# Mathematical Model of Hepatitis B Disease with Optimal Control and Cost-Effectiveness Analysis

Authors: Malede Atnaw Belay; Okelo Jeconia Abonyo; David Mwangi Theuri
Journal: Computational and Mathematical Methods in Medicine
 (2023)

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## Abstract

In this paper, a mathematical model of hepatitis B disease with a two-dose vaccine series has been formulated and analyzed. We demonstrated that the model’s disease-free equilibrium is globally asymptotically stable when the basic reproduction numberR0 is less than one, whereas the model’s endemic equilibrium is locally asymptotically stable when R0 is greater than one. Sensitivity analysis is performed, and based on its results, the model is extended to an optimal control problem by incorporating two control interventions, namely, prevention and enhanced newborn vaccination. Finally, simulation analyses of the model are conducted to illustrate the theoretical findings and effectiveness of each strategy, which indicates that the use of prevention efforts is the most cost-saving strategy.


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## Body

## 1. Introduction

Hepatitis is an irritation of the liver. It can be motivated by drugs, autoimmune disease, excessive alcohol consumption, bacteria, and viruses. Viral hepatitis is a type of contagious disease that mostly affects the liver and causes serious sickness and death. It is caused by infection with one of the five known hepatotropic viruses, which are named hepatitis A virus (HAV), hepatitis B virus (HBV), hepatitis C virus (HCV), hepatitis D virus (HDV), and hepatitis E virus (HEV). Mortality due to viral hepatitis has increased by 63% since 1990 and is now ranked as the seventh leading cause of mortality worldwide; moreover, this disease is responsible for approximately 1.4 million deaths per year from chronic infection and hepatitis-related liver cancer and cirrhosis. Out of those deaths, around 47% are due to the hepatitis B virus [1].Viral hepatitis B is a blood-borne infection; the virus is most commonly transmitted from mother to child during birth, but it can also be transmitted through sexual contact; sharing needles, syringes, or other drug injections; or exposure to sharp instruments [2]. The acute stage of hepatitis B virus disease occurs within the first six months of infection. In this stage, a few infected people are capable of opposing and clearing the infection; for others, the infection transfers to a chronic or lifelong disease and develops into a serious health problem. Hepatitis B disease obtained in adulthood causes chronic hepatitis in less than 5% of cases, but disease in infancy and early childhood causes chronic hepatitis in about 95% of cases [3]. It is particularly difficult in the West Pacific and African regions, where 116 million and 81 million people, respectively, are chronically infected. Thus, Africa has the second-highest number of chronic HBV infections next to the West Pacific, and it is considered an area of high endemicity. Moreover, 5%-15% of individuals in developing and low-income countries are chronic carriers of the hepatitis B infection [4].There is no specific medication for acute hepatitis B. But chronic hepatitis B infection can be treated by nucleotide analogues (NAS) and pegylated interferon (PEG-IFN). The purpose of these medications is to suppress HBV reproduction, protecting the growth of liver disease and its progression to cirrhosis and hepatocellular carcinoma. The main means of preventing infection and complications brought on by the hepatitis B virus is hepatitis B immunization. The Advisory Committee on Immunization Practices (ACIP) endorsed Heplisav-B (HepB-CpG), a yeast-derived vaccine made with a new adjuvant and given as a 2-dose series for use in anyone older than 18 years. 90% to 100% of individuals who received the Heplisav-B (HepB-CpG) vaccine developed seroprotective antibodies to the hepatitis B surface antigen (anti-HBs) [5]. Heplisav-B is a better option for people with diabetes or for those who may have had difficulty responding to the vaccine in the past, according to data released by Dynavax, which has shown faster and higher response rates compared to other three-dose vaccine brands. Thus, an important option for the prevention of hepatitis B is administering Heplisav-B (HepB-CpG) at least four weeks apart to get the second dose, in addition to getting all infants the universal hepatitis B vaccine as soon as possible after birth.A mathematical model is an essential tool for comprehending the transmission and spread of diseases. It can also suggest effective control and prevention measures and forecast the severity and potential scope of epidemic diseases. Many mathematical models have been performed in order to study the dynamics of hepatitis B disease. Among those models, the work done by Khan et al. [6] developed a deterministic compartmental model for hepatitis B with migrated class effect. The system simulation revealed that the rate of flow from migrated class is directly proportional to the number of infected individuals. [7] studied a compartmental SEICR model of hepatitis B disease by administrated vaccination and treatment control strategies. Pontryagin’s maximum approach and forward-backward iterative scheme were used for the necessary conditions of optimal values and to solve the optimal system numerically, respectively. Results from simulation showed that mixing the use of both vaccination and treatment was the most successful strategy to manage hepatitis B infection. Similarly, [8] proposed a deterministic compartmental model of hepatitis B disease with vaccination and treatment interventions. A sensitivity analysis of the basic reproductive number suggested that an increase in both effective vaccination at birth and the successful rate of treatment for hepatitis B patients had a great influence on decreasing the magnitude of the corresponding basic reproduction number. They concluded from their simulation that, in addition to these interventions, other measures may have a significant impact on hepatitis B control because the disease did not eradicate within the time frame specified. The study in [9] formulated a mathematical model for hepatitis B with isolation, awareness, and treatment time-dependent controls. According to their simulation, integrating all three time-dependent controls at the same time can reduce disease in the population. [10] developed an optimal control model of hepatitis B disease using isolation, enhanced vaccines, and treatment interventions. And simulation of their results showed that combining these three interventions at the same time is beneficial and efficient for controlling hepatitis B infection. Other researchers such as [11–15] have also studied on dynamics of hepatitis B disease with optimal controls.In this work, we suggested a model for hepatitis B disease adopted from [8], including a two-dose vaccine series class, and extended it to an optimal control model to study the cost-effectiveness analysis, which was not studied by the mentioned researchers above. In the optimal control model, we proposed two time-dependent control measures, namely, prevention (avoid direct contact with blood and bodily fluids, use a condom with sexual partners, and avoid sharing sharp items) and increasing newborn vaccination.The rest of this work is organized as follows: In Section2, the description of state variables and parameters and model development are done. In Section 3, we study the qualitative analysis, which includes positivity and boundedness of the model, stability of equilibria, and sensitivity analysis of the basic reproduction number. Section 4 contains the simulation results of the autonomous system analysis. The development of an optimal control model, its simulation results, and cost-effectiveness analysis are presented in Section 5. Lastly, brief conclusions are given in Section 6.


## 2. Model Formulation

We developed a compartmental mathematical model of hepatitis B disease based on its transmission routes.Nt is the total population, which is divided into six classes, St,V1t,V2t,It,Ct, and Rt. Thus, Nt is defined as follows:
(1)Nt=St+V1t+V2t+At+Ct+Rt.Moreover, descriptions of the state variables of the model are in Table1. In the development of the model, the following assumptions are made.
(i)
Hepatitis B infection is transmitted through vertical and horizontal(ii)
Since the acute stage of hepatitis B infection is comparatively short compared to a pregnancy period, the infection from acutely infected mothers is not taken into account(iii)
We assume the disease-induced death rate exists in the chronic class(iv)
There is a natural death rate for each compartment(v)
We assume that susceptible individuals are generated by the birth at the constant rate ofΛ(vi)
All parameters are nonnegativeTable 1
Description of state variables of the bacterial meningitis disease model.
State variableDescriptionStIndividuals who are not yet infected but will be infected easilyV1tPeople who have taken the first dose of vaccineV2tIndividuals who have received second dose of vaccine after one month of the first doseAtInfectious individuals in acute stage (the first six months after someone is infected by HBV)CtInfectious individuals at chronic stageRtIndividuals who recovered from hepatitis B diseaseSusceptible individuals become infected horizontally with the force of infection,λ=ξκA+C, where κ is the degree of infectiousness of acute infections in comparison to chronic infections and ξ represents the effective contact rate. If κ>1, acute infection is more likely susceptible than chronic. If κ=1, both chronic and acute have an equal risk of infecting the susceptible, but if κ<1, chronic infections are more likely to infect the susceptible than acute infections. In the case of vertical transmission, we assume that newborn babies are successfully vaccinated at the rate Λω or unsuccessfully vaccinated at the rate Λ1−ω, where Λ is the birth rate, ω is the proportion of successfully vaccinated, and 0<ω<1. As a result, Λω joining the vaccinated first dose class, the unsuccessfully vaccinated rate Λ1−ωτC moving into chronic class, and the remaining Λ1−ω1−τC entering into the susceptible class where τ denotes the percentage of unimmunized babies born to carrier mothers developing the chronic state. We also assumed that the portion of unimmunized babies developing to chronic is less than the sum of death rates of hepatitis B disease and recovery rate from the chronic class [16].Further explanations of parameters in the model are described in Table2. From Figure 1, we can obtain the following nonlinear first-order system of an ordinary differential equation:
(2)dSdt=Λ1−ω1−τC+φ2V1−λ+μ+φ1S,dV1dt=φ1S+Λω−α+μ+φ2V1,dV2dt=αV1−ε+μV2,dAdt=λS−δ+σ+μA,dCdt=δA+Λ1−ωτC−ϑ+μ+ψC,dRdt=σA+ϑC+εV2−μR.Table 2
Description of parameters of bacterial meningitis disease model.
ParameterDescriptionΛRate of birthμNatural death rateσNatural recovery rate from acute classϑRate of recovery from chronic class due to treatmentδRate of acute moves to chronic class by developing symptomsαThe rate of receiving second dose of vaccine from vaccinated first dose classεRate of recovery from vaccinated second dose classψThe death rate due to the disease in chronic classφ1Rate of first dose of vaccine from susceptibleφ2Vaccine waning rate from vaccinated first dose classFigure 1
Compartmental flow diagram of hepatitis B disease transmission.With the initial conditionsS0=S0>0,V10=V10>0,A0=A0>0,V20=V20>0,C0=C0>0, and R0=R0>0.


