# A Mathematical Evaluation of the Cost-Effectiveness of Self-Protection, Vaccination, and Disinfectant Spraying for COVID-19 Control

Authors: Sacrifice Nana-Kyere; Baba Seidu
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

The world is on its path from the post-COVID period, but a fresh wave of the coronavirus infection engulfing most European countries makes the pandemic catastrophic. Mathematical models are of significant importance in unveiling strategies that could stem the spread of the disease. In this paper, a deterministic mathematical model of COVID-19 is studied to characterize a range of feasible control strategies to mitigate the disease. We carried out an analytical investigation of the model’s dynamic behaviour at its equilibria and observed that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number,R0 is less than unity. The endemic equilibrium is also shown to be globally asymptotically stable when R0>1. Further, we showed that the model exhibits forward bifurcation around R0=1. Sensitivity analysis was carried out to determine the impact of various factors on the basic reproduction number R0 and consequently, the spread of the disease. An optimal control problem was formulated from the sensitivity analysis. Cost-effectiveness analysis is conducted to determine the most cost-effective strategy that can be adopted to control the spread of COVID-19. The investigation revealed that combining self-protection and environmental control is the most cost-effective control strategy among the enlisted strategies.


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## Body

## 1. Introduction

The world is on its path from the post-COVID fiscal year. However, a fresh wave of the coronavirus infection engulfing the European countries makes the pandemic catastrophic, with most regions grappling with their worst outbreaks since the pandemic began [1]. Since December 2019, when the coronavirus epidemic was reported in Hubei in China, the epidemic has threatened the global well-being of humans by significantly affecting human resources through sickness and death [2]. Asian countries such as India, Sri Lanka, and Thailand are battling COVID-19 containment. The rapid resurgence of the virus has put enormous pressure on these countries’ health systems and medical supplies, with some calling for foreign assistance amid the escalating crisis. Generally, the availability of vaccines and the introduction of various vaccination programs have lowered the global daily infection and death counts, yet the deaths and infections have not entirely abated [3]. Mathematical models have made an indelible mark on the fight against infectious diseases by identifying protocols to respond to hikes in the transmission dynamics of epidemics [4–12]. Notable works in the case of COVID-19 have been considered by several authors. Yang et al. [13] made a generalized prediction of the coronavirus epidemic in mainland China by considering an SEIR model that utilized the available data. Zuo et al. [14] presented a data analysis and comparison method that concerned institutions could employ for proper decision making.calibrated the Brazil coronavirus pandemic that worked on determining the peak time of the epidemic. Wu et al. [2] used the available data from China to establish the possibility of symptomatic people dying from the coronavirus epidemic. Götz and Heidrich [15] considered a mathematical model of the COVID-19 epidemic that used data from Germany to estimate the model’s parameters and established the ascertainment of official infected cases. In [16], a nonlinear coronavirus model was considered and utilized to fit China’s data to estimate the model’s parameters. Abbasi et al. developed a mathematical model for the novel coronavirus disease and proved through stability analysis studies that, mitigating measures alone could not eradicate the disease. Their conclusion was based on the fact that the endemic equilibrium was globally asymptotically stable. Gu et al. [17] proposed a mathematical model that primarily focused on isolation, quarantine, and hospitalization to study the transmission dynamics of the novel coronavirus epidemic. They exploited nonpharmaceutical intervention measures: social distancing, isolation, contact tracing, and quarantine to mitigate the spreading of the disease. The authors further rebuilt the model into a stochastic model and compared the stochastic model analysis to the deterministic one. Currie et al. [18] developed a mathematical COVID-19 model that identified how mathematical models could influence management decision-making. In a related study, Huang et al. [19] constructed a mathematical model with several compartments that calibrated the model’s parameters with the available data and predicted the transmissibility of COVID-19. Senapati et al. [20] designed a nonlinear model that considered India’s data to estimate the model’s parameters and established the effectiveness of various protocols ushered in to suppress the disease. Zhou et al. [21] considered a SEIR COVID-19 model and applied the available data to estimate the reproduction number. In [22], a SEIR model was calibrated to the available data to derive some vital model parameters and determine the propagation trend of the epidemic. Hou et al. [23] proposed a nonlinear SEIR compartmental model that assessed the potency of Wuhan’s quarantine measure during the peaks of the epidemic and estimated the model’s parameters by fitting the model to Wuhan’s data. The model proposed by Du and Yuan [24] examined the propagation trend of the SARS-CoV2 and validated the model with existing Influenza data. Petropoulos and Makridakis [25] made a generalized prediction of the SARS-CoV2 pandemic pattern in Cyprus and underlaid strategies for containing the epidemic.Optimal control models have received credit from the scientific community for their contribution in determining essential measures for pushing down cases of infections and the spike of new variants [26–33]. In December 2019, when WHO ascertained the existence of the pandemic, intriguing optimal control studies have been carried out that helped to wrestle the epidemic from spreading. Nana-Kyere et al. [34] presented a coronavirus model that mainly examined the epidemic’s stability and constructed an optimal control problem to deduce potent strategies to curb the disease. Asamoah et al. [35] calibrated their model to Ghana’s data to estimate some essential parameters that influence the dynamics of the disease. They extended the model to an optimal control problem to pinpoint strategies that will boost lowering the infected cases of the pandemic with minimal cost. Tsay et al. [36] studied the dynamical behaviour of the epidemic by designing a compartmental model for the disease. They redesigned the model into the optimal control problem to find strategies to help manage the virus. Kouidere et al. [37] characterized some feasible control strategies that come with desirable impact in containing the 2019 coronavirus epidemic. Perkins and Guido [38] considered an unorthodox method of handling epidemic without involving drugs to determine how the coronavirus infection could be managed. They reformulated the model into an optimal control problem to deduce strategies paramount for curtailing the disease. A nonlinear COVID-2019 model was developed to calibrate the model’s parameters by fitting the model to Ethiopia’s data [39]. The mode was extended into an optimal control problem in order to determine the effectiveness of treatment controls, personal protection, and public health education’s in suppressing the epidemic. Srivastav et al. [40] categorized a compartmental COVID-19 model into two main groups of the aged and young and used the data of Italy and Spain to estimate some of the model’s parameters. The authors then considered optimal control methods that developed strategies for managing the epidemic. Seidu [41] proposed a SARS-CoV2 mathematical model that captured the respective roles of mildly symptomatic, severely symptomatic, and exposed individuals in the transmission of the disease. He modified the model to optimal control and determined the most cost-effective strategy among the considered strategies through the cost-effectiveness method. In another related study, Nana-Kyere et al. [42] proposed a nonlinear mathematical model for the COVID-19 virus that examined the transmission dynamics of the disease. They modified the model into an optimal control problem and solved it to determine whether the proposed control strategies were potent enough to mitigate the epidemic from spreading. Agossou et al. [43] assessed the effects of pharmaceutical and nonpharmaceutical control strategies on the transmission dynamics of the COVID-19 epidemic.The COVID-19 epidemic has attracted a voluminous amount of information from mathematical models since the time the disease was confirmed by WHO. Nevertheless, a significant amount of these pieces of information cannot be adequately relied on by public health to contain the virus. The unreliability of these models is because they are not epidemiologically well-composed and do not accurately describe the propagation trends of the disease for the correct management decision to be taken. This research mainly focuses on extending the model of Khan et al. [44] into an optimal control problem by identifying control strategies for the COVID epidemic and carrying out a socioeconomic analysis of the identified strategies to boost management decision-making on the containment of the virus. This has become necessary as the control model analysis would provide policy direction for public health. The paper will add to the existing knowledge on the epidemic dynamics by assessing the combined effect of treatment, vaccination, and disinfection of the environment on the transmission dynamics of the disease.The rest of the research is presented as follows: Section2 “The Model Development” presents the model formulation, model’s parameter definition, invariant region, boundedness, and stationary points. Section 3 “Qualitative Analysis” deals with the stability analysis, sensitivity and bifurcation analysis. In Section 4 “Development of Optimal Control Model”, we present an optimal control problem formulation and its qualitative analysis. Section 5 “Numerical Experimentation and Discussion” provides the numerical solution for the models. In Section 6 “Economic Cost Analysis of the Control Implementation”, a cost-effectiveness analysis is carried out to determine the most cost-effective strategy among the considered strategies. The section also discusses the simulated results of the work with a recommended conclusion.


## 2. The Model Development

The aggressiveness of the COVID-19 disease on the population means that newly designed models need to identify measures that would help clamp down on the disease. In this research, we use the COVID-19 model studied by Khan et al. [44]. The work segregates the model into eight compartments of Susceptible S, Exposed E, Infected I, asymptomatically infected A, Quarantined Q, Hospitalized H, Recovered R, and virus concentration in the environment M. The model assumes that Susceptible individuals are recruited into the population at a rate of Λ persons per unit time. The contact between a naive susceptible individual on the one hand and the infected or the asymptomatically infected on the other hand is given by η1. The parameter ψ represents the transmissibility factor associated with the Asymptomatically infected persons. The model assumes that, the mortality that occurs naturally in individuals is given by μ. The parameter θ denotes the infection following exposure. The parameters ω and ρ are the incubation periods of the disease. The infected, asymptomatically infected, quarantined, and hospitalized individuals have recovery rates, respectively, given by τ1,τ2,ϕ1,ϕ2. The infected and quarantined individuals are hospitalized at rates γ and δ2. The COVID-19-induced death rates in the infected and hospitalized are ξ1 and ξ2. Susceptible individuals contract the disease at a rate η2 due to a visit to the seafood market and contacting a viral source there or eating from a Covid-19-infected animal. The model assumes that the infected and asymptomatically infected individuals contribute to viral deposits in the seafood market at rate q1 and q2, respectively. The parameter q3 denotes the rate of removal of coronaviruses from the seafood market. Finally, the exposed individuals are quarantined at a rate δ1. Based on these assumptions, the proposed dynamical COVID-19 model is of the form:
(1)dSdt=Λ−μS−η2MS−η1ISN−ψASNη1,dEdt=η2MS+η1ISN+ψASNη1−1−θω+θρ+δ1+μE,dIdt=1−θωE−τ1+ξ1+γ+μI,dAdt=θρE−τ2+μA,dQdt=δ1E−ϕ1+δ2+μQ,dHdt=γI+δ2Q−ϕ2+ξ2+μH,dRdt=τ1I+τ2A+ϕ1Q+ϕ2H−μR,dMdt=q1I+q2A−q3M,withS0≥0,E0≥0,I0≥0,A0≥0,Q0≥0,H0≥0,R0≥0andM0≥0.


### 2.1. Qualitative Analysis: Disease-Free Equilibrium and theR0

In an epidemiological setting, the basic reproduction numberR0, crystallizes the ability of the infection to spread or die out in the population. The R0 is a measure of the number of people, on average, each sick person will infect throughout its period of infectivity. When R0 is above one, the epidemic is generally recognized to be growing, and when the number is less than one, the outbreak is said to be declining. According to Khan et al. [44], R0, the basic reproduction number is given by
(2)R0=η2Λ1−θωq1μq3R1R2+θρη2Λq2μq3R1R3+ψη1θρR1R3+η11−θωR1R2,where
(3)R1=1−θω+θρ+δ1+μ,R2=τ1+ξ1+γ+μ,R3=τ2+μ.


