Int. J. Anal. Appl. (2022), 20:53 Mathematical Model of Hepatitis B Virus With Effect of Vaccination and Treatments Saif H. Elkhadir1,∗, Ali E. M. Saeed2, Abdelfatah Abasher3 1Mathematics Department, Omdurman Islamic University, Sudan 2Mathematics Department, Alzaiem Alazhari University, Sudan 3Mathematics Department, Jazan University, Saudi Arabia ∗Corresponding author: safemath@gmail.com Abstract. In this paper, a mathematical model of hepatitis B virus with vaccination and treatments is studied, Stability analysis discussed and the disease-free equilibrium and endemic equilibrium points obtained, the basic reproductive number R0 determined and became the threshold for equilibrium points stability. The study showed when R0 < 1 the disease-free equilibrium point was stable, whereas R0 > 1 the virus is endemic and the endemic equilibrium point is stable. The sensitivity analysis for the parameters that could reduce the spread of hepatitis B virus is studied. Finally the numerical simulation are established by using SageMath software package to show the effect of vaccination and treatments. We found that vaccination and also treatments give an effect on value of R0. Increasing the value of the vaccine in the immunized compartment or in the suspected compartment may decrease the value of R0 which mean reduce the spread of the disease. 1. Introduction Infectious diseases are disorders caused by small organisms such as bacteria, viruses, fungi or para- sites. Some infectious diseases transmitted from infected person to another by direct physical contact, airborne droplets, water or food, disease vectors, or mother to newborn. Hepatitis B is an infectious disease of the liver caused by the hepatitis B virus (HBV). In some people hepatitis B can become chronic leading to liver failure, liver cancer or cirrhosis. Most healthy adults who are newly infected will recover without any problems. But babies and young children may not be able to successfully get rid of the virus. 90% of healthy adults will get rid of the virus and recover without any problems; 10% will develop chronic hepatitis B. Young Children – Up to 50% of young children between 1 and 5 years Received: Aug. 31, 2022. 2010 Mathematics Subject Classification. 92C60. Key words and phrases. epidemic disease; HBV model; hepatitis B virus. https://doi.org/10.28924/2291-8639-20-2022-53 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-53 2 Int. J. Anal. Appl. (2022), 20:53 who are infected will develop a chronic hepatitis B infection. Infants – 90% will become chronically infected; only 10% will be able to get rid of the virus [1]. Mathematical Model of Hepatitis B virus based on MSEIR model, the population divided into five compartments Immunized(M), Susceptible (S), Exposed (E), Infected (I) and Recovered (R). There are many previous studies on the spread of hepatitis B, Zhang (2015) study the application and optimal control for an HBV model with vaccination and treatments [2], Aniji (2019) developed the model by adding the Carrier compartment [3], Beay (2017) developed the model by adding migration compart- ments [4], Wiraningsih (2015) discussed the model with vaccination effect [5], cao (2015) studied the Global stability of an epidemic model with carrier state in heterogeneous network [6]. In this paper we use SEIR model and apply it to hepatitis B virus. We extend the model by introducing the immunized class. The paper outline as follows, in section 2, we present the description and assumption of the model, in section 3 we discus the stability analysis for disease free equilibrium and endemic equilibrium, in section 4 we show the relation between the basic reproductive number and the model parameters, in section 5 the explore the numerical simulation to show the dynamical behaviour of our results, we finished the paper with a conclusion in section 6. 