameters and z is the vector of dependent variables. The sum of squares error (SSE) used to measure the error is represented by(39)SSEθ=argmin∑i=1zit−z¯it2,where n is the number of real data points that are available and . represents the Euclidean norm in the variable Rn. The expression zit is the corresponding model solution at time ti, and the z¯it represents the actual total confirmed cases of COVID-19. During least-squares fitting, we seek a value for the model parameter θ such that the squared sum of errors is at its lowest. This value is θ¯. Given that the dependence of a solution zt,θ on the parameter θ is through a highly non-linear system of differential equations, it is clear that this problem is a non-linearleast-squares problem.As shown in Figure2, the model parameters of system (1) are estimated using least-square fitting methods, which results in a better fit for the model solution to the real data. The corresponding parameter values are shown in Table 4.Figure 2
SEIHR model fit with real data on the number of COVID-19 cases in Ethiopia.Table 4
Parameters from literature value and estimated values used for model (1).
ParametersEstimated valueReferenceμ0.0000301Estimatedπ4995Estimatedδ0.000011618[16]σ0.0218Estimatedω1.2981Estimatedβ0.38974Estimatedτ10.03232Assumedτ20.6450Estimatedθ0.06813/day[33]As shown in Figure2, the model solution fits the real data better when the model parameters of the model system (1) are estimated using least-square fitting methods. The corresponding parameter values are shown in the color blue table Table 4 below. The model’s basic reproduction number estimate is given by the expression R0=2.64196>1. The prevalence of COVID-19 will cause an epidemic because R0>1.


## 5. Optimal Control Model

In this section, optimal control is a powerful mathematical tool that can be used to make decisions involving complex biological situations or reduce the number of exposed and infected people in the populations [34]. We extended a model (1) by incorporating three control measures to reduce COVID-19 transmissions. Thus, the following optimal control variables are provided: the control variable u1t represents the personal protection through the use of surgical face masks, social distance, isolation, and awareness of disease transmission whereas the control variable u2t, represents the vaccination. Treatment for infected people is represented by the control variable u3t. After incorporating the controls into the model (1), the optimal control model is formulated as follows:(40)dSdt=π−1−u1βIS−u2+μS+ωR,dEdt=1−u1βIS−τ1+δ+σ+μE,dIdt=δE−θ+σ+μ+u3I,dHdt=θ+u3I−τ2+σ+μH,dRdt=τ1E+τ2H−μ+ωR.In this optimal control problem, our main objective is to reduce the overall numbers of humans who have been exposed to and infected with COVID-19 in the population, while also reducing the overall cost of controlling the disease dynamics. The minimization problem’s cost functional is what we refer to as:(41)Ju1,u2,u3=minΔu1,u2,u3∫0tfdt,A1E+A2I+12B1u12+B2u22+B3u32,Subject to the terms of the model system (40). The relative weight constant Bi measures the relative cost interventions associated with the control ui for i = 1,2,3 and balances the units of integrand to reduce the dominance of any term in the integral. The expression 1/2Biui2 stands for the cost function that corresponds to the controls uit, which is quadratic in accordance with the literature [27, 35, 36]. The positive constants A1 and A2 measure the relative importance of reducing the associated classes on the spread of the disease. To find the best possible control,(42)Ju1∗,u2∗,u3∗=minJu1,u2,u3,u1,u2,u3∈Ω,where Ω the set of admissible control function as(43)Ω=u1.u2.u3.;0≤u1t,u2t,u3t≤1∀t∈0,tf,is the control set subject to the model (40). All of the controls in this situation are bounded and measurable.