### 2.2. Endemic Equilibrium

Any system that is not characterized by an absence of infection but rather a high transmissibility rate will eventually result in an endemic state. The model Equation (1) has a unique endemic equilibrium given as
(4)S∗=ΛR0,Q∗=δ1E∗ϕ1+δ2+μ,A∗=θρE∗,I∗=1−θωE∗,M∗=q1I∗+q2A∗q3,H∗=γI∗+δ2Q∗ϕ2+ξ2+μ,R∗=τ1I∗+τ2A∗+ϕ1Q∗+ϕ2H∗μ,E∗=Λq3ΛR0+μτ1+ξ1+γ+μτ2+μμη2q1g1+η2q2g2+η1q3g3+η1ψg4,with g1=N1−θωτ2+μ, g2=Nθρτ1+ξ1+γ+μ,g3=1−θωτ2+μ, and g4=θρτ1+ξ1+γ+μ.Using the analysis from [44] with the notation λ=η2MS+η1IS/N+ψAS/Nη1. the endemic equilibrium can be characterised in terms of R0 as
(5)Fλ∗∗=l1λ∗∗2+l2λ∗∗+l3,where
(6)l1=k1k2k3q3k3δ1k2δ2μ+ϕ2+k5μ+ϕ1+k4k6+θk2k4k5ρμ+τ2,l2=k1k2k3μq3k3δ1k2δ2μ+ϕ2+k5μ+ϕ1+k4k8+θk2k4k5ρ−η1ψ+μ+τ2+η2k7Λ1−θk3q1ω−θk2ρq2+k12k22k4k5k32μq3,l3=k12k22k32k4k5μ2q31−R0,k1=δ1+θρ+1−θω+μ,k2=γ+μ+ξ1+τ1,k3=μ+τ2,k4=δ2+μ+ϕ1,k5=μ+ξ2+ϕ2,k6=γ1−θωμ+ϕ2+k51−θωμ+τ1+k2μ,k7=k3−δ1k2δ2μ+ϕ2+k5μ+ϕ1−k4k6−θk2k4k5ρμ+τ2,k8=γ1−θωμ+ϕ2+k51−θω−η1+μ+τ1+k2μ.The Equation (5) is used to generate the bifurcation plot in Figure 1, which was not done in [44].Figure 1
Forward bifurcation diagram.


## 2.1. Qualitative Analysis: Disease-Free Equilibrium and theR0

In an epidemiological setting, the basic reproduction numberR0, crystallizes the ability of the infection to spread or die out in the population. The R0 is a measure of the number of people, on average, each sick person will infect throughout its period of infectivity. When R0 is above one, the epidemic is generally recognized to be growing, and when the number is less than one, the outbreak is said to be declining. According to Khan et al. [44], R0, the basic reproduction number is given by
(2)R0=η2Λ1−θωq1μq3R1R2+θρη2Λq2μq3R1R3+ψη1θρR1R3+η11−θωR1R2,where
(3)R1=1−θω+θρ+δ1+μ,R2=τ1+ξ1+γ+μ,R3=τ2+μ.


## 2.2. Endemic Equilibrium

Any system that is not characterized by an absence of infection but rather a high transmissibility rate will eventually result in an endemic state. The model Equation (1) has a unique endemic equilibrium given as
(4)S∗=ΛR0,Q∗=δ1E∗ϕ1+δ2+μ,A∗=θρE∗,I∗=1−θωE∗,M∗=q1I∗+q2A∗q3,H∗=γI∗+δ2Q∗ϕ2+ξ2+μ,R∗=τ1I∗+τ2A∗+ϕ1Q∗+ϕ2H∗μ,E∗=Λq3ΛR0+μτ1+ξ1+γ+μτ2+μμη2q1g1+η2q2g2+η1q3g3+η1ψg4,with g1=N1−θωτ2+μ, g2=Nθρτ1+ξ1+γ+μ,g3=1−θωτ2+μ, and g4=θρτ1+ξ1+γ+μ.Using the analysis from [44] with the notation λ=η2MS+η1IS/N+ψAS/Nη1. the endemic equilibrium can be characterised in terms of R0 as
(5)Fλ∗∗=l1λ∗∗2+l2λ∗∗+l3,where
(6)l1=k1k2k3q3k3δ1k2δ2μ+ϕ2+k5μ+ϕ1+k4k6+θk2k4k5ρμ+τ2,l2=k1k2k3μq3k3δ1k2δ2μ+ϕ2+k5μ+ϕ1+k4k8+θk2k4k5ρ−η1ψ+μ+τ2+η2k7Λ1−θk3q1ω−θk2ρq2+k12k22k4k5k32μq3,l3=k12k22k32k4k5μ2q31−R0,k1=δ1+θρ+1−θω+μ,k2=γ+μ+ξ1+τ1,k3=μ+τ2,k4=δ2+μ+ϕ1,k5=μ+ξ2+ϕ2,k6=γ1−θωμ+ϕ2+k51−θωμ+τ1+k2μ,k7=k3−δ1k2δ2μ+ϕ2+k5μ+ϕ1−k4k6−θk2k4k5ρμ+τ2,k8=γ1−θωμ+ϕ2+k51−θω−η1+μ+τ1+k2μ.The Equation (5) is used to generate the bifurcation plot in Figure 1, which was not done in [44].Figure 1
Forward bifurcation diagram.


## 3. Qualitative Analysis: Global Stability-Disease Free Equilibrium

In performing the global qualitative analysis, we employed the method of Castillo-Chavez et al. [45] to investigate the global stability of the state Equation (1) at the disease-free equilibrium. Following Castillo-Chavez et al. [45], the COVID-19 model Equation (1) can be restructured as follows:
(7)dy1dt=W1y1,y2,dy2dt=W2y1,y2,W2y1,0=0,with y1=S,R representing the uninfected population, and y2=E,I,A,Q,H,M denoting the infected. The disease-free equilibrium point is given by.If the following criteria are satisfied, the pointU1=y10,0 is a globally asymptotically stable equilibrium for the model (1).F1: Givendy1/dt=W1y1,0, y10 is globally asymptotically stable.F2:Wy1,y2=Dy2−W~2y1,y2, where W~2y1,y2≥0 for y1,y2∈ζ.Theorem 1.
The pointU1=y10,0 is globally asymptotically stable equilibrium whenever R0<1.Proof.
Considering the model Equation (1), we generate W1y1,y2 and W2y1,y2 as;
(8)W1y1,y2=Λ−μS−η2MS−η1ISN−ψASNη1τ1I+τ2A+ϕ1Q+ϕ2H−μR,W2y1,y2=η2MS+η1ISN+ψASNη1−1−θω+θρ+δ1+μE1−θωE−τ1+ξ1+γ+μIθρE−τ2+μAδ1E−ϕ1+δ2+μQγI+δ2Q−ϕ2+ξ2+μHq1I+q2A−q3M.
WithS=S0, I=I0, A=A0, R=R0 and M=M0 then W1y1,0 becomes
(9)W1y1,y2=Λ−μS0−η2M0S0−η1I0S0N−ψA0S0Nη1τ1I0+τ2A0+ϕ1Q0+ϕ2H0−μR0.
It can be shown from (9) that as t⟶∞, y1⟶y10. Hence y1=y10 is globally asymptotically stable, which satisfies condition one. Now, ascertaining whether condition F2 would be satisfied, we consider Wy1,y2=Dy2−W~2y1,y2. Thus, we have
(10)W1y1,y2=−k1η1NS0ψη1NS000η2S01−θω−k20000θρ0−k3000δ100−k4000γ0δ2−k500q1q200−q3EIAQHM−W~2y1,y200000,where matrix D given by
(11)D=−k1η1NS0ψη1NS000η2S01−θω−k20000θρ0−k3000δ100−k4000γ0δ2−k500q1q200−k66withW~2y1,y2=S0−Sη2Mη1INψη1AN000000000000000,
It can be deduced from model Equation (1) that the total population is bounded by S0. Hence, η2MS≤η2MS0, η1I/NS≤η1I/NS0 and ψη1A/NS≤ψη1A/NS0 which implies W~2y1,y2 is positive definite. Additionally, matrix D is undoubtedly an M-matrix, with the off-diagonal entries positive. Hence, the requirement of the two conditions is met, proving the global asymptotic stability of U1.


### 3.1. Qualitative Analysis: Global Stability-Endemic Equilibrium

In assessing global asymptotic stability of the endemic equilibrium of the model system (1), we capitalize on the geometrical method [46] which uses Bendixson criterion as sufficient condition for determining model system (1) global stability.Theorem 2.
GivenR0>1, the endemic equilibrium of model Equation (1) is globally asymptotically stable; otherwise, it is unstable.Proof.
In proving the global stability of the endemic equilibrium, the study of ([47], with referencing to Lemma 3, Lemma 4 and Theorem 5.) is considered. we consider the subsystem of model Equation (1) as
(12)ddtS=Λ−μS−η2MS−η1ISN−ψASNη1,ddtE=η2MS+η1ISN+ψASNη1−1−θω+θρ+δ1+μE,ddtI=1−θωE−τ1+ξ1+γ+μI.
Considering subsystem (12) the Jacobian matrix becomes,
(13)Jv=−w110−η1S/Nw21−w22η1S/N01−θω−w33,with
(14)w11=Λ−η2M−η1IN−ψANη1−μ,w21=η2M−η1IN−ψANη1,w22=−1−θω+θρ+δ1+μ,w33=−τ1+ξ1+γ+μ.
Then,Jv2, the second additive matrix of Jv is given by
(15)Jv2=−w11+w22η1SN−η1SN1−θω−w11+w3300η2M−η1IN−ψANη1−w22+w33.
Now, we denote the functionPy=PS,E,I=diag1,E/I,E/I, hence we deduce P−′y=diag1,I/E,I/E. Then, the time derivative of Pfy becomes Pfy=diag0,E~/I−EI~/I2,E~/I−EI~/I2 and
PfyP−′y=diag0,E~/E−I~/I,E~/E−I~/I. It follows that
(16)PfyJv2P−′y=−w11+w22η1SINE−η1SINE1−θωEI−w11+w3300η2M−η1IN−ψANη1−w22+w33.
Hence, matrixC=PfyJv2P−′y+PfyP−′y can be determined and written as;
(17)C=C11C12C21C22,where
(18)C11=−η2M−η1IN−ψANη1−μ−1−θω+θρ+δ1−2μ,C12=η1SINE,−η1SINE,C21=1−θωEI,0andC22=w2+E~E−I~I0η2M−η1IN−ψANη1w3+E~E−I~I,with
(19)w2=−η2M−η1IN−ψANη1−τ1+ξ1+γ−2μ,w3=−1−θω+θρ+δ1+μ−τ1+ξ1+γ−2μ.
Considern1,n2,n3 to be a vector in ℝ3 and the norm • given by n1,n2,n3=maxn1,n2,n3. Let ρ1C represents the Lozinki measure with respect to this norm; we adopt the method of approximating the ρ1C as explored in [48], and choose ρ1C≤supp1,p2, where supp1,p2=ρ1C11+C12,ρ1C22+C21 and C12 and C21 are matrix norms with respect to vector norms ρ1, and ρ represents the Lozinki measure with respect to the norm ρ1.
Then,ρ1C11=−η2MS−η1I/NS−ψAS/Nη1−μ−1−θω+θρ+δ1−2μ(20)C12=maxη1SNE,−η1SNE=η1SINE,C12=max1−θωEI,0=1−θωEI,ρ1C22=E~E−I~I−τ1+ξ1+γ−2μ.
Hence,(21)p1=−η2MS−η1INS−ψASNη1−1−θω+θρ+δ1−2μ+η1SINE,p2=−E~E−I~I−τ1+ξ1+γ−2μ+1−θωEI.
It could be verified that(22)E~E=η2MSE+η1ISNE+ψASNEη1−1−θω+θρ+δ1+μ,I~I=1−θωEI−τ1+ξ1+γ+μ,which provides the following inequalities when substituted into P1 and P2:P1≤η1SI/NE+E~/E−μ,P2≤E~/E−μ.
Further,(23)ρ1C≤supp1,p2≤supη1SINE+E~E−μ,E~E−μ,=E~E−μ+Supη1SINE,0,=E~E−μ+η1SINE.
Hence,(24)ρ1C≤E~E−μ+η1SINE,substitute (24) into the below equation
(25)q~a=limt⟶∞sup1t∫0tρ1Cdt.
Now from Equation (25), we denote λ1=μ and λ2=η1SI/NE, then if λ2<λ1, it follows that ρ1C≤E~/E−∈ where ∈=μ−η1SI/NE. Now, for t>T, it follows that
(26)q~a=limt⟶∞sup1t∫0tρ1Cdt<−∈<0.
Therefore, from (26), we get q~a=limt⟶∞sup1/t∫0tρ1Cdt<0. This verifies that the system of nonlinear differential equation of submodel (12) is GAS around the interior equilibrium of S∗,E∗,I∗. Therefore, considering the remaining subsystem of model (1).
(27)ddtA=θρE−τ2+μA,ddtQ=δ1E−ϕ1+δ2+μQ,ddtH=γI+δ2Q−ϕ2+ξ2+μH,ddtR=τ1I+τ2A+ϕ1Q+ϕ2H−μR,ddtM=q1I+q2A−q3M.
The limit of (27) is given by
(28)ddtA=θρE∗−τ2+μA∗,ddtQ=δ1E∗−ϕ1+δ2+μQ∗,ddtH=γI∗+δ2Q∗−ϕ2+ξ2+μH∗,ddtR=τ1I∗+τ2A∗+ϕ1Q∗+ϕ2H∗−μR∗,ddtM=q1I∗+q2A∗−q3M∗.
When system (28) is solved with the initial conditions of A0,Q0,H0,R0,M0, then as t⟶∞, A⟶A∗, Q⟶Q∗, H⟶H∗, R⟶R∗, and M⟶M∗ which is the GAS of the endemic equilibrium.


