# Modeling the Effect of Vaccination and Treatment on the Transmission Dynamics of Hepatitis B Virus and HIV/AIDS Coinfection

Authors: Engida Endriyas Endashaw; Temesgen Tibebu Mekonnen
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

Hepatitis B and HIV/AIDS coinfections are common globally due to their similar mode of transmission. Since HIV infection modifies the course of HBV infection by increasing the rate of chronicity, prolonging HBV viremia, and increasing liver disease-associated deaths, individuals with coinfection of both diseases have a higher tendency of developing cirrhosis of the liver, higher levels of HBV DNA, reduced rate of clearance of the hepatitis B e antigen (HBeAg), and more likely to die than an individual with a single infection. This nature of HBV-HIV/AIDS coinfection motivated us to conduct this study. In this paper, we proposed and rigorously analyzed a deterministic mathematical model with the aim of investigating the effect of vaccination against hepatitis B virus and treatment for all infections on the transmission dynamics of HBV-HIV/AIDS coinfection in a population. We proved that the solutions of the submodels and the coinfection model are positive and bounded. The models are studied qualitatively using the stability theory of differential equations, and the effective reproduction numbers of the models are derived using the next generation matrix method. Stability of the equilibria of the submodels and the coinfection model is analyzed using Routh-Hurwitz criteria. The disease-free and endemic equilibria of the submodels and the coinfection model are computed, and both local and global asymptotic stability conditions for those equilibria are discussed. We performed a sensitivity analysis to illustrate the influence of different parameters on the effective reproduction number of HBV-HIV/AIDS coinfection model, and we identified the most sensitive parameters areωB and ωH, which are the effective contact rates for HBV and HIV transmission, respectively. The numerical simulation of the model is done using MATLAB, and the findings from the simulations are discussed. It is observed that if the vaccination and treatment rates are increased, then the number of individuals susceptible to both infections and HBV-HIV/AIDS coinfection decreases and even falls to zero over time. Hence, it is concluded that vaccination against hepatitis B virus infection, treatment of hepatitis B and HIV/AIDS infections, and HBV-HIV/AIDS infection at the highest possible rate is very essential to control the spread of HBV-HIV/AIDS coinfection as an important public health problem.


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## Body

## 1. Introduction

Globally, more in the developing countries, there are a number of deadly infectious diseases that are severely affecting the lifespan of the human population. Hepatitis B is one of such deadly diseases that is caused by hepatitis B virus (HBV). Currently, despite the use of a preventive vaccine for several decades as well as the use of effective and well-tolerated viral suppressive medications since 1998, approximately 250 million people are chronically infected with the virus and a lot of people die every year from hepatitis B-related liver failure and liver cancer [1, 2]. The virus is highly infectious, and the main routes of transmission are mother to neonate (vertical transmission), sexual contact with infectious partners, transplantation of organs from infected donors, exposure to improperly sterilized health care instruments, and the administration of contaminated blood products in poorer countries [1, 3]. The virus assaults liver cells and causes acute and chronic liver disease [4]. Acute hepatitis B infection is usually self-limiting; however, chronic hepatitis B infection is life-long [5]. On the other hand, human immunodeficiency virus (HIV) is also a global public health problem affecting many people worldwide causing large morbidity and mortality [6, 7]. It is a retrovirus which weakens the body immunity by removing CD4 T-cells, and if untreated, it continues to multiply in the host till it reaches the highest level leading to a very serious stage called AIDS [8, 9]. Everyone diagnosed with HIV should start ART regardless of his/her stage of infection or complications to lower the viral load, to improve the quality of life, and to lower the possibility of transmitting the virus to [10, 11]. HBV and HIV/AIDS coinfections are common globally due to their similar mode of transmission. Although there is a lack of accurate data for coinfection in many parts of the world, its spread is increasing in parts of West and South Africa [12]. The viruses have similar properties such as transmission, using a reverse transcriptase enzyme in replication, likelihood to develop chronic infection, and excessive chance of mutation in their genome, sometimes resulting in resistance to extensively used antiviral agents [13, 14]. Coinfection of HBV and HIV/AIDS occurs when individuals get infections concurrently or individuals who are positive for one of the strains are infected with the other strain within a brief period of time before an infection with the first strain has been established and an immune response has developed. It is common to note that HIV infection modifies the course of HBV infection by increasing the rate of chronicity, prolonging HBV viremia, and increasing liver disease-associated deaths [12]. Individuals with coinfection of both diseases have a higher tendency of developing cirrhosis of the liver, higher levels of HBV DNA, reduced rate of clearance of hepatitis B e antigen (HBeAg), and more likely to die than the ones with hepatitis B [15, 16]. Antiretroviral therapy (ART) is challenging when coinfection is present as HIV-infected individuals are usually less responsive to treatment for HBV and have a high risk of hepatotoxicity and drug interactions; however, the availability of ART with activity against HBV and HIV/AIDS coinfection, particularly tenofovir disoproxil fumarate (TDF) and tenofovir alafenamide fumarate (TAF), allows for simplification of treatment regimen and results in significant improvements in outcomes [12]. The study of infectious disease coepidemics is critical to understand how the diseases are related and how prevention and treatment efforts can be most effective. Thus, in the study of infectious diseases, mathematical models give us a fascinating insight into the very complicated infection dynamics and effective control measures. In recent years, most mathematical epidemic models are formulated based on a single disease, although a growing number of studies have considered coepidemics. Shen et al. [17] formulated a mathematical model for coronavirus infection with an optimal control analysis when vaccination is present. Using the suggested controls, they developed an optimal control model and derived mathematical results from it. Through optimal control, they obtained results showing that controls can be useful for minimizing infected individuals and improving population health. Kumar et al. [18] studied an efficient numerical scheme for a fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy. They solved the nonlinear mathematical models with fractional derivative by the aid of Legendre wavelet operational matrix method and noticed that the suggested schemes are more efficient and effective. Li et al. [19] developed a mathematical model to study the third wave of coronavirus infection in Pakistan through a new mathematical model and observed that the suggested parameters can decrease efficiently the infection cases of the third wave in Pakistan. Bonyah et al. [20] proposed a deterministic coinfection Zika-dengue model and analyzed the transmission of the diseases, which incorporates the prevention and treatment of the infectious nature of each disease. Their numerical optimal control analysis indicates that effective prevention and treatment of each disease will help in the effective control and eradication of the disease. They obtained results showing that the control of coinfection of dengue Zika requires that communities should combine both prevention and treatment associated with each disease at the same time. Okosun et al. [21] examined a coinfection of a cryptosporidiosis-HIV/AIDS deterministic model by incorporating time-dependent control preventions and treatments. They studied the model basic properties and presented the numerical results, which show that the best strategy to control coinfection is to combine all controls at the same time. The authors in [22] conducted a study on cholera-schistosomiasis coinfection dynamics. Based on their findings, they suggested to the public health department that the only possible cost-effective strategy for the elimination of schistosomiasis and cholera coinfection is the combination of both diseases’ preventive measures and the treatment of schistosomiasis. Mathematical models formulated to study the dynamics of HBV-HIV/AIDS coinfection are rare in the literature, although the coexistence between the two infections exists. In our review of the literature, we have got only two mathematical models of HBV-HIV/AIDS coinfection and we used them as the initial literature reviewed as follows: Nampala et al. [23] developed a mathematical model of hepatotoxicity and antiretroviral therapeutic effects in HIV/HBV coinfection. They used numerical simulations to study the therapeutic as well as toxic effect of the currently used HIV/HBV therapy and consequently derived an optimal combination for treating coinfection. In their model, they did not consider protection against all infections rather focused on hepatotoxicity and antiretroviral therapeutic effects in HIV/HBV coinfection. They obtained results showing that emtricitabine, tenofovir, and efavirenz are the optimal combination that maximises the therapeutic effect of therapy and minimizes the toxic response to medication in HBV/HIV coinfection. Bowong et al. [13] formulated and analyzed a realistic mathematical model for HBV-HIV coinfection, which incorporates the key epidemiological and biological features of each of the two diseases in a population. In their model, they did not consider treatment for all infections rather focused only on vaccination against HBV infection. The above two studies motivated us to develop our new model. Therefore, in this paper, we proposed and rigorously analyzed a deterministic mathematical model incorporating vaccination against hepatitis B virus and treatment for all infections on the transmission dynamics of HBV-HIV/AIDS coinfection in a population. We have checked this case has never been done before. This new model will be used to evaluate the effect of vaccination against hepatitis B virus and treatment for all infections on the transmission dynamics of HBV-HIV/AIDS coinfection as control strategies for minimizing the spread of coinfection in a population. The rest of this paper is organized as follows. The model is formulated in Section 2 and is analyzed in Sections 3 and 4. Sensitivity analysis and numerical simulation were carried out in Sections 5 and 6. Finally, the results and discussion and conclusion of the study are carried out in Sections 7 and 8, respectively.


## 2. Model Assumptions and Formulation

The model subdivides the total population at timet denoted by Nt, into nine mutually exclusive compartments depending on their disease status: susceptible individuals to both diseases St, immune individuals following vaccination to HBV infection (Vt), HBV-only infected class (IBt), HIV-only infected class (IHt), HBV-HIV/AIDS coinfected class (IBHt), HBV-HIV/AIDS-treated class (TBHt), HBV-treated class (TBt), HIV-treated class (THt), and suppressed viral load class (SVt). The integration among people is homogeneous and become infected with HIV and HBV by the force of infection λH and λB, respectively. Individuals are recruited into the population (vaccinated class) at a constant rate b with the proportion ρ of which are immunized to protect them from HBV infection. The size of the vaccinated class decreases due to the natural death rate μ, the transfer of HIV-infected individuals, and individuals whose vaccine efficacy wanes to IH and susceptible classes, respectively. The efficacy of the vaccine wanes out at a rate q and the natural recovery rate of individuals from HBV infection in HBV-only infected class is σ. The susceptible class increases due to the recruitment of unvaccinated individuals, the transfer of naturally recovered individuals from HBV-only infected class, and the transfer of individuals whose vaccine efficacy wanes. Because of the natural mortality rate μ and the movement of HBV and HIV-infected individuals to HBV-only and HIV-only infected classes, respectively, the susceptible class decreases. Individuals in S and V compartments get HIV infection by the force of infection λH, whereas individuals in S get HBV infection by the force of infection λB. Individuals in HBV-only infected class get HIV infection by force of infection λH within a brief period of time before an infection with the first strain (HBV) has been established and an immune response has developed. Similarly, individuals in HIV-only infected class get HBV infection by force of infection λB within a brief period of time before an infection with the first strain (HIV) has been established and an immune response has developed. Individuals in IB and IBH progress to TB and TBH classes, respectively, after getting treatment. The proportion π of individuals in IH progress to TH due to the treatment rate γ and the remaining proportion (1−π) of individuals progress to IBH by force of infection λB before an infection with the first strain (HIV) has been established and an immune response has developed. Naturally recovered patients from HBV infection in HBV-HIV/AIDS-coinfected class enter in to HIV-only infected class at a rate r. Since the effective treatment reduces the viral load of the infected individuals in TB, TH, and TBH classes to the required undetectable level, individuals in these compartments progress to a suppressed viral load class at the progress rate 𝜙, θ, and ε, respectively. If the vaccine efficacy not wanes, individuals who are vaccinated against HBV infection are susceptible to HIV-only. Due to the fact that there may be no immunity to loss life whether or not one is unwell or healthy, the natural death rate for individuals in different classes is the same. We take no consideration of disease-related deaths in all treated classes due to the effective treatment. HBV-infected individuals get treatment at chronic stage only. The immunity acquired naturally by HBV-infected individuals is lifelong. No individual gets infected with both diseases at once, rather he/she gets infected with one of the two diseases first and the second afterwards, but within a brief period of time before infection with the first strain is established. Individuals in HBV-only infected class who are able to clear the infection due to natural immunity enter into the susceptible class at rate σ, but they are not susceptible to HBV infection again since they develop antibodies and protected from HBV infection for lifelong. Vertical transmission for both HBV and HIV has not been considered here.Using parameters and their description in Table1 and the model assumptions and formulation given in Section 2, the schematic diagram for the transmission dynamics of HBV-HIV/AIDS coinfection is given by the following:Table 1
Parameters and their description of the model.
ParametersDescriptionsμNatural death ratebRecruitment rate of population into immunized classδHHIV/AIDS-induced mortalityδBHBV-induced mortalityδBHHBV-HIV/AIDS-induced mortalityαTreatment rate for HBV-infected classγTreatment rate forπ proportion of HIV/AIDS-infected individualsβTreatment rate for HBV-HIV/AIDS-coinfected classλBForce of infection associated with HBV infectionλHForce of infection associated with HIV/AIDS infectionrRecovery rate of individuals from HBV infection naturally in theIBH classσRecovery rate of individuals from HBV infection naturally fromIB classρProportion of immunized populationπThe proportion of HIV/AIDS-infected individuals progressing to treatment classϕThe progress rate of HB-treated individuals to suppressed viral load classθThe progress rate of HIV/AIDS-treated individuals to suppressed viral load classεThe progress rate of HBV-HIV/AIDS-treated individuals to suppressed viral load classωBThe effective contact rate for HBV transmissionωHThe effective contact rate for HIV transmissionqThe vaccine efficacy wanes rateFrom the flow diagram of the model in Figure1, the dynamical system of the model is as follows:
(1)dsdt=1−ρb+qV+σIB−λB+λH+μS,dVdt=ρb−λH+q+μV,dIBdt=λBS−σ+α+λH+δB+μIB,dIHdt=λHS+V+rIBH−πγ+1−πλB+δH+μIH,dIBHdt=1−πλBIH+λHIB−r+β+δBH+μIBH,dTBdt=αIB−ϕ+μTB,dTHdt=πγIH−θ+μTH,dTBHdt=βIBH−ε+μTBH,dSVdt=εTBH+ϕTB+θTH−μSV,Figure 1
Flow diagram of the model.where the force of infection associated with HBV infection is given by the following:(2)λB=ωBIB+IBHN,and the force of infection associated with HIV infection is given by the following:
(3)λH=ωHIH+IBHN.Initially,S0>0, V0≥0, IB0≥0, IH0≥0, IBH0≥0, TB0≥0, TH0≥0, TBH0≥0, and SV0≥0.