### 5.1. Existence of Optmal Control Problem

In this section, we establish the existence of an optimal control triples that optimize the objective functional specified in (41). As a consequence, we get the result below.Theorem 6.
Given a cost functionalJu1,u2,u3 subject to control induced state system (40), then there exist an optimal control triples u∗ = u1∗,u2∗,u3∗ and the corresponding to the optimal solution S∗,E∗,I∗,H∗,R∗ that minimize J (u) over Ω such that(44)Ju1∗,u2∗,u3∗=minJu1,u2,u3:u1,u2,u3∈Ω.Proof.
In order to show the existence of optimal controls triples, we must confirm the five terms listed below based on the result in [37].(i)
The set of controls and related state variables is non-empty.(ii)
The closed and convex measurable control set.(iii)
A linearized function in the state and control variables, depending on time and state variables, bounded the right side of the state system.(iv)
The cost functional integrand,gx,u, is convex on Ω.(v)
There exist constantsb1, b2 > 0 and b3 > 1 such that the integrand of the objective functional satisfies gx,u ≥ b1u12,u22,u32b3/2−b2.(a)
Take note that we have demonstrated the boundedness of the Covid-19 model (1). It follows that for all valid control functions in Ω, the state system solutions are continuous and bounded. Additionally, the Lipschitz condition is satisfied with regard to state variables by the right-hand side functions of the model equation (40). As a result, the system’s (40) solutions are cited in [38]. As a result, the set Ω is not empty.(b)
Given that the control setΩ = 0,13, then Ω is closed by definition. Moreover, for any two points y, z∈Ω such that y = (y1, y2, y3) and z = (z1, z2, z3). By the definition of a convex set [39], for any λ∈0,1, it follows that λyi+1−λzi∈Ωi,i=1,2,3. Hence, λyi+1−λzi∈Ωi implies that Ω is convex set.(c)
The state system (41) is clearly linear in control variables u1, u2 and u3 with coefficients depending on state variables. With this condition (iii) is satisfied.(d)
The integrandgt,x,u of the cost functional is given by(45)gt,x,u=A1E+A2I⏟ft,x+12B1u12+B2u22+B3u32⏟ht,u.Lety = (y1,y2,y3) and z = (z1,z2,z3) ∈ω and ϖ∈0,1. Then we have(46)ht,1−ϖy+ϖz−1−ϖht,y+ϖht,z=B121−ϖ2y12+2ϖ1−ϖy1z1+ϖ2z12+B221−ϖ2y22+2ϖ1−ϖy2z2+ϖ2z22+B321−ϖ2y32+2ϖ1−ϖy3z3+ϖ2z32=ϖ2−ϖB12y1−z12+B22y2−z22+B32y3−z32≤0.Hence,(47)ht,1−ϖy+ϖz≤1−ϖht,y+ϖh1t,z,which proves thatht,u is a convex function [40]. Since the sum of two convex functions is convex, so that gt,x,u is convex function.(e)
Lastly, we have(48)gx,u=A1E+A2I+12B1u12+B2u22+B3u32,≥12B1u12+B2u22+B3u32,≥12B1u12+B2u22+B3u32b3/2−b2,whereb1 = 1/2minB1,B2,B3, b2 > 0 and b3 = 2.
Hence, there exists an optimal control that minimizes the objective function. This completes the proof.