### 3.2. Bifurcation Analysis

It has been observed from stability analysis of various epidemic models that theR0<1 is the sufficient condition for wiping out infections in a population. This condition usually holds when the system exhibits forward bifurcation. However, in considering a control intervention measure such as vaccination of the susceptibles, it has been observed that the condition R0<1 may not be sufficient to control the disease. Thus, the model transmissibility asymptomatic pattern is now determined by a quantity R0 defined as the effective reproduction number. It has been observed in some studies [49–53] that, there can be coexistence of the endemic and disease-free equilibrium, which are both stable for R0<1, which represents backwards bifurcation. In controlling diseases in the presence of backward bifurcation, the R0 has to be lowered to a value less than one. This section employs the theorem propounded by Castillo-Chavez et al. [54] to explore the type of bifurcation exhibited by the model system (1). To determine whether the phenomenon of bifurcation exists in model system (1), the centre manifold theory [54] is employed. The following necessary substitutions are taken to conform to the theorem’s requirement for easier computation of the bifurcation coefficients.We denotex1=S, x2=E, x3=I, x4=A, x5=Q, x6=H, x7=R, and x8=M such that the total population here is given by N=x1+x2+x3+x4+x5+x6+x7+x8. With vector notation of X=x1,x2,⋯,x8T, the model system (1) can be redesigned in the form dX/dt=FX, with F=f1,f2,⋯,f8T given by
(29)ddtx1=Λ−μx1−η2x8x1−η1x3x1N−ψx4x1Nη1,ddtx2=η2x8x1+η1x3x1N+ψx4x1Nη1−1−θω+θρ+δ1+μx2,ddtx3=1−θωx2−τ1+ξ1+γ+μx3,ddtx4=θρx2−τ2+μx4,ddtx5=δ1x2−ϕ1+δ2+μx5,ddtx6=γx3+δ2x5−ϕ2+ξ2+μx6,ddtx7=τ1x3+τ2x4+ϕ1x5+ϕ2x6−μx7,ddtx8=q1x3+q2x4−q3x8.The Jacobian of the new system (29) evaluated at the DFE, denoted by JDE0 is given by,
(30)JDE0=−μ0−η1−ψη1∗000−η2Λμ0−h11η1ψη1∗000η2Λμ01−θω−h22000000θρ0−τ2+μ00000δ100−h3300000γ0δ2−h440000τ1τ2ϕ1ϕ2−μ000q1q2000−q3,where h11=1−θω+θρ+δ1+μ, h22=τ1+ξ1+γ+μ, h33=ϕ1+δ2+μ, h44=ϕ2+ξ2+μ. Given that η1=η1∗ is considered as the bifurcation parameter in the case when R0=1, then expressing η1 in terms of the other parameters when R0=1 becomes,
(31)η1=η1∗=μq3R1R2R3−η2Λ1−θωq1q3+θρη2Λq2R2μq3ψθρR2+1−θωR3.HenceJDE0η1=η1∗ has a right eigenvector denoted by W=w1,w2,⋯,w8T, with
(32)w1=−η11−θωμq3τ2+μ+ψη1θρμq3τ1+ξ1+γ+μ+b1μ2q3τ1+ξ1+γ+μτ2+μw2,w3=1−θωτ1+ξ1+γ+μw2,w2>0,w4=θρτ2+μw2,w5=δ1ϕ1+δ2+μw2,w6=γ1−θωϕ1+δ2+μ+δ1δ2τ1+ξ1+γ+μτ1+ξ1+γ+μϕ1+δ2+μϕ2+ξ2+μw2,w7=τ11−θωϕ1+δ2+μϕ2+ξ2+μτ2+μ+b2+b3+b4μτ1+ξ1+γ+μϕ1+δ2+μϕ2+ξ2+μτ2+μw2,w8=1−θωτ2+μ+q2θρτ1+ξ1+γ+μq1τ1+ξ1+γ+μτ2+μw2,where
(33)b1=η2Λ1−θωτ2+μ+q2θρτ1+ξ1+γ+μ,b2=τ2θρτ1+ξ1+γ+μϕ1+δ2+μϕ2+ξ2+μ,b3=ϕ1δ1τ1+ξ1+γ+μϕ2+ξ2+μτ2+μ,b4=γ1−θωϕ1+δ2+μ+δ1δ2τ1+ξ1+γ+μ.Further,JDE0η1=η1∗ has a left eigenvector V=v1,v2,⋯,v8T, where v1=v5=v6=v7=0, v2>0, v3=η1q3μ+q1η2Λ/q3τ1+ξ1+γ+μμv2, v4=ψη1q3μ+q2η2Λ/q3τ2+μμv2,v8=η2Λ/q3μv2.The newly translated model Equation (29) with the bifurcation parameter η1=η1∗ has at least one eigenvalue zero, and it affirms the existence of bifurcation in the transformed model (29). As expounded by Castillo-Chavez and Song ([54], refer to Theorem 5.1.), the equations
(34)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xjη1∗,0,b=∑k,i,j=1nvkwi∂2fk∂xi∂η1η1∗,0,are the principal determinants of the direction of bifurcation (whether backwards or forward bifurcation) in the transformed system (29). When a>0 and b>0, then the bifurcation is backward and when a<0 and b>0, the bifurcation is forward.The bifurcation coefficients are obtained as(35)a=v2w1w21−θωτ2+μq3η1+θρψτ1+ξ1+γ+μq3η1+1−θωτ2+μ+q2θρτ1+ξ1+γ+μη2q3τ1+ξ1+γ+μτ2+μ,b=v2Λw21−θωτ2+μ+ψθρτ1+ξ1+γ+μμτ1+ξ1+γ+μτ2+μ.Assuredly,a<0, since w1<0. b is always positive. Hence, it can be concluded that model (29) exhibits forward bifurcation as shown in Figure 1.


### 3.3. Local Sensitivity Analysis

Due to the fact that epidemic model’s parameter values change, qualitative sensitivity analysis is necessary in order to determine the influence of parameters on model’s behaviour. The analysis sensitivity reveals the criticality of each parameter ofR0 to the model system (1). Sensitivity analysis is usually employed to bring to bear, the model’s parameters with strong influence on the R0 that needs to be worked on in ushering in control measures. The indices from sensitivity computation assist us in quantifying the relative change in a variable with a change in the parameter. Hence, we will consider the normalized forward sensitivity index for model system (1), defined as the ratio of the relative change in the variable to the change in the parameter.Definition 3.
For a given parameterσ, the normalized forward sensitivity index of R0 is computed using the formula discussed in [55, 56] as;
(36)ςσR0=∂R0∂σσR0.We will use Equation (36) and parameter values from Table 1 to derive the sensitivity indices of the parameters of the model system (1) as presented in Table 2.Table 1
COVID-19 model parameters and description.
ParameterValuesSourceΛμN(0)Assumedη10.003[1]ψ0.004[1]ω0.00001111[1]τ10.00023[1]q10.00101[1]q30.23008[1]ξ10.0002[1]ϕ10.1[1]δ20.06[1]μ0.000035[1]η23.4002×10−7[1]θ0.21003[1]ρ0.0180322[1]τ20.19[1]q20.0214[1]δ10.1233[1]γ0.0005[1]ϕ20.2[1]ξ20.01[1]Table 2
Parameters forR0 and their sensitivity index for model (1).
ParameterSensitivity indexParameterSensitivity indexΛ0.9956ω8.1966×10−5η10.0044τ20.9998η20.9956τ13.2276×10−6ψ0.0044γ7.0166×10−6q15.5293×10−7μ-0.9951q20.9956q30.9956ρ1.0295ξ12.8067×10−6θ1.0295δ10.9613The sensitivity indices computation from Table1 gives fascinating values for different parameters, and we can make some inferences. Thus, parameters with positive indices have the strength to make the number of the COVID-19 cases rise since they increase the R0. Additionally, the negative sensitivity indices’ parameters help lower the COVID-19 epidemic because they lower the R0. Further, parameters with +1 sensitivity indices increase or decrease R0 by the same percentage as the parameters’ percentages. Figures 2 and 3 further presents the influence of various combinations of parameters on the basic reproduction number, R0.Figure 2
Sensitivity plot ofR0 in terms of Λ, ω, θ, ρ, and δ1 using values in Table 2.
(a)
Sensitivity plot ofR0 in terms of Λ and ω(b)
Sensitivity plot ofR0 in terms of Λ and θ(c)
Sensitivity plot ofR0 in terms of Λ and ρ(d)
Sensitivity plot ofR0 in terms of Λ and δ1Figure 3
Sensitivity plot ofR0 in terms of Λ, q1, q2, ψ and δ1 using values in Table 2.
(a)
Sensitivity plot ofR0 in terms of Λ and q2(b)
Sensitivity plot ofR0 in terms of Λ and q1(c)
Sensitivity plot ofR0 in terms of q3 and ψ(d)
Sensitivity plot ofR0 in terms of ψ and δ1