## 3. Model Analysis

### 3.1. Positivity and Boundedness of Solutions

Theorem 1.
LetΩh0=S0,V10,V20,A0,C0,R0∈R+6 be the initial condition for model (2). Then, the set of solutions St,V1t,V2t,At,Ct,Rt of the model is nonnegative for all t>0. Furthermore, N≤Λ/μ.Proof.
To prove this, let us define,t1=supt>0:St0>0,V1t0>0,V2t0>0,At0>0,Ct0>0,Rt0>0,∀t0in0,t. Since S0,V10,V20,A0,C0, and R0 are nonnegative, then t1>0. If t1<∞, by using the variation of the constant formula to the first equation of model (2) at t1 given by
(3)dSdt+λt+μ+φ1S=Λ1−ω1−τCt+φ2V1t,which can be rewritten as
(4)ddtSteμ+φ1t+∫0tλmdm=Λ1−ω1−τCt+φ2V1teμ+φ1t+∫0tλmdm.
Thus,(5)St1eμ+φ1t1+∫0t1λmdm−S0=∫0t1Λ1−ω1−τCz+φ2V1zeμ+φ1z+∫0zλmdmdz.
Therefore,(6)St1=S0e−μ−φ1t1−∫0t1λmdm+e−μ−φ1t1−∫0t1λmdm×∫0t1Λ1−ω1−τCz+φ2V1zeμ+φ1z+∫0zλmdmdz≥0.
It can be shown in the same manner for other state variables. This contradicts the fact thatt1 is the supremum because at least one of the state variables should be equal to zero at t1. Therefore, t1=∞.
Moreover, the total population at any timet is given by
(7)Nt=St+V1t+V2t+At+Ct+Rt.
After differentiating both sides ofNt and substituting the expressions into (2), we obtain
(8)dNdt=Λ−μN−ψC.
Since the parameterψ and the state variable C are positive in (8), we have
(9)dNdt≤Λ−μN.
Using the separation of variables rule, equation (9) becomes
(10)dNΛ−μN≤dt.
With the help of [17] integrating both sides of equation (10), it gives
(11)∫dNΛ−μN≤∫dt.
After further simplification and applying the initial conditionN0=N0, we obtain
(12)N≤Λμ−1μΛ−μN0e−μt.
Ast⟶∞ in equation (12), the total population size Nt⟶Λ/μ, which implies that 0<N≤Λ/μ. Hence,
(13)Ωh=S,V1,V2,A,C,RεR+6:0<N≤Λμis a feasible region of the system. Thus, the model is epidemiologically feasible as well as mathematically well posed. Therefore, it is sufficient to study the dynamics of the basic model in Ωh.


### 3.2. Disease-Free Equilibrium

The disease-free equilibrium (DFE) of model (2) only exists in the absence of infections. Solve for DFE by setting the left-hand side of system (2) equal to zero. Since A=C=0 at disease-free equilibrium, system (2) becomes
(14)Λ1−ω+φ2V1−μ+φ1S=0,φ1S+Λω−α+μ+φ2V1=0,αV1−ε+μV2=0,εV2−μR=0.Thus, from system (14), the DFE of model (2) is given by
(15)E0=S0,V10,V20,A0,C0,R0=Λι−ωνμι+φ1ν,Λφ1+ωμμι+φ1ν,Λαφ1+ωμε+μμι+φ1ν,0,0,Λεαφ1+μωμε+μμι+φ1ν,where
(16)ι=α+μ+φ2,ν=α+μ,α+μ<ωα+μ.To figure out the basic reproduction numberR0 of system (2), we use the method presented in [18]. It is defined as R0=ρFV−1, where ρFV−1 denotes the spectral radius of FV−1. We consider only the infectious classes in the system of differential equation (2) as follows:
(17)dAdt=λS−δ+σ+μA,dCdt=δA+Λ1−ωτC−ϑ+μ+ψC.Supposef is the rate of new infection coming into the system and v is the difference in the rate of immigration into the compartments and the rate of transfer from compartments:
(18)f=λS0,v=δ+σ+μAϑ+μ+ψC−Λ1−ωτC−δA,where
(19)λ=ζκA+C.Next, obtain the Jacobian matrix off and v with respect to A and C at the disease-free equilibrium E0,
(20)F=ζS0κ100,V=δ+σ+μ0−δϑ+μ+ψ−Λ1−ωτ.Then, by computing the inverse of matrixV, we get
(21)V−1=1δ+σ+μϑ+μ+ψ−Λ1−ωτϑ+μ+ψ−Λ1−ωτ0δδ+σ+μ.The product ofFV−1 gives as
(22)FV−1=κζS0δ+σ+μ+ζδS0δ+σ+μϑ+μ+ψ−Λ1−ωτζS0ϑ+μ+ψ−Λ1−ωτ00.The basic reproduction numberR0 is the spectral radius of the matrix in (22) which is
(23)R0=ζS0κδ+σ+μ+δδ+σ+μϑ+μ+ψ−Λ1−ωτ,where ϑ+μ+ψ−Λ1−ωτ>0, because of the assumption in Section 2. Thus, the basic reproduction number of system (2) is the sum of two terms. The first term represents the average number of secondary infections produced by one acute individual, and the second term gives the average number of secondary infections produced by one chronic individual in the totally susceptible class.Theorem 2.
The disease-free equilibrium point is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
The Jacobian matrix of system (2) at the disease-free equilibrium E0 is given by
(24)JE0=−μ+φ1φ20−ζκS0−Λ1−ωτ−ζS00φ1−α+μ+φ200000α−μ+ε000000ζκS0−δ+σ+μζS00000δΛ1−ωτ−ϑ+μ+ψ000εσϑ−μ.
Then, by continuously expanding the determinant of the characteristic equationJE0−λ=0, we obtained λ1=−μ<0,λ2=−μ+ε<0; moreover, λ3 and λ4 are negative with the help of Routh-Hurwiz criteria. The remaining two eigenvalues are eigenvalues of
(25)J1=ζκS0−δ+σ+μζS0δΛ1−ωτ−ϑ+μ+ψ.
Now, we can apply the usual conditions that guarantee that the eigenvalues ofJ1 have a negative real part. In particular, we want a trace of J1<0 and determinant of J1>0. Using the second inequalities, we obtained the following condition:
(26)ζκS0−δ+σ+μΛ1−ωτ−ϑ+μ+ψ−δζS0>0,−δ+σ+μΛ1−ωτ−ϑ+μ+ψ>ϑ+μ+ψ−Λ1−ωτζκS0+δζS0,δ+σ+μϑ+μ+ψ−Λ1−ωτ>ϑ+μ+ψ−Λ1−ωτζκS0+δζS0,1>ζS0κδ+σ+μ+δδ+σ+μϑ+μ+ψ−Λ1−ωτ=R0.
We can see from conditionR0<1 the trace of J1<0 and the determinant of J1>0, implying that the remaining two eigenvalues are negative real parts. Therefore, for R0<1, the disease-free equilibrium point is locally asymptotically stable, while for R0>1, the disease-free equilibrium point is unstable.Theorem 3.
The disease-free equilibrium pointE0 is globally asymptotically stable whenever R0<1.Proof.
To prove this, we use the method of the Lyapunov function defined as(27)L=k1A+k2C,where k1=q1κ+δ/q2,k2=1, q1=ϑ+μ+ψ−Λ1−ωτ, and q2=δ+σ+μ. Differentiating (27) with respect to t gives
(28)dLdt=κ1Ȧ+κ2C.̇
SubstitutingȦ and Ċ from (2) into dL/dt yields
(29)dLdt=q1κ+δq1q2q1ζSκA+C−q2A+δA−q1C,=κq1q1κ+δζASq1q2−A+q1q1κ+δζSCq1q2−C,=κq1q1κ+δζSq1q2−1A+q1q1κ+δζSq1q2−1C≤κq1q1κ+δζS0q1q2−1A+q1q1κ+δζS0q1q2−1C=κq1R0−1A+q1R0−1C
Since,S≤S0 and R0=q1κ+δζS0/q1q2.
Thus,dL/dt<0 if R0<1. Therefore, according to [19] invariant principle DFE, E0 is globally asymptotically stable in Ωh if R0<1.