### 2.1. Positivity of the Solutions

For (1) to be epidemiologically meaningful, it is important to prove that all its state variables are nonnegative for all time t≥0. In other words, the solutions of (1) with positive initial data remain positive for all time t≥0 because the model deals with the human population. This can be verified as follows:Theorem 1.
LetΩ =S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:S0>0,V0≥0,IB0≥0,IH0≥0,IBH0≥0,TB0≥0,TH0≥0,TBH0≥0,andSV0≥0, then the solutions of S,V,IB,IH,IBH,TB,TH,TBH,SV are positive for t≥0.Proof.
Takingds/dt=1−ρb+qV+σIB−λB+λH+μS from (1) and making the left side of the equation zero, we have a solution:
(4)St=S0e−λB+λH+μt+1−ρb+qV+σIBλB+λH+μ1−e−λB+λH+μt.
SinceS0>0,e−λB+λH+μt>0 and 1−ρb+qV+σIB/λB+λH+μ>0 for t≥0, St=S0e−λB+λH+μt+1−ρb+qV+σIB/λB+λH+μ1−e−λB+λH+μt>0.
TakingdV/dt=ρb−λH+q+μV from (1), we have a solution
(5)Vt=V0e−λH+q+μt+ρbλH+q+μ1−e−λH+q+μt.
SinceV0>0,e−λH+q+μt>0 and ρb/λH+q+μ>0 for t≥0, Vt=V0e−λH+q+μt+ρb/λH+q+μ1−e−λH+q+μt>0.
TakingdIB/dt=λBS−σ+α+λH+δB+μIB from (1), we have a solution
(6)IBt=IB0e−σ+α+λH+δB+μt+λBSσ+α+λH+δB+μ1−e−σ+α+λH+δB+μt.
SinceIB0>0,e−σ+α+λH+δB+μt>0 and λBS/σ+α+λH+δB+μ>0 for t≥0, IBt>0.
TakingdIH/dt=λHS+λHV+rIBH−πγ+1−πλB+δH+μIHfrom2, we have a solution
(7)IHt=IH0e−πγ+1−πλB+δH+μt+λHS+λHV+rIBHπγ+1−πλB+δH+μ1−e−πγ+1−πλB+δH+μt.Since IH0>0,e−πγ+1−πλB+δH+μt>0 and λHS+λHV+rIBH/πγ+1−πλB+δH+μ>0,
(8)IHt=IH0e−πγ+1−πλB+δH+μt+λHS+λHV+rIBHπγ+1−πλB+δH+μ1−e−πγ+1−πλB+δH+μt>0.
TakingdIBH/dt=1−πλBIH+λHIB−r+β+δBH+μIBH from 2, from (1) we have a solution
(9)IBHt=IBH0e−r+β+δBH+μt+1−πλBIH+λHIBr+β+δBH+μ1−e−r+β+δBH+μt.
SinceIBH0>0,e−r+β+δBH+μt>0 and 1−πλBIH+λHIB/r+β+δBH+μ>0 for t≥0,
(10)IBHt=IBH0e−r+β+δBH+μt+1−πλBIH+λHIBr+β+δBH+μ1−e−r+β+δBH+μt>0.
TakingdTB/dt=αIB−ϕ+μTB from (1), we have a solution
(11)TBt=TB0e−ϕ+μt+αIBϕ+μ1−e−ϕ+μt.
SinceTB0>0,e−r+β+δBH+μt>0 and αIB/ϕ+μ>0 for t≥0, TBt=TB0e−ϕ+μt+αIB/ϕ+μ1−e−ϕ+μt>0.
TakingdTH/dt=πγIH−θ+μTH from (1), we have a solution
(12)THt=TH0e−θ+μt+πγIHθ+μ1−e−θ+μt.
SinceTH0>0,e−θ+μt>0 and πγIH/θ+μ>0,
(13)THt=TH0e−θ+μt+πγIHθ+μ1−e−θ+μt>0.
TakingdTBH/dt=βIBH−ε+μTBH from (1), we have a solution
(14)TBHt=TBH0e−ε+μt+βIBHε+μ1−e−ε+μt+βIBHε+μ.
SinceTBH0>0,e−ε+μt>0 and βIBH/ε+μ>0,
(15)TBHt=TBH0e−ε+μt+βIBHε+μ1−e−ε+μt+βIBHε+μ>0.
TakingdSV/dt=εTBH+ϕTB+θTH−μSV from (1), we have a solution
(16)SVt=SV0e−μt+εTBH+ϕTB+θTHμ1−e−μt.
SinceSV0>0,e−μt>0and εTBH+ϕTB+θTH/μ>0,
(17)SVt=SV0e−μt+εTBH+ϕTB+θTHμ1−e−μt>0.
This completes the proof of the theorem. Therefore, the solutions of the model are positive.


### 2.2. Boundedness of the Solution Region

Theorem 2.
The solution trajectories of (1) evolve in a positive invariant region.
(18)Ω=S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:N≤bμ.Proof.
LetNt represent the whole population at time t.
Thus,(19)N=S+V+IB+IH+IBH+TB+TH+TBH+SV⟹dNdt=dsdt+dvdt+dIBdt+dIHdt+dIBHdt+dTBdt+dTHdt+dTBHdt+dSVdt=b−μS+V+IB+IH+IBH+TB+TH+TBH+SV−IB+TBδB+IH+THδH+IBH+TBHδBH⟹dNdt=b−μN−IB+TBδB+IH+THδH+IBH+TBHδBH.
IfN>b/μ, dN/dt<0.
In the absence of mortality due to HBV infection, HIV/AIDS, and HBV-HIV/AIDS coinfection and comparing both sides of the equation using the standard comparison theorem, (19) becomes
(20)dNdt≤b−μN.
Integrating (20) on both sides, we get the following:
(1)
−1/μlnb−μN≤t+C where C is a constant(2)
b−μN≥C1e−μt, where C1=e−μt is a constant(3)
b−μN0≥C1⟺N0≤b/μ
Now takingb−μN0=C1, the inequality b−μN≥C1e−μt becomes the following:
(21)Nt≤N0e−μt+bμ1−e−μt.
Ast⟶∞, the population size Nt⟶b/μ whenever N0≤b/μ is as follows:
(1)
0≤Nt≤b/μ(2)
All feasible solutions of components of the dynamical system with initial conditions enter the regionΩ=S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:N≤b/μ for all t≥0
Thus, the regionΩ is positively invariant.
After checking the positivity of the solutions and the boundedness of the solution region of the full model, we considered the dynamics of the two submodels, namely, HBV-only and HIV/AIDS-only submodels. This would help us to lay down the foundation for the qualitative analysis of the full mode.


## 2.1. Positivity of the Solutions

For (1) to be epidemiologically meaningful, it is important to prove that all its state variables are nonnegative for all time t≥0. In other words, the solutions of (1) with positive initial data remain positive for all time t≥0 because the model deals with the human population. This can be verified as follows:Theorem 1.
LetΩ =S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:S0>0,V0≥0,IB0≥0,IH0≥0,IBH0≥0,TB0≥0,TH0≥0,TBH0≥0,andSV0≥0, then the solutions of S,V,IB,IH,IBH,TB,TH,TBH,SV are positive for t≥0.Proof.
Takingds/dt=1−ρb+qV+σIB−λB+λH+μS from (1) and making the left side of the equation zero, we have a solution:
(4)St=S0e−λB+λH+μt+1−ρb+qV+σIBλB+λH+μ1−e−λB+λH+μt.
SinceS0>0,e−λB+λH+μt>0 and 1−ρb+qV+σIB/λB+λH+μ>0 for t≥0, St=S0e−λB+λH+μt+1−ρb+qV+σIB/λB+λH+μ1−e−λB+λH+μt>0.
TakingdV/dt=ρb−λH+q+μV from (1), we have a solution
(5)Vt=V0e−λH+q+μt+ρbλH+q+μ1−e−λH+q+μt.
SinceV0>0,e−λH+q+μt>0 and ρb/λH+q+μ>0 for t≥0, Vt=V0e−λH+q+μt+ρb/λH+q+μ1−e−λH+q+μt>0.
TakingdIB/dt=λBS−σ+α+λH+δB+μIB from (1), we have a solution
(6)IBt=IB0e−σ+α+λH+δB+μt+λBSσ+α+λH+δB+μ1−e−σ+α+λH+δB+μt.
SinceIB0>0,e−σ+α+λH+δB+μt>0 and λBS/σ+α+λH+δB+μ>0 for t≥0, IBt>0.
TakingdIH/dt=λHS+λHV+rIBH−πγ+1−πλB+δH+μIHfrom2, we have a solution
(7)IHt=IH0e−πγ+1−πλB+δH+μt+λHS+λHV+rIBHπγ+1−πλB+δH+μ1−e−πγ+1−πλB+δH+μt.Since IH0>0,e−πγ+1−πλB+δH+μt>0 and λHS+λHV+rIBH/πγ+1−πλB+δH+μ>0,
(8)IHt=IH0e−πγ+1−πλB+δH+μt+λHS+λHV+rIBHπγ+1−πλB+δH+μ1−e−πγ+1−πλB+δH+μt>0.
TakingdIBH/dt=1−πλBIH+λHIB−r+β+δBH+μIBH from 2, from (1) we have a solution
(9)IBHt=IBH0e−r+β+δBH+μt+1−πλBIH+λHIBr+β+δBH+μ1−e−r+β+δBH+μt.
SinceIBH0>0,e−r+β+δBH+μt>0 and 1−πλBIH+λHIB/r+β+δBH+μ>0 for t≥0,
(10)IBHt=IBH0e−r+β+δBH+μt+1−πλBIH+λHIBr+β+δBH+μ1−e−r+β+δBH+μt>0.
TakingdTB/dt=αIB−ϕ+μTB from (1), we have a solution
(11)TBt=TB0e−ϕ+μt+αIBϕ+μ1−e−ϕ+μt.
SinceTB0>0,e−r+β+δBH+μt>0 and αIB/ϕ+μ>0 for t≥0, TBt=TB0e−ϕ+μt+αIB/ϕ+μ1−e−ϕ+μt>0.
TakingdTH/dt=πγIH−θ+μTH from (1), we have a solution
(12)THt=TH0e−θ+μt+πγIHθ+μ1−e−θ+μt.
SinceTH0>0,e−θ+μt>0 and πγIH/θ+μ>0,
(13)THt=TH0e−θ+μt+πγIHθ+μ1−e−θ+μt>0.
TakingdTBH/dt=βIBH−ε+μTBH from (1), we have a solution
(14)TBHt=TBH0e−ε+μt+βIBHε+μ1−e−ε+μt+βIBHε+μ.
SinceTBH0>0,e−ε+μt>0 and βIBH/ε+μ>0,
(15)TBHt=TBH0e−ε+μt+βIBHε+μ1−e−ε+μt+βIBHε+μ>0.
TakingdSV/dt=εTBH+ϕTB+θTH−μSV from (1), we have a solution
(16)SVt=SV0e−μt+εTBH+ϕTB+θTHμ1−e−μt.
SinceSV0>0,e−μt>0and εTBH+ϕTB+θTH/μ>0,
(17)SVt=SV0e−μt+εTBH+ϕTB+θTHμ1−e−μt>0.
This completes the proof of the theorem. Therefore, the solutions of the model are positive.