## 3.1. Qualitative Analysis: Global Stability-Endemic Equilibrium

In assessing global asymptotic stability of the endemic equilibrium of the model system (1), we capitalize on the geometrical method [46] which uses Bendixson criterion as sufficient condition for determining model system (1) global stability.Theorem 2.
GivenR0>1, the endemic equilibrium of model Equation (1) is globally asymptotically stable; otherwise, it is unstable.Proof.
In proving the global stability of the endemic equilibrium, the study of ([47], with referencing to Lemma 3, Lemma 4 and Theorem 5.) is considered. we consider the subsystem of model Equation (1) as
(12)ddtS=Λ−μS−η2MS−η1ISN−ψASNη1,ddtE=η2MS+η1ISN+ψASNη1−1−θω+θρ+δ1+μE,ddtI=1−θωE−τ1+ξ1+γ+μI.
Considering subsystem (12) the Jacobian matrix becomes,
(13)Jv=−w110−η1S/Nw21−w22η1S/N01−θω−w33,with
(14)w11=Λ−η2M−η1IN−ψANη1−μ,w21=η2M−η1IN−ψANη1,w22=−1−θω+θρ+δ1+μ,w33=−τ1+ξ1+γ+μ.
Then,Jv2, the second additive matrix of Jv is given by
(15)Jv2=−w11+w22η1SN−η1SN1−θω−w11+w3300η2M−η1IN−ψANη1−w22+w33.
Now, we denote the functionPy=PS,E,I=diag1,E/I,E/I, hence we deduce P−′y=diag1,I/E,I/E. Then, the time derivative of Pfy becomes Pfy=diag0,E~/I−EI~/I2,E~/I−EI~/I2 and
PfyP−′y=diag0,E~/E−I~/I,E~/E−I~/I. It follows that
(16)PfyJv2P−′y=−w11+w22η1SINE−η1SINE1−θωEI−w11+w3300η2M−η1IN−ψANη1−w22+w33.
Hence, matrixC=PfyJv2P−′y+PfyP−′y can be determined and written as;
(17)C=C11C12C21C22,where
(18)C11=−η2M−η1IN−ψANη1−μ−1−θω+θρ+δ1−2μ,C12=η1SINE,−η1SINE,C21=1−θωEI,0andC22=w2+E~E−I~I0η2M−η1IN−ψANη1w3+E~E−I~I,with
(19)w2=−η2M−η1IN−ψANη1−τ1+ξ1+γ−2μ,w3=−1−θω+θρ+δ1+μ−τ1+ξ1+γ−2μ.
Considern1,n2,n3 to be a vector in ℝ3 and the norm • given by n1,n2,n3=maxn1,n2,n3. Let ρ1C represents the Lozinki measure with respect to this norm; we adopt the method of approximating the ρ1C as explored in [48], and choose ρ1C≤supp1,p2, where supp1,p2=ρ1C11+C12,ρ1C22+C21 and C12 and C21 are matrix norms with respect to vector norms ρ1, and ρ represents the Lozinki measure with respect to the norm ρ1.
Then,ρ1C11=−η2MS−η1I/NS−ψAS/Nη1−μ−1−θω+θρ+δ1−2μ(20)C12=maxη1SNE,−η1SNE=η1SINE,C12=max1−θωEI,0=1−θωEI,ρ1C22=E~E−I~I−τ1+ξ1+γ−2μ.
Hence,(21)p1=−η2MS−η1INS−ψASNη1−1−θω+θρ+δ1−2μ+η1SINE,p2=−E~E−I~I−τ1+ξ1+γ−2μ+1−θωEI.
It could be verified that(22)E~E=η2MSE+η1ISNE+ψASNEη1−1−θω+θρ+δ1+μ,I~I=1−θωEI−τ1+ξ1+γ+μ,which provides the following inequalities when substituted into P1 and P2:P1≤η1SI/NE+E~/E−μ,P2≤E~/E−μ.
Further,(23)ρ1C≤supp1,p2≤supη1SINE+E~E−μ,E~E−μ,=E~E−μ+Supη1SINE,0,=E~E−μ+η1SINE.
Hence,(24)ρ1C≤E~E−μ+η1SINE,substitute (24) into the below equation
(25)q~a=limt⟶∞sup1t∫0tρ1Cdt.
Now from Equation (25), we denote λ1=μ and λ2=η1SI/NE, then if λ2<λ1, it follows that ρ1C≤E~/E−∈ where ∈=μ−η1SI/NE. Now, for t>T, it follows that
(26)q~a=limt⟶∞sup1t∫0tρ1Cdt<−∈<0.
Therefore, from (26), we get q~a=limt⟶∞sup1/t∫0tρ1Cdt<0. This verifies that the system of nonlinear differential equation of submodel (12) is GAS around the interior equilibrium of S∗,E∗,I∗. Therefore, considering the remaining subsystem of model (1).
(27)ddtA=θρE−τ2+μA,ddtQ=δ1E−ϕ1+δ2+μQ,ddtH=γI+δ2Q−ϕ2+ξ2+μH,ddtR=τ1I+τ2A+ϕ1Q+ϕ2H−μR,ddtM=q1I+q2A−q3M.
The limit of (27) is given by
(28)ddtA=θρE∗−τ2+μA∗,ddtQ=δ1E∗−ϕ1+δ2+μQ∗,ddtH=γI∗+δ2Q∗−ϕ2+ξ2+μH∗,ddtR=τ1I∗+τ2A∗+ϕ1Q∗+ϕ2H∗−μR∗,ddtM=q1I∗+q2A∗−q3M∗.
When system (28) is solved with the initial conditions of A0,Q0,H0,R0,M0, then as t⟶∞, A⟶A∗, Q⟶Q∗, H⟶H∗, R⟶R∗, and M⟶M∗ which is the GAS of the endemic equilibrium.


## 3.2. Bifurcation Analysis

It has been observed from stability analysis of various epidemic models that theR0<1 is the sufficient condition for wiping out infections in a population. This condition usually holds when the system exhibits forward bifurcation. However, in considering a control intervention measure such as vaccination of the susceptibles, it has been observed that the condition R0<1 may not be sufficient to control the disease. Thus, the model transmissibility asymptomatic pattern is now determined by a quantity R0 defined as the effective reproduction number. It has been observed in some studies [49–53] that, there can be coexistence of the endemic and disease-free equilibrium, which are both stable for R0<1, which represents backwards bifurcation. In controlling diseases in the presence of backward bifurcation, the R0 has to be lowered to a value less than one. This section employs the theorem propounded by Castillo-Chavez et al. [54] to explore the type of bifurcation exhibited by the model system (1). To determine whether the phenomenon of bifurcation exists in model system (1), the centre manifold theory [54] is employed. The following necessary substitutions are taken to conform to the theorem’s requirement for easier computation of the bifurcation coefficients.We denotex1=S, x2=E, x3=I, x4=A, x5=Q, x6=H, x7=R, and x8=M such that the total population here is given by N=x1+x2+x3+x4+x5+x6+x7+x8. With vector notation of X=x1,x2,⋯,x8T, the model system (1) can be redesigned in the form dX/dt=FX, with F=f1,f2,⋯,f8T given by
(29)ddtx1=Λ−μx1−η2x8x1−η1x3x1N−ψx4x1Nη1,ddtx2=η2x8x1+η1x3x1N+ψx4x1Nη1−1−θω+θρ+δ1+μx2,ddtx3=1−θωx2−τ1+ξ1+γ+μx3,ddtx4=θρx2−τ2+μx4,ddtx5=δ1x2−ϕ1+δ2+μx5,ddtx6=γx3+δ2x5−ϕ2+ξ2+μx6,ddtx7=τ1x3+τ2x4+ϕ1x5+ϕ2x6−μx7,ddtx8=q1x3+q2x4−q3x8.The Jacobian of the new system (29) evaluated at the DFE, denoted by JDE0 is given by,
(30)JDE0=−μ0−η1−ψη1∗000−η2Λμ0−h11η1ψη1∗000η2Λμ01−θω−h22000000θρ0−τ2+μ00000δ100−h3300000γ0δ2−h440000τ1τ2ϕ1ϕ2−μ000q1q2000−q3,where h11=1−θω+θρ+δ1+μ, h22=τ1+ξ1+γ+μ, h33=ϕ1+δ2+μ, h44=ϕ2+ξ2+μ. Given that η1=η1∗ is considered as the bifurcation parameter in the case when R0=1, then expressing η1 in terms of the other parameters when R0=1 becomes,
(31)η1=η1∗=μq3R1R2R3−η2Λ1−θωq1q3+θρη2Λq2R2μq3ψθρR2+1−θωR3.HenceJDE0η1=η1∗ has a right eigenvector denoted by W=w1,w2,⋯,w8T, with
(32)w1=−η11−θωμq3τ2+μ+ψη1θρμq3τ1+ξ1+γ+μ+b1μ2q3τ1+ξ1+γ+μτ2+μw2,w3=1−θωτ1+ξ1+γ+μw2,w2>0,w4=θρτ2+μw2,w5=δ1ϕ1+δ2+μw2,w6=γ1−θωϕ1+δ2+μ+δ1δ2τ1+ξ1+γ+μτ1+ξ1+γ+μϕ1+δ2+μϕ2+ξ2+μw2,w7=τ11−θωϕ1+δ2+μϕ2+ξ2+μτ2+μ+b2+b3+b4μτ1+ξ1+γ+μϕ1+δ2+μϕ2+ξ2+μτ2+μw2,w8=1−θωτ2+μ+q2θρτ1+ξ1+γ+μq1τ1+ξ1+γ+μτ2+μw2,where
(33)b1=η2Λ1−θωτ2+μ+q2θρτ1+ξ1+γ+μ,b2=τ2θρτ1+ξ1+γ+μϕ1+δ2+μϕ2+ξ2+μ,b3=ϕ1δ1τ1+ξ1+γ+μϕ2+ξ2+μτ2+μ,b4=γ1−θωϕ1+δ2+μ+δ1δ2τ1+ξ1+γ+μ.Further,JDE0η1=η1∗ has a left eigenvector V=v1,v2,⋯,v8T, where v1=v5=v6=v7=0, v2>0, v3=η1q3μ+q1η2Λ/q3τ1+ξ1+γ+μμv2, v4=ψη1q3μ+q2η2Λ/q3τ2+μμv2,v8=η2Λ/q3μv2.The newly translated model Equation (29) with the bifurcation parameter η1=η1∗ has at least one eigenvalue zero, and it affirms the existence of bifurcation in the transformed model (29). As expounded by Castillo-Chavez and Song ([54], refer to Theorem 5.1.), the equations
(34)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xjη1∗,0,b=∑k,i,j=1nvkwi∂2fk∂xi∂η1η1∗,0,are the principal determinants of the direction of bifurcation (whether backwards or forward bifurcation) in the transformed system (29). When a>0 and b>0, then the bifurcation is backward and when a<0 and b>0, the bifurcation is forward.The bifurcation coefficients are obtained as(35)a=v2w1w21−θωτ2+μq3η1+θρψτ1+ξ1+γ+μq3η1+1−θωτ2+μ+q2θρτ1+ξ1+γ+μη2q3τ1+ξ1+γ+μτ2+μ,b=v2Λw21−θωτ2+μ+ψθρτ1+ξ1+γ+μμτ1+ξ1+γ+μτ2+μ.Assuredly,a<0, since w1<0. b is always positive. Hence, it can be concluded that model (29) exhibits forward bifurcation as shown in Figure 1.