### 5.2. Hamiltonian System

The Hamiltonian (H), which is made up of the state (40) and integrand of the objective functional (41), is required to drive the necessary condition. Pontryagin’s minimum principle for the ideal control pair (u1∗, u2∗, u2∗) is given as follows.(49)H=A1E+A2I+12∑i=13Biui2+λidSdt+dEdt+dIdt+dHdt+dRdt.Substituting (40) and (41) into (49) with respect to controls u1, u2, u3 is given by(50)H=A1E+A2I+12∑i=13Biui2+λ1Λ−1−u1βIS−u2+μS+ωR,+λ21−u1βIS−τ1+δ+σ+μE,+λ3δE−θ+σ+μ+u3I,+λ4θ+u3I−τ2+σ+μH,+λ5τ1E+τ2H−μ+ωR,where λ1, λ2, λ3, λ4, and λ5 are adjoint variables.Theorem 7.
Letu = (u1,u2,u3) is optimal control with a unique optimal solution (S^,E^,I^,H^,R^) of the optimal control problem (41) with a fixed time ttf for all t∈ [0, ttf]. Moreover, there exist adjoint function λi for i = 1,2,3,4,5 such that(51)dλ1dt=βI1−u1λ1−λ2+λ1u2+μ,dλ2dt=λ2τ1+δ+σ+μ−λ3δ−λ5τ1−A1,dλ3dt=1−u1βSλ1−λ2+λ3θ+σ+μ+u3−λ4θ+u3−A2,dλ4dt=λ4τ2+σ+μ−λ5τ2,dλ5dt=−λ1ω+λ5ω+μ.
With transiversality conditionsλitf=0 for i = 1, 2, 3, 4, 5. Furthermore, the optimal controls (u1∗, u2∗, u3∗) are given by(52)u1∗=max0,min1,βSIλ2−λ1B1u2∗=max0,min1,λ1SB2,u3∗=max0,min1,−Iλ4−λ3B3.Proof.
The Hamiltonian function’s derivative with respect toSt,Et,It,Ht, and Rt, respectively, is used to provide the form of the adjoint equation, which is then used to demonstrate the theorem. The adjoint equation is then provided by(53)dλ1dt=−∂H∂S=βI1−u1λ1−λ2+λ1u2+μ,dλ2dt=−∂H∂E=λ2τ1+δ+σ+μ−λ3δ−λ5τ1−A1,dλ3dt=−∂H∂I=1−u1βSλ1−λ2+λ3θ+σ+μ+u3−λ4θ+u3−A2,dλ4dt=−∂H∂H=λ4τ2+σ+μ−λ5τ2,dλ5dt=−∂H∂R=−λ1ω+λ5ω+μ,with transiversality conditions λitf=0 for i = 1, 2, 3, 4, 5.
The optimal controlui∗ is obtained by the optimality condition given by ∂H/ui = 0,i = 1,2,3,4,5. Which implies that(54)∂H∂u1=βSIλ1−λ2+B1u1=0,∂H∂u2=B2u2−λ1S=0,∂H∂u3=λ4−λ3I+B3u3=0.
The compact form of optimal controlsu = (u1∗, u2∗, u3∗) can be written as(55)u1∗=max0,min1,βSIλ2−λ1B1,u2∗=max0,min1,λ1SB2,u3∗=max0,min1,−Iλ4−λ3B3.
In the next part, we will see the numerical simulation of optimality system to identify an optimum and least cost strategy for controlling the dynamics of COVID-19.


## 5.1. Existence of Optmal Control Problem

In this section, we establish the existence of an optimal control triples that optimize the objective functional specified in (41). As a consequence, we get the result below.Theorem 6.
Given a cost functionalJu1,u2,u3 subject to control induced state system (40), then there exist an optimal control triples u∗ = u1∗,u2∗,u3∗ and the corresponding to the optimal solution S∗,E∗,I∗,H∗,R∗ that minimize J (u) over Ω such that(44)Ju1∗,u2∗,u3∗=minJu1,u2,u3:u1,u2,u3∈Ω.Proof.
In order to show the existence of optimal controls triples, we must confirm the five terms listed below based on the result in [37].(i)
The set of controls and related state variables is non-empty.(ii)
The closed and convex measurable control set.(iii)
A linearized function in the state and control variables, depending on time and state variables, bounded the right side of the state system.(iv)
The cost functional integrand,gx,u, is convex on Ω.(v)
There exist constantsb1, b2 > 0 and b3 > 1 such that the integrand of the objective functional satisfies gx,u ≥ b1u12,u22,u32b3/2−b2.(a)
Take note that we have demonstrated the boundedness of the Covid-19 model (1). It follows that for all valid control functions in Ω, the state system solutions are continuous and bounded. Additionally, the Lipschitz condition is satisfied with regard to state variables by the right-hand side functions of the model equation (40). As a result, the system’s (40) solutions are cited in [38]. As a result, the set Ω is not empty.(b)
Given that the control setΩ = 0,13, then Ω is closed by definition. Moreover, for any two points y, z∈Ω such that y = (y1, y2, y3) and z = (z1, z2, z3). By the definition of a convex set [39], for any λ∈0,1, it follows that λyi+1−λzi∈Ωi,i=1,2,3. Hence, λyi+1−λzi∈Ωi implies that Ω is convex set.(c)
The state system (41) is clearly linear in control variables u1, u2 and u3 with coefficients depending on state variables. With this condition (iii) is satisfied.(d)
The integrandgt,x,u of the cost functional is given by(45)gt,x,u=A1E+A2I⏟ft,x+12B1u12+B2u22+B3u32⏟ht,u.Lety = (y1,y2,y3) and z = (z1,z2,z3) ∈ω and ϖ∈0,1. Then we have(46)ht,1−ϖy+ϖz−1−ϖht,y+ϖht,z=B121−ϖ2y12+2ϖ1−ϖy1z1+ϖ2z12+B221−ϖ2y22+2ϖ1−ϖy2z2+ϖ2z22+B321−ϖ2y32+2ϖ1−ϖy3z3+ϖ2z32=ϖ2−ϖB12y1−z12+B22y2−z22+B32y3−z32≤0.Hence,(47)ht,1−ϖy+ϖz≤1−ϖht,y+ϖh1t,z,which proves thatht,u is a convex function [40]. Since the sum of two convex functions is convex, so that gt,x,u is convex function.(e)
Lastly, we have(48)gx,u=A1E+A2I+12B1u12+B2u22+B3u32,≥12B1u12+B2u22+B3u32,≥12B1u12+B2u22+B3u32b3/2−b2,whereb1 = 1/2minB1,B2,B3, b2 > 0 and b3 = 2.
Hence, there exists an optimal control that minimizes the objective function. This completes the proof.