2. Description of the model The Population of our model is divided into five compartments: Immunized (M), Susceptible (S), Exposed (E), Infected (I) and Recovered (R). The interaction between the five compartments is shown in the following diagram: Figure 1. The transmission of hepatitis B virus model The basic assumptions of the model: • birth rate equal to the death rate. • The population size is constant N. • The rate of transmission from exposed to infective is η. • The rate of transmission from infected to recovered compartment is ψ. • The rate of transmission from susceptible to infected class is ν. • The individual who take vaccine transfer from Susceptible class to Recovered class with rate δ. Int. J. Anal. Appl. (2022), 20:53 3 • The individual who lost immune transfer from Recovered class to Susceptible class with rate γ The differential equations that govern the model are dM dt = αB −φM −µM dS dt =(1−α)B +φM +γR− (νI +µ+δ)S dE dt = νSI −ηE −µE (1) dI dt = ηE −ψI −βI −µI dR dt = δS +ψI −γR−µR where, M +S +E + I +R = N (total population). The description of the parameters are shown in the following table Parameter Description α Immunized newborns β HB induced mortality γ Rate of transmission from R to S µ Rate of natural mortality ν Rate of transmission from S to E η Rate of transmission from E to I δ Rate of transmission from S to R ψ Rate of transmission from I to R φ Rate expiration of vaccine efficacy By assumption that the total population is constant, (the birth rate = the death rate) then the model can be expressed in a simpler term in the form dM dt = αµ−φM −µM dS dt =(1−α)µ+φM +γR− (νI +µ+δ)S dE dt = νSI −ηE −µE (2) dI dt = ηE −ψI −βI −µI dR dt = δS +ψI −γR−µR 4 Int. J. Anal. Appl. (2022), 20:53 3. Stability Analysis 3.1. Equilibrium Solutions: Let E(M,S,E,I,R) be the equilibrium point of the system (2), at the equilibrium state, we have dM dt = dS dt = dE dt = dI dt = dR dt =0 that gives, αµ−φM −µM =0 (1−α)µ+φM +γR− (νI +µ+δ)S =0 νSI −ηE −µE =0 (3) ηE −ψI −βI −µI =0 δS +ψI −γR−µR =0 3.2. Disease-Free Equilibrium DFE:. Let E0(M0,S0,E0, I0,R0) be a solution of the system (3), suppose both I and E must be zero, (I0 = 0,E0 = 0). By substitute (I = 0,E = 0) into the system (3) we get αµ− (φ+µ)M =0 (1−α)µ+φM +γR− (ν +δ)S =0 (4) δS +−(γ +µ)R =0 By solving these equations we get M0 = αµ φ+µ , R0 = δ(φ+(1−α)µ) φγ +δφ+µφ+µγ +µδ +µ2 , and S0 = φγ +µγ +αµγ +φµ+µ2 −αµ2 φγ +δφ+µφ+µγ +µδ +µ2 therefore the disease - free equilibrium of the model is E0 ( αµ φ+µ , φγ +µγ +αµγ +φµ+µ2 −αµ2 φγ +δφ+µφ+µγ +µδ +µ2 ,0,0, δ(φ+(1−α)µ) φγ +δφ+µφ+µγ +µδ +µ2 ) Int. J. Anal. Appl. (2022), 20:53 5 3.3. Basic reproductive number (R0) for DFE:. According to calculation method of R0 [12], Let X = [E,I]T and set F = ( νSI 0 ) , V = ( (η +µ)E −ηE +(ψ +µ+β)I ) then we can write dX dt =F −V, Compute F and V where, F =   ∂F1 ∂E ∂F1 ∂I ∂F2 ∂E ∂F2 ∂I   E0 = ( 0 νS0 0 0 ) , V =   ∂V1 ∂E ∂V1 ∂I ∂V2 ∂E ∂V2 ∂I   E0 = ( η +µ 0 −η ψ +µ+β ) , the next generation matrix K = FV −1 K = 1 (η +µ)(ψ +µ+β) ( 0 νS0 0 0 )( ψ +µ+β 0 η η +µ ) , K = ( νηS0 (η+µ)(ψ+µ+β) νS0 (ψ+µ+β) 0 0 ) The eigen values of K are λ1 =0 and λ2 = νηS0 (η +µ)(ψ +µ+β) = νη(φγ +µγ +αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +µ+β)(φγ +δφ+µφ+µγ +µδ +µ2) , Hence, R0 = ρ(K)= νη(φγ +µγ +αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +µ+β)(φγ +δφ+µφ+µγ +µδ +µ2) , (5) Theorem 3.1. The disease free-equilibrium point E0 of the system (2) is asymptotically stable if R0 < 1 and unstable if R0 > 1. The value of R0 measures whether the disease will spread and become endemic or will disappear from the population. When R0 < 1, the disease will disappear and the exposed and infected compartments tend to zero as time goes on. When R0 > 1, the disease will spread and become endemic. This means that each compartments will be positive valued for a long time. In another words, in the case of R0 > 1 the endemic equilibrium point E∗ exists and stable. 