## 3.3. Local Sensitivity Analysis

Due to the fact that epidemic model’s parameter values change, qualitative sensitivity analysis is necessary in order to determine the influence of parameters on model’s behaviour. The analysis sensitivity reveals the criticality of each parameter ofR0 to the model system (1). Sensitivity analysis is usually employed to bring to bear, the model’s parameters with strong influence on the R0 that needs to be worked on in ushering in control measures. The indices from sensitivity computation assist us in quantifying the relative change in a variable with a change in the parameter. Hence, we will consider the normalized forward sensitivity index for model system (1), defined as the ratio of the relative change in the variable to the change in the parameter.Definition 3.
For a given parameterσ, the normalized forward sensitivity index of R0 is computed using the formula discussed in [55, 56] as;
(36)ςσR0=∂R0∂σσR0.We will use Equation (36) and parameter values from Table 1 to derive the sensitivity indices of the parameters of the model system (1) as presented in Table 2.Table 1
COVID-19 model parameters and description.
ParameterValuesSourceΛμN(0)Assumedη10.003[1]ψ0.004[1]ω0.00001111[1]τ10.00023[1]q10.00101[1]q30.23008[1]ξ10.0002[1]ϕ10.1[1]δ20.06[1]μ0.000035[1]η23.4002×10−7[1]θ0.21003[1]ρ0.0180322[1]τ20.19[1]q20.0214[1]δ10.1233[1]γ0.0005[1]ϕ20.2[1]ξ20.01[1]Table 2
Parameters forR0 and their sensitivity index for model (1).
ParameterSensitivity indexParameterSensitivity indexΛ0.9956ω8.1966×10−5η10.0044τ20.9998η20.9956τ13.2276×10−6ψ0.0044γ7.0166×10−6q15.5293×10−7μ-0.9951q20.9956q30.9956ρ1.0295ξ12.8067×10−6θ1.0295δ10.9613The sensitivity indices computation from Table1 gives fascinating values for different parameters, and we can make some inferences. Thus, parameters with positive indices have the strength to make the number of the COVID-19 cases rise since they increase the R0. Additionally, the negative sensitivity indices’ parameters help lower the COVID-19 epidemic because they lower the R0. Further, parameters with +1 sensitivity indices increase or decrease R0 by the same percentage as the parameters’ percentages. Figures 2 and 3 further presents the influence of various combinations of parameters on the basic reproduction number, R0.Figure 2
Sensitivity plot ofR0 in terms of Λ, ω, θ, ρ, and δ1 using values in Table 2.
(a)
Sensitivity plot ofR0 in terms of Λ and ω(b)
Sensitivity plot ofR0 in terms of Λ and θ(c)
Sensitivity plot ofR0 in terms of Λ and ρ(d)
Sensitivity plot ofR0 in terms of Λ and δ1Figure 3
Sensitivity plot ofR0 in terms of Λ, q1, q2, ψ and δ1 using values in Table 2.
(a)
Sensitivity plot ofR0 in terms of Λ and q2(b)
Sensitivity plot ofR0 in terms of Λ and q1(c)
Sensitivity plot ofR0 in terms of q3 and ψ(d)
Sensitivity plot ofR0 in terms of ψ and δ1


## 4. Development of Optimal Control Model

To generate control strategies to wrestle the disease from existence, we employ optimal control theory. For the assessment of the conditions necessary to determine the feasible optimal control in the COVID-19 model (1), we apply Pontryagin’s maximum principle [57]. We note that, personal protection, vaccination of the susceptible population, and spraying the environment with disinfectant have significantly been encouraged during the COVID era. Hence, to add control strategies to the study, we rebuild the model (1) by reducing the transmission rate by a factor 1−u1. We consider further the vaccination control denoted by u2 for the susceptibles and a control for disinfection of the environment represented by u3. Hence, the new formulated optimal control model of model system (1) becomes;
(37)ddtS=Λ−μS−1−u1η2MS−1−u1η1ISN−1−u1ψASNη1−u2S,ddtE=1−u1η2MS+1−u1η1ISN+1−u1ψASNη1−1−θω+θρ+δ1+μE,ddtI=1−θωE−τ1+ξ1+γ+μI,ddtA=θρE−τ2+μA,ddtQ=δ1E−ϕ1+δ2+μQ,ddtH=γI+δ2Q−ϕ2+ξ2+μH,ddtR=τ1I+τ2A+ϕ1Q+ϕ2H+u2S−μR,ddtM=q1I+q2A−q3M−u3M,withS≥0,E≥0,I≥0,A≥0,Q≥0,H≥0,R≥0andM≥0.The controlsut=ui∈U are bounded and Lebegue measurable.The objective here is to minimizeJ given by
(38)J=∫0tfB1E+B2I+B3A+12h1u12+12h2u22+12h3u32dt,subject to model system (37). We assign factors B1,B2,B3,h1,h2,andh3 in the integrand of (18) as coefficients that serve as weight that balance Equation (38). The coefficients B1,B2,B3,h1,h2,andh3 estimate the cost of ushered in interventions that need to be taken over the time of 0,T. We determine an optimal control u1∗,u2∗,u3∗ such that
(39)Ju1∗,u2∗,u3∗=minUJu1,u2,u3,withU subject to the model system (37). We use the most explored method in optimal control modelling, Pontryagin maximum principle [57], and find the solution and the necessary conditions of the optimal control problem (37).


### 4.1. Existence of the Optimal Controls

Related to the result of Fleming and Rishel as provided in [58], the existence of optimal control triple that minimizes (38) subject to (37) is shown.Theorem 4.
There exists an optimal controlu∗=u1∗,u2∗,u3∗ that minimizes the objective functional (38).
The existence of the optimal control can be proved by showing that, the following conditions hold.
(D1) The set of controls is convex and closed.
(D2) The system (37) is bounded by a linear function in both the state and control variables.
(D3) The integrand in the objective functional (38) is convex with respect to the control.
(D4) There exist constantsk1,k2≥0andk3≥1 that make the integrand in the objective functional (38) bounded by k1∑i=12ui2k3/2−k2.Proof.
We construct the proof in steps as follows:
(D1): The control setU=0,13 is closed. Again, we let v,y∈U, so that v=v1,v2,v3 and y=y1,y2,y3. Then, for every π∈0,1, we have πvi+1−πyi∈U, confirming the convexity property of the control set.
(D2): The control system (37) and its related solution are represented by Φ and Θ such that Φt,Θ1,u−Φt,Θ2,u becomes
(40)−u2S1−S2−μS1−S2−1−u1η2M1S1−M2S2−1−u1η1I1S1N−I2S2N−1−u1ψA1S1N−A2S2Nη1+1−u1η2M1S1+M2S2+1−u1η1I1S1N−I2S2N+1−u1ψA1S1N−A2S2Nη1−1−θω+θρ+δ1+μE1−E2+1−θωE1−E2−τ1+ξ1+γ+μI1−I2+θρE1−E2−τ2+μA1−A2+δ1E1−E2−ϕ1+δ2+μQ1−Q2+γI1−I2+δ2Q1−Q2−ϕ2+ξ2+μH1−H2+τ1I1−I2+τ2A1−A2+ϕ1Q1−Q2+ϕ2H1−H2+u2S1−S2−μR1−R2+q1I1−I2+q2A1−A2−q3M1−M2−u3M1−M2≤21−u1η2M1S1−M2S2+21−u1η1I1S1N−I2S2N+2ψη11−u1A1S1N−A2S2N+μS1−S2+21−θω+2θρ+2δ1+μE1−E2+2τ1+2γ+ξ1+q1+μI1−I2+2τ2+q2+μA1−A2+2ϕ1+2δ2+μQ1−Q2+2ϕ2+ξ2+μH1−H2+2u2S1−S2+μR1−R2+q3+u3M1−M2,≤21−u1η2S1M1−M2+M2S1−S2+21−u1η1S1I1−I2+I2S1−S2+2ψη11−u1S1A1−A2+A2S1−S2+21−θω+2θρ+2δ1+μE1−E2+μS1−S2+2τ1+2γ+ξ1+q1+μI1−I2+2τ2+q2+μA1−A2+2ϕ2+ξ2+μH1−H2+μR1−R2+q3+u3M1−M2+2ϕ1+2δ2+μQ1−Q2,≤Y1S1−S2+Y2E1−E2+Y3I1−I2+Y4A1−A2+Y5Q1−Q2+Y6H1−H2+Y7R1−R2+Y8M1−M2,
whereY1=2u2+μ, Y2=21−θω+2θρ+2δ1+μ, Y3=21−u1η1Λ/μ+2τ1+2γ+ξ1+q1+μ, Y4=2ψη11−u1Λ/μ+2τ2+q2+μ, Y5=2ϕ1+2δ2+μ, Y6=2ϕ2+ξ2+μ, Y7=μ, Y8=21−u1η2Λ/μ+q3+u3 and Y=maxY1,Y2,Y3,Y4,Y5,Y6,Y7,Y8.
Hence the Lipschitz continuous uniformity ofΦ is proved.
(D3): We let the Langrangian of the objective functional (38) be Lt,n,u. Then,
(41)Lt,n,u=ν1t,n+ν2t,n,with, ν1t,n=B1E+B2I+B3A and ν2t,n=1/2∑j=13hjuj. The convexity of ν2t,n, which is a linear combination of the control function 1/2∑j=13hjuj has to be shown. We let hu=1/2u2 for λ:0,12⟶R. Then, ∀t1,t2∈0,12,ρ2∈0,1. Hence, it follows that the below inequality holds.
(42)ρ2λt1+1−ρ2λt2≥λρ2t1+1−ρ2t2.
This confirms the convexity of the Lagrangian with respect to the control.
(D4): By inspection, we noticed in (41) that Lt,n,u≥ν1t,n. Hence, grounded on this, we conclude that
(43)Lt,n,u≥12h1u12+12h2u22+12h3u32≥k1∑i=12ui2k3/2−k2,with k1=1/2minh1,h2,h3, k2>0 and k3=2. This completes the proof.