## 5.2. Hamiltonian System

The Hamiltonian (H), which is made up of the state (40) and integrand of the objective functional (41), is required to drive the necessary condition. Pontryagin’s minimum principle for the ideal control pair (u1∗, u2∗, u2∗) is given as follows.(49)H=A1E+A2I+12∑i=13Biui2+λidSdt+dEdt+dIdt+dHdt+dRdt.Substituting (40) and (41) into (49) with respect to controls u1, u2, u3 is given by(50)H=A1E+A2I+12∑i=13Biui2+λ1Λ−1−u1βIS−u2+μS+ωR,+λ21−u1βIS−τ1+δ+σ+μE,+λ3δE−θ+σ+μ+u3I,+λ4θ+u3I−τ2+σ+μH,+λ5τ1E+τ2H−μ+ωR,where λ1, λ2, λ3, λ4, and λ5 are adjoint variables.Theorem 7.
Letu = (u1,u2,u3) is optimal control with a unique optimal solution (S^,E^,I^,H^,R^) of the optimal control problem (41) with a fixed time ttf for all t∈ [0, ttf]. Moreover, there exist adjoint function λi for i = 1,2,3,4,5 such that(51)dλ1dt=βI1−u1λ1−λ2+λ1u2+μ,dλ2dt=λ2τ1+δ+σ+μ−λ3δ−λ5τ1−A1,dλ3dt=1−u1βSλ1−λ2+λ3θ+σ+μ+u3−λ4θ+u3−A2,dλ4dt=λ4τ2+σ+μ−λ5τ2,dλ5dt=−λ1ω+λ5ω+μ.
With transiversality conditionsλitf=0 for i = 1, 2, 3, 4, 5. Furthermore, the optimal controls (u1∗, u2∗, u3∗) are given by(52)u1∗=max0,min1,βSIλ2−λ1B1u2∗=max0,min1,λ1SB2,u3∗=max0,min1,−Iλ4−λ3B3.Proof.
The Hamiltonian function’s derivative with respect toSt,Et,It,Ht, and Rt, respectively, is used to provide the form of the adjoint equation, which is then used to demonstrate the theorem. The adjoint equation is then provided by(53)dλ1dt=−∂H∂S=βI1−u1λ1−λ2+λ1u2+μ,dλ2dt=−∂H∂E=λ2τ1+δ+σ+μ−λ3δ−λ5τ1−A1,dλ3dt=−∂H∂I=1−u1βSλ1−λ2+λ3θ+σ+μ+u3−λ4θ+u3−A2,dλ4dt=−∂H∂H=λ4τ2+σ+μ−λ5τ2,dλ5dt=−∂H∂R=−λ1ω+λ5ω+μ,with transiversality conditions λitf=0 for i = 1, 2, 3, 4, 5.
The optimal controlui∗ is obtained by the optimality condition given by ∂H/ui = 0,i = 1,2,3,4,5. Which implies that(54)∂H∂u1=βSIλ1−λ2+B1u1=0,∂H∂u2=B2u2−λ1S=0,∂H∂u3=λ4−λ3I+B3u3=0.
The compact form of optimal controlsu = (u1∗, u2∗, u3∗) can be written as(55)u1∗=max0,min1,βSIλ2−λ1B1,u2∗=max0,min1,λ1SB2,u3∗=max0,min1,−Iλ4−λ3B3.
In the next part, we will see the numerical simulation of optimality system to identify an optimum and least cost strategy for controlling the dynamics of COVID-19.