## 2.2. Boundedness of the Solution Region

Theorem 2.
The solution trajectories of (1) evolve in a positive invariant region.
(18)Ω=S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:N≤bμ.Proof.
LetNt represent the whole population at time t.
Thus,(19)N=S+V+IB+IH+IBH+TB+TH+TBH+SV⟹dNdt=dsdt+dvdt+dIBdt+dIHdt+dIBHdt+dTBdt+dTHdt+dTBHdt+dSVdt=b−μS+V+IB+IH+IBH+TB+TH+TBH+SV−IB+TBδB+IH+THδH+IBH+TBHδBH⟹dNdt=b−μN−IB+TBδB+IH+THδH+IBH+TBHδBH.
IfN>b/μ, dN/dt<0.
In the absence of mortality due to HBV infection, HIV/AIDS, and HBV-HIV/AIDS coinfection and comparing both sides of the equation using the standard comparison theorem, (19) becomes
(20)dNdt≤b−μN.
Integrating (20) on both sides, we get the following:
(1)
−1/μlnb−μN≤t+C where C is a constant(2)
b−μN≥C1e−μt, where C1=e−μt is a constant(3)
b−μN0≥C1⟺N0≤b/μ
Now takingb−μN0=C1, the inequality b−μN≥C1e−μt becomes the following:
(21)Nt≤N0e−μt+bμ1−e−μt.
Ast⟶∞, the population size Nt⟶b/μ whenever N0≤b/μ is as follows:
(1)
0≤Nt≤b/μ(2)
All feasible solutions of components of the dynamical system with initial conditions enter the regionΩ=S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:N≤b/μ for all t≥0
Thus, the regionΩ is positively invariant.
After checking the positivity of the solutions and the boundedness of the solution region of the full model, we considered the dynamics of the two submodels, namely, HBV-only and HIV/AIDS-only submodels. This would help us to lay down the foundation for the qualitative analysis of the full mode.


## 3. Analysis of the Submodels

The analysis of the full model will be followed by analyzing the dynamics of the hepatitis B (HB) only and HIV-only submodels.


### 3.1. HB-Only Submodel

The HB-only model is obtained by settingIH=IBH=TH=TBH=SV=0. Thus, we have the following dynamical system:
(22)dsdt=1−ρb+qV+σIB−λB+μS,dVdt=ρb−q+μV,dIBdt=λBS−σ+α+δB+μIB,dTBdt=αIB−μTB,whereλB=ωBIB/N is the force of infection.The solutions of the model are positive, and the trajectories evolve in a positive invariant region(23)ΩB=S,V,IB,TB∈ℝ4>0:N≤bμ.


#### 3.1.1. Disease-Free Equilibrium Point (DFE) of HBV-Only Submodel

The DFE of the dynamical system in (22) represented by E0B is equal to E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0. It is obtained by making the right-hand side of (22) and the infectious and treatment class zero.


#### 3.1.2. Effective and Basic Reproduction Numbers of the Model

The effective reproduction number of hepatitis B-infected individuals, denoted byRBeff that of the dynamical system (22), is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0B/∂xj∂viE0B/∂xj−1 of the dynamical system (22) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infection from one compartment i to another, and E0B is the disease-free equilibrium point.In model (22), F=λBS and V=σ+α+δB+μIB.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are as follows:
(24)F=ωBq+μ1−ρq+μandV=σ+α+δB+μ⟹V−1=1σ+α+δB+μ⟹FV−1=ωBq+μ1−ρq+μσ+α+δB+μ⟹ρFV−1=ωBq+μ1−ρq+μσ+α+δB+μ⟹RBeff=ωBq+μ1−ρq+μσ+α+δB+μ.On the other hand, the basic reproduction number of the dynamical system (22), denoted by RB0, is derived when initially the entire population is susceptible [27, 28]. That is, when there is no vaccination and treatment (ρ=α=0).In model (22), F=λBS and V=σ+δB+μIB.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are as follows:
(25)F=ωBandV=σ+δB+μ⟹V−1=1σ+δB+μ⟹FV−1=ωBσ+δB+μ⟹ρFV−1=ωBσ+δB+μ⟹RB0=ωBσ+δB+μ.


#### 3.1.3. Stability of the Disease-Free Equilibrium (DFE) of HB-Only Submodel

Theorem 3.
The disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is locally asymptotically stable if the effective reproduction number RBeff<1 and is unstable otherwise.Proof.
The Jacobean matrix of (22) at the DFE point E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 is as follows:
(26)JE0B=∂f1E0B∂S∂f1E0B∂V∂f1E0B∂IB∂f1E0B∂TB∂f2E0B∂S∂f2E0B∂V∂f2E0B∂IB∂f2E0B∂TB∂f3E0B∂S∂f3E0B∂V∂f3E0B∂IB∂f3E0B∂TB∂f4E0B∂S∂f4E0B∂V∂f4E0B∂IB∂f4E0B∂TB=−μqσ−ωBq+μ1−ρq+μ00−q+μ0000ωBq+μ1−ρq+μ−σ+α+δB+μ000α−μ.
The corresponding characteristic equation is as follows:(27)−μ+λqσ−ωBq+μ1−ρq+μ00−q+μ+λ0000ωBq+μ1−ρq+μ−σ+α+δB+μ−λ000α−μ+λ=0.
Thus, after some steps, the roots of the characteristic equation areλ1=λ2=−μ,λ3=−q+μ and
(28)λ4=−qα−qμ−αμ−μ2−qσ−μσ−qδB−μδB+qωB+μωB−μρωBq+μ.
But the fourth eigenvalueλ4 can be simplified as σ+α+δB+μRBeff−1.
(1)
λ4 is negative if RBeff<1(2)
All the eigenvalues of the Jacobean matrix have negative real parts whenRBeff<1
Hence, the disease-free equilibrium
E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is locally asymptotically stable if the effective reproduction number RBeff<1 and unstable otherwise.Theorem 4.
The disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is globally asymptotically stable in the feasible region ΩB if the effective reproduction number RBeff<1 and is unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(29)LB=b1IB,where
(30)b1=1σ+α+δB+μ.
LB is a continuous function for all S,V,IB,TB∈ℝ4>0, and it has first-order partial derivative and has minimum at
(31)E0B=bq+μ1−ρμq+μ,ρbq+μ,0,0.
Its time derivative along the solution path yields the following:(32)LḂ=b1İB=b1ωBIBSN−σ+α+δB+μIB=b1ωBSN−σ+α+δB+μIB.
Since the state variables of the model when the HBV is endemic in the population do not exceed the state variables of the model in a population free of HBV, at the disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 with
(33)RBeff=ωBq+μ1−ρq+μσ+α+δB+μ,
LḂ can be simplified as follows:
(34)LḂ≤b1ωBS0N0−σ+α+δB+μIB,LḂ=b1ωBS0S0+V0−σ+α+δB+μIB=b1ωBq+μ1−ρq+μ−σ+α+δB+μIB=b1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ−σ+α+δB+μIB=b1RBeffσ+α+δB+μ−σ+α+δB+μIB=b1σ+α+δB+μRBeff−1IB=RBeff−1IB.
This impliesLḂ≤0 when RBeff<1. Furthermore, L̇=0 if and only if IB=0. Thus, by LaSalle’s invariance principle [29], the largest invariant set in ΩB contained in S,V,IB,TB∈ℝ4>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0B on ΩB if the effective reproduction number RBeff<1 and unstable otherwise.


#### 3.1.4. Existence and Stability of Endemic Equilibrium (EE) of HB-Only Submodel

Let an arbitrary equilibrium point of HB-only submodel (22) be denoted by E∗B=S∗,V∗,I∗B,T∗B and λB∗=ωBIB∗/N∗ be the associated infection rate (“force of infection”) at the endemic equilibrium point. After some calculations, we got λB∗=μq+μσ+α+δB+μRBeff−1/α+μq+μ+μρδB.This impliesλB∗>0 if RBeff>1, and hence, an endemic equilibrium point E∗B=S∗,V∗,I∗B,T∗B) of HB-only submodel (22) exists whenever RBeff>1.
(35)S∗=bα+μq+μ1−ρ+μρα+δB+μσ+α+δB+μμq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μ,V∗=ρbq+μ,I∗B=μωBbα+μq+μ1−ρ+μρα+δB+μ/μq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μq+μωB−σ+α+δB+μ−ωBρbωBq+μα+μ,T∗B=αμωBbα+μq+μ1−ρ+μρα+δB+μ/μq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μq+μωB−σ+α+δB+μ−ωBρbωBμq+μα+μ.Theorem 5.
The positive endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is locally asymptotically stable if RBeff>1.Proof.
The Jacobian matrix of (22) is as follows:
(36)JB=−ωBI∗BN∗+μqσ−ωBS∗N∗00−q+μ00ωBI∗BN∗0ωBS∗N∗−α+σ+δB+μ000α−μ.
The Jacobian matrix of3(22) at the endemic equilibrium E∗B=S∗,V∗,I∗B,T∗B is as follows:
(37)JBE∗B=−q+μωB−α+σ+δB+μ+μρα+δB+μ−ωBρα+δB+μq−α+δB+μ00−q+μ00q+μωB−α+σ+δB+μ−ωBμρρα+δB+μ00000α−μ.
Letλi be the eigenvalues of the Jacobian matrix JBE∗B where i=1,2,3,4.
Then, the eigenvalues of the Jacobian matrixJBE∗B are λ1=−μ<0 or λ2=−q+μ<0 or k2λ2+k1λ+k0=0 where
(38)k2=1,k1=q+μα+σ+δB+μRBeff−1+μρα+δB+μρα+δB+μ,k0=q+μα+σ+δB+μRBeff−1ρ.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationk2λ2+k1λ+k0=0, we can apply Routh-Hurwitz stability criteria. Since k2=1>0, both k1 and k0 should be positive.
Clearly,k1 and k0 are positive if RBeff>1.
This impliesλ3<0 and λ4<0.
Hence, since all eigenvalues are negative, the endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is locally asymptotically stable if RBeff>1.Theorem 6.
The endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is globally asymptotically stable if RBeff>1.Proof.
Let us take the Lyapunov functionLB=b1S−S∗2+b2IB−IB∗2, b1>0,b2>0.
The time derivative ofLBS,IB is
(39)L̇B=2b1S−S∗Ṡ+2b2IB−IB∗İB=2b1S−S∗1−ρb+qV+σIB−ωBIBN+μS+2b2IB−IB∗ωBIBNS−σ+α+δB+μIB.
Using expressions at the endemic equilibrium1−ρb+qV+σIB=ωBI∗B/N∗+μS∗, σ+α+δB+μ=ωBS∗/N∗, and N≈N∗,
(40)L̇B=2b1S−S∗ωBI∗BN∗S∗+μS∗−ωBIBNS−μS+2b2IB−IB∗ωBIBNS−ωBIBN∗S∗=2b1S−S∗μS∗−S+ωBNI∗BS∗−IBS+2b2IB−IB∗ωBIBNS−S∗=2b1S−S∗μS∗−S+ωBNI∗BS∗−IBS∗+IBS∗−IBS+2b2ωBIBNIB−IB∗S−S∗=2b1S−S∗μS∗−S+ωBNS∗I∗B−IB+IBS∗−S+2b2ωBIBNIB−IB∗S−S∗=−2b1μS−S∗S−S∗+2b1ωBNS∗I∗B−IBS−S∗−2b1ωBNIBS−S∗S−S∗−2b2ωBIBNIB∗−IBS−S∗=−2b1μS−S∗2−2b1ωBNIBS−S∗2+2b1ωBNS∗I∗B−IBS−S∗−2b2ωBIBNIB∗−IBS−S∗=−2b1μS−S∗2−2b1ωBNIBS−S∗2+2ωBNIB∗−IBS−S∗b1S∗−b2IB≤0ifIB≤b1S∗b2,I∗B≤IBandS≥S∗.
This impliesL̇B≤0, and it vanishes only at the endemic equilibrium E∗B.
Hence, by LaSalle’s invariance principle [30], all model solutions approach to E∗B as t⟶∞ whenever RBeff>1.
This proves the global asymptotic stability of the endemic equilibrium pointE∗B on ΩB if the effective reproduction number RBeff>1.