6 Int. J. Anal. Appl. (2022), 20:53 3.4. Stability Analysis of DFE:. we examine the behavior of the model near E0, linearize the system near E0, suppose the left hand side of system (2) be F1, . . . ,Fn respectively, then the Jacobin matrix is given as J ( F1,F2,F3,F4,F5 M,S,E,I,R ) =  −(φ+µ) 0 0 0 0 φ −(νI +µ+δ) 0 −νS γ 0 νI −(η +µ) νS 0 0 0 η −(ψ +β +µ) 0 0 δ 0 ψ −(γ +µ)   the Jacobian matrix of the disease-free equilibrium is given by: J(E0)=   −(φ+µ) 0 0 0 0 φ −(µ+δ) 0 −νS0 γ 0 νI −(η +µ) νS0 0 0 0 η −(ψ +β +µ) 0 0 δ 0 ψ −(γ +µ)   now let J(E0)= [ J0 J2 J1 J3 ] , where J0 = [ −(φ+µ) 0 φ −(µ+δ) ] ,J2 = [ 0 0 0 0 −νS0 γ ] J1 =   o 0 0 0 0 δ   and J3 =   −(η +µ) νS0 0 η −(ψ +β +µ) 0 0 ψ −(γ +µ)   The eigen values given by: |J(E0)−λ|=0 implies that, ∣∣∣∣∣ −(φ+µ)−λ 0φ −(µ+δ)−λ ∣∣∣∣∣ =0 (6) and ∣∣∣∣∣∣∣∣ −(η +µ)−λ νS0 0 η −(ψ +β +µ)−λ 0 0 ψ −(γ +µ)−λ ∣∣∣∣∣∣∣∣ =0 (7) from (6) we obtain λ1 =−φ−µ, Int. J. Anal. Appl. (2022), 20:53 7 λ2 =−µ−δ, and from (7), λ3 =−γ −µ, (η +µ+λ)(ψ +β +µ+λ)−νηS0 =0, that yields λ2 +(η +ψ +β +2µ)λ+(η +µ)(ψ +β +µ)−νηS0 =0, or, λ2 +ω1λ+ω2 =0 (8) where, ω1 = η +ψ +β +2µ, and ω2 =(η +µ)(ψ +β +µ)−νηS0 from Routh-Hurwitz criterion [13], equation (8) have negative real part if and only if: ω1 > 0 and ω2 > 0, since all the parameters η,ψ,β,µ are positive then ω1 > 0, and the condition (ω2 > 0) yields (η +µ)(ψ +β +µ) > νηS0 It can be seen that all the eigenvalues have negative real parts and therefore the disease-free equilibrium is Locally Asymptotically Stable. 3.5. Stability Analysis of Endemic Equilibrium: Denote the endemic equilibrium of the system (2) is E∗ =(M∗,S∗,E∗, I∗,R∗), which can obtain by putting dM dt = dS dt = dE dt = dI dt = dR dt =0 or, αµ−φM −µM =0 (1−α)µ+φM +γR− (νI +µ+δ)S =0 νSI −ηE −µE =0 (9) ηE −ψI −βI −µI =0 δS +ψI −γR−µR =0 by solving these equations we obtain, M∗ = αµ (φ+µ) = σ0, E∗ = (ψ +µ+β) η I = σ1I, S∗ = (η +µ)σ1 ν = σ2, 8 Int. J. Anal. Appl. (2022), 20:53 R∗ = ψ (γ +µ) I∗ + δσ2 (γ +µ) = I∗σ3 +σ4, and I∗ = (µ+δ)σ2 −γσ4 − (1−α)µ−φσ0 (σ3γ −νσ2) where, σ0 = αµ (φ+µ) ,σ1 = (ψ +µ+β) η ,σ2 = (η +µ)σ1 ν ,σ3 = ψ (γ +µ) and σ4 = δσ2 (γ +µ) the Jacobian matrix of the system (4.2) at E∗ J(E∗)=   −(φ+µ) 0 0 0 0 φ −(νI∗ +µ+δ) 0 −νS∗ γ 0 νI∗ −(η +µ) νS∗ 0 0 0 η −(ψ +β +µ) 0 0 δ 0 ψ −(γ +µ)   now let J(E0)= [ J0 J2 J1 J3 ] , where J0 = [ −(φ+µ) 0 φ −(νI∗ +µ+δ) ] ,J2 = [ 0 0 0 0 −νS∗ γ ] J1 =   0 νI∗ 0 0 0 δ   and J3 =   −(η +µ) νS∗ 0 η −(ψ +β +µ) 0 0 ψ −(γ +µ)   The eigen values given by: |J(E∗)−λ|=0 that implies that ∣∣∣∣∣ −(φ+µ)−λ 0φ −(νI∗ +µ+δ)−λ ∣∣∣∣∣ =0 (10) and ∣∣∣∣∣∣∣∣ −(η +µ)−λ νS∗ 0 η −(ψ +β +µ)−λ 0 0 ψ −(γ +µ)−λ ∣∣∣∣∣∣∣∣ =0 (11) from (10) we obtain λ1 =−φ−µ, and λ2 =−(νI∗ +µ+δ), since I∗ is positive implies that λ1,λ2 are negative. and from (11), −(η +µ+λ)[(ψ +β +µ+λ)(γ +µ+λ)]−νS∗(γ +µ+λ)η =0 Int. J. Anal. Appl. (2022), 20:53 9 or, (γ +µ+λ)[(η +µ+λ)(ψ +β +µ+λ)+νηS∗] = 0, that yields , λ3 =−γ −µ, and λ4,5 are given from λ2 +ω1λ+ω2 =0, where ω1 = η +2µ+ψ +β ω2 =(η +µ)(ψ +β +µ)−νηS∗ =0 thus, λ2 +ω1λ =0, λ4 =0, λ5 =−ω1 It can be seen that all the eigenvalues have negative real parts and therefore the endemic equilibrium is locally asymptotically stable. 