### 4.2. Characterization of Optimal Controls

Using the Principle translates the COVID model (37) and its associated objective functional (38) into a problem of minimizing a Hamiltonian H, with respect to the controls ujt, for j=1,2,3. The Hamiltonian for system (17) is given by;
(44)H=Lt,n,u+∑k=18ςkfk.where thefk in Equation (44) are the right-hand sides of system (37) and ςk are the respective adjoint variables for the state S,E,I,A,Q,H,R,andM.Theorem 5.
Consider the triple optimal controlu1∗,u2∗,u3∗ satisfying the Condition (39), then there exist adjoint variables ςk satisfying adjoint system below;
(45)ddtς1=ς1μ+ς1−ς2η2M1−u1+ς1−ς21−u1η1IN+ς1−ς21−u1ψANη1−u2ς7,ddtς2=ς2μ+ς2−ς31−θω+ς2−ς4θρ+ς2−ς5δ1−B1,ddtς3=ς1−ς21−u1η1SN+ς3−ς7τ1+ς3−ς6γ+δ1+μς3−q1ς8−B2,ddtς4=ς1−ς21−u1ψSNη1+ς4−ς7τ2−q2ς8+μς4−B3,ddtς5=ς5−ς7ϕ1+ς5−ς6δ2+μς5,ddtς6=ς6−ς7ϕ2+ξ2ς6+μς6,ddtς7=μς7,ddtς8=ς1−ς21−u1η2S+q3ς8−u3ς8,with transversality conditions(46)ςjT=0,j=1,2,⋯,8.The control functionsu1∗,u2∗,u3∗ satisfies the optimality condition and is given by
(47)ui∗t=min1,max0,Γi,i=1,2,3.where
(48)Γ1=ς2−ς1MSh1η2+ς2−ς1ISNh1η1+ς2−ς1ASNh1η1,Γ2=ς1−ς7Sh2,Γ3=Mh3ς8.Proof.
Referring to the Hamiltonian (49), given by
(49)H=B1E+B2I+B3A+12h1u12+12h2u22+12h3u32+ς1Λ−μS−1−u1η2MS−1−u1η1ISN−1−u1ψASNη1−u2S+ς21−u1η2MS+1−u1η1ISN+1−u1ψASNη1−1−θω+θρ+δ1+μE+ς31−θωE−τ1+ξ1+γ+μI+ς4θρE−τ2+μA+ς5δ1E−ϕ1+δ2+μQ+ς6γI+δ2Q−ϕ2+ξ2+μH+ς7τ1I+τ2A+ϕ1Q+ϕ2H+u2S−μR+ς8q1I+q2A−q3M−u3M,
we generate the adjoint system (45) by differentiating partially the Hamiltonain (49) with respect to the corresponding state variables S,E,I,A,Q,H,R,andM as
dς1/dt=−∂L/∂S,dς2/dt=−∂L/∂E,dς3/dt=−∂L/∂I,dς4/dt=−∂L/∂A,dς5/dt=−∂L/∂Q,dς6/dt=−∂L/∂H,dς7/dt=−∂L/∂R,dς8/dt=−∂L/∂M The characterization of the controls u1∗,u2∗,u3∗ as in (39) is determined by applying the equation below;
(50)∂L∂ui=0,i=1,2,3.
Utilizing bounds on the controls by standard argument, we determine the characterization(51)ui∗=0ifφi∗≤0,φi∗if0≤φi∗≤1,1ifφi∗≥1,where
(52)φ1∗=ς2−ς1MSh1η2+ς2−ς1ISNh1η1+ς2−ς1ASNh1η1,φ2∗=ς1−ς7Sh2,φ3∗=Mh3ς8,
This completes the proof.


## 4.1. Existence of the Optimal Controls

Related to the result of Fleming and Rishel as provided in [58], the existence of optimal control triple that minimizes (38) subject to (37) is shown.Theorem 4.
There exists an optimal controlu∗=u1∗,u2∗,u3∗ that minimizes the objective functional (38).
The existence of the optimal control can be proved by showing that, the following conditions hold.
(D1) The set of controls is convex and closed.
(D2) The system (37) is bounded by a linear function in both the state and control variables.
(D3) The integrand in the objective functional (38) is convex with respect to the control.
(D4) There exist constantsk1,k2≥0andk3≥1 that make the integrand in the objective functional (38) bounded by k1∑i=12ui2k3/2−k2.Proof.
We construct the proof in steps as follows:
(D1): The control setU=0,13 is closed. Again, we let v,y∈U, so that v=v1,v2,v3 and y=y1,y2,y3. Then, for every π∈0,1, we have πvi+1−πyi∈U, confirming the convexity property of the control set.
(D2): The control system (37) and its related solution are represented by Φ and Θ such that Φt,Θ1,u−Φt,Θ2,u becomes
(40)−u2S1−S2−μS1−S2−1−u1η2M1S1−M2S2−1−u1η1I1S1N−I2S2N−1−u1ψA1S1N−A2S2Nη1+1−u1η2M1S1+M2S2+1−u1η1I1S1N−I2S2N+1−u1ψA1S1N−A2S2Nη1−1−θω+θρ+δ1+μE1−E2+1−θωE1−E2−τ1+ξ1+γ+μI1−I2+θρE1−E2−τ2+μA1−A2+δ1E1−E2−ϕ1+δ2+μQ1−Q2+γI1−I2+δ2Q1−Q2−ϕ2+ξ2+μH1−H2+τ1I1−I2+τ2A1−A2+ϕ1Q1−Q2+ϕ2H1−H2+u2S1−S2−μR1−R2+q1I1−I2+q2A1−A2−q3M1−M2−u3M1−M2≤21−u1η2M1S1−M2S2+21−u1η1I1S1N−I2S2N+2ψη11−u1A1S1N−A2S2N+μS1−S2+21−θω+2θρ+2δ1+μE1−E2+2τ1+2γ+ξ1+q1+μI1−I2+2τ2+q2+μA1−A2+2ϕ1+2δ2+μQ1−Q2+2ϕ2+ξ2+μH1−H2+2u2S1−S2+μR1−R2+q3+u3M1−M2,≤21−u1η2S1M1−M2+M2S1−S2+21−u1η1S1I1−I2+I2S1−S2+2ψη11−u1S1A1−A2+A2S1−S2+21−θω+2θρ+2δ1+μE1−E2+μS1−S2+2τ1+2γ+ξ1+q1+μI1−I2+2τ2+q2+μA1−A2+2ϕ2+ξ2+μH1−H2+μR1−R2+q3+u3M1−M2+2ϕ1+2δ2+μQ1−Q2,≤Y1S1−S2+Y2E1−E2+Y3I1−I2+Y4A1−A2+Y5Q1−Q2+Y6H1−H2+Y7R1−R2+Y8M1−M2,
whereY1=2u2+μ, Y2=21−θω+2θρ+2δ1+μ, Y3=21−u1η1Λ/μ+2τ1+2γ+ξ1+q1+μ, Y4=2ψη11−u1Λ/μ+2τ2+q2+μ, Y5=2ϕ1+2δ2+μ, Y6=2ϕ2+ξ2+μ, Y7=μ, Y8=21−u1η2Λ/μ+q3+u3 and Y=maxY1,Y2,Y3,Y4,Y5,Y6,Y7,Y8.
Hence the Lipschitz continuous uniformity ofΦ is proved.
(D3): We let the Langrangian of the objective functional (38) be Lt,n,u. Then,
(41)Lt,n,u=ν1t,n+ν2t,n,with, ν1t,n=B1E+B2I+B3A and ν2t,n=1/2∑j=13hjuj. The convexity of ν2t,n, which is a linear combination of the control function 1/2∑j=13hjuj has to be shown. We let hu=1/2u2 for λ:0,12⟶R. Then, ∀t1,t2∈0,12,ρ2∈0,1. Hence, it follows that the below inequality holds.
(42)ρ2λt1+1−ρ2λt2≥λρ2t1+1−ρ2t2.
This confirms the convexity of the Lagrangian with respect to the control.
(D4): By inspection, we noticed in (41) that Lt,n,u≥ν1t,n. Hence, grounded on this, we conclude that
(43)Lt,n,u≥12h1u12+12h2u22+12h3u32≥k1∑i=12ui2k3/2−k2,with k1=1/2minh1,h2,h3, k2>0 and k3=2. This completes the proof.


## 4.2. Characterization of Optimal Controls

Using the Principle translates the COVID model (37) and its associated objective functional (38) into a problem of minimizing a Hamiltonian H, with respect to the controls ujt, for j=1,2,3. The Hamiltonian for system (17) is given by;
(44)H=Lt,n,u+∑k=18ςkfk.where thefk in Equation (44) are the right-hand sides of system (37) and ςk are the respective adjoint variables for the state S,E,I,A,Q,H,R,andM.Theorem 5.
Consider the triple optimal controlu1∗,u2∗,u3∗ satisfying the Condition (39), then there exist adjoint variables ςk satisfying adjoint system below;
(45)ddtς1=ς1μ+ς1−ς2η2M1−u1+ς1−ς21−u1η1IN+ς1−ς21−u1ψANη1−u2ς7,ddtς2=ς2μ+ς2−ς31−θω+ς2−ς4θρ+ς2−ς5δ1−B1,ddtς3=ς1−ς21−u1η1SN+ς3−ς7τ1+ς3−ς6γ+δ1+μς3−q1ς8−B2,ddtς4=ς1−ς21−u1ψSNη1+ς4−ς7τ2−q2ς8+μς4−B3,ddtς5=ς5−ς7ϕ1+ς5−ς6δ2+μς5,ddtς6=ς6−ς7ϕ2+ξ2ς6+μς6,ddtς7=μς7,ddtς8=ς1−ς21−u1η2S+q3ς8−u3ς8,with transversality conditions(46)ςjT=0,j=1,2,⋯,8.The control functionsu1∗,u2∗,u3∗ satisfies the optimality condition and is given by
(47)ui∗t=min1,max0,Γi,i=1,2,3.where
(48)Γ1=ς2−ς1MSh1η2+ς2−ς1ISNh1η1+ς2−ς1ASNh1η1,Γ2=ς1−ς7Sh2,Γ3=Mh3ς8.Proof.
Referring to the Hamiltonian (49), given by
(49)H=B1E+B2I+B3A+12h1u12+12h2u22+12h3u32+ς1Λ−μS−1−u1η2MS−1−u1η1ISN−1−u1ψASNη1−u2S+ς21−u1η2MS+1−u1η1ISN+1−u1ψASNη1−1−θω+θρ+δ1+μE+ς31−θωE−τ1+ξ1+γ+μI+ς4θρE−τ2+μA+ς5δ1E−ϕ1+δ2+μQ+ς6γI+δ2Q−ϕ2+ξ2+μH+ς7τ1I+τ2A+ϕ1Q+ϕ2H+u2S−μR+ς8q1I+q2A−q3M−u3M,
we generate the adjoint system (45) by differentiating partially the Hamiltonain (49) with respect to the corresponding state variables S,E,I,A,Q,H,R,andM as
dς1/dt=−∂L/∂S,dς2/dt=−∂L/∂E,dς3/dt=−∂L/∂I,dς4/dt=−∂L/∂A,dς5/dt=−∂L/∂Q,dς6/dt=−∂L/∂H,dς7/dt=−∂L/∂R,dς8/dt=−∂L/∂M The characterization of the controls u1∗,u2∗,u3∗ as in (39) is determined by applying the equation below;
(50)∂L∂ui=0,i=1,2,3.
Utilizing bounds on the controls by standard argument, we determine the characterization(51)ui∗=0ifφi∗≤0,φi∗if0≤φi∗≤1,1ifφi∗≥1,where
(52)φ1∗=ς2−ς1MSh1η2+ς2−ς1ISNh1η1+ς2−ς1ASNh1η1,φ2∗=ς1−ς7Sh2,φ3∗=Mh3ς8,
This completes the proof.


## 5. Numerical Experimentation and Discussion

With the aid of the forward-backwards sweep method as derived by Lenhart and Workman [59], a numerical solution to the optimality system of the control model was obtained. We offer the scenarios of different feasible control strategies to identify their effect on the dynamics of the COVID model. The backwards scheme solves the adjoint system (45) backwards in time, and the forward scheme solves (37) forward in time. In addition, the structured scheme considered the set of initial conditions related to the state variables and the boundary condition to solve the control optimality system in a simulated interval of t∈0,250. With this set in play, we consider the balancing factors of the objective functional (38) as: B1=10,B2=10,B3=10,h1=5,h2=10,andh3=8, and initial population N0=100000000 for the simulation purpose. We concluded that h2>h1andh3, as we assume that the cost of vaccines and implementing a vaccination program would be higher than the cost of personal protection and disinfecting the environment. Even though the model in [44] is a Fractional-order model, the authors estimated the model’s parameters at α1=α2=1. However, at α1=α2=1, the model becomes an integer order model. Furthermore, since the study modifies model 1 to optimal control which simulated results are always compared to the noncontrol model, it becomes necessary that we consider the same parameter values as used in model 1. Hence, we would consider model [44] parameter values for the simulation.