### 3.2. HIV/AIDS-Only Submodel

The HIV-only model is obtained by settingV=IB=IBH=TB=TBH=SV=0. Thus, we have the following dynamical system:
(41)dsdt=b−λH+μS,dIHdt=λHS−πγ+δH+μIH,dTHdt=πγIH−μTH,where(42)λH=ωHIHN.The solutions of the model are positive, and the trajectories evolve in a positive invariant regionΩH=S,IH,TH∈ℝ3>0:N≤b/μ.


#### 3.2.1. Disease-Free Equilibrium Point (DFE) of HIV/AIDS-Only Submodel

The DFE of the dynamical system in (41), denoted by E0H, is equal to E0H=b/μ,0,0. It is obtained by making the right-hand side of (41) zero and the infectious and treatment classes as well.


#### 3.2.2. Effective and Basic Reproduction Numbers of HIV/AIDS-Only Submodels

The effective reproduction number of HIV/AIDS-only infected individuals, denoted byRHeff that of the dynamical system (41), is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0H/∂xj∂viE0H/∂xj−1 of the dynamical system (41) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infections from one compartment i to another, and E0H is the disease-free equilibrium point.In model (41), F=λHS and V=πγ+δH+μIH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(43)F=ωHandV=πγ+δH+μ⟹V−1=1πγ+δH+μ⟹FV−1=ωHπγ+δH+μ⟹RHeff=ωHπγ+δH+μ.On the other hand, the basic reproduction number of HIV/AIDS-only infected individuals, denoted byRH0 that of the dynamical system (41), is defined as the average number of secondary infections produced by a single HIV/AIDS infectious individual introduced in a completely susceptible population during his or her entire infectious period [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0H/∂xj∂viE0H/∂xj−1 of the dynamical system (41) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infection from one compartment i to another, and E0H is the disease-free equilibrium point.In model (41), F=λHS and V=δH+μIH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(44)F=ωHandV=δH+μ⟹V−1=1δH+μ⟹FV−1=ωHδH+μ⟹RH0=ωHδH+μ.


#### 3.2.3. Stability of the Disease-Free Equilibrium (DFE) of HIV/AIDS-Only Submodel

Theorem 7.
The disease-free equilibriumE0H=b/μ,0,0 of (41) is locally asymptotically stable if the basic reproduction number RH0<1 and is unstable otherwise.Proof.
The Jacobean matrix of (41) at the DFE point E0H=b/μ,0,0 is as follows:
(45)JE0H=∂f1E0H∂S∂f1E0H∂IH∂f1E0H∂TH∂f2E0H∂S∂f2E0H∂IH∂f2E0H∂TH∂f3E0H∂S∂f3E0H∂IH∂f3E0H∂TH=−μ−ωH00ωH−πγ+δH+μ00πγ−μ.
The corresponding characteristic equation is as follows:(46)−μ+λ−ωH00−πγ+δH+μ−ωH+λ00πγ−μ+λ=0.
After some necessary steps, the roots of the characteristic equation areλ1=λ2=−μ or λ3=−μ+πγ+δH+ωH.
But, to show that all eigenvalues are negative, it is enough to show thatλ3 is negative for RH0<1.
Now(47)λ3=−μ+πγ+δH+ωH=−μ+πγ+δH+RH0μ+πγ+δH=μ+πγ+δHRH0−1.(1)
λ3 is negative if RH0<1(2)
All eigenvalues of the Jacobean matrix have negative real parts whenRH0<1
Hence, the disease-free equilibriumE0H of (41) is locally asymptotically stable if RH0<1 and unstable otherwise.Theorem 8.
The disease-free equilibrium pointE0H of (41) is globally asymptotically stable in the feasible region ΩH if RH0<1 and unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(48)LH=h1IH,where h1=1/πγ+δH+μ.
LH is a continuous function, for all S,IH,TH∈ℝ3>0, and it has first-order partial derivative and has a minimum at E0H=b/μ,0,0.
Its time derivative along the solution path yields the following:(49)LḢ=h1İH=h1ωHIHSN−πγ+δH+μIH=h1ωHSN−πγ+δH+μIH.
Since the state variables of the model when HIV/AIDS is endemic in the population do not exceed the state variables of the model in a population free of HIV/AIDS, at the disease-free equilibriumE0H=b/μ,0,0 with RH0<1,
LḢ can be simplified as follows:
(50)LḢ≤h1ωHS0N0−πγ+δH+μIH=h1ωH−πγ+δH+μIH=h1RH0πγ+δH+μ−πγ+δH+μIH=h1πγ+δH+μRH0−1IH=RH0−1IH.
This impliesLḢ≤0 when RH0<1. Furthermore, LḢ=0 if and only if IH=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in ΩH contained in S,IH,TH∈ℝ3>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0H on ΩH if the basic reproduction number RH0<1 and unstable otherwise.


#### 3.2.4. Existence and Stability of the Endemic Equilibrium Point (EE) of HIV/AIDS-Only Submodel

Let an arbitrary equilibrium point of HIV/AIDS-only submodel (41) is denoted by E∗H=S∗,I∗H,T∗H and λH∗=ωHIH∗/N∗ be the associated infection rate (“force of infection”) at the endemic equilibrium point. After some calculations, we got λH∗=μπγ+δH+μRHeff−1/πγ+μ. This implies λH∗>0 if RHeff>1, and hence, an endemic equilibrium point E∗H=S∗,I∗H,T∗H of a dynamical system (41) exists whenever RH0>1, where
(51)E∗H=bπγ+μμωH−πγ+δH+μ+μπγ+μ,bωH−πγ+δH+μπγ+δH+μωH−πγ+δH+μ+πγ+μ,πγbωH−πγ+δH+μμπγ+δH+μωH−πγ+δH+μ+πγ+μ.Theorem 9.
The endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is locally asymptotically stable if RHeff>1.Proof.
The Jacobian matrix of (41) is as follows:
(52)JH=−ωHI∗HN∗+μ−ωHS∗N∗0ωHI∗HN∗ωHS∗N∗−πγ+δH+μ00πγ−μ.
The Jacobian matrix of (41) at the endemic equilibrium point is as follows:
(53)JHE∗H=−μωH−πγ+δH+μ+πγ+μπγ+μ−πγ+δH+μ0μωH−πγ+δH+μπγ+μ000πγ−μ.
Letλi be the eigenvalues of (53) where i=1,2,3.
Then, the eigenvalues of (53) are λ1=−μ<0 or m2λ2+m1λ+m0=0 where
(54)m2=1,m1==μπγ+δH+μRHeff−1+πγ+μπγ+μ,m0==μπγ+δH+μ2RHeff−1πγ+μ.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationm2λ2+m1λ+m0=0, we can apply Routh-Hurwitz stability criteria. Since m2=1>0, both m1 and m0 should be positive.
Clearly,m1 and m0 are positive if RHeff>1.
This impliesλ2<0 and λ3<0.
Hence, since all eigenvalues are negative, the endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is locally asymptotically stable if RHeff>1.Theorem 10.
The endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is globally asymptotically stable if RHeff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(55)LHS,IH=h1S−S∗2+h2IH−IH∗2,h1>0,h2>0.
DifferentiatingLHS,IH with respect to time gives the following:
(56)L̇HS,IH=2h1S−S∗Ṡ+2h2IH−IH∗İH=2h1S−S∗b−ωHIHN+μS+2h2IH−IH∗ωHIHNS−πγ+δH+μIH.
Substitutingb=ωHI∗H/N∗+μS∗ and πγ+δH+μ=ωHS∗/N∗ with N≈N∗in 10 yields the following:
(57)L̇HS,IH=2h1S−S∗ωHI∗HN∗S∗+μS∗−ωHIHNS−μS+2h2IH−IH∗ωHIHNS−ωHIHN∗S∗=2h1S−S∗μS∗−S+ωHNI∗HS∗−IHS+2h2IH−IH∗ωHIHNS−S∗=2h1S−S∗μS∗−S+ωHNI∗HS∗−IHS∗+IHS∗−IHS+2h2ωHIHNIH−IH∗S−S∗=2h1S−S∗μS∗−S+ωHNS∗I∗H−IH+IHS∗−S+2h2ωHIHNIH−IH∗S−S∗=−2h1μS−S∗S−S∗+2h1ωHNS∗I∗H−IHS−S∗−2h1ωHNIHS−S∗S−S∗−2h2ωHIHNIH∗−IHS−S∗=−2h1μS−S∗2−2h1ωHNIHS−S∗2+2h1ωHNS∗I∗H−IHS−S∗−2h2ωHIHNIH∗−IHS−S∗=−2h1μS−S∗2−2h1ωHNIHS−S∗2+2ωHNIH∗−IHS−S∗h1S∗−h2IH≤0ifIH≥h1S∗h2,S∗≤S,I∗H≥IH.
This impliesL̇HS,IH≤0, and it vanishes only at the endemic equilibrium E∗H.
Hence, by LaSalle’s invariance principle [30], all model solutions approach to E∗H as t⟶∞ whenever RH0>1.
This proves the global asymptotic stability of the endemic equilibrium pointE∗H on ΩH if the basic reproduction number RHeff>1.


## 3.1. HB-Only Submodel

The HB-only model is obtained by settingIH=IBH=TH=TBH=SV=0. Thus, we have the following dynamical system:
(22)dsdt=1−ρb+qV+σIB−λB+μS,dVdt=ρb−q+μV,dIBdt=λBS−σ+α+δB+μIB,dTBdt=αIB−μTB,whereλB=ωBIB/N is the force of infection.The solutions of the model are positive, and the trajectories evolve in a positive invariant region(23)ΩB=S,V,IB,TB∈ℝ4>0:N≤bμ.


### 3.1.1. Disease-Free Equilibrium Point (DFE) of HBV-Only Submodel

The DFE of the dynamical system in (22) represented by E0B is equal to E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0. It is obtained by making the right-hand side of (22) and the infectious and treatment class zero.


### 3.1.2. Effective and Basic Reproduction Numbers of the Model

The effective reproduction number of hepatitis B-infected individuals, denoted byRBeff that of the dynamical system (22), is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0B/∂xj∂viE0B/∂xj−1 of the dynamical system (22) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infection from one compartment i to another, and E0B is the disease-free equilibrium point.In model (22), F=λBS and V=σ+α+δB+μIB.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are as follows:
(24)F=ωBq+μ1−ρq+μandV=σ+α+δB+μ⟹V−1=1σ+α+δB+μ⟹FV−1=ωBq+μ1−ρq+μσ+α+δB+μ⟹ρFV−1=ωBq+μ1−ρq+μσ+α+δB+μ⟹RBeff=ωBq+μ1−ρq+μσ+α+δB+μ.On the other hand, the basic reproduction number of the dynamical system (22), denoted by RB0, is derived when initially the entire population is susceptible [27, 28]. That is, when there is no vaccination and treatment (ρ=α=0).In model (22), F=λBS and V=σ+δB+μIB.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are as follows:
(25)F=ωBandV=σ+δB+μ⟹V−1=1σ+δB+μ⟹FV−1=ωBσ+δB+μ⟹ρFV−1=ωBσ+δB+μ⟹RB0=ωBσ+δB+μ.