4. Sensitivity Analysis of R0 In this section, we show the relation between the basic reproductive number R0 and the parameters that can reduce the spread of the disease α,δ and ψ. the value of the parameters are shown in the table 1 Parameter Description Value References α Immunized newborns 0.05-1 [10] β HBV induced mortality 0.015 [10] γ Rate of transmission from R to S 0.06 [10] µ Rate of natural mortality 0.0121 [10] ν Rate of transmission from S to E 0.08 Assume η Rate of transmission from E to I 0.75 Assume δ Rate of transmission from S to R 0.1-1 [10] ψ Rate of transmission from I to R 0.1-1 [10] φ Rate expiration of vaccine efficacy 0.05 [10] Table 1. Parameters Values of the model 10 Int. J. Anal. Appl. (2022), 20:53 4.1. The relation between R0 and α: We can write the basic reproductive number as R0 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +β +µ)(φγ +δφ+µφ+µγ +µδ +µ2) = K1 −αK2 where, K1 = νη(φγ+µγ+φµ+µ2) (η+µ)(ψ+β+µ)(φγ+δφ+µφ+µγ+µδ+µ2) and K2 = (µγ+µ2)νη (η+µ)(ψ+β+µ)(φγ+δφ+µφ+µγ+µδ+µ2) Suppose the values of the parameters are given as β = 0.015,γ = 0.07,µ = 0.0125,ν = 0.08,η = 0.75,δ = 0.02,ψ =0.03 and φ =0.05. Table 2 and figure 2 show the relation between α and R0 for six values of α namely α = 0.15,α = 0.25,α =0.35,α =0.45,α =0.55, and α =0.65. Parameter α R0 Free/Endemic 0.15 1.068428 Endemic 0.25 1.046399 Endemic 0.35 1.024369 Endemic 0.45 1.00234 Endemic 0.55 0.98031 Free 0.65 0.958281 Free Table 2. The relation between α and R0 Figure 2. The relation between α and R0 4.2. The relation between R0 and δ: We can write the basic reproductive number as R0 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +β +µ)(φγ +δφ+µφ+µγ +µδ +µ2) = C1 C2 +δC3 where, C1 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2 C2 =(η +µ)(ψ +β +µ)(φγ +µφ+µγ +µ 2) C3 =(η +µ)(ψ +β +µ)(φ+µ). Suppose that the values of parameter are given as α = 0.05,β = 0.015,γ = 0.07,µ = 0.0125,ν = 0.08,η = 0.75,ψ = 0.03 and φ = 0.05. Table 3 and figure 3 show the effect of immunization for adults to the value of R0. For six values of δ, namely δ = 0.01,δ = 0.05,δ = 0.1,δ = 0.15,δ = 0.2, and δ =0.25. Int. J. Anal. Appl. (2022), 20:53 11 Parameter δ R0 Free/Endemic 0.01 2.265647 Endemic 0.05 1.581678 Endemic 0.1 1.148342 Endemic 0.15 0.901386 Free 0.2 0.741849 Free 0.25 0.630293 Free Table 3. The relation between δ and R0 Figure 3. The relation between δ and R0 4.3. The relation between R0 and ψ: We can write the basic reproductive number as R0 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +β +µ)(φγ +δφ+µφ+µγ +µδ +µ2) = A1 (A2 +A3ψ) where, A1 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2 A2 =(η +µ)(φγ +δφ+µφ+µγ +µδ +µ 2)(β +µ) A3 =(η +µ)(φγ +δφ+µφ+µγ +µδ +µ 2) we suppose that the values of parameter are given as α =0.05,β =0.015,γ =0.07,µ =0.0125,ν = 0.08,η =0.75,δ =0.02 and φ =0.05. Table 4 and figure 4 show the effect of treatments on infected individuals(ψ) to the value of R0. For six values of ψ, namely ψ =0.01,ψ =0.05,ψ =0.1,ψ =0.15,ψ =0.2, and ψ =0.25. Parameter ψ R0 Free/Endemic 0.01 2.42524 Endemic 0.05 1.173503 Endemic 0.1 0.713306 Free 0.15 0.512375 Free 0.2 0.399765 Free 0.25 0.327735 Free Table 4. The relation between ψ and R0 Figure 4. The relation between ψ and R0 12 Int. J. Anal. Appl. (2022), 20:53 5. Results and Discussion 5.1. Effect of immunization of newborns (α). We suppose that the values of parameter of the model are given as β=0.015, δ=0.05, γ=0.09, µ=0.0125 φ=0.05, ν=0.08 ,η=0.55 ,ψ=0.3. There are five values of α, 0.05, 0.25, 0.45, 0.65, 0.85 and 1. The figures 5 show the effect of increasing value of α gives an increasing in immunized and recovered curves and a decreasing in infectious curve. (a) The population compartments α = 0.05 (b) The population compartments α = 0.25 (c) The population