### 5.1. Strategy1 (Implementing Controls u1,u2, and u3)

The strategy1 considered the controls u1,u2,andu3. The simulation results are presented in Figures 4(a)–4(d) of Figure 4. We observe that, the exposed population (28) without control swiftly rose in the first five days and reached almost 8.5×105 within the next five days. The dynamics of the exposed graph changed after day 11. The graph began to slowly decrease until day 80 when the population seemed to have been wiped out. The exposed without control graph maintained this new level for the next 70 days and began to surge to a height of 0.6×105 for the next 65 days when it began to decrease for the rest of the simulated time smoothly. Also, the infected population without control (See Figure 4(b)) gradually rose in the first 10 days until day 50 when it began to decrease with an estimated infected population of 250. It decreased slowly to 240 in 170 days and picked up after the 170. The asymptomatically infected graph of (See Figure 4(c)) sharply increased in 10 days to reach a population of about 14500 and then decreased slowly after day 25 and reached the lower bound at 90 days. The level was maintained until 150 days when it smoothly increased for the remaining simulated time. The exposed population under controls (Figure 4(a)) decreased slowly and was expected to wipe out in 20 days completely. The infected population with control (See Figure 4(b)) increased in a similar pattern slowly in the first 10 days and decreased slightly, with the population recognized to be a little below 90 at the final time. The asymptomatically infected population with control (See Figure 4(c)) was the same as that without controls for the first 5 days but is reduced after 40 days. From the control profile graph (Figure 4(d)), u1, which represents the personal protection control, starts from the upper bound but drops suddenly to the lower bound at the initial zero days until 60 when it swiftly moves to the upper bound. The next 190 days remained at the upper bound and then returned to the lower bound at the final time of 250 days. The vaccination control, u2, remained at the lower bound until 59 days when it moved to the upper bound. It stayed at the upper bound until 200 days and then dropped to the lower bound and remained there for the rest of the time. u3, denoting disinfectant control, started at the lower bound and picked to the upper bound after 60 days. It remained at the upper bound for the remaining time until 249 days, when it dropped to the lower bound.Figure 4
Graphs of simulation results for strategy 1 (withu1≠0, u2≠0, and u3≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profile


### 5.2. Strategy2 (Implementing Controls u1 and u2)

The strategy2 sets u3, the disinfectant control to zero and experiments with the controls u1 and u2. The simulation results are presented in Figure 5. The exposed population without control (See Figure 5(a)) shows a sudden rise in the first 10 days. The graph rises to a peak of about 8.5×105 and after 10 days, decreases steeply to a minimal population of about 1×105 within the next 90 days and maintains this new level until 150 days when it begins to increase smoothly. The infected population without control (See Figure 5(b)) gently increases for the first 25 days to reach a maximum height of 250. The new height of the infected became unstable after the next 25 days, and that resulted in the population slowly decreasing to 240 in the next 150 days, after which it began to rise for the remaining time. With the asymptotically infected without control (See Figure 5(c)), we noted a quick rise of the curve in the first 10 days and reached the pinnacle at 14500 in 25 days at the infection hotspots.Figure 5
Graphs of simulation results for strategy 2 (withu1≠0 and u2≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profileThe exposed population yielded to the design experiment strategy2. The exposed population starts with 1.9×105 population and gently reduces until 25 days when it seems to have been wiped from the system. The infected population under control strategy 2 slightly increased in the early days and began to lower after 10 days. The graph decreased in that pattern and reached a lower height of 90 at the final time. Also, the asymptotically infected population increased to about 2000 in the first 5 days, decreased slowly to zero in 30 days, and maintained that level for the remaining period of the simulation time. Figure 5(d) presents the control profile for Strategy 2. By inspection, we observed that the personal protection control u1 falls to the lower bound at the initial zero time. It remained on the lower bound for 60 days when it suddenly moved to the upper bound. It then remained at the upper bound until 250 days when it dropped to the lower bound again. Control u2, representing the vaccination control, remained at the lower bound for the first 59 days when it swiftly moved to the upper bound. It stayed at the upper bound for the next 120 days when it dropped to the lower bound and remained there for the remaining time.


### 5.3. Strategy3 (Implementing Controls u1 and u3)

Withu2 set to zero, strategy 3 uses the controls u1 and u3 for the experimentation by simulating the optimality system with the structured interval. The simulation results for this strategy are presented in Figure 6. We observe that, the exposed population without control (See Figure 6(a)) moved sharply to 8.5×105 in day 10 of the simulated time and then decreased gradually to almost 0.1×105 and maintained the level until 150 days when it began to rise for the remaining time. The infected population without control (See Figure 6(b)) increased slowly and moved to a height of 250 for 50 days. The population then decreased slowly for the next 100 days and began to pick up after the 100 days. The asymptomatically infected population without control (See Figure 6(c)) also shoots quickly in the first 25 days to its maximum of 14500. The dynamics changed, and the population began to reverse slowly to a level of about 100, and then picked up after 150 days and smoothly increased to the final time. The optimal control plot of the exposed produced a graph that seemed to wane in the early days but began to rise after 30 days to meet the noncontrol graph at 70 days. The level of the controlled exposed population was maintained until the final time. The infected population with control Strategy 3 (See Figure 6(b)) got lowered in the early days and continued till the final time of the simulation. The controlled population of asymptotically infected persons (See Figure 6(c)) got depleted in the first 30 days. However, for the uncontrolled problem, the population resurged and maintained the level for the rest of the simulation time. From Figure 6(d), u1 denoting personal protection control drops suddenly from the upper bound to the lower bound at time t=0. It then remained at the lower bound for 15 days and shot to the upper bound, where it remained for 15 days and dropped to the lower bound. It remained in the lower bound for another 15 days and moved to the upper bound. It then stayed at the upper bound for 15 days and dropped to the lower bound. It picked after 5 days to the upper bound and stayed there until it dropped to the lower bound. u3 denoting disinfectant control started from the lower bound and moved to the upper bound after 15 days. It dropped to the lower bound after 15 days and again moved to the upper bound after 15. It stayed at the upper bound for another 15 days and then dropped to the lower bound for 5 days. It picked to the upper bound of 70 days and remained until the final time before dropping.Figure 6
Graphs of simulation results for strategy 3 (withu1≠0 and u3≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profile


### 5.4. Strategy4 (Implementing Controls u2 and u3)

The strategy4 set u1=0 and utilized the control u2 and u3 in the numerical simulation for assigned interval. The simulation results are presented in Figures 7(a)–7(d). The Strategy 4 is observed to also do well in reducing the exposed, infected and asymptomatically infected populations. From the control profile (Figure 7(d)), the vaccination control, u2, shot to the upper bound at time t=50, stayed at the upper bound for 150 days, and then dropped to the lower bound. It remained at the lower bound for the rest of the simulated time. The disinfectant spraying control u3 stayed at the lower bound from day 0 to 50, rocketed to the upper bound and remained there until 249 days when it returned to the lower bound.Figure 7
Graphs of simulation results for strategy 4 (withu2≠0 and u3≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profile


## 5.1. Strategy1 (Implementing Controls u1,u2, and u3)

The strategy1 considered the controls u1,u2,andu3. The simulation results are presented in Figures 4(a)–4(d) of Figure 4. We observe that, the exposed population (28) without control swiftly rose in the first five days and reached almost 8.5×105 within the next five days. The dynamics of the exposed graph changed after day 11. The graph began to slowly decrease until day 80 when the population seemed to have been wiped out. The exposed without control graph maintained this new level for the next 70 days and began to surge to a height of 0.6×105 for the next 65 days when it began to decrease for the rest of the simulated time smoothly. Also, the infected population without control (See Figure 4(b)) gradually rose in the first 10 days until day 50 when it began to decrease with an estimated infected population of 250. It decreased slowly to 240 in 170 days and picked up after the 170. The asymptomatically infected graph of (See Figure 4(c)) sharply increased in 10 days to reach a population of about 14500 and then decreased slowly after day 25 and reached the lower bound at 90 days. The level was maintained until 150 days when it smoothly increased for the remaining simulated time. The exposed population under controls (Figure 4(a)) decreased slowly and was expected to wipe out in 20 days completely. The infected population with control (See Figure 4(b)) increased in a similar pattern slowly in the first 10 days and decreased slightly, with the population recognized to be a little below 90 at the final time. The asymptomatically infected population with control (See Figure 4(c)) was the same as that without controls for the first 5 days but is reduced after 40 days. From the control profile graph (Figure 4(d)), u1, which represents the personal protection control, starts from the upper bound but drops suddenly to the lower bound at the initial zero days until 60 when it swiftly moves to the upper bound. The next 190 days remained at the upper bound and then returned to the lower bound at the final time of 250 days. The vaccination control, u2, remained at the lower bound until 59 days when it moved to the upper bound. It stayed at the upper bound until 200 days and then dropped to the lower bound and remained there for the rest of the time. u3, denoting disinfectant control, started at the lower bound and picked to the upper bound after 60 days. It remained at the upper bound for the remaining time until 249 days, when it dropped to the lower bound.Figure 4
Graphs of simulation results for strategy 1 (withu1≠0, u2≠0, and u3≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profile


## 5.2. Strategy2 (Implementing Controls u1 and u2)

The strategy2 sets u3, the disinfectant control to zero and experiments with the controls u1 and u2. The simulation results are presented in Figure 5. The exposed population without control (See Figure 5(a)) shows a sudden rise in the first 10 days. The graph rises to a peak of about 8.5×105 and after 10 days, decreases steeply to a minimal population of about 1×105 within the next 90 days and maintains this new level until 150 days when it begins to increase smoothly. The infected population without control (See Figure 5(b)) gently increases for the first 25 days to reach a maximum height of 250. The new height of the infected became unstable after the next 25 days, and that resulted in the population slowly decreasing to 240 in the next 150 days, after which it began to rise for the remaining time. With the asymptotically infected without control (See Figure 5(c)), we noted a quick rise of the curve in the first 10 days and reached the pinnacle at 14500 in 25 days at the infection hotspots.Figure 5
Graphs of simulation results for strategy 2 (withu1≠0 and u2≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profileThe exposed population yielded to the design experiment strategy2. The exposed population starts with 1.9×105 population and gently reduces until 25 days when it seems to have been wiped from the system. The infected population under control strategy 2 slightly increased in the early days and began to lower after 10 days. The graph decreased in that pattern and reached a lower height of 90 at the final time. Also, the asymptotically infected population increased to about 2000 in the first 5 days, decreased slowly to zero in 30 days, and maintained that level for the remaining period of the simulation time. Figure 5(d) presents the control profile for Strategy 2. By inspection, we observed that the personal protection control u1 falls to the lower bound at the initial zero time. It remained on the lower bound for 60 days when it suddenly moved to the upper bound. It then remained at the upper bound until 250 days when it dropped to the lower bound again. Control u2, representing the vaccination control, remained at the lower bound for the first 59 days when it swiftly moved to the upper bound. It stayed at the upper bound for the next 120 days when it dropped to the lower bound and remained there for the remaining time.