### 3.3. Endemic Equilibrium (EE)

The endemic equilibrium point of the system (2) is the steady-state solution where the hepatitis B disease persists in the population, and it is denoted by E∗=S∗,V1∗,V2∗,A∗,C∗,R∗. This can be obtained by equating the left side of system (2) to zero, and expressing the other state variables in terms of A∗ gives
(30)C∗=δA∗q1,S∗=α+μ+φ2Λ1−ωq1−δτA∗+q1φ2Λωα+μ+φ2ζq1κ+δA∗+q1μ+φ1α+μ+φ2q1μ,V1∗=φ1S∗+Λωα+μ+φ2,V2∗=αφ1S∗+Λωα+μ+φ2ε+μ,R∗=εαφ1S∗+Λωμα+μ+φ2ε+μ+σq1+ϑδA∗μq1.Then, substitute the expression ofC∗ and S∗ above into the fourth equation of system (2), and after further simplification, we obtain
(31)A∗=q12q2μα+μ+φ2+φ1α+μζq1κ+δτδ+q1q2α+μ+φ2ζq1κ+δR0−1,where
(32)q1=ϑ+μ+ψ−Λ1−ωτ>0,q2=δ+σ+μ.Therefore, system (2) has a unique endemic equilibrium point if and only if R0>1 and q1>δτA∗.Theorem 4.
The endemic equilibriumE∗ of system (2) is locally asymptotically stable for R0>1 (but near to one).Proof.
To investigate the possibility of backward or forward bifurcation of model (2), we use the method introduced by [20]. This is done by assigning the variables as follows:
(33)S=x1,V1=x2,V2=x3,A=x4,C=x5,R=x6.
Then, system (2) can be written as
(34)dx1dt=Λ1−ω1−τx5+φ2x2−λ+μ+φ1x1,dx2dt=φ1x1+Λω−α+μ+φ2x2,dx3dt=αx2−ε+μx3,dx4dt=λx1−δ+σ+μx4,dx5dt=δx4+Λ1−ωτx5−ϑ+μ+ψx5,dRdt=σx4+ϑx5+εx3−μx6.
Choosingζ as the bifurcation parameter and solving for ζ=ζ∗ in (23) when R0=1 yield
(35)ζ∗=q1q2S0κq1+δ,where
(36)q1=ϑ+μ+ψ−Λ1−ωτ,q2=δ+σ+μ.
The linearization matrix of system (34) around the disease-free equilibrium when ζ=ζ∗ is
(37)JE0,ζ∗=−m1φ20−m6−m70φ1−m200000α−m3000000m4ζ∗S00000δm5000εσϑ−μ,where m1=μ+φ1,m2=α+μ+φ2,m3=μ+ε,m4=ζ∗κS0−δ+σ+μ,m5=−q1,m6=ζ∗κS0, and m7=Λ1−ωτ+ζ∗S0.
The Jacobian matrix (37) has simple zero eigenvalues; hence, the center manifold theory will be used to analyze the dynamic of the system (34) near ζ=ζ∗. Next, we obtain the left and the right eigenvectors of JE0,ζ∗ associated with the zero eigenvalue. The right eigenvector is given by W=w1,w2,w3,w4,w5,w6T from (37):
(38)−m1φ20−m6−m70φ1−m200000α−m3000000m4ζ∗S00000δm5000εσϑ−μw1w2w3w4w5w6=000000.
Then, equation (38) can be expressed as
(39)−m1w1+φ2w2−m6w3−m7w5=0,φ1w1−m2w2=0,αw2−m3w3=0,m4w4+ζ∗S0w5=0,δw4+m5w5=0,εw3+σw4+ϑw5−μw6=0.
For the computing system of (39), we obtain
(40)w1=−m2m3δκq1+δΛ1−ωτ+q1q2w4q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1α,w2=−m3αδφ1ακq1+δΛ1−ωτ+q1q2q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1αw4,w3=−δφ1ακq1+δΛ1−ωτ+q1q2w4q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1α,w4=w4>0,w5=δw4q1,w6=w4μ−δφ1ακq1+δΛ1−ωτ+q1q2q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1α+δq1+σ.
Similarly, the left eigenvectorV=v1,v2,v3,v4,v5,v6T of equation (37) associated with the zero eigenvalue is calculated as
(41)−m1φ10000φ2−m2α00000−m300ε−m600m4δσ−m700ζ∗S0m5ϑ00000−μv1v2v3v4v5v6=000000.
Then, equation (41) becomes
(42)−m1v1+φ1v2=0,φ2v1−m2v2+αv3=0,−m3v3+εv6=0,−m6v1+m4v4+δv5+σv6=0,−m7v1+ζ∗S0v4+m5v5+ϑv6=0,−μv6=0.
Solving the system of equation (42), we have
(43)V=0,0,0,v4,q2v4κq1+δ,0,where v4>0. Furthermore, using the property W·V=1, we can determine the expressions for v4 and w4 as
(44)w4=1,v4=q1κq1+δq1κq1+δ+δq1.
Now, the local stability near the beatification pointζ=ζ∗ is ascertained by the signs of a and b which are defined on the center manifold theorem 4.1 in [20] as follows:
(45)a=∑k,i,j=16vkwiwj∂2fk∂xi∂xjE0,b=∑k,i=16vkwi∂2fk∂xi∂ζE0,where
(46)f1=Λ1−ω1−τx5+φ2x2−λ+μ+φ1x1,f2=φ1x1+Λω−α+μ+φ2x2,f3=αx2−ε+μx3,f4=λx1−δ+σ+μx4,f5=δx4+Λ1−ωτx5−ϑ+μ+ψx5,f6=σx4+ϑx5+εx3−μx6.
We obtained by examining only the nonzero components of the left eigenvectorsv4 and v5 and taking into account the nonzero second-order partial derivatives of equation (46), which yields
(47)a=−2ζm2m3δκq1+δΛ1−ωτ+q1q2q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1ακq1+δq1+δq1κq1+δ+δq2<0,b=q1κq1+δq1+δq1κq1+δ+δq2>0.
The coefficientb is positive, and in contrast, the coefficient a is negative; thus, with the help of center manifold theorem 4.1 in [20], system (2) exhibits a forward bifurcation at R0=1, and hence, the endemic equilibrium point is locally asymptotically stable for R0>1 but adequately near 1.


### 3.4. Sensitivity Analysis

Here, we studied the sensitivity parameters involved in the basic reproduction number of the model (2) using the idea presented in [21]. This helped us determine the parameters that have a significant impact on hepatitis B disease. To evaluate the sensitivity index of R0 to a given parameter Γ, we use the following relation:
(48)ΩΓR0=∂R0∂ΓΓR0.Thus, we evaluated the sensitivity index ofR0 corresponding to each parameter in (23) as follows:
(49)ΩζR0=1,ΩκR0=κρκρ+δ,ΩΛR0=κρ+δ−Λκ1−ωτκρ+δ+Λ1−ωτρ,ΩωR0=ω−α+μκρ+δ+q3κΛτq3κρ+δ−Λτρ,ΩϑR0=ϑκκρ+δ−1ρ,ΩψR0=ψκκρ+δ−1ρ,ΩτR0=τΛ1−ω1ρ−κκρ+δ,ΩδR0=δ1κρ+δ−1q2,Ωφ1R0=−φα+μq1,Ωφ2R0=φ2q1−q3μq1q3,ΩαR0=αq11−ω−q3φ1+μq3q1,ΩμR0=μ1−ωκρ+δ+q3κq3κρ+δ−α+φ1+φ2+2μq2ρ+q1ρ+q1q2q1q2ρ,ΩσR0=−σq2,where
(50)ρ=ϑ+μ+ψ−Λ1−ωτ,q1=μα+μ+φ2+φ1α+μ,q2=δ+σ+μ,q3=α+μ+φ21−ω+φ2ω.The sensitivity indices of the basic reproduction number ofR0 with respect to each of the parameters involved in R0 are given in Table 3. The positive value of the sensitivity indices of the basic reproduction number R0 to the parameters, Λ,ζ,τ,κ, and φ2, in Table 3 indicates that an increase in the values of each parameter leads to an increase in the burden of hepatitis B disease in the community, while a decrease in the values of each parameter also leads to minimizing the persistence of the disease. For instance, increasing or decreasing the values of ζ,Λ,κ, and τ by 10% will produce a corresponding 10%,15.9%,6.3%,and5.3% increase or decrease in the basic reproduction number of the model, respectively. On the other hand, the negative value of the sensitivity indices of R0 to the parameters, μ,ω,ψ,φ1,σ, and α, in Table 3 shows that an increase in the value of each of the parameters forces to decrease in the basic reproduction number, while decreasing the values of each parameter leads to an increase in the basic reproduction of the model. Moreover, maximizing the value of ω by 10% will produce 8.9% minimization on the basic reproduction number due to newborn vaccination. Figure 2 also shows that the parameters that have the most influence on the basic reproduction number are birth rate Λ, successful vaccination rate ω, contact rate ζ, and infectiousness rate relative to acute κ. Thus, the identification of these critical parameters is important to the development of an optimal control model of the system (2).Table 3
Sensitivity indices of parameters involved inR0.
ParameterValueSourceSensitivity indicesΛ12Assumed1.46ζ0.24Assumed1τ0.11[22]0.466δ0.3Assumed0.1448κ0–1[23]0.628φ20.03–0.06[24]0.04ω0.4Assumed-0.932μ0.9Assumed-0.257ψ0.5Assumed-0.294ϑ0.025[16]-0.0147φ10.52Assumed-0.356σ0.12Assumed-0.1448α0.72Assumed-0.0196Figure 2
Sensitivity plot.


## 3.1. Positivity and Boundedness of Solutions

Theorem 1.
LetΩh0=S0,V10,V20,A0,C0,R0∈R+6 be the initial condition for model (2). Then, the set of solutions St,V1t,V2t,At,Ct,Rt of the model is nonnegative for all t>0. Furthermore, N≤Λ/μ.Proof.
To prove this, let us define,t1=supt>0:St0>0,V1t0>0,V2t0>0,At0>0,Ct0>0,Rt0>0,∀t0in0,t. Since S0,V10,V20,A0,C0, and R0 are nonnegative, then t1>0. If t1<∞, by using the variation of the constant formula to the first equation of model (2) at t1 given by
(3)dSdt+λt+μ+φ1S=Λ1−ω1−τCt+φ2V1t,which can be rewritten as
(4)ddtSteμ+φ1t+∫0tλmdm=Λ1−ω1−τCt+φ2V1teμ+φ1t+∫0tλmdm.
Thus,(5)St1eμ+φ1t1+∫0t1λmdm−S0=∫0t1Λ1−ω1−τCz+φ2V1zeμ+φ1z+∫0zλmdmdz.
Therefore,(6)St1=S0e−μ−φ1t1−∫0t1λmdm+e−μ−φ1t1−∫0t1λmdm×∫0t1Λ1−ω1−τCz+φ2V1zeμ+φ1z+∫0zλmdmdz≥0.
It can be shown in the same manner for other state variables. This contradicts the fact thatt1 is the supremum because at least one of the state variables should be equal to zero at t1. Therefore, t1=∞.
Moreover, the total population at any timet is given by
(7)Nt=St+V1t+V2t+At+Ct+Rt.
After differentiating both sides ofNt and substituting the expressions into (2), we obtain
(8)dNdt=Λ−μN−ψC.
Since the parameterψ and the state variable C are positive in (8), we have
(9)dNdt≤Λ−μN.
Using the separation of variables rule, equation (9) becomes
(10)dNΛ−μN≤dt.
With the help of [17] integrating both sides of equation (10), it gives
(11)∫dNΛ−μN≤∫dt.
After further simplification and applying the initial conditionN0=N0, we obtain
(12)N≤Λμ−1μΛ−μN0e−μt.
Ast⟶∞ in equation (12), the total population size Nt⟶Λ/μ, which implies that 0<N≤Λ/μ. Hence,
(13)Ωh=S,V1,V2,A,C,RεR+6:0<N≤Λμis a feasible region of the system. Thus, the model is epidemiologically feasible as well as mathematically well posed. Therefore, it is sufficient to study the dynamics of the basic model in Ωh.