## 6. Numerical Simulation

In this section, we will see the numerical simulation to show the model’s dynamical characteristics, the stability of the equilibrium points, and a sensitivity analysis. We perform numerical simulation on the model (1) using MATLAB software, estimating the values of the model’s basic parameters. Next, we graphically display the simulation results. We demonstrate numerically how to solve the optimal control problem proposed in system (40) and explain how a control strategy affects the spread of corona virus diseases. We employ a Runge-Kutta fourth order and the forward-backward sweep technique described in Lenhart and Workman’s book [34]. The cost coefficients for the control variables are predicted to be B1 = 60,B2 = 100, and B3 = 80, while the relative importance of reducing the associated classes on the spread of the disease is predicted to be A1 = 80 and A2 = 60. We analyze and compare the numerical results of the impact of controls on the dissemination of COVID-19 in populations using all pertinent information from the aforementioned sources.


### 6.1. Strategy A: Combination of Protective (u1) and Vaccination (u2)

The controls for personal protectionu1 and vaccination u2 in this strategy in order to maximize the objective functional J (u), with the value of the treatment of infected individuals (u3) set to zero. The exposed and infected plots demonstrated the effectiveness of the strategy under consideration. The numerical results shown in Figure 3(a) below show that the number of COVID-19-infected populations is increasing quickly in the absence of control strategies, while it is decreasing and finally reaching a halt in the presence of u1 and u2 controls. We deduced from the second numerical result shown in Figure 3(b) below experimented that the number of COVID-19 exposed individuals is rapidly increasing in the absence of control strategies, while the number of exposed populations is decreasing at the end of the experiment in the presence of u1 and u2 controls. As a result, while this strategy is effective at reducing the number of exposed people, it is less effective at doing so. Figure 3(c) also shows the control profiles u1 and u2, which come to the conclusion that personal protective control u1 controls maintain upper bound (100%) for 175 days while vaccination control u2 controls reach maximum level (95%) in 150 days.Figure 3
Simulation result with strategy of personal protectionu1 and vaccination u2.
(a)(b)(c)


### 6.2. Strategy B: Combination of Protective (u1) and Treatment (u3)

In this case, personal protection control (u1) and infection treatment (u3) have been used to reduce the objective functional J (u), while the control vaccination (u2) has been set to zero. According to the numerical results in Figure 4(a) blow, the overall number of COVID-19-infected humans is decreasing in the presence of control while rapidly increasing in the absence of control. According to the second numerical result in Figure 4(b), COVID-19 exposed rates are decreasing in the presence of controls while rising in the absence of controls. The control profiles u1 and u3 are plotted in Figure 4(c), and they indicate that controls on personal protective measures u1 and the treatment of infected people u3 should maintain an upper bound of 100% throughout the intervention period.Figure 4
Simulation strategy of personal protectionu1 and treatment of infected u3.
(a)(b)(c)


### 6.3. Strategy C: Combination of Vaccination (u2) and Treatment of Infected (u3)

In order to reduce the objective functionalJ (u), we simulated the model using control vaccination (u2) and treatment of infected (u3), with personal protection (u1) set to zero. From the numerical results shown in Figure 5(a), we deduced that the number of COVID-19-infected populations is rapidly rising in the absence of control strategies, while it is falling in the presence of controls. From the second numerical result displayed in Figure 5(b), we deduced that the number of exposed populations of COVID-19 are increasing in the absence of controls, while decreasing in the presence of controls. The control profiles u2 and u3 are also shown in Figure 5(c). The control profiles u2 and u3 come to the conclusion that the vaccination control u2 is at its maximum level (95%) in 150 days and the treatment of infected u3 is maintained at its maximum level (100%) until the end of the implementation.Figure 5
Simulation result with strategy of vaccinationu2 and treatment of infected u3.
(a)(b)(c)


### 6.4. Strategy D: Applying All Control Strategies

To reduce the objective functionalJ (u), we have used all three control interventions in this instance on each compartment. Figures 6(a) and 6(b) show that the total of infected and exposed population decreases in the presence of controls while increasing in the absence of controls. Additionally, the control profile shown in Figure 6(c) demonstrates that the control u1 and u3 are at their highest level (100%) in every days and the control u2 is at maximum level (95%) in 150 days. In all of the above strategies, this control profile’s u1 and u3 interpretation is the same.Figure 6
Simulation result with all control strategiesu1,u2 and u3.
(a)(b)(c)