## 5.3. Strategy3 (Implementing Controls u1 and u3)

Withu2 set to zero, strategy 3 uses the controls u1 and u3 for the experimentation by simulating the optimality system with the structured interval. The simulation results for this strategy are presented in Figure 6. We observe that, the exposed population without control (See Figure 6(a)) moved sharply to 8.5×105 in day 10 of the simulated time and then decreased gradually to almost 0.1×105 and maintained the level until 150 days when it began to rise for the remaining time. The infected population without control (See Figure 6(b)) increased slowly and moved to a height of 250 for 50 days. The population then decreased slowly for the next 100 days and began to pick up after the 100 days. The asymptomatically infected population without control (See Figure 6(c)) also shoots quickly in the first 25 days to its maximum of 14500. The dynamics changed, and the population began to reverse slowly to a level of about 100, and then picked up after 150 days and smoothly increased to the final time. The optimal control plot of the exposed produced a graph that seemed to wane in the early days but began to rise after 30 days to meet the noncontrol graph at 70 days. The level of the controlled exposed population was maintained until the final time. The infected population with control Strategy 3 (See Figure 6(b)) got lowered in the early days and continued till the final time of the simulation. The controlled population of asymptotically infected persons (See Figure 6(c)) got depleted in the first 30 days. However, for the uncontrolled problem, the population resurged and maintained the level for the rest of the simulation time. From Figure 6(d), u1 denoting personal protection control drops suddenly from the upper bound to the lower bound at time t=0. It then remained at the lower bound for 15 days and shot to the upper bound, where it remained for 15 days and dropped to the lower bound. It remained in the lower bound for another 15 days and moved to the upper bound. It then stayed at the upper bound for 15 days and dropped to the lower bound. It picked after 5 days to the upper bound and stayed there until it dropped to the lower bound. u3 denoting disinfectant control started from the lower bound and moved to the upper bound after 15 days. It dropped to the lower bound after 15 days and again moved to the upper bound after 15. It stayed at the upper bound for another 15 days and then dropped to the lower bound for 5 days. It picked to the upper bound of 70 days and remained until the final time before dropping.Figure 6
Graphs of simulation results for strategy 3 (withu1≠0 and u3≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profile


## 5.4. Strategy4 (Implementing Controls u2 and u3)

The strategy4 set u1=0 and utilized the control u2 and u3 in the numerical simulation for assigned interval. The simulation results are presented in Figures 7(a)–7(d). The Strategy 4 is observed to also do well in reducing the exposed, infected and asymptomatically infected populations. From the control profile (Figure 7(d)), the vaccination control, u2, shot to the upper bound at time t=50, stayed at the upper bound for 150 days, and then dropped to the lower bound. It remained at the lower bound for the rest of the simulated time. The disinfectant spraying control u3 stayed at the lower bound from day 0 to 50, rocketed to the upper bound and remained there until 249 days when it returned to the lower bound.Figure 7
Graphs of simulation results for strategy 4 (withu2≠0 and u3≠0).
(a)
The solution path of the exposed(b)
The solution path of the infected(c)
The solution path of the asymptomatically infected(d)
Control profile


## 6. Economic Cost Analysis of the Control Implementation

With the simulation results of the optimality control problem, the present section seeks to identify the strategy among the considered strategies that minimize the cost involved in executing the strategy yet give a satisfying performance output. To identify the most cost-effective strategy, we compare strategies 1, 2, 3, and 4. The approach outlined in related previous studies [60–63] is adopted to examine the average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER) of strategies 1, 2, 3, and 4. The ICER is the ratio of the differences in cost of two control strategies to the differences in infection averted by executing that strategy. On the other hand, the ACER is the ratio of the total cost involved in executing the strategy to the total infected prevented. The mathematical formulation
(53)ICER=Differenceincostsofcontrolstrategiese1ande2Differenceininfectionavertedbythestrategies,is used to exhibit the number of infections averted, and the related cost of executing the strategies under consideration in increasing order of magnitude of the infection averted is presented in Table3.Table 3
Total number of infections averted from the least to the highest.
StrategiesTotal infection averted (×107)Total cost (×103)ACERICERStrategy 15.03266.15231.2225×10−41.2225×10−4Strategy 25.08263.82810.7532×10−4−4.6484×10−3Strategy 45.09604.00480.7859×10−41.3186×10−3Strategy 35.69574.96060.8709×10−41.5936×10−4The ICER representations are derived as follows;(54)ICER1=6.1523×1035.0326×107=1.2225×10−4,ICER2=3.8281−6.1523×1035.0826−5.0326×107=−4.6484×10−3,ICER4=4.0048−3.8281×1035.0960−5.0826×107=1.3186×10−3,ICER3=4.9606−4.0048×1035.69566−5.09600×107=1.5936×10−4.From Table3, we notice that the ACER value of strategy 2 is the least among the strategies considered. The ACER values insinuate that strategy 2 is the most cost-effective. Further, we observe that strategy 1 has the highest ACER value, which is the least cost-effective measure. However, an in-depth analysis of the least cost-effective measure was required. The ICER of the strategies was then computed. The ICER computations depicted that strategy 1 is more costly but less effective than strategy 2. The analysis confirmed that ICER of strategy 1 is higher than ICER of strategy 2. Hence, the control strategy 1 is deleted from the considered strategies, and the remaining strategies ICERs are recalculated for further analysis on the three. The analysis of the remaining strategies is presented in Table 4.Table 4
ICER computation of strategy 2, 4, and 3.
StrategiesTotal infection averted (×107)Total cost (×103)ICERStrategy 25.08263.82810.7532×10−4Strategy 45.09604.00481.3187×10−3Strategy 35.69574.96061.5938×10−4From the computed ICERs of Table5, we see that strategy 4 ICER is higher than strategy 2. The cross-examination of the two ICER shows the dominance of strategy 2 over strategy 4. Thus, strategy 2 costs less and performs best in managing the disease. Hence, strategy 4 is removed from the remaining strategies, and the ICER of strategy 2 and 3 are recalculated and analyzed, as depicted in Table 5.Table 5
ICER computation of strategy 2 and 3.
StrategiesTotal infection averted (×107)Total cost (×103)ICERStrategy 25.08263.82810.7532×10−4Strategy 35.69574.96061.8472×10−4The ICER computation from Table5 indicated that strategy 3 is higher than strategy 2. The analysis further proves that strategy 2 is the most cost-effective control strategy among the listed strategies.


### 6.1. Conclusion

The research considered a dynamical study on a nonlinear autonomous COVID-19 model to unveil strategies that could assist in stemming the spreading of the disease. With the introduction of the control strategies: personal protection, vaccination, and disinfecting of the environment into the state model (1), the basic model was redesigned into an optimal control problem. We utilized Pontryagin’s maximum principle’s [57] analytic method to characterize possible controls that may help stem the epidemic from spreading. We supported the qualitative analyses with a numerical solution that employed the fourth-order Runge-Kutta method to generate numerical solutions of the control problem. The results from the executed strategies unveiled a satisfying impact on the dynamics of the exposed, asymptomatically infected, and infected individual. The strategy 1 considered the three controls of personal protection, vaccination, and disinfectant of the environment. Interestingly, the generated graphs showed the retrospective impact of the strategy on the disease’s dynamics. Thus, we see substantially minimized exposed, infected, and asymptomatically infected individuals. Strategy 2 employed personal protection and vaccination control as its intervention strategy. A similar result to using strategy 1 was obtained. Regarding strategy 3, personal protection and disinfectant controls were used. The simulation produced a pleasing effect on the dynamics of the infected. Thus, the graph depicts a convincing minimized exposed, infected, and asymptomatically infected with the strategy. The strategy of 4 chose the vaccination and disinfectant of the environment. The result minimized the exposed, infected, and asymptomatically infected. A careful examination of the strategies that employed vaccination as part of the controls substantially minimized the exposed, infected, and asymptomatically infected graphs. For instance, strategy 3, which did not consider vaccination control experienced an early rise in the asymptomatically infected graph but was minimized in the long run. The results suggest that vaccines are essential pharmaceutical control strategies needed to mitigate the disease but cannot be wholly relied on due to the resistance imposed by the mutation of the virus and the vaccines efficacies waning after a short while. This is a result of the leakiness of vaccines and limited vaccination capacity. The finding is consistent with the findings of the work [43, 64, 65], which recommended that policymakers should be pragmatic in relaxing other COVID-19 mitigating protocols since the vaccine efficacies are not 100%, as evidence of reinfection after vaccination has been proved in several studies [66–68]. Finally, we assessed the various strategies’ economic costs with the cost-effectiveness analysis method. The method examined the effectiveness and cost of the listed strategies considered. The analysis unveiled that the strategy 2: personal protection and vaccination control strategy is the most cost-effective method among the considered strategies. From our investigations, we recommend that a deliberate attempt should be made by public health and all stakeholders to strengthen the personal protection protocols as recommended by WHO and encourage and educate society about the benefits of being vaccinated, as vaccines reduce hospitalization [66].


## 6.1. Conclusion

The research considered a dynamical study on a nonlinear autonomous COVID-19 model to unveil strategies that could assist in stemming the spreading of the disease. With the introduction of the control strategies: personal protection, vaccination, and disinfecting of the environment into the state model (1), the basic model was redesigned into an optimal control problem. We utilized Pontryagin’s maximum principle’s [57] analytic method to characterize possible controls that may help stem the epidemic from spreading. We supported the qualitative analyses with a numerical solution that employed the fourth-order Runge-Kutta method to generate numerical solutions of the control problem. The results from the executed strategies unveiled a satisfying impact on the dynamics of the exposed, asymptomatically infected, and infected individual. The strategy 1 considered the three controls of personal protection, vaccination, and disinfectant of the environment. Interestingly, the generated graphs showed the retrospective impact of the strategy on the disease’s dynamics. Thus, we see substantially minimized exposed, infected, and asymptomatically infected individuals. Strategy 2 employed personal protection and vaccination control as its intervention strategy. A similar result to using strategy 1 was obtained. Regarding strategy 3, personal protection and disinfectant controls were used. The simulation produced a pleasing effect on the dynamics of the infected. Thus, the graph depicts a convincing minimized exposed, infected, and asymptomatically infected with the strategy. The strategy of 4 chose the vaccination and disinfectant of the environment. The result minimized the exposed, infected, and asymptomatically infected. A careful examination of the strategies that employed vaccination as part of the controls substantially minimized the exposed, infected, and asymptomatically infected graphs. For instance, strategy 3, which did not consider vaccination control experienced an early rise in the asymptomatically infected graph but was minimized in the long run. The results suggest that vaccines are essential pharmaceutical control strategies needed to mitigate the disease but cannot be wholly relied on due to the resistance imposed by the mutation of the virus and the vaccines efficacies waning after a short while. This is a result of the leakiness of vaccines and limited vaccination capacity. The finding is consistent with the findings of the work [43, 64, 65], which recommended that policymakers should be pragmatic in relaxing other COVID-19 mitigating protocols since the vaccine efficacies are not 100%, as evidence of reinfection after vaccination has been proved in several studies [66–68]. Finally, we assessed the various strategies’ economic costs with the cost-effectiveness analysis method. The method examined the effectiveness and cost of the listed strategies considered. The analysis unveiled that the strategy 2: personal protection and vaccination control strategy is the most cost-effective method among the considered strategies. From our investigations, we recommend that a deliberate attempt should be made by public health and all stakeholders to strengthen the personal protection protocols as recommended by WHO and encourage and educate society about the benefits of being vaccinated, as vaccines reduce hospitalization [66].



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*Source: 1715414-2022-12-29.xml*