## 3.2. Disease-Free Equilibrium

The disease-free equilibrium (DFE) of model (2) only exists in the absence of infections. Solve for DFE by setting the left-hand side of system (2) equal to zero. Since A=C=0 at disease-free equilibrium, system (2) becomes
(14)Λ1−ω+φ2V1−μ+φ1S=0,φ1S+Λω−α+μ+φ2V1=0,αV1−ε+μV2=0,εV2−μR=0.Thus, from system (14), the DFE of model (2) is given by
(15)E0=S0,V10,V20,A0,C0,R0=Λι−ωνμι+φ1ν,Λφ1+ωμμι+φ1ν,Λαφ1+ωμε+μμι+φ1ν,0,0,Λεαφ1+μωμε+μμι+φ1ν,where
(16)ι=α+μ+φ2,ν=α+μ,α+μ<ωα+μ.To figure out the basic reproduction numberR0 of system (2), we use the method presented in [18]. It is defined as R0=ρFV−1, where ρFV−1 denotes the spectral radius of FV−1. We consider only the infectious classes in the system of differential equation (2) as follows:
(17)dAdt=λS−δ+σ+μA,dCdt=δA+Λ1−ωτC−ϑ+μ+ψC.Supposef is the rate of new infection coming into the system and v is the difference in the rate of immigration into the compartments and the rate of transfer from compartments:
(18)f=λS0,v=δ+σ+μAϑ+μ+ψC−Λ1−ωτC−δA,where
(19)λ=ζκA+C.Next, obtain the Jacobian matrix off and v with respect to A and C at the disease-free equilibrium E0,
(20)F=ζS0κ100,V=δ+σ+μ0−δϑ+μ+ψ−Λ1−ωτ.Then, by computing the inverse of matrixV, we get
(21)V−1=1δ+σ+μϑ+μ+ψ−Λ1−ωτϑ+μ+ψ−Λ1−ωτ0δδ+σ+μ.The product ofFV−1 gives as
(22)FV−1=κζS0δ+σ+μ+ζδS0δ+σ+μϑ+μ+ψ−Λ1−ωτζS0ϑ+μ+ψ−Λ1−ωτ00.The basic reproduction numberR0 is the spectral radius of the matrix in (22) which is
(23)R0=ζS0κδ+σ+μ+δδ+σ+μϑ+μ+ψ−Λ1−ωτ,where ϑ+μ+ψ−Λ1−ωτ>0, because of the assumption in Section 2. Thus, the basic reproduction number of system (2) is the sum of two terms. The first term represents the average number of secondary infections produced by one acute individual, and the second term gives the average number of secondary infections produced by one chronic individual in the totally susceptible class.Theorem 2.
The disease-free equilibrium point is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
The Jacobian matrix of system (2) at the disease-free equilibrium E0 is given by
(24)JE0=−μ+φ1φ20−ζκS0−Λ1−ωτ−ζS00φ1−α+μ+φ200000α−μ+ε000000ζκS0−δ+σ+μζS00000δΛ1−ωτ−ϑ+μ+ψ000εσϑ−μ.
Then, by continuously expanding the determinant of the characteristic equationJE0−λ=0, we obtained λ1=−μ<0,λ2=−μ+ε<0; moreover, λ3 and λ4 are negative with the help of Routh-Hurwiz criteria. The remaining two eigenvalues are eigenvalues of
(25)J1=ζκS0−δ+σ+μζS0δΛ1−ωτ−ϑ+μ+ψ.
Now, we can apply the usual conditions that guarantee that the eigenvalues ofJ1 have a negative real part. In particular, we want a trace of J1<0 and determinant of J1>0. Using the second inequalities, we obtained the following condition:
(26)ζκS0−δ+σ+μΛ1−ωτ−ϑ+μ+ψ−δζS0>0,−δ+σ+μΛ1−ωτ−ϑ+μ+ψ>ϑ+μ+ψ−Λ1−ωτζκS0+δζS0,δ+σ+μϑ+μ+ψ−Λ1−ωτ>ϑ+μ+ψ−Λ1−ωτζκS0+δζS0,1>ζS0κδ+σ+μ+δδ+σ+μϑ+μ+ψ−Λ1−ωτ=R0.
We can see from conditionR0<1 the trace of J1<0 and the determinant of J1>0, implying that the remaining two eigenvalues are negative real parts. Therefore, for R0<1, the disease-free equilibrium point is locally asymptotically stable, while for R0>1, the disease-free equilibrium point is unstable.Theorem 3.
The disease-free equilibrium pointE0 is globally asymptotically stable whenever R0<1.Proof.
To prove this, we use the method of the Lyapunov function defined as(27)L=k1A+k2C,where k1=q1κ+δ/q2,k2=1, q1=ϑ+μ+ψ−Λ1−ωτ, and q2=δ+σ+μ. Differentiating (27) with respect to t gives
(28)dLdt=κ1Ȧ+κ2C.̇
SubstitutingȦ and Ċ from (2) into dL/dt yields
(29)dLdt=q1κ+δq1q2q1ζSκA+C−q2A+δA−q1C,=κq1q1κ+δζASq1q2−A+q1q1κ+δζSCq1q2−C,=κq1q1κ+δζSq1q2−1A+q1q1κ+δζSq1q2−1C≤κq1q1κ+δζS0q1q2−1A+q1q1κ+δζS0q1q2−1C=κq1R0−1A+q1R0−1C
Since,S≤S0 and R0=q1κ+δζS0/q1q2.
Thus,dL/dt<0 if R0<1. Therefore, according to [19] invariant principle DFE, E0 is globally asymptotically stable in Ωh if R0<1.


## 3.3. Endemic Equilibrium (EE)

The endemic equilibrium point of the system (2) is the steady-state solution where the hepatitis B disease persists in the population, and it is denoted by E∗=S∗,V1∗,V2∗,A∗,C∗,R∗. This can be obtained by equating the left side of system (2) to zero, and expressing the other state variables in terms of A∗ gives
(30)C∗=δA∗q1,S∗=α+μ+φ2Λ1−ωq1−δτA∗+q1φ2Λωα+μ+φ2ζq1κ+δA∗+q1μ+φ1α+μ+φ2q1μ,V1∗=φ1S∗+Λωα+μ+φ2,V2∗=αφ1S∗+Λωα+μ+φ2ε+μ,R∗=εαφ1S∗+Λωμα+μ+φ2ε+μ+σq1+ϑδA∗μq1.Then, substitute the expression ofC∗ and S∗ above into the fourth equation of system (2), and after further simplification, we obtain
(31)A∗=q12q2μα+μ+φ2+φ1α+μζq1κ+δτδ+q1q2α+μ+φ2ζq1κ+δR0−1,where
(32)q1=ϑ+μ+ψ−Λ1−ωτ>0,q2=δ+σ+μ.Therefore, system (2) has a unique endemic equilibrium point if and only if R0>1 and q1>δτA∗.Theorem 4.
The endemic equilibriumE∗ of system (2) is locally asymptotically stable for R0>1 (but near to one).Proof.
To investigate the possibility of backward or forward bifurcation of model (2), we use the method introduced by [20]. This is done by assigning the variables as follows:
(33)S=x1,V1=x2,V2=x3,A=x4,C=x5,R=x6.
Then, system (2) can be written as
(34)dx1dt=Λ1−ω1−τx5+φ2x2−λ+μ+φ1x1,dx2dt=φ1x1+Λω−α+μ+φ2x2,dx3dt=αx2−ε+μx3,dx4dt=λx1−δ+σ+μx4,dx5dt=δx4+Λ1−ωτx5−ϑ+μ+ψx5,dRdt=σx4+ϑx5+εx3−μx6.
Choosingζ as the bifurcation parameter and solving for ζ=ζ∗ in (23) when R0=1 yield
(35)ζ∗=q1q2S0κq1+δ,where
(36)q1=ϑ+μ+ψ−Λ1−ωτ,q2=δ+σ+μ.
The linearization matrix of system (34) around the disease-free equilibrium when ζ=ζ∗ is
(37)JE0,ζ∗=−m1φ20−m6−m70φ1−m200000α−m3000000m4ζ∗S00000δm5000εσϑ−μ,where m1=μ+φ1,m2=α+μ+φ2,m3=μ+ε,m4=ζ∗κS0−δ+σ+μ,m5=−q1,m6=ζ∗κS0, and m7=Λ1−ωτ+ζ∗S0.
The Jacobian matrix (37) has simple zero eigenvalues; hence, the center manifold theory will be used to analyze the dynamic of the system (34) near ζ=ζ∗. Next, we obtain the left and the right eigenvectors of JE0,ζ∗ associated with the zero eigenvalue. The right eigenvector is given by W=w1,w2,w3,w4,w5,w6T from (37):
(38)−m1φ20−m6−m70φ1−m200000α−m3000000m4ζ∗S00000δm5000εσϑ−μw1w2w3w4w5w6=000000.
Then, equation (38) can be expressed as
(39)−m1w1+φ2w2−m6w3−m7w5=0,φ1w1−m2w2=0,αw2−m3w3=0,m4w4+ζ∗S0w5=0,δw4+m5w5=0,εw3+σw4+ϑw5−μw6=0.
For the computing system of (39), we obtain
(40)w1=−m2m3δκq1+δΛ1−ωτ+q1q2w4q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1α,w2=−m3αδφ1ακq1+δΛ1−ωτ+q1q2q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1αw4,w3=−δφ1ακq1+δΛ1−ωτ+q1q2w4q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1α,w4=w4>0,w5=δw4q1,w6=w4μ−δφ1ακq1+δΛ1−ωτ+q1q2q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1α+δq1+σ.
Similarly, the left eigenvectorV=v1,v2,v3,v4,v5,v6T of equation (37) associated with the zero eigenvalue is calculated as
(41)−m1φ10000φ2−m2α00000−m300ε−m600m4δσ−m700ζ∗S0m5ϑ00000−μv1v2v3v4v5v6=000000.
Then, equation (41) becomes
(42)−m1v1+φ1v2=0,φ2v1−m2v2+αv3=0,−m3v3+εv6=0,−m6v1+m4v4+δv5+σv6=0,−m7v1+ζ∗S0v4+m5v5+ϑv6=0,−μv6=0.
Solving the system of equation (42), we have
(43)V=0,0,0,v4,q2v4κq1+δ,0,where v4>0. Furthermore, using the property W·V=1, we can determine the expressions for v4 and w4 as
(44)w4=1,v4=q1κq1+δq1κq1+δ+δq1.
Now, the local stability near the beatification pointζ=ζ∗ is ascertained by the signs of a and b which are defined on the center manifold theorem 4.1 in [20] as follows:
(45)a=∑k,i,j=16vkwiwj∂2fk∂xi∂xjE0,b=∑k,i=16vkwi∂2fk∂xi∂ζE0,where
(46)f1=Λ1−ω1−τx5+φ2x2−λ+μ+φ1x1,f2=φ1x1+Λω−α+μ+φ2x2,f3=αx2−ε+μx3,f4=λx1−δ+σ+μx4,f5=δx4+Λ1−ωτx5−ϑ+μ+ψx5,f6=σx4+ϑx5+εx3−μx6.
We obtained by examining only the nonzero components of the left eigenvectorsv4 and v5 and taking into account the nonzero second-order partial derivatives of equation (46), which yields
(47)a=−2ζm2m3δκq1+δΛ1−ωτ+q1q2q1κq1+δm3μα+μ+φ2+φ1α+μ+q1q2κφ1ακq1+δq1+δq1κq1+δ+δq2<0,b=q1κq1+δq1+δq1κq1+δ+δq2>0.
The coefficientb is positive, and in contrast, the coefficient a is negative; thus, with the help of center manifold theorem 4.1 in [20], system (2) exhibits a forward bifurcation at R0=1, and hence, the endemic equilibrium point is locally asymptotically stable for R0>1 but adequately near 1.