### 3.1.3. Stability of the Disease-Free Equilibrium (DFE) of HB-Only Submodel

Theorem 3.
The disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is locally asymptotically stable if the effective reproduction number RBeff<1 and is unstable otherwise.Proof.
The Jacobean matrix of (22) at the DFE point E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 is as follows:
(26)JE0B=∂f1E0B∂S∂f1E0B∂V∂f1E0B∂IB∂f1E0B∂TB∂f2E0B∂S∂f2E0B∂V∂f2E0B∂IB∂f2E0B∂TB∂f3E0B∂S∂f3E0B∂V∂f3E0B∂IB∂f3E0B∂TB∂f4E0B∂S∂f4E0B∂V∂f4E0B∂IB∂f4E0B∂TB=−μqσ−ωBq+μ1−ρq+μ00−q+μ0000ωBq+μ1−ρq+μ−σ+α+δB+μ000α−μ.
The corresponding characteristic equation is as follows:(27)−μ+λqσ−ωBq+μ1−ρq+μ00−q+μ+λ0000ωBq+μ1−ρq+μ−σ+α+δB+μ−λ000α−μ+λ=0.
Thus, after some steps, the roots of the characteristic equation areλ1=λ2=−μ,λ3=−q+μ and
(28)λ4=−qα−qμ−αμ−μ2−qσ−μσ−qδB−μδB+qωB+μωB−μρωBq+μ.
But the fourth eigenvalueλ4 can be simplified as σ+α+δB+μRBeff−1.
(1)
λ4 is negative if RBeff<1(2)
All the eigenvalues of the Jacobean matrix have negative real parts whenRBeff<1
Hence, the disease-free equilibrium
E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is locally asymptotically stable if the effective reproduction number RBeff<1 and unstable otherwise.Theorem 4.
The disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is globally asymptotically stable in the feasible region ΩB if the effective reproduction number RBeff<1 and is unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(29)LB=b1IB,where
(30)b1=1σ+α+δB+μ.
LB is a continuous function for all S,V,IB,TB∈ℝ4>0, and it has first-order partial derivative and has minimum at
(31)E0B=bq+μ1−ρμq+μ,ρbq+μ,0,0.
Its time derivative along the solution path yields the following:(32)LḂ=b1İB=b1ωBIBSN−σ+α+δB+μIB=b1ωBSN−σ+α+δB+μIB.
Since the state variables of the model when the HBV is endemic in the population do not exceed the state variables of the model in a population free of HBV, at the disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 with
(33)RBeff=ωBq+μ1−ρq+μσ+α+δB+μ,
LḂ can be simplified as follows:
(34)LḂ≤b1ωBS0N0−σ+α+δB+μIB,LḂ=b1ωBS0S0+V0−σ+α+δB+μIB=b1ωBq+μ1−ρq+μ−σ+α+δB+μIB=b1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ−σ+α+δB+μIB=b1RBeffσ+α+δB+μ−σ+α+δB+μIB=b1σ+α+δB+μRBeff−1IB=RBeff−1IB.
This impliesLḂ≤0 when RBeff<1. Furthermore, L̇=0 if and only if IB=0. Thus, by LaSalle’s invariance principle [29], the largest invariant set in ΩB contained in S,V,IB,TB∈ℝ4>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0B on ΩB if the effective reproduction number RBeff<1 and unstable otherwise.


### 3.1.4. Existence and Stability of Endemic Equilibrium (EE) of HB-Only Submodel

Let an arbitrary equilibrium point of HB-only submodel (22) be denoted by E∗B=S∗,V∗,I∗B,T∗B and λB∗=ωBIB∗/N∗ be the associated infection rate (“force of infection”) at the endemic equilibrium point. After some calculations, we got λB∗=μq+μσ+α+δB+μRBeff−1/α+μq+μ+μρδB.This impliesλB∗>0 if RBeff>1, and hence, an endemic equilibrium point E∗B=S∗,V∗,I∗B,T∗B) of HB-only submodel (22) exists whenever RBeff>1.
(35)S∗=bα+μq+μ1−ρ+μρα+δB+μσ+α+δB+μμq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μ,V∗=ρbq+μ,I∗B=μωBbα+μq+μ1−ρ+μρα+δB+μ/μq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μq+μωB−σ+α+δB+μ−ωBρbωBq+μα+μ,T∗B=αμωBbα+μq+μ1−ρ+μρα+δB+μ/μq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μq+μωB−σ+α+δB+μ−ωBρbωBμq+μα+μ.Theorem 5.
The positive endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is locally asymptotically stable if RBeff>1.Proof.
The Jacobian matrix of (22) is as follows:
(36)JB=−ωBI∗BN∗+μqσ−ωBS∗N∗00−q+μ00ωBI∗BN∗0ωBS∗N∗−α+σ+δB+μ000α−μ.
The Jacobian matrix of3(22) at the endemic equilibrium E∗B=S∗,V∗,I∗B,T∗B is as follows:
(37)JBE∗B=−q+μωB−α+σ+δB+μ+μρα+δB+μ−ωBρα+δB+μq−α+δB+μ00−q+μ00q+μωB−α+σ+δB+μ−ωBμρρα+δB+μ00000α−μ.
Letλi be the eigenvalues of the Jacobian matrix JBE∗B where i=1,2,3,4.
Then, the eigenvalues of the Jacobian matrixJBE∗B are λ1=−μ<0 or λ2=−q+μ<0 or k2λ2+k1λ+k0=0 where
(38)k2=1,k1=q+μα+σ+δB+μRBeff−1+μρα+δB+μρα+δB+μ,k0=q+μα+σ+δB+μRBeff−1ρ.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationk2λ2+k1λ+k0=0, we can apply Routh-Hurwitz stability criteria. Since k2=1>0, both k1 and k0 should be positive.
Clearly,k1 and k0 are positive if RBeff>1.
This impliesλ3<0 and λ4<0.
Hence, since all eigenvalues are negative, the endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is locally asymptotically stable if RBeff>1.Theorem 6.
The endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is globally asymptotically stable if RBeff>1.Proof.
Let us take the Lyapunov functionLB=b1S−S∗2+b2IB−IB∗2, b1>0,b2>0.
The time derivative ofLBS,IB is
(39)L̇B=2b1S−S∗Ṡ+2b2IB−IB∗İB=2b1S−S∗1−ρb+qV+σIB−ωBIBN+μS+2b2IB−IB∗ωBIBNS−σ+α+δB+μIB.
Using expressions at the endemic equilibrium1−ρb+qV+σIB=ωBI∗B/N∗+μS∗, σ+α+δB+μ=ωBS∗/N∗, and N≈N∗,
(40)L̇B=2b1S−S∗ωBI∗BN∗S∗+μS∗−ωBIBNS−μS+2b2IB−IB∗ωBIBNS−ωBIBN∗S∗=2b1S−S∗μS∗−S+ωBNI∗BS∗−IBS+2b2IB−IB∗ωBIBNS−S∗=2b1S−S∗μS∗−S+ωBNI∗BS∗−IBS∗+IBS∗−IBS+2b2ωBIBNIB−IB∗S−S∗=2b1S−S∗μS∗−S+ωBNS∗I∗B−IB+IBS∗−S+2b2ωBIBNIB−IB∗S−S∗=−2b1μS−S∗S−S∗+2b1ωBNS∗I∗B−IBS−S∗−2b1ωBNIBS−S∗S−S∗−2b2ωBIBNIB∗−IBS−S∗=−2b1μS−S∗2−2b1ωBNIBS−S∗2+2b1ωBNS∗I∗B−IBS−S∗−2b2ωBIBNIB∗−IBS−S∗=−2b1μS−S∗2−2b1ωBNIBS−S∗2+2ωBNIB∗−IBS−S∗b1S∗−b2IB≤0ifIB≤b1S∗b2,I∗B≤IBandS≥S∗.
This impliesL̇B≤0, and it vanishes only at the endemic equilibrium E∗B.
Hence, by LaSalle’s invariance principle [30], all model solutions approach to E∗B as t⟶∞ whenever RBeff>1.
This proves the global asymptotic stability of the endemic equilibrium pointE∗B on ΩB if the effective reproduction number RBeff>1.


## 3.1.1. Disease-Free Equilibrium Point (DFE) of HBV-Only Submodel

The DFE of the dynamical system in (22) represented by E0B is equal to E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0. It is obtained by making the right-hand side of (22) and the infectious and treatment class zero.


## 3.1.2. Effective and Basic Reproduction Numbers of the Model

The effective reproduction number of hepatitis B-infected individuals, denoted byRBeff that of the dynamical system (22), is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0B/∂xj∂viE0B/∂xj−1 of the dynamical system (22) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infection from one compartment i to another, and E0B is the disease-free equilibrium point.In model (22), F=λBS and V=σ+α+δB+μIB.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are as follows:
(24)F=ωBq+μ1−ρq+μandV=σ+α+δB+μ⟹V−1=1σ+α+δB+μ⟹FV−1=ωBq+μ1−ρq+μσ+α+δB+μ⟹ρFV−1=ωBq+μ1−ρq+μσ+α+δB+μ⟹RBeff=ωBq+μ1−ρq+μσ+α+δB+μ.On the other hand, the basic reproduction number of the dynamical system (22), denoted by RB0, is derived when initially the entire population is susceptible [27, 28]. That is, when there is no vaccination and treatment (ρ=α=0).In model (22), F=λBS and V=σ+δB+μIB.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are as follows:
(25)F=ωBandV=σ+δB+μ⟹V−1=1σ+δB+μ⟹FV−1=ωBσ+δB+μ⟹ρFV−1=ωBσ+δB+μ⟹RB0=ωBσ+δB+μ.


## 3.1.3. Stability of the Disease-Free Equilibrium (DFE) of HB-Only Submodel

Theorem 3.
The disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is locally asymptotically stable if the effective reproduction number RBeff<1 and is unstable otherwise.Proof.
The Jacobean matrix of (22) at the DFE point E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 is as follows:
(26)JE0B=∂f1E0B∂S∂f1E0B∂V∂f1E0B∂IB∂f1E0B∂TB∂f2E0B∂S∂f2E0B∂V∂f2E0B∂IB∂f2E0B∂TB∂f3E0B∂S∂f3E0B∂V∂f3E0B∂IB∂f3E0B∂TB∂f4E0B∂S∂f4E0B∂V∂f4E0B∂IB∂f4E0B∂TB=−μqσ−ωBq+μ1−ρq+μ00−q+μ0000ωBq+μ1−ρq+μ−σ+α+δB+μ000α−μ.
The corresponding characteristic equation is as follows:(27)−μ+λqσ−ωBq+μ1−ρq+μ00−q+μ+λ0000ωBq+μ1−ρq+μ−σ+α+δB+μ−λ000α−μ+λ=0.
Thus, after some steps, the roots of the characteristic equation areλ1=λ2=−μ,λ3=−q+μ and
(28)λ4=−qα−qμ−αμ−μ2−qσ−μσ−qδB−μδB+qωB+μωB−μρωBq+μ.
But the fourth eigenvalueλ4 can be simplified as σ+α+δB+μRBeff−1.
(1)
λ4 is negative if RBeff<1(2)
All the eigenvalues of the Jacobean matrix have negative real parts whenRBeff<1
Hence, the disease-free equilibrium
E0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is locally asymptotically stable if the effective reproduction number RBeff<1 and unstable otherwise.Theorem 4.
The disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 of (22) is globally asymptotically stable in the feasible region ΩB if the effective reproduction number RBeff<1 and is unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(29)LB=b1IB,where
(30)b1=1σ+α+δB+μ.
LB is a continuous function for all S,V,IB,TB∈ℝ4>0, and it has first-order partial derivative and has minimum at
(31)E0B=bq+μ1−ρμq+μ,ρbq+μ,0,0.
Its time derivative along the solution path yields the following:(32)LḂ=b1İB=b1ωBIBSN−σ+α+δB+μIB=b1ωBSN−σ+α+δB+μIB.
Since the state variables of the model when the HBV is endemic in the population do not exceed the state variables of the model in a population free of HBV, at the disease-free equilibriumE0B=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0 with
(33)RBeff=ωBq+μ1−ρq+μσ+α+δB+μ,
LḂ can be simplified as follows:
(34)LḂ≤b1ωBS0N0−σ+α+δB+μIB,LḂ=b1ωBS0S0+V0−σ+α+δB+μIB=b1ωBq+μ1−ρq+μ−σ+α+δB+μIB=b1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ−σ+α+δB+μIB=b1RBeffσ+α+δB+μ−σ+α+δB+μIB=b1σ+α+δB+μRBeff−1IB=RBeff−1IB.
This impliesLḂ≤0 when RBeff<1. Furthermore, L̇=0 if and only if IB=0. Thus, by LaSalle’s invariance principle [29], the largest invariant set in ΩB contained in S,V,IB,TB∈ℝ4>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0B on ΩB if the effective reproduction number RBeff<1 and unstable otherwise.