## 6.1. Strategy A: Combination of Protective (u1) and Vaccination (u2)

The controls for personal protectionu1 and vaccination u2 in this strategy in order to maximize the objective functional J (u), with the value of the treatment of infected individuals (u3) set to zero. The exposed and infected plots demonstrated the effectiveness of the strategy under consideration. The numerical results shown in Figure 3(a) below show that the number of COVID-19-infected populations is increasing quickly in the absence of control strategies, while it is decreasing and finally reaching a halt in the presence of u1 and u2 controls. We deduced from the second numerical result shown in Figure 3(b) below experimented that the number of COVID-19 exposed individuals is rapidly increasing in the absence of control strategies, while the number of exposed populations is decreasing at the end of the experiment in the presence of u1 and u2 controls. As a result, while this strategy is effective at reducing the number of exposed people, it is less effective at doing so. Figure 3(c) also shows the control profiles u1 and u2, which come to the conclusion that personal protective control u1 controls maintain upper bound (100%) for 175 days while vaccination control u2 controls reach maximum level (95%) in 150 days.Figure 3
Simulation result with strategy of personal protectionu1 and vaccination u2.
(a)(b)(c)


## 6.2. Strategy B: Combination of Protective (u1) and Treatment (u3)

In this case, personal protection control (u1) and infection treatment (u3) have been used to reduce the objective functional J (u), while the control vaccination (u2) has been set to zero. According to the numerical results in Figure 4(a) blow, the overall number of COVID-19-infected humans is decreasing in the presence of control while rapidly increasing in the absence of control. According to the second numerical result in Figure 4(b), COVID-19 exposed rates are decreasing in the presence of controls while rising in the absence of controls. The control profiles u1 and u3 are plotted in Figure 4(c), and they indicate that controls on personal protective measures u1 and the treatment of infected people u3 should maintain an upper bound of 100% throughout the intervention period.Figure 4
Simulation strategy of personal protectionu1 and treatment of infected u3.
(a)(b)(c)


## 6.3. Strategy C: Combination of Vaccination (u2) and Treatment of Infected (u3)

In order to reduce the objective functionalJ (u), we simulated the model using control vaccination (u2) and treatment of infected (u3), with personal protection (u1) set to zero. From the numerical results shown in Figure 5(a), we deduced that the number of COVID-19-infected populations is rapidly rising in the absence of control strategies, while it is falling in the presence of controls. From the second numerical result displayed in Figure 5(b), we deduced that the number of exposed populations of COVID-19 are increasing in the absence of controls, while decreasing in the presence of controls. The control profiles u2 and u3 are also shown in Figure 5(c). The control profiles u2 and u3 come to the conclusion that the vaccination control u2 is at its maximum level (95%) in 150 days and the treatment of infected u3 is maintained at its maximum level (100%) until the end of the implementation.Figure 5
Simulation result with strategy of vaccinationu2 and treatment of infected u3.
(a)(b)(c)


## 6.4. Strategy D: Applying All Control Strategies

To reduce the objective functionalJ (u), we have used all three control interventions in this instance on each compartment. Figures 6(a) and 6(b) show that the total of infected and exposed population decreases in the presence of controls while increasing in the absence of controls. Additionally, the control profile shown in Figure 6(c) demonstrates that the control u1 and u3 are at their highest level (100%) in every days and the control u2 is at maximum level (95%) in 150 days. In all of the above strategies, this control profile’s u1 and u3 interpretation is the same.Figure 6
Simulation result with all control strategiesu1,u2 and u3.
(a)(b)(c)