## 3.4. Sensitivity Analysis

Here, we studied the sensitivity parameters involved in the basic reproduction number of the model (2) using the idea presented in [21]. This helped us determine the parameters that have a significant impact on hepatitis B disease. To evaluate the sensitivity index of R0 to a given parameter Γ, we use the following relation:
(48)ΩΓR0=∂R0∂ΓΓR0.Thus, we evaluated the sensitivity index ofR0 corresponding to each parameter in (23) as follows:
(49)ΩζR0=1,ΩκR0=κρκρ+δ,ΩΛR0=κρ+δ−Λκ1−ωτκρ+δ+Λ1−ωτρ,ΩωR0=ω−α+μκρ+δ+q3κΛτq3κρ+δ−Λτρ,ΩϑR0=ϑκκρ+δ−1ρ,ΩψR0=ψκκρ+δ−1ρ,ΩτR0=τΛ1−ω1ρ−κκρ+δ,ΩδR0=δ1κρ+δ−1q2,Ωφ1R0=−φα+μq1,Ωφ2R0=φ2q1−q3μq1q3,ΩαR0=αq11−ω−q3φ1+μq3q1,ΩμR0=μ1−ωκρ+δ+q3κq3κρ+δ−α+φ1+φ2+2μq2ρ+q1ρ+q1q2q1q2ρ,ΩσR0=−σq2,where
(50)ρ=ϑ+μ+ψ−Λ1−ωτ,q1=μα+μ+φ2+φ1α+μ,q2=δ+σ+μ,q3=α+μ+φ21−ω+φ2ω.The sensitivity indices of the basic reproduction number ofR0 with respect to each of the parameters involved in R0 are given in Table 3. The positive value of the sensitivity indices of the basic reproduction number R0 to the parameters, Λ,ζ,τ,κ, and φ2, in Table 3 indicates that an increase in the values of each parameter leads to an increase in the burden of hepatitis B disease in the community, while a decrease in the values of each parameter also leads to minimizing the persistence of the disease. For instance, increasing or decreasing the values of ζ,Λ,κ, and τ by 10% will produce a corresponding 10%,15.9%,6.3%,and5.3% increase or decrease in the basic reproduction number of the model, respectively. On the other hand, the negative value of the sensitivity indices of R0 to the parameters, μ,ω,ψ,φ1,σ, and α, in Table 3 shows that an increase in the value of each of the parameters forces to decrease in the basic reproduction number, while decreasing the values of each parameter leads to an increase in the basic reproduction of the model. Moreover, maximizing the value of ω by 10% will produce 8.9% minimization on the basic reproduction number due to newborn vaccination. Figure 2 also shows that the parameters that have the most influence on the basic reproduction number are birth rate Λ, successful vaccination rate ω, contact rate ζ, and infectiousness rate relative to acute κ. Thus, the identification of these critical parameters is important to the development of an optimal control model of the system (2).Table 3
Sensitivity indices of parameters involved inR0.
ParameterValueSourceSensitivity indicesΛ12Assumed1.46ζ0.24Assumed1τ0.11[22]0.466δ0.3Assumed0.1448κ0–1[23]0.628φ20.03–0.06[24]0.04ω0.4Assumed-0.932μ0.9Assumed-0.257ψ0.5Assumed-0.294ϑ0.025[16]-0.0147φ10.52Assumed-0.356σ0.12Assumed-0.1448α0.72Assumed-0.0196Figure 2
Sensitivity plot.


## 4. Simulation of Autonomous System Analysis

In this section, we perform simulations to illustrate the previous theoretical analysis of model (2). The fourth-order Runge-Kutta method is used to compute the system, and the results are represented graphically. The initial conditions S0,V10,V20,A0,C0,R0=13,9,3,4,5,8 and ε=0.86 are used for the simulation of hepatitis B disease in addition to the parameter values in Table 3. Figure 3 shows that the simulations of different population groups (acute and chronic) of individuals converge to the disease-free equilibrium using different initial conditions, Λ=9 (in this case, R0=0.8249). All other parameters are as in Table 3. These results indicate that hepatitis B disease can be eradicated from the community as long as basic reproduction number R0<1. Figure 4 demonstrates that the trajectories of infectious (acute and chronic) individuals converge to the endemic equilibrium using different initial conditions, and Λ=12 (in this case, R0=1.2069). All other parameters are as in Table 3. These results show that the unique endemic equilibrium point is locally asymptotically stable when the basic reproduction number is near one. The effective contact rate significantly contributes to the spread of the disease by raising the incidence rate of individuals in the infectious population. Figure 5(a) shows that increasing the effective contact rate ζ value will significantly increase the number of susceptible individuals getting infectious, and it results in decreasing the number of individuals in the susceptible class. On the contrary, decreasing the effective contact rate results in decreasing the size of acute and chronic individuals, and vice versa as shown in Figures 5(b) and 5(c), respectively. Thus, decreasing the contact rate of hepatitis B using appropriate mechanisms is better to decrease the spread of hepatitis B disease in the community. Figure 6 demonstrates the simulation of the first series vaccinated, second series vaccinated, chronic, and acute individuals for various values of newborn vaccination rate ω. It is observed that enhancing newborn vaccinations leads to increases and stabilizes the sizes of first series vaccinated and second series vaccinated individuals as shown in Figures 6(a) and 6(b) accordingly but decreases the size of acute and chronic individuals indicated in Figures 6(c) and 6(d), respectively. It leads the health sectors to concentrate on enhancing newborn vaccination to control the spread of hepatitis B disease. The change in first dose vaccinated, second dose vaccinated, and recovered individuals is shown in Figure 7 for various values of second dose vaccine uptake rate α. It detected that as the second dose vaccine uptake rate increases, the second dose vaccinated and recovered individuals in Figures 7(b) and 7(c) increase accordingly, while the first dose vaccinated individuals decrease and stabilize as shown in Figure 7(a), which corresponds to the second dose vaccine uptake rate.Figure 3
Simulation of system (2) showing the total number of HBV infectious individuals as a function of time, using different initial conditions for Λ=9 (in this case, R0=0.8249). All other parameters are as in Table 3.Figure 4
Simulation of system (2) showing the total number of hepatitis B infectious individuals as a function of time, using different initial conditions for Λ=12 (in this case, R0=1.2069). All other parameters are as in Table 3.Figure 5
The impact of effective contact rate on susceptible, acute, and chronic individuals.
(a)(b)(c)Figure 6
The impact of newborn vaccination on first series vaccinated, second series vaccinated, acute, and chronic individuals.
(a)(b)(c)(d)Figure 7
The effect of second dose vaccine uptake rate.
(a)(b)(c)