## 3.1.4. Existence and Stability of Endemic Equilibrium (EE) of HB-Only Submodel

Let an arbitrary equilibrium point of HB-only submodel (22) be denoted by E∗B=S∗,V∗,I∗B,T∗B and λB∗=ωBIB∗/N∗ be the associated infection rate (“force of infection”) at the endemic equilibrium point. After some calculations, we got λB∗=μq+μσ+α+δB+μRBeff−1/α+μq+μ+μρδB.This impliesλB∗>0 if RBeff>1, and hence, an endemic equilibrium point E∗B=S∗,V∗,I∗B,T∗B) of HB-only submodel (22) exists whenever RBeff>1.
(35)S∗=bα+μq+μ1−ρ+μρα+δB+μσ+α+δB+μμq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μ,V∗=ρbq+μ,I∗B=μωBbα+μq+μ1−ρ+μρα+δB+μ/μq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μq+μωB−σ+α+δB+μ−ωBρbωBq+μα+μ,T∗B=αμωBbα+μq+μ1−ρ+μρα+δB+μ/μq+μα+μσ+α+δB+μ+α+δB+μωB−σ+α+δB+μq+μωB−σ+α+δB+μ−ωBρbωBμq+μα+μ.Theorem 5.
The positive endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is locally asymptotically stable if RBeff>1.Proof.
The Jacobian matrix of (22) is as follows:
(36)JB=−ωBI∗BN∗+μqσ−ωBS∗N∗00−q+μ00ωBI∗BN∗0ωBS∗N∗−α+σ+δB+μ000α−μ.
The Jacobian matrix of3(22) at the endemic equilibrium E∗B=S∗,V∗,I∗B,T∗B is as follows:
(37)JBE∗B=−q+μωB−α+σ+δB+μ+μρα+δB+μ−ωBρα+δB+μq−α+δB+μ00−q+μ00q+μωB−α+σ+δB+μ−ωBμρρα+δB+μ00000α−μ.
Letλi be the eigenvalues of the Jacobian matrix JBE∗B where i=1,2,3,4.
Then, the eigenvalues of the Jacobian matrixJBE∗B are λ1=−μ<0 or λ2=−q+μ<0 or k2λ2+k1λ+k0=0 where
(38)k2=1,k1=q+μα+σ+δB+μRBeff−1+μρα+δB+μρα+δB+μ,k0=q+μα+σ+δB+μRBeff−1ρ.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationk2λ2+k1λ+k0=0, we can apply Routh-Hurwitz stability criteria. Since k2=1>0, both k1 and k0 should be positive.
Clearly,k1 and k0 are positive if RBeff>1.
This impliesλ3<0 and λ4<0.
Hence, since all eigenvalues are negative, the endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is locally asymptotically stable if RBeff>1.Theorem 6.
The endemic equilibriumE∗B=S∗,V∗,I∗B,T∗B of (22) is globally asymptotically stable if RBeff>1.Proof.
Let us take the Lyapunov functionLB=b1S−S∗2+b2IB−IB∗2, b1>0,b2>0.
The time derivative ofLBS,IB is
(39)L̇B=2b1S−S∗Ṡ+2b2IB−IB∗İB=2b1S−S∗1−ρb+qV+σIB−ωBIBN+μS+2b2IB−IB∗ωBIBNS−σ+α+δB+μIB.
Using expressions at the endemic equilibrium1−ρb+qV+σIB=ωBI∗B/N∗+μS∗, σ+α+δB+μ=ωBS∗/N∗, and N≈N∗,
(40)L̇B=2b1S−S∗ωBI∗BN∗S∗+μS∗−ωBIBNS−μS+2b2IB−IB∗ωBIBNS−ωBIBN∗S∗=2b1S−S∗μS∗−S+ωBNI∗BS∗−IBS+2b2IB−IB∗ωBIBNS−S∗=2b1S−S∗μS∗−S+ωBNI∗BS∗−IBS∗+IBS∗−IBS+2b2ωBIBNIB−IB∗S−S∗=2b1S−S∗μS∗−S+ωBNS∗I∗B−IB+IBS∗−S+2b2ωBIBNIB−IB∗S−S∗=−2b1μS−S∗S−S∗+2b1ωBNS∗I∗B−IBS−S∗−2b1ωBNIBS−S∗S−S∗−2b2ωBIBNIB∗−IBS−S∗=−2b1μS−S∗2−2b1ωBNIBS−S∗2+2b1ωBNS∗I∗B−IBS−S∗−2b2ωBIBNIB∗−IBS−S∗=−2b1μS−S∗2−2b1ωBNIBS−S∗2+2ωBNIB∗−IBS−S∗b1S∗−b2IB≤0ifIB≤b1S∗b2,I∗B≤IBandS≥S∗.
This impliesL̇B≤0, and it vanishes only at the endemic equilibrium E∗B.
Hence, by LaSalle’s invariance principle [30], all model solutions approach to E∗B as t⟶∞ whenever RBeff>1.
This proves the global asymptotic stability of the endemic equilibrium pointE∗B on ΩB if the effective reproduction number RBeff>1.


## 3.2. HIV/AIDS-Only Submodel

The HIV-only model is obtained by settingV=IB=IBH=TB=TBH=SV=0. Thus, we have the following dynamical system:
(41)dsdt=b−λH+μS,dIHdt=λHS−πγ+δH+μIH,dTHdt=πγIH−μTH,where(42)λH=ωHIHN.The solutions of the model are positive, and the trajectories evolve in a positive invariant regionΩH=S,IH,TH∈ℝ3>0:N≤b/μ.


### 3.2.1. Disease-Free Equilibrium Point (DFE) of HIV/AIDS-Only Submodel

The DFE of the dynamical system in (41), denoted by E0H, is equal to E0H=b/μ,0,0. It is obtained by making the right-hand side of (41) zero and the infectious and treatment classes as well.


### 3.2.2. Effective and Basic Reproduction Numbers of HIV/AIDS-Only Submodels

The effective reproduction number of HIV/AIDS-only infected individuals, denoted byRHeff that of the dynamical system (41), is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0H/∂xj∂viE0H/∂xj−1 of the dynamical system (41) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infections from one compartment i to another, and E0H is the disease-free equilibrium point.In model (41), F=λHS and V=πγ+δH+μIH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(43)F=ωHandV=πγ+δH+μ⟹V−1=1πγ+δH+μ⟹FV−1=ωHπγ+δH+μ⟹RHeff=ωHπγ+δH+μ.On the other hand, the basic reproduction number of HIV/AIDS-only infected individuals, denoted byRH0 that of the dynamical system (41), is defined as the average number of secondary infections produced by a single HIV/AIDS infectious individual introduced in a completely susceptible population during his or her entire infectious period [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0H/∂xj∂viE0H/∂xj−1 of the dynamical system (41) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infection from one compartment i to another, and E0H is the disease-free equilibrium point.In model (41), F=λHS and V=δH+μIH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(44)F=ωHandV=δH+μ⟹V−1=1δH+μ⟹FV−1=ωHδH+μ⟹RH0=ωHδH+μ.


### 3.2.3. Stability of the Disease-Free Equilibrium (DFE) of HIV/AIDS-Only Submodel

Theorem 7.
The disease-free equilibriumE0H=b/μ,0,0 of (41) is locally asymptotically stable if the basic reproduction number RH0<1 and is unstable otherwise.Proof.
The Jacobean matrix of (41) at the DFE point E0H=b/μ,0,0 is as follows:
(45)JE0H=∂f1E0H∂S∂f1E0H∂IH∂f1E0H∂TH∂f2E0H∂S∂f2E0H∂IH∂f2E0H∂TH∂f3E0H∂S∂f3E0H∂IH∂f3E0H∂TH=−μ−ωH00ωH−πγ+δH+μ00πγ−μ.
The corresponding characteristic equation is as follows:(46)−μ+λ−ωH00−πγ+δH+μ−ωH+λ00πγ−μ+λ=0.
After some necessary steps, the roots of the characteristic equation areλ1=λ2=−μ or λ3=−μ+πγ+δH+ωH.
But, to show that all eigenvalues are negative, it is enough to show thatλ3 is negative for RH0<1.
Now(47)λ3=−μ+πγ+δH+ωH=−μ+πγ+δH+RH0μ+πγ+δH=μ+πγ+δHRH0−1.(1)
λ3 is negative if RH0<1(2)
All eigenvalues of the Jacobean matrix have negative real parts whenRH0<1
Hence, the disease-free equilibriumE0H of (41) is locally asymptotically stable if RH0<1 and unstable otherwise.Theorem 8.
The disease-free equilibrium pointE0H of (41) is globally asymptotically stable in the feasible region ΩH if RH0<1 and unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(48)LH=h1IH,where h1=1/πγ+δH+μ.
LH is a continuous function, for all S,IH,TH∈ℝ3>0, and it has first-order partial derivative and has a minimum at E0H=b/μ,0,0.
Its time derivative along the solution path yields the following:(49)LḢ=h1İH=h1ωHIHSN−πγ+δH+μIH=h1ωHSN−πγ+δH+μIH.
Since the state variables of the model when HIV/AIDS is endemic in the population do not exceed the state variables of the model in a population free of HIV/AIDS, at the disease-free equilibriumE0H=b/μ,0,0 with RH0<1,
LḢ can be simplified as follows:
(50)LḢ≤h1ωHS0N0−πγ+δH+μIH=h1ωH−πγ+δH+μIH=h1RH0πγ+δH+μ−πγ+δH+μIH=h1πγ+δH+μRH0−1IH=RH0−1IH.
This impliesLḢ≤0 when RH0<1. Furthermore, LḢ=0 if and only if IH=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in ΩH contained in S,IH,TH∈ℝ3>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0H on ΩH if the basic reproduction number RH0<1 and unstable otherwise.


### 3.2.4. Existence and Stability of the Endemic Equilibrium Point (EE) of HIV/AIDS-Only Submodel

Let an arbitrary equilibrium point of HIV/AIDS-only submodel (41) is denoted by E∗H=S∗,I∗H,T∗H and λH∗=ωHIH∗/N∗ be the associated infection rate (“force of infection”) at the endemic equilibrium point. After some calculations, we got λH∗=μπγ+δH+μRHeff−1/πγ+μ. This implies λH∗>0 if RHeff>1, and hence, an endemic equilibrium point E∗H=S∗,I∗H,T∗H of a dynamical system (41) exists whenever RH0>1, where
(51)E∗H=bπγ+μμωH−πγ+δH+μ+μπγ+μ,bωH−πγ+δH+μπγ+δH+μωH−πγ+δH+μ+πγ+μ,πγbωH−πγ+δH+μμπγ+δH+μωH−πγ+δH+μ+πγ+μ.Theorem 9.
The endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is locally asymptotically stable if RHeff>1.Proof.
The Jacobian matrix of (41) is as follows:
(52)JH=−ωHI∗HN∗+μ−ωHS∗N∗0ωHI∗HN∗ωHS∗N∗−πγ+δH+μ00πγ−μ.
The Jacobian matrix of (41) at the endemic equilibrium point is as follows:
(53)JHE∗H=−μωH−πγ+δH+μ+πγ+μπγ+μ−πγ+δH+μ0μωH−πγ+δH+μπγ+μ000πγ−μ.
Letλi be the eigenvalues of (53) where i=1,2,3.
Then, the eigenvalues of (53) are λ1=−μ<0 or m2λ2+m1λ+m0=0 where
(54)m2=1,m1==μπγ+δH+μRHeff−1+πγ+μπγ+μ,m0==μπγ+δH+μ2RHeff−1πγ+μ.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationm2λ2+m1λ+m0=0, we can apply Routh-Hurwitz stability criteria. Since m2=1>0, both m1 and m0 should be positive.
Clearly,m1 and m0 are positive if RHeff>1.
This impliesλ2<0 and λ3<0.
Hence, since all eigenvalues are negative, the endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is locally asymptotically stable if RHeff>1.Theorem 10.
The endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is globally asymptotically stable if RHeff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(55)LHS,IH=h1S−S∗2+h2IH−IH∗2,h1>0,h2>0.
DifferentiatingLHS,IH with respect to time gives the following:
(56)L̇HS,IH=2h1S−S∗Ṡ+2h2IH−IH∗İH=2h1S−S∗b−ωHIHN+μS+2h2IH−IH∗ωHIHNS−πγ+δH+μIH.
Substitutingb=ωHI∗H/N∗+μS∗ and πγ+δH+μ=ωHS∗/N∗ with N≈N∗in 10 yields the following:
(57)L̇HS,IH=2h1S−S∗ωHI∗HN∗S∗+μS∗−ωHIHNS−μS+2h2IH−IH∗ωHIHNS−ωHIHN∗S∗=2h1S−S∗μS∗−S+ωHNI∗HS∗−IHS+2h2IH−IH∗ωHIHNS−S∗=2h1S−S∗μS∗−S+ωHNI∗HS∗−IHS∗+IHS∗−IHS+2h2ωHIHNIH−IH∗S−S∗=2h1S−S∗μS∗−S+ωHNS∗I∗H−IH+IHS∗−S+2h2ωHIHNIH−IH∗S−S∗=−2h1μS−S∗S−S∗+2h1ωHNS∗I∗H−IHS−S∗−2h1ωHNIHS−S∗S−S∗−2h2ωHIHNIH∗−IHS−S∗=−2h1μS−S∗2−2h1ωHNIHS−S∗2+2h1ωHNS∗I∗H−IHS−S∗−2h2ωHIHNIH∗−IHS−S∗=−2h1μS−S∗2−2h1ωHNIHS−S∗2+2ωHNIH∗−IHS−S∗h1S∗−h2IH≤0ifIH≥h1S∗h2,S∗≤S,I∗H≥IH.
This impliesL̇HS,IH≤0, and it vanishes only at the endemic equilibrium E∗H.
Hence, by LaSalle’s invariance principle [30], all model solutions approach to E∗H as t⟶∞ whenever RH0>1.
This proves the global asymptotic stability of the endemic equilibrium pointE∗H on ΩH if the basic reproduction number RHeff>1.


## 3.2.1. Disease-Free Equilibrium Point (DFE) of HIV/AIDS-Only Submodel

The DFE of the dynamical system in (41), denoted by E0H, is equal to E0H=b/μ,0,0. It is obtained by making the right-hand side of (41) zero and the infectious and treatment classes as well.


## 3.2.2. Effective and Basic Reproduction Numbers of HIV/AIDS-Only Submodels

The effective reproduction number of HIV/AIDS-only infected individuals, denoted byRHeff that of the dynamical system (41), is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0H/∂xj∂viE0H/∂xj−1 of the dynamical system (41) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infections from one compartment i to another, and E0H is the disease-free equilibrium point.In model (41), F=λHS and V=πγ+δH+μIH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(43)F=ωHandV=πγ+δH+μ⟹V−1=1πγ+δH+μ⟹FV−1=ωHπγ+δH+μ⟹RHeff=ωHπγ+δH+μ.On the other hand, the basic reproduction number of HIV/AIDS-only infected individuals, denoted byRH0 that of the dynamical system (41), is defined as the average number of secondary infections produced by a single HIV/AIDS infectious individual introduced in a completely susceptible population during his or her entire infectious period [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0H/∂xj∂viE0H/∂xj−1 of the dynamical system (41) [25, 26], where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infection from one compartment i to another, and E0H is the disease-free equilibrium point.In model (41), F=λHS and V=δH+μIH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(44)F=ωHandV=δH+μ⟹V−1=1δH+μ⟹FV−1=ωHδH+μ⟹RH0=ωHδH+μ.


## 3.2.3. Stability of the Disease-Free Equilibrium (DFE) of HIV/AIDS-Only Submodel

Theorem 7.
The disease-free equilibriumE0H=b/μ,0,0 of (41) is locally asymptotically stable if the basic reproduction number RH0<1 and is unstable otherwise.Proof.
The Jacobean matrix of (41) at the DFE point E0H=b/μ,0,0 is as follows:
(45)JE0H=∂f1E0H∂S∂f1E0H∂IH∂f1E0H∂TH∂f2E0H∂S∂f2E0H∂IH∂f2E0H∂TH∂f3E0H∂S∂f3E0H∂IH∂f3E0H∂TH=−μ−ωH00ωH−πγ+δH+μ00πγ−μ.
The corresponding characteristic equation is as follows:(46)−μ+λ−ωH00−πγ+δH+μ−ωH+λ00πγ−μ+λ=0.
After some necessary steps, the roots of the characteristic equation areλ1=λ2=−μ or λ3=−μ+πγ+δH+ωH.
But, to show that all eigenvalues are negative, it is enough to show thatλ3 is negative for RH0<1.
Now(47)λ3=−μ+πγ+δH+ωH=−μ+πγ+δH+RH0μ+πγ+δH=μ+πγ+δHRH0−1.(1)
λ3 is negative if RH0<1(2)
All eigenvalues of the Jacobean matrix have negative real parts whenRH0<1
Hence, the disease-free equilibriumE0H of (41) is locally asymptotically stable if RH0<1 and unstable otherwise.Theorem 8.
The disease-free equilibrium pointE0H of (41) is globally asymptotically stable in the feasible region ΩH if RH0<1 and unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(48)LH=h1IH,where h1=1/πγ+δH+μ.
LH is a continuous function, for all S,IH,TH∈ℝ3>0, and it has first-order partial derivative and has a minimum at E0H=b/μ,0,0.
Its time derivative along the solution path yields the following:(49)LḢ=h1İH=h1ωHIHSN−πγ+δH+μIH=h1ωHSN−πγ+δH+μIH.
Since the state variables of the model when HIV/AIDS is endemic in the population do not exceed the state variables of the model in a population free of HIV/AIDS, at the disease-free equilibriumE0H=b/μ,0,0 with RH0<1,
LḢ can be simplified as follows:
(50)LḢ≤h1ωHS0N0−πγ+δH+μIH=h1ωH−πγ+δH+μIH=h1RH0πγ+δH+μ−πγ+δH+μIH=h1πγ+δH+μRH0−1IH=RH0−1IH.
This impliesLḢ≤0 when RH0<1. Furthermore, LḢ=0 if and only if IH=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in ΩH contained in S,IH,TH∈ℝ3>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0H on ΩH if the basic reproduction number RH0<1 and unstable otherwise.


## 3.2.4. Existence and Stability of the Endemic Equilibrium Point (EE) of HIV/AIDS-Only Submodel

Let an arbitrary equilibrium point of HIV/AIDS-only submodel (41) is denoted by E∗H=S∗,I∗H,T∗H and λH∗=ωHIH∗/N∗ be the associated infection rate (“force of infection”) at the endemic equilibrium point. After some calculations, we got λH∗=μπγ+δH+μRHeff−1/πγ+μ. This implies λH∗>0 if RHeff>1, and hence, an endemic equilibrium point E∗H=S∗,I∗H,T∗H of a dynamical system (41) exists whenever RH0>1, where
(51)E∗H=bπγ+μμωH−πγ+δH+μ+μπγ+μ,bωH−πγ+δH+μπγ+δH+μωH−πγ+δH+μ+πγ+μ,πγbωH−πγ+δH+μμπγ+δH+μωH−πγ+δH+μ+πγ+μ.Theorem 9.
The endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is locally asymptotically stable if RHeff>1.Proof.
The Jacobian matrix of (41) is as follows:
(52)JH=−ωHI∗HN∗+μ−ωHS∗N∗0ωHI∗HN∗ωHS∗N∗−πγ+δH+μ00πγ−μ.
The Jacobian matrix of (41) at the endemic equilibrium point is as follows:
(53)JHE∗H=−μωH−πγ+δH+μ+πγ+μπγ+μ−πγ+δH+μ0μωH−πγ+δH+μπγ+μ000πγ−μ.
Letλi be the eigenvalues of (53) where i=1,2,3.
Then, the eigenvalues of (53) are λ1=−μ<0 or m2λ2+m1λ+m0=0 where
(54)m2=1,m1==μπγ+δH+μRHeff−1+πγ+μπγ+μ,m0==μπγ+δH+μ2RHeff−1πγ+μ.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationm2λ2+m1λ+m0=0, we can apply Routh-Hurwitz stability criteria. Since m2=1>0, both m1 and m0 should be positive.
Clearly,m1 and m0 are positive if RHeff>1.
This impliesλ2<0 and λ3<0.
Hence, since all eigenvalues are negative, the endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is locally asymptotically stable if RHeff>1.Theorem 10.
The endemic equilibriumE∗H=S∗,I∗H,T∗H of (41) is globally asymptotically stable if RHeff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(55)LHS,IH=h1S−S∗2+h2IH−IH∗2,h1>0,h2>0.
DifferentiatingLHS,IH with respect to time gives the following:
(56)L̇HS,IH=2h1S−S∗Ṡ+2h2IH−IH∗İH=2h1S−S∗b−ωHIHN+μS+2h2IH−IH∗ωHIHNS−πγ+δH+μIH.
Substitutingb=ωHI∗H/N∗+μS∗ and πγ+δH+μ=ωHS∗/N∗ with N≈N∗in 10 yields the following:
(57)L̇HS,IH=2h1S−S∗ωHI∗HN∗S∗+μS∗−ωHIHNS−μS+2h2IH−IH∗ωHIHNS−ωHIHN∗S∗=2h1S−S∗μS∗−S+ωHNI∗HS∗−IHS+2h2IH−IH∗ωHIHNS−S∗=2h1S−S∗μS∗−S+ωHNI∗HS∗−IHS∗+IHS∗−IHS+2h2ωHIHNIH−IH∗S−S∗=2h1S−S∗μS∗−S+ωHNS∗I∗H−IH+IHS∗−S+2h2ωHIHNIH−IH∗S−S∗=−2h1μS−S∗S−S∗+2h1ωHNS∗I∗H−IHS−S∗−2h1ωHNIHS−S∗S−S∗−2h2ωHIHNIH∗−IHS−S∗=−2h1μS−S∗2−2h1ωHNIHS−S∗2+2h1ωHNS∗I∗H−IHS−S∗−2h2ωHIHNIH∗−IHS−S∗=−2h1μS−S∗2−2h1ωHNIHS−S∗2+2ωHNIH∗−IHS−S∗h1S∗−h2IH≤0ifIH≥h1S∗h2,S∗≤S,I∗H≥IH.
This impliesL̇HS,IH≤0, and it vanishes only at the endemic equilibrium E∗H.
Hence, by LaSalle’s invariance principle [30], all model solutions approach to E∗H as t⟶∞ whenever RH0>1.
This proves the global asymptotic stability of the endemic equilibrium pointE∗H on ΩH if the basic reproduction number RHeff>1.


## 4. Analysis of the Full Model

The solutions of the HBV-HIV/AIDS coinfection model are positive, and the trajectories evolve in a positive invariant regionΩ=S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0:N≤b/μ as we proved in Sections 2.1 and 2.2.


### 4.1. Disease-Free Equilibrium (DFE) of HBV-HIV/AIDS Coinfection Model

The disease-free equilibrium of the full model represented byE0BH in (1) is obtained by setting IB=IH=IBH=TB=TH=TBH=SV=0, and it is equal to E0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0.


### 4.2. Basic Reproduction NumberRBH0 of the Full Model

The basic reproduction number of HBV-HIV/AIDS coinfectious individuals, denoted byRBH0 of the dynamical system (1), is defined as the average number of secondary infections produced by a single HBV-HIV/AIDS coinfection infectious individual introduced in a completely susceptible population during his or her entire infectious period [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0BH/∂xj∂viE0BH/∂xj−1 of the dynamical system (1) [25, 26] where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infections from one compartment i to another, and E0BH is the disease-free equilibrium point.In model (1), we have three infection classes. Therefore,
(58)F=f1f2f3=λBSλHS1−πλBIH+λHIBandV=v1v2v3=σ+λH+δB+μIB1−πλB+δH+μIH−rIBHr+δBH+μIBH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(59)F=ωB0ωB0ωHωH000andV=σ+δB+μ000δH+μ−r00r+δBH+μ⟹V−1=1μ+σ+δB0001μ+δHrr+μ+δBHμ+δH001r+μ+δBH⟹FV−1=ωBμ+σ+δB0ωBr+μ+δBH0ωHμ+δHωHr+μ+δBH+rωHr+μ+δBHμ+δH000.The corresponding characteristic equation is as follows:(60)ωBμ+σ+δB−λ0ωBr+μ+δBH0ωHμ+δH−λωHr+μ+δBH+rωHr+μ+δBHμ+δH00−λ=0⟹ωBμ+σ+δB−λωHμ+δH−λωHr+μ+δBH+rωHr+μ+δBHμ+δH0−λ=0⟹λωBμ+σ+δB−λωHμ+δH−λ=0⟹λ1=0,λ2=ωBμ+σ+δB,andλ3=ωHμ+δH.Now,ρFV−1=maxωB/μ+σ+δB,ωH/μ+δH.Hence,RBH0=maxωB/μ+σ+δB,ωH/μ+δH.