compartments α = 0.45 (d) The population compartments α = 0.65 (e) The population compartments α = 0.85 (f) The population compartments α = 1 Figure 5. The variation of the parameter α Int. J. Anal. Appl. (2022), 20:53 13 5.2. Effect of immunization of adults (δ). We suppose that the values of the parameters are given as α=0.1,β=0.015, γ=0.09, µ=0.0125, φ=0.07, ν=0.08 ,η=0.5 and ψ=0.3, There are five values of δ are 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6. The figures 6 show the effect of increasing value of δ gives an increasing in immunized and recovered curves and a decreasing in infectious curve. (a) The population compartments δ =0.1 (b) The population compartments δ =0.2 (c) The population compartments δ =0.3 (d) The population compartments δ =0.4 (e) The population compartments δ =0.5 (f) The population compartments δ =0.6 Figure 6. The variation of the parameter δ 14 Int. J. Anal. Appl. (2022), 20:53 5.3. Effect of treatment on infected to the spread of hepatitis B virus (ψ). We suppose that the values of parameter of the model are given as α = 0.1,β = 0.015, ,γ = 0.09,µ = 0.0125,φ = 0.07,ν =0.08,η =0.65 and δ =0.05, There are five values of ψ, are ψ =0.1,ψ =0.2,ψ =0.3,ψ = 0.4,ψ = 0.5 and ψ = 0.6. The figure 7 show the effect of increasing value of ψ gives an increasing in immunized and recovered curves and a decreasing in infectious curve. (a) The population compartments ψ =0.1 (b) The population compartments ψ =0.2 (c) The population compartments ψ =0.3 (d) The population compartments ψ =0.4 (e) The population compartments ψ =0.5 (f) The population compartments ψ =0.6 Figure 7. The variation of the parameter ψ Int. J. Anal. Appl. (2022), 20:53 15 6. Conclusion In the model of hepatitis B virus using the standard SEIR model has been developed by considering vaccination and treatments in compartments. This strategy aims to reduce the spread of hepatitis B virus. The model was divided into five compartments which include immunized, exposed, infected, and recovered compartments. The model has two cases free - equilibrium point and endemic - equilibrium point. The existence of endemic equilibrium point depends on the value of reproductive number R0. From the table 2 and table 3 we notice that increasing the value of α and δ leads to decreasing in the value of R0, which its becomes from value greater than one to value less than one, that means giving more vaccines for newborns and adults change the disease from endemic to free condition. From table 4 we notice that increasing in the value of ψ leads to decreasing in R0, which its becomes from value greater than one to value less than one, which means the treatments on infected has a big effect on changing from endemic case to free case. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] Hepatitis B Foundation, https://www.hepb.org, accessed 20 August 2022. [2] J. Zhang, S. Zhang, Application and Optimal Control for an HBV Model with Vaccination and Treatment, Discrete Dyn. Nat. Soc. 2018 (2018), 2076983. https://doi.org/10.1155/2018/2076983. [3] M. Aniji, N. Kavitha, S. 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Stability Analysis 3.1. Equilibrium Solutions: 3.2. Disease-Free Equilibrium DFE: 3.3. Basic reproductive number (R0) for DFE: 3.4. Stability Analysis of DFE: 3.5. Stability Analysis of Endemic Equilibrium: 4. Sensitivity Analysis of R0 4.1. The relation between R0 and : 4.2. The relation between R0 and : 4.3. The relation between R0 and : 5. Results and Discussion 5.1. Effect of immunization of newborns () 5.2. Effect of immunization of adults () 5.3. Effect of treatment on infected to the spread of hepatitis B virus () 6. Conclusion References