## 5. Extension of the Model into an Optimal Control

In this section, we incorporate optimal control interventions on model (2); this aided us in determining the most effective intervention options for eradicating the disease in the period allocated. The sensitivity index results from Table 3 are taken into account when deciding the control measures. The first successful intervention which will be efficient to reduce hepatitis B incidence is to minimize the probability of transmission per contact of infectious individuals with susceptible class. This measure will reduce contact rate ζ and infectiousness rate κ. The second intervention measure is to enhance newborn vaccination ω. This strategy will minimize the proportion of unimmunized babies rates τ. To develop the mathematical control model in order to analyze the impact of the mentioned controls on the disease dynamics, we incorporate those time-dependent variables in (2). The first control variable is prevention (avoid direct contact with blood and bodily fluids, use a condom with sexual partners, and avoid sharing sharp items) denoted by u1t. It is assumed that a fraction of the susceptible population are being infectious at a rate of 1−u1λh, and the remaining are still in the susceptible class. Moreover, increase the provision of newborn vaccination denoted by u2t. It suggested that the fraction of newborn babies vaccinated successfully joined the first vaccinated class at a rate of Λω+u2t, those who are unsuccessfully immunized move into chronic class at rate of Λ1−ω+u2tτC, and the remaining are in the susceptible class at a rate of Λ1−ω+u2t1−τC. As a result of incorporating these two control measures into (2), the nonlinear first-order system of the ordinary differential is formed as in (51):
(51)dSdt=Λ1−ω+u21−τC+φ2V1−1−u1λhS−μ+φ1S,dV1dt=φ1S+Λω+u2−α+μ+φ2V1,dV2dt=αV1−ε+μV2,dAdt=1−u1λhS−δ+σ+μA,dCdt=δA+Λ1−ω+u2τC−ϑ+μ+ψC,dRdt=σA+ϑC+εV2−μR.The goal is to diminish the number of infectious individuals and the cost of implementation of measures in that period. Hence, we propose the optimal control problem for the system (51) to diminish the objective functional given as
(52)J=MinU∫0tfl1A+l2C+12w1u12+w2u22dt,where l1,l2,w1, and w2 are positive weight constants. The expressions 1/2w1u12 and 1/2w2u22 represent costs which are associated with the controls u1 and u2, respectively. Our goal is to diminish the number of acute, chronic, and costs of controls. Hence, we seek to find optimal double controls u1∗,u2∗ such that Ju1∗,u2∗=minJu1,u2,ui∈U, where U=u1,u2, such that ui is Lebesgue measurable on 0≤ui≤1 for 0≤t≤tf,i=1,2.Theorem 5.
GivenJu subject to system (51) with S0,V10,V20,A0,C0,R0≥0,0,0,0,0,0, then there exists an optimal control u∗ and corresponding state variables S∗,V1∗,V2∗,A∗,C∗,R∗, which minimizes Ju over U that means
(53)Ju1∗,u2∗=minJu1,u2/u1,u2∈U.The justification of the above theorem is based on the following assumptions given in [25]:
(1)
The state of controls and corresponding state variables are nonempty(2)
The measurable control set is convex and closed(3)
Each right-hand side of the state system is continuous, bounded above by a sum of the bounded control and the state, and can be written as a linear function ofU with coefficients depending on time and the state.(4)
The integrandfx,u of the objective functional is convex on U.(5)
There exist constantsz1,z2>0, and ζ∗>1 such that the integrand of the objective functional satisfies
(54)f≥z1u12+u22ζ∗/2−z2Proof.
(1)
U is a nonempty set of measurable functions on 0≤tf with values in real numbers. The system (51) has bounded coefficients, and thus, any solutions are bounded on 0,tf. Therefore, the corresponding solutions for the system (51) exist with the help of [26].(2)
It is enough to writeU=U1×U2. Therefore, U=U1×U2 is bounded and convex ∀t∈0,tf.(3)
By definition, each right-hand side of system (51) is continuous. All variables S,V1,V2,A,C,R and U are bounded on 0,tf. To prove the boundedness, we use the method by [27]. To end this, we use the fact that the super solutions of system equation (51) given by
(55)dS^dt=Λ1+ω+u2τC^+φ2V^1,dV^1dt=φ1S^+Λω+u2,dV^2dt=αV^1,dA^dt=λhS^,dC^dt=δA^+ΛτC^,dR^dt=σA^+ϑC^+εV^2are bounded on a finite time interval. System (55) can be written as
(56)x=dS^dtdV^1dtdV^2dtdA^dtdC^dtdR^dt=0φ200Λω+u20φ1000000α0000λh00000000δΛτ000εσϑ0S^V^1V^2A^C^R^+ΛΛω+u20000.The system is linear in finite time with bounded coefficients, and therefore, the super solutionsS^,V^1,V^2,A^,C^, and R^ are uniformly bounded. Since the solution to each state equation is bounded, we see that
(57)ft,x,u≤0φ200Λω0φ1000000α0000λh00000000δΛτ000εσϑ0S^V^1V^2A^C^R^+ΛΛω0000+0Λu20000+CΛu200000≤M1x+u2CM2+M3+M4,where M1,M2,M3, and M4 depend on the coefficients of the system. Thus, the assumption holds.(4)
The integrandfx,u of the objective functional is given by
(58)fx,u=l1C+l2I+12w1u12+w2u22.Lett∈0,1, ϖ=ϖ1,ϖ2 and v=v1,v2∈U, and tϖ+1−tv=tϖ1+1−tv1,tϖ+1−tv2. Then, we have
(59)fx,tϖ+1−tv−tfx,ϖ+1−tfx,v=w12t2ϖ12+2t1−tϖ1v1+1−t2v12+w22t2ω22+2t1−tω2v2+1−t2v22−tw12ϖ12+w22ϖ22−1−tw12v12+w22v22=t2−tw12v1−ϖ12+w22v2−ϖ22=t2−t2∑i=13wivi−ϖi2≤0.Hence,fx,tϖ+1−tv≤tfx,ϖ+1−tfx,v. Thus, this assumption is justified.(5)
There exist constantsz1 and z2>0 and ζ∗>1 such that the integrand fx,u of the objective function
(60)fx,u=l1A+l2C+12w1u12+w2u22≥z12u12+u22−z2,where d1=minw1,w2, ζ∗=2,z2>0. Thus, this assumption is explained. Therefore, the optimal control u exists


### 5.1. The Hamiltonian and Optimality System

We used Pontryagin’s maximum principle presented in [28] to compute the necessary conditions for which the objective function would be optimized. Hence, by this principle, we obtained a Hamiltonian H defined as
(61)HS,V1,V2,A,C,R=l1A+l2C+12w1u12+w2u22+λ1dSdt+λ2dV1dt+λ3dV2dt+λ4dAdt+λ5Cdt+λ6dRdt,where λi=1,⋯,6 are the adjoint variables associated with S,V1,V2,A,C, and R to be determined.Theorem 6.
For an optimal controlu1∗ and u2∗ that minimize J over U, there are adjoint variables λ1,⋯,λ6 such that
(62)dλ1dt=1−u1λh+μ+φ1λ1−φ1λ2−1−u1λhλ4,dλ2dt=−φ2λ1+α+μ+φ2λ2−αλ3,dλ3dt=ε+μλ3−ελ6,dλ4dt=1−u1ζκSλ1−1−u1ζκSλ4+δ+σ+μλ4−δλ5−σλ6−l1,dλ5dt=−Λ1−ω+u2τλ1+1−u1ζSλ1−1−u1ζSλ4−Λ1−ω+u2τλ5+μ+ψ+ϑλ5−ϑλ6−l2,dλ6dt=μλ6.With the transversality conditionsλitf=0,i=1,⋯,6. Furthermore, we obtain the control set u1∗,u2∗ characterized by
(63)u1∗t=max0,min1,ϕ1,u2∗t=max0,min1,ϕ2,where
(64)ϕ1=λhSλ4−λhλ1Sw1,ϕ2=ΛτCλ5+Λ1−τCλ1−Λλ2w2.Proof.
The adjoint equations governed by the nonautonomous system (51) are obtained by taking partial derivatives of the Hamiltonian H given by equation (61) with respect to the associated state variables, so that
(65)dλ1dt=−∂H∂S=1−u1λh+μ+φ1λ1−φ1λ2−1−u1λhλ4,dλ2dt=−∂H∂V1=−φ2λ1+α+μ+φ2λ2−αλ3,dλ3dt=−∂H∂V3=ε+μλ3−ελ6,dλ4dt=−∂H∂A=1−u1ζκSλ1−1−u1ζκSλ4+δ+σ+μλ4−δλ5−σλ6−l1,dλ5dt=−∂H∂C=−Λ1−ω+u2τλ1+1−u1ζSλ1−1−u1ζSλ4−Λ1−ω+u2τλ5+μ+ψ+ϑλ5,−ϑλ6−l2,dλ6dt=−∂H∂R=μλ6.
Similarly, to get the controls, we solved the equation∂H/∂ui=0 at ui∗ for i=1,2 and obtain
(66)u1∗=λhSλ4−λhλ1Sw1,u2∗=ΛτCλ5+Λ1−τCλ1−Λλ2w2.
When we write by using standard control arguments involving the bounds on the controls, we conclude:(67)u1∗=ϕ1,if0<ϕ1<1,0,ifϕ1≤0,1,ifϕ1≥1,u2∗=ϕ2,if0<ϕ2<1,0,ifϕ2≤0,1,ifϕ2≥1.
Alternately, in compact form, each optimal control can be written as(68)u1∗t=max0,min1,ϕ1,u2∗t=max0,min1,ϕ2,where
(69)ϕ1=λhSλ4−λhλ1Sw1,ϕ2=ΛτCλ5+Λ1−τCλ1−Λλ2w2.
This completes the proof.The optimality system is formed from the optimal control system (51) and the adjoint variable system (62) by incorporating the characterized control set and initial and transversal conditions as
(70)dSdt=Λ1−ω+u21−τC+φ2V1−1−u1λhS−μ+φ1S,dV1dt=φ1S+Λω+u2−α+μ+φ2V1,dV2dt=αV1−ε+μV2,dAdt=1−u1λhS−δ+σ+μA,dCdt=δA+Λ1−ω+u2τC−ϑ+μ+ψC,dRdt=σA+ϑC+εV2−μR,dλ1dt=1−u1λh+μ+φ1λ1−φ1λ2−1−u1λhλ4,dλ2dt=−φ2λ1+α+μ+φ2λ2−αλ3,dλ3dt=ε+μλ3−ελ6,dλ4dt=1−u1ζκSλ1−1−u1ζκSλ4+δ+σ+μλ4−δλ5−σλ6−l1,dλ5dt=−Λ1−ω+u2τλ1+1−u1ζSλ1−1−u1ζSλ4−Λ1−ω+u2τλ5+μ+ψ+ϑλ5−ϑλ6−l2,dλ6dt=μλ6,u1∗t=max0,min1,λhSλ4−λhλ1Sw1,u2∗t=max0,min1,ΛτCλ5+Λ1−τCλ1−Λλ2w2.With the conditions,λitf=0,i=1,⋯,6,S0=S0,V10=V10,V20=V20,A0=A0,C0=C0,R0=R0.