### 4.3. Effective Reproduction NumberRBHeff of the Full Model

The effective reproduction number of HBV-HIV/AIDS-coinfected individuals, denoted byRBHeff, is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. To obtain RBHeff, we used the next generation matrix method that was formulated by [26, 27].In model (1), we have three infection classes. Therefore,
(61)F=f1f2f3=λBSλHS+λHV1−πλBIH+λHIBandV=v1v2v3=σ+α+λH+δB+μIBπγ+1−πλB+δH+μIH−rIBHr+β+δBH+μIBH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(62)F=ωBq+μ1−ρq+μ0ωBq+μ1−ρq+μ0ωHωH000andV=σ+α+δB+μ000πγ+δH+μ−r00r+β+δBH+μ⟹V−1=1α+μ+σ+δB0001μ+πγ+δHrr+β+μ+δBHμ+πγ+δH001r+β+μ+δBH⟹FV−1=ωBq+μ1−ρq+μα+μ+σ+δB0ωBq+μ1−ρq+μr+β+μ+δBH0ωHμ+πγ+δHωHr+β+μ+δBH+rωHr+β+μ+δBHμ+πγ+δH000.After some steps, the eigenvalues of (62) are λ1=0,λ2=q+μ−μρωB/q+μα+μ+σ+δB,and λ3=ωH/μ+πγ+δH.Now,(63)ρFV−1=maxωBq+μ−μρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH⟹RBHeff=maxωBq+μ−μρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH.


### 4.4. Stability of the Disease-Free Equilibrium (DFE) of the HBV-HIV/AIDS Coinfection Model

Theorem 11.
The disease-free equilibrium pointE0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 of (1) is locally asymptotically stable if RBHeff<1.Proof.
The Jacobean matrix of (1) at the DFE point E0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 is as follows:
(64)JE0BH=−μqσ−ωBd−ωHd−ωB+ωHd00000−q+μ0−ωHμρq+μ−ωHμρq+μ000000ωBd−σ+α+δB+μ0ωB0000000ωH−πγ+δH+μωH+r00000000−r+β+δBH+μ000000α00−ϕ+μ000000πγ00−θ+μ000000β00−ϵ+μ000000ϕθϵ−μ,
where(65)k1=q+μ,k2=σ+α+δB+μ,k3=πγ+δH+μ,k4=r+β+δBH+μ,k5=ϕ+μ,k6=θ+μ,k7=ϵ+μ,d=q+μ1−ρq+μ.
Letλi be eigenvalues of JE0BH where i=1,⋯,9.
Then, the eigenvalues ofJE0BH are λ1=λ2=−μ<0 or λ3=−q+μ<0 or λ4=−ϵ+μ<0 or λ5=−θ+μ<0 or λ6=−μ+ϕ<0 or λ7=−r+β+μ+δBH<0 or l2λ2+l1λ+l0=0 where l2=1>0,(66)l1=σ+α+δB+μ1−RBeff+πγ+δH+μ1−RHeff,andl0=σ+α+δB+μ1−RBeffπγ+δH+μ1−RHeff.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationl2λ2+l1λ+l0=0, we can apply Routh-Hurwitz stability criteria. Since l2=1>0, both l1 and l0 should be positive.
Clearly,l1 and l0 are positive if RBeff<1 and RHeff<1.
This impliesl1 and l0 are positive, if RBHeff=maxRBeff,RHeff<1.
This impliesλ8<0 and λ9<0.
This implies all eigenvalues are negative.
Hence, the disease-free equilibrium pointE0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 of (1) is locally asymptotically stable if RBHeff=maxRBeff,RHeff<1.Theorem 12.
The disease-free equilibrium pointE0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 of (1) is globally asymptotically stable in the feasible region Ω if the effective reproduction number RBHeff<1 and is unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(67)L=α1IB+α2IH+IBH,where
(68)α1=1/2r+β+δBH+μσ+α+δB+μ,α2=1/2r+β+δBH+μπγ+δH+μ.
L is a continuous function for all S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0, and it has first-order partial derivative and has a minimum at
(69)E0BH=bq+μ1−ρμq+μ,ρbq+μ,0,0,0,0,0,0,0.
Its time derivative along the solution path yields the following:(70)L̇=α1İB+α2İH+İBH=α1ωBIB+IBHSN−ωHIH+IBHIBN−σ+α+δB+μIB+α2ωHIH+IBHS+VN+rIBH−ωB1−πIB+IBHIHN−πγ+δH+μIH+ωB1−πIB+IBHIHN+ωHIH+IBHIBN−r+β+δBH+μIBH.
Since the state variables of the model when the HBV-HIV/AIDS is endemic in the population do not exceed the state variables of the model in a population free of HBV-HIV/AIDS, at the disease-free equilibrium(71)E0BH=bq+μ1−ρμq+μ,ρbq+μ,0,0,0,0,0,0,0,
withRBHeff=maxωBq+μ−μρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH, L̇ can be simplified as follows:
(72)L≤̇α1ωBIBS0S0+V0+ωBIBHS0S0+V0—σ+α+δB+μIB+α2ωHIHS0+V0N+ωHIBHS0+V0N−πγ+δH+μIH−r+β+δBH+μIBH=α1ωBS0S0+V0−σ+α+δB+μIB+α2ωH−πγ+δH+μIH+α1ωBS0S0+V0+α2ωH−r+β+δBH+μIBH=α1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ−σ+α+δB+μIB+α2RH0πγ+δH+μ−πγ+δH+μIH+α1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ+α2RH0πγ+δH+μ−r+β+δBH+μIBH=α1σ+α+δB+μRBeff−1IB+α2πγ+δH+μRH0−1IH+α1RBeffσ+α+δB+μ+α2RH0πγ+δH+μ−r+β+δBH+μIBH.
Substitutingα1 and α2, we get
(73)12r+β+δBH+μRBeff−1IB+12r+β+δBH+μRH0−1IH+r+β+δBH+μRBeff2+RH02−1IBH≤r+β+δBH+μRBHeff2+RBHeff2−1IBH=12r+β+δBH+μRBHeff−1IB+12r+β+δBH+μRBHeff−1IH+r+β+δBH+μRBHeff2+RBHeff2−1IBH=12r+β+δBH+μRBHeff−1IB+12r+β+δBH+μRBHeff−1IH+r+β+δBH+μRBHeff−1IBH
This impliesL̇≤0 when RBHeff<1. Furthermore,
L̇=0 if and only if IB=IH=IBH=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in Ω contained in S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0BH on Ω if the effective reproduction number RBHeff<1 and is unstable otherwise.


### 4.5. Existence and Stability of Endemic Equilibrium (EE) of HBV-HIV/AIDS Coinfection Model

Let the unique endemic equilibrium point of a dynamical system in (1) which exists when the disease persists in the community is denoted by E∗BH=S∗,V∗,I∗B,I∗H,I∗BH,T∗B,T∗H,T∗BH,S∗V. From the analysis of HBV-only submodel (22) and HIV/AIDS-only submodel (41), there exists an endemic equilibrium point for HBV-only submodel and HIV/AIDS-only submodel if RB0>1 and RHeff>1, respectively.This implies the endemic equilibriumE∗BH=S∗,V∗,I∗B,I∗H,I∗BH,T∗B,T∗H,T∗BH,S∗V for HBV-HIV/AIDS coinfection exists because the effective reproduction number for HBV- HIV/AIDS coinfection is the maximum of RBeff,RH0, which is greater than one where
(74)S∗=bq+1−ρλ∗H+μσ+α+λ∗H+δB+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,V∗=ρbλ∗H+q+μ,I∗B=λ∗Bb1−ρλ∗H+μ+qλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,I∗H=λ∗Hbλ∗H+q+μq+1−ρλ∗H+μσ+α+λ∗H+δB+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μλ∗B1−πβ+δBH+μ+πγ+δH+μr+β+δBH+μ+λ∗Hρbλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μλ∗B1−πβ+δBH+μ+πγ+δH+μr+β+δBH+μ+λ∗Bλ∗Hrbq+1−ρλ∗H+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μλ∗B1−πβ+δBH+μ+πγ+δH+μr+β+δBH+μ,I∗BH=λ∗B1−πλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μI∗H+bλ∗H1−ρλ∗H+μ+qr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,T∗B=λ∗Bαb1−ρλ∗H+μ+qϕ+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,T∗H=πγI∗Hθ+μ,T∗BH=βλ∗B1−πλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μI∗H+bλ∗H1−ρλ∗H+μ+qε+μr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,S∗V=εβλ∗B1−πλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μI∗H+bλ∗H1−ρλ∗H+μ+qμε+μr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ+λ∗Bαbϕ1−ρλ∗H+μ+qμϕ+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ+θπγI∗Hμθ+μ.The stability analysis of the endemic equilibrium of the full model (1) in terms of the model parameters analytically is difficult. Hence, we will give an explanation of the stability analysis of the endemic equilibrium E∗BH of the full model (1) under the section numerical simulation.


## 4.1. Disease-Free Equilibrium (DFE) of HBV-HIV/AIDS Coinfection Model

The disease-free equilibrium of the full model represented byE0BH in (1) is obtained by setting IB=IH=IBH=TB=TH=TBH=SV=0, and it is equal to E0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0.


## 4.2. Basic Reproduction NumberRBH0 of the Full Model

The basic reproduction number of HBV-HIV/AIDS coinfectious individuals, denoted byRBH0 of the dynamical system (1), is defined as the average number of secondary infections produced by a single HBV-HIV/AIDS coinfection infectious individual introduced in a completely susceptible population during his or her entire infectious period [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE0BH/∂xj∂viE0BH/∂xj−1 of the dynamical system (1) [25, 26] where Fi is the rate of appearance of a new infection in the compartment i, vi is the transfer of infections from one compartment i to another, and E0BH is the disease-free equilibrium point.In model (1), we have three infection classes. Therefore,
(58)F=f1f2f3=λBSλHS1−πλBIH+λHIBandV=v1v2v3=σ+λH+δB+μIB1−πλB+δH+μIH−rIBHr+δBH+μIBH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(59)F=ωB0ωB0ωHωH000andV=σ+δB+μ000δH+μ−r00r+δBH+μ⟹V−1=1μ+σ+δB0001μ+δHrr+μ+δBHμ+δH001r+μ+δBH⟹FV−1=ωBμ+σ+δB0ωBr+μ+δBH0ωHμ+δHωHr+μ+δBH+rωHr+μ+δBHμ+δH000.The corresponding characteristic equation is as follows:(60)ωBμ+σ+δB−λ0ωBr+μ+δBH0ωHμ+δH−λωHr+μ+δBH+rωHr+μ+δBHμ+δH00−λ=0⟹ωBμ+σ+δB−λωHμ+δH−λωHr+μ+δBH+rωHr+μ+δBHμ+δH0−λ=0⟹λωBμ+σ+δB−λωHμ+δH−λ=0⟹λ1=0,λ2=ωBμ+σ+δB,andλ3=ωHμ+δH.Now,ρFV−1=maxωB/μ+σ+δB,ωH/μ+δH.Hence,RBH0=maxωB/μ+σ+δB,ωH/μ+δH.


## 4.3. Effective Reproduction NumberRBHeff of the Full Model

The effective reproduction number of HBV-HIV/AIDS-coinfected individuals, denoted byRBHeff, is defined as the expected number of secondary cases produced by one typical infection joining in a population made up of both susceptible and nonsusceptible hosts during its infectious period [24, 25]. To obtain RBHeff, we used the next generation matrix method that was formulated by [26, 27].In model (1), we have three infection classes. Therefore,
(61)F=f1f2f3=λBSλHS+λHV1−πλBIH+λHIBandV=v1v2v3=σ+α+λH+δB+μIBπγ+1−πλB+δH+μIH−rIBHr+β+δBH+μIBH.The associated Jacobian matrices ofF and V evaluated at DFE point, respectively, are
(62)F=ωBq+μ1−ρq+μ0ωBq+μ1−ρq+μ0ωHωH000andV=σ+α+δB+μ000πγ+δH+μ−r00r+β+δBH+μ⟹V−1=1α+μ+σ+δB0001μ+πγ+δHrr+β+μ+δBHμ+πγ+δH001r+β+μ+δBH⟹FV−1=ωBq+μ1−ρq+μα+μ+σ+δB0ωBq+μ1−ρq+μr+β+μ+δBH0ωHμ+πγ+δHωHr+β+μ+δBH+rωHr+β+μ+δBHμ+πγ+δH000.After some steps, the eigenvalues of (62) are λ1=0,λ2=q+μ−μρωB/q+μα+μ+σ+δB,and λ3=ωH/μ+πγ+δH.Now,(63)ρFV−1=maxωBq+μ−μρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH⟹RBHeff=maxωBq+μ−μρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH.


## 4.4. Stability of the Disease-Free Equilibrium (DFE) of the HBV-HIV/AIDS Coinfection Model

Theorem 11.
The disease-free equilibrium pointE0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 of (1) is locally asymptotically stable if RBHeff<1.Proof.
The Jacobean matrix of (1) at the DFE point E0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 is