## 7. Cost Effectiveness Analysis

In this section, in order to demonstrate the intervention that was used to minimize the spread of COVID-19, we must demonstrate the most efficient and less costly strategy with cost-effective analysis. The incremental cost effectiveness ratio (ICER) for two strategies,i and j, as calculated by [41–43] is used to investigate cost effectiveness analysis and givens as(56)ICER=Change in  total  averted costs between two strategies i and jChange in total number of infections averted between two strategies i and j.The total number of people averted is calculated by subtracting the total number of new cases with control from the total number of new cases without control using numerical simulation results from the COVID-19 model (40). Utilizing each strategy’s specific cost functions 1/2B1u12, 1/2B2u22 and 1/2B3u32 [23, 44], it is possible to determine the total cost of each strategy over the course of the intervention. The estimated total number of people averted in increasing order and the total cost are listed in Table 5 based on simulations of the system (31) using the values of parameters in Table 4.Table 5
Total amount of infection averted and total cost for all strategies.
StrategiesTotal infections avertedTotal cost ($)Strategy A26,406.9250,797.317Strategy C35,404.7962,900.205Strategy D35,659.4366,683.279Strategy B35,906.1969,862.7575Applying ICER techniques, we used the value in Table5 the number of people averted and total cost of each strategy as obtained to compare two intervention strategies. The incremental cost effectiveness ratio (ICER) for the each competing strategies is estimated as obtained as(57)ICERA=50,797.31726,406.92=1.924,ICERC=62,900.205−50,797.31735,404.79−26,406.92=1.345,ICERD=66,683.279−62,900.20535,659.43−35,404.79=14.8566,ICERB=69,862.7575−66,683.27935,906.19−35,659.43=12.884.The total number of individuals averted by Strategies A, C, D, and B in increasing order along with the total cost are shown in Table6.Table 6
Total amount of infection averted and total cost for all strategies.
StrategiesTotal infections avertedTotal cost ($)ICERStrategy A26,406.9250,797.3171.924Strategy C35,404.7962,900.2051.345Strategy D35,659.4366,683.27914.8566Strategy B35,906.1969,862.757512.884When A and C are compared in Table6, ICER (C) is lower than ICER (A). This suggests that strategy A is expensive and less effective at saving people. As a result, strategy C dominates more than strategy A. As a result, we eliminated strategy A from the competing strategies. Then, as shown in Table 7, compute the ICER once more for the other strategies C, D, and B.Table 7
Amount of the total infection averted and total cost used with ICER.
StrategiesAmount of infections savedTotal cost ($)ICERC35,404.7962,900.2051.345D35,659.4366,683.27914.8566B35,906.1969,862.757512.884The comparison between the intervention strategies C and D shows that the ICERC is lower than the ICERD in Table7. This demonstrates that strategy C saves more people than strategy D. As a result, we have omitted the strategy d from a group of competing strategies as well as the ICER as shown in Table 8.Table 8
Amount of the total infection averted and total cost used with ICER.
StrategiesAmount of infections savedTotal cost ($)ICERC35,404.7962,900.2051.345B35,906.1969,862.757512.884The comparison of the two intervention strategies, C and B, in Table8, demonstrates that ICER (C) is less than ICER (B). This suggests that strategy C more dominates than strategy B. So that the least expensive overall and most effective strategy was C. As a result of the analysis, we advise that the best and least expensive method for reducing the spread of COVID-19 is intervention C, which implies a combination of vaccination and treatment of infected individuals.


## 8. Conclusion

In this paper, we analyzed a deterministic mathematical model of the COVID-19 vaccine epidemic with optimal control model. We first obtained the feasible region where the model is epidemiologically and mathematically well-posed. Then the basic reproduction number with respect to the disease-free equilibrium point is computed using the next-generation matrix method. We then analyzed both local and global stability of the disease free equilibrium point based on basic reproduction number. Using the Jacobian matrix and the Lyapunov method, respectively, the disease-free equilibrium point is both locally and universally stable if the basic reproduction number is less than unity. The endemic equilibrium point exists when the basic reproduction number is greater than one. The model’s sensitivity analysis has been investigated. In Parameter estimation, the model parameters are fitted using COVID-19 infected data reported from October 1, 2022 to October 30, 2022 in Ethiopia. The model was expanded to the problem of optimal control using three controls, namely personal protection, immunization, and treatment infection for reducing novel coronavirus diseases. Besides, with all combinations of the three controls, the optimal control model applies the Pontraygin’s maximum principle and uses cost effectiveness analysis to determine the minimal cost using incremental cost effectiveness ratio (ICER). The outcome of the numerical simulation indicated that every control factor that was taken into account in the simulation helped to lower COVID-19 infections. The combination of vaccination and treatment for infected individuals is the best strategy, to effectively reduce the COVID-19 disease, according to the results of simulation with optimal control and cost effectiveness analysis.



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*Source: 2050684-2023-02-16.xml*