### 5.2. Simulation of the Optimal Control Analysis

In this section, numerical results of model (51) are discussed to show the effect of different control approaches like avoiding direct contact with blood and bodily fluids, sharing sharp items, and increasing the rate of newborn vaccination. The solution to the optimal control problem was obtained by solving the optimality system of state and adjoint variables through the forward-backward sweep method. Lenhart and Workman [29] gave a rough outline of this method, which is reproduced here for convenience.Step 1.
Make an initial guess foru1&u2 over the interval 0,tf.Step 2.
Using the initial conditionxt0=x0, and the stored values for u1&u2, solve x forward in time (using a 4th-order Runge-Kutta scheme) for system (51).Step 3.
Using the transversality conditionλtf=0, and the stored values for u1,u2&x, solve λ backward in time (using a 4th-order Runge-Kutta scheme) for the system (62).Step 4.
Updateu1&u2 by entering the new x and λ values into the characterization of the optimal controls.Step 5.
If values of the variables in this iteration and the last iteration are negligibly small, output current values as solutions. If values are not small, return to Step2.We assume thatw2>w1; this assumption is based on the fact that the cost associated with a vaccine for hepatitis B disease is costlier than the prevention of acute and chronic individuals from susceptible populations. S0,V10,V20,A0,C0,R0=13,9,3,4,5,8,l1=3,l2=5,w1=0.6,w2=9, and u1max=u2max are used for the simulation of the optimal control model (51) in addition to parameter values in Table 3.


#### 5.2.1. Strategy One: Implementing Preventionu1 Only

Using the control prevention of susceptible individuals from hepatitis B disease, we can minimize the objective functionalJ by setting newborn vaccination intervention to zero. Hence, the number of acute individuals in Figure 8(a) and chronic individuals in Figure 8(b) has decreased significantly as a result of the control strategy used, compared to the numbers of acute and chronic individuals without control. To reach here, the implementation of the control u1 in Figure 8(c) is at maximum capacity for 6.7 months and then drops after month 6.7.Figure 8
The effect of preventionu1 on hepatitis B dynamics.
(a)(b)(c)


#### 5.2.2. Strategy Two: Implementing Newborn Vaccinationu2 Only

In this strategy, we set the prevention intervention equal to zero and consider only a newborn vaccination to illustrate the dynamics of hepatitis B disease. Figures9(a) and 9(b) show that optimized intervention by administering vaccines to newborn babies decreases the number of acute and chronic individuals, but the acceleration of minimization of those infectious individuals is slower than the use of prevention only and the combined use of strategies. Furthermore, Figure 9(c) shows that to accomplish these results, the control u2 is implemented at a maximum rate only for 1.5 months.Figure 9
The effect of newborn vaccinationu2 on hepatitis B dynamics.
(a)(b)(c)


#### 5.2.3. Strategy Three: Implementing Preventionu1 and Newborn Vaccination u2 Controls

Figures10(a) and 10(b) depict the implementation of both the prevention of susceptible individuals from infections as well as the maximization of vaccine administration for newborn babies within twenty-four hours. It demonstrates that when both controls are used, individuals in the acute and chronic classes can be reduced more quickly than when only one intervention is used. To obtain these results, the control profile u2 is executed at its maximum level for only half a month, and the control u1 is also executed at its maximum level for only 3.7 months, before being reduced after half a month and 3.7 months, respectively. Therefore, the optimal control trajectories from the simulation indicate that the combined use of prevention and newborn vaccination helps to eradicate hepatitis B disease in a specified period of time.Figure 10
The impact of combined application of control variablesu1 and u2 on hepatitis B dynamics.
(a)(b)(c)


### 5.3. Cost-Effectiveness Analysis

Through a cost-effectiveness analysis, the cost benefits associated with the implementation of the control intervention can be compared. Consequently, adopting the methodology employed in numerous earlier investigations [30–32], the incremental cost-effectiveness ratio (ICER) is calculated to identify the less costly but more effective intervention among the listed strategies considered in this study. Most frequently, ICER is used to compare the costs and health advantages of any two alternative control intervention techniques, i and j, which are competing for the same scarce resources. The ICER is defined mathematically as
(71)ICER=DifferenceincostproducedbystrategiesiandjDifferenceinthetotalnumberofinfectionsavertedinstrategiesiandj,where the total infections averted is the difference between the total numbers of infections (acute and chronic) without controls and with controls during the infectious period. Moreover, the total cost (Tc) for each strategy obtained from Figures 11(a)–11(c) during the period is given by
(72)Tc=12∫0tfw1u12t+w2u22tdt.Figure 11
Cost functions.
(a)(b)(c)To analyze the cost-effectiveness of the strategies using ICER, we primarily ranked the control strategies in increasing order of the total number of infections averted as presented in Table4. Thus, we first compared the ICER value for strategies one and two:
(73)ICERone=23.3167259=0.09,ICERtwo=112.6487−23.3167268−259=9.9.Table 4
Incremental cost-effective ratio for the most effective strategies.
StrategiesTotal infections avertedTotal cost ($)ICERStrategy one25923.31670.09Strategy two268112.64879.9Strategy three31673.38790.88From Table4, we observed that the value of the ICER of strategy two is greater than the ICER of strategy one. As a result, strategy two is more expensive and less effective than strategy one. So, strategy two is eliminated from the list of control interventions; then, we compared the ICER of strategies one and three as follows:
(74)ICERone=23.3167259=0.09,ICERthree=73.3879−23.3167316−259=0.88.Obviously, we find that strategy one has the lowest ICER value. Therefore, we conclude that strategy one (avoid direct contact with blood and bodily fluids, use condoms with sexual partners, and avoid sharing sharp items) is the most cost-saving and effective control intervention among the listed strategies.


## 5.1. The Hamiltonian and Optimality System

We used Pontryagin’s maximum principle presented in [28] to compute the necessary conditions for which the objective function would be optimized. Hence, by this principle, we obtained a Hamiltonian H defined as
(61)HS,V1,V2,A,C,R=l1A+l2C+12w1u12+w2u22+λ1dSdt+λ2dV1dt+λ3dV2dt+λ4dAdt+λ5Cdt+λ6dRdt,where λi=1,⋯,6 are the adjoint variables associated with S,V1,V2,A,C, and R to be determined.Theorem 6.
For an optimal controlu1∗ and u2∗ that minimize J over U, there are adjoint variables λ1,⋯,λ6 such that
(62)dλ1dt=1−u1λh+μ+φ1λ1−φ1λ2−1−u1λhλ4,dλ2dt=−φ2λ1+α+μ+φ2λ2−αλ3,dλ3dt=ε+μλ3−ελ6,dλ4dt=1−u1ζκSλ1−1−u1ζκSλ4+δ+σ+μλ4−δλ5−σλ6−l1,dλ5dt=−Λ1−ω+u2τλ1+1−u1ζSλ1−1−u1ζSλ4−Λ1−ω+u2τλ5+μ+ψ+ϑλ5−ϑλ6−l2,dλ6dt=μλ6.With the transversality conditionsλitf=0,i=1,⋯,6. Furthermore, we obtain the control set u1∗,u2∗ characterized by
(63)u1∗t=max0,min1,ϕ1,u2∗t=max0,min1,ϕ2,where
(64)ϕ1=λhSλ4−λhλ1Sw1,ϕ2=ΛτCλ5+Λ1−τCλ1−Λλ2w2.Proof.
The adjoint equations governed by the nonautonomous system (51) are obtained by taking partial derivatives of the Hamiltonian H given by equation (61) with respect to the associated state variables, so that
(65)dλ1dt=−∂H∂S=1−u1λh+μ+φ1λ1−φ1λ2−1−u1λhλ4,dλ2dt=−∂H∂V1=−φ2λ1+α+μ+φ2λ2−αλ3,dλ3dt=−∂H∂V3=ε+μλ3−ελ6,dλ4dt=−∂H∂A=1−u1ζκSλ1−1−u1ζκSλ4+δ+σ+μλ4−δλ5−σλ6−l1,dλ5dt=−∂H∂C=−Λ1−ω+u2τλ1+1−u1ζSλ1−1−u1ζSλ4−Λ1−ω+u2τλ5+μ+ψ+ϑλ5,−ϑλ6−l2,dλ6dt=−∂H∂R=μλ6.
Similarly, to get the controls, we solved the equation∂H/∂ui=0 at ui∗ for i=1,2 and obtain
(66)u1∗=λhSλ4−λhλ1Sw1,u2∗=ΛτCλ5+Λ1−τCλ1−Λλ2w2.
When we write by using standard control arguments involving the bounds on the controls, we conclude:(67)u1∗=ϕ1,if0<ϕ1<1,0,ifϕ1≤0,1,ifϕ1≥1,u2∗=ϕ2,if0<ϕ2<1,0,ifϕ2≤0,1,ifϕ2≥1.
Alternately, in compact form, each optimal control can be written as(68)u1∗t=max0,min1,ϕ1,u2∗t=max0,min1,ϕ2,where
(69)ϕ1=λhSλ4−λhλ1Sw1,ϕ2=ΛτCλ5+Λ1−τCλ1−Λλ2w2.
This completes the proof.The optimality system is formed from the optimal control system (51) and the adjoint variable system (62) by incorporating the characterized control set and initial and transversal conditions as
(70)dSdt=Λ1−ω+u21−τC+φ2V1−1−u1λhS−μ+φ1S,dV1dt=φ1S+Λω+u2−α+μ+φ2V1,dV2dt=αV1−ε+μV2,dAdt=1−u1λhS−δ+σ+μA,dCdt=δA+Λ1−ω+u2τC−ϑ+μ+ψC,dRdt=σA+ϑC+εV2−μR,dλ1dt=1−u1λh+μ+φ1λ1−φ1λ2−1−u1λhλ4,dλ2dt=−φ2λ1+α+μ+φ2λ2−αλ3,dλ3dt=ε+μλ3−ελ6,dλ4dt=1−u1ζκSλ1−1−u1ζκSλ4+δ+σ+μλ4−δλ5−σλ6−l1,dλ5dt=−Λ1−ω+u2τλ1+1−u1ζSλ1−1−u1ζSλ4−Λ1−ω+u2τλ5+μ+ψ+ϑλ5−ϑλ6−l2,dλ6dt=μλ6,u1∗t=max0,min1,λhSλ4−λhλ1Sw1,u2∗t=max0,min1,ΛτCλ5+Λ1−τCλ1−Λλ2w2.With the conditions,λitf=0,i=1,⋯,6,S0=S0,V10=V10,V20=V20,A0=A0,C0=C0,R0=R0.


## 5.2. Simulation of the Optimal Control Analysis

In this section, numerical results of model (51) are discussed to show the effect of different control approaches like avoiding direct contact with blood and bodily fluids, sharing sharp items, and increasing the rate of newborn vaccination. The solution to the optimal control problem was obtained by solving the optimality system of state and adjoint variables through the forward-backward sweep method. Lenhart and Workman [29] gave a rough outline of this method, which is reproduced here for convenience.Step 1.
Make an initial guess foru1&u2 over the interval 0,tf.Step 2.
Using the initial conditionxt0=x0, and the stored values for u1&u2, solve x forward in time (using a 4th-order Runge-Kutta scheme) for system (51).Step 3.
Using the transversality conditionλtf=0, and the stored values for u1,u2&x, solve λ backward in time (using a 4th-order Runge-Kutta scheme) for the system (62).Step 4.
Updateu1&u2 by entering the new x and λ values into the characterization of the optimal controls.Step 5.
If values of the variables in this iteration and the last iteration are negligibly small, output current values as solutions. If values are not small, return to Step2.We assume thatw2>w1; this assumption is based on the fact that the cost associated with a vaccine for hepatitis B disease is costlier than the prevention of acute and chronic individuals from susceptible populations. S0,V10,V20,A0,C0,R0=13,9,3,4,5,8,l1=3,l2=5,w1=0.6,w2=9, and u1max=u2max are used for the simulation of the optimal control model (51) in addition to parameter values in Table 3.


### 5.2.1. Strategy One: Implementing Preventionu1 Only

Using the control prevention of susceptible individuals from hepatitis B disease, we can minimize the objective functionalJ by setting newborn vaccination intervention to zero. Hence, the number of acute individuals in Figure 8(a) and chronic individuals in Figure 8(b) has decreased significantly as a result of the control strategy used, compared to the numbers of acute and chronic individuals without control. To reach here, the implementation of the control u1 in Figure 8(c) is at maximum capacity for 6.7 months and then drops after month 6.7.Figure 8
The effect of preventionu1 on hepatitis B dynamics.
(a)(b)(c)


### 5.2.2. Strategy Two: Implementing Newborn Vaccinationu2 Only

In this strategy, we set the prevention intervention equal to zero and consider only a newborn vaccination to illustrate the dynamics of hepatitis B disease. Figures9(a) and 9(b) show that optimized intervention by administering vaccines to newborn babies decreases the number of acute and chronic individuals, but the acceleration of minimization of those infectious individuals is slower than the use of prevention only and the combined use of strategies. Furthermore, Figure 9(c) shows that to accomplish these results, the control u2 is implemented at a maximum rate only for 1.5 months.Figure 9
The effect of newborn vaccinationu2 on hepatitis B dynamics.
(a)(b)(c)


### 5.2.3. Strategy Three: Implementing Preventionu1 and Newborn Vaccination u2 Controls

Figures10(a) and 10(b) depict the implementation of both the prevention of susceptible individuals from infections as well as the maximization of vaccine administration for newborn babies within twenty-four hours. It demonstrates that when both controls are used, individuals in the acute and chronic classes can be reduced more quickly than when only one intervention is used. To obtain these results, the control profile u2 is executed at its maximum level for only half a month, and the control u1 is also executed at its maximum level for only 3.7 months, before being reduced after half a month and 3.7 months, respectively. Therefore, the optimal control trajectories from the simulation indicate that the combined use of prevention and newborn vaccination helps to eradicate hepatitis B disease in a specified period of time.Figure 10
The impact of combined application of control variablesu1 and u2 on hepatitis B dynamics.
(a)(b)(c)


## 5.2.1. Strategy One: Implementing Preventionu1 Only

Using the control prevention of susceptible individuals from hepatitis B disease, we can minimize the objective functionalJ by setting newborn vaccination intervention to zero. Hence, the number of acute individuals in Figure 8(a) and chronic individuals in Figure 8(b) has decreased significantly as a result of the control strategy used, compared to the numbers of acute and chronic individuals without control. To reach here, the implementation of the control u1 in Figure 8(c) is at maximum capacity for 6.7 months and then drops after month 6.7.Figure 8
The effect of preventionu1 on hepatitis B dynamics.
(a)(b)(c)


## 5.2.2. Strategy Two: Implementing Newborn Vaccinationu2 Only

In this strategy, we set the prevention intervention equal to zero and consider only a newborn vaccination to illustrate the dynamics of hepatitis B disease. Figures9(a) and 9(b) show that optimized intervention by administering vaccines to newborn babies decreases the number of acute and chronic individuals, but the acceleration of minimization of those infectious individuals is slower than the use of prevention only and the combined use of strategies. Furthermore, Figure 9(c) shows that to accomplish these results, the control u2 is implemented at a maximum rate only for 1.5 months.Figure 9
The effect of newborn vaccinationu2 on hepatitis B dynamics.
(a)(b)(c)


## 5.2.3. Strategy Three: Implementing Preventionu1 and Newborn Vaccination u2 Controls

Figures10(a) and 10(b) depict the implementation of both the prevention of susceptible individuals from infections as well as the maximization of vaccine administration for newborn babies within twenty-four hours. It demonstrates that when both controls are used, individuals in the acute and chronic classes can be reduced more quickly than when only one intervention is used. To obtain these results, the control profile u2 is executed at its maximum level for only half a month, and the control u1 is also executed at its maximum level for only 3.7 months, before being reduced after half a month and 3.7 months, respectively. Therefore, the optimal control trajectories from the simulation indicate that the combined use of prevention and newborn vaccination helps to eradicate hepatitis B disease in a specified period of time.Figure 10
The impact of combined application of control variablesu1 and u2 on hepatitis B dynamics.
(a)(b)(c)


## 5.3. Cost-Effectiveness Analysis

Through a cost-effectiveness analysis, the cost benefits associated with the implementation of the control intervention can be compared. Consequently, adopting the methodology employed in numerous earlier investigations [30–32], the incremental cost-effectiveness ratio (ICER) is calculated to identify the less costly but more effective intervention among the listed strategies considered in this study. Most frequently, ICER is used to compare the costs and health advantages of any two alternative control intervention techniques, i and j, which are competing for the same scarce resources. The ICER is defined mathematically as
(71)ICER=DifferenceincostproducedbystrategiesiandjDifferenceinthetotalnumberofinfectionsavertedinstrategiesiandj,where the total infections averted is the difference between the total numbers of infections (acute and chronic) without controls and with controls during the infectious period. Moreover, the total cost (Tc) for each strategy obtained from Figures 11(a)–11(c) during the period is given by
(72)Tc=12∫0tfw1u12t+w2u22tdt.Figure 11
Cost functions.
(a)(b)(c)To analyze the cost-effectiveness of the strategies using ICER, we primarily ranked the control strategies in increasing order of the total number of infections averted as presented in Table4. Thus, we first compared the ICER value for strategies one and two:
(73)ICERone=23.3167259=0.09,ICERtwo=112.6487−23.3167268−259=9.9.Table 4
Incremental cost-effective ratio for the most effective strategies.
StrategiesTotal infections avertedTotal cost ($)ICERStrategy one25923.31670.09Strategy two268112.64879.9Strategy three31673.38790.88From Table4, we observed that the value of the ICER of strategy two is greater than the ICER of strategy one. As a result, strategy two is more expensive and less effective than strategy one. So, strategy two is eliminated from the list of control interventions; then, we compared the ICER of strategies one and three as follows:
(74)ICERone=23.3167259=0.09,ICERthree=73.3879−23.3167316−259=0.88.Obviously, we find that strategy one has the lowest ICER value. Therefore, we conclude that strategy one (avoid direct contact with blood and bodily fluids, use condoms with sexual partners, and avoid sharing sharp items) is the most cost-saving and effective control intervention among the listed strategies.


## 6. Conclusions

In this paper, a deterministic compartmental hepatitis B epidemic model with a two-shot vaccination series has been suggested and studied. As confirmed in Figures3 and 4, respectively, hepatitis B disease is eradicated if R0<1; otherwise, the disease persists in the community. From numerical findings, we found that as contact rates decrease (see Figures 5(a) and 5(b)) and newborn vaccine rates increase (see Figures 6(c) and 6(d)), populations of infectious individuals (both acute and chronic) decrease. Additionally, Figure 7(c) shows that the number of recovery individuals increases as the second dose uptake rate increases. We also extended the model to an optimal control model by incorporating two time-dependent control variables based on the result of a sensitivity analysis. We solved the optimality system numerically using a forward-backward fourth-order Runge-Kutta scheme in an attempt to understand how to eradicate hepatitis B disease from the community using different strategies in the most cost-effective way. With 100% use of prevention efforts for 6.7 months, the total cost of prevention to prevent susceptible individuals was $23.3167. Thus, strategy one (the use of only prevention efforts) is the most cost-saving and effective control strategy to eradicate hepatitis B disease in a resource-limited environment, even though, in terms of the infection averted, we saw that strategies two and three were better than strategy one.



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*Source: 5215494-2023-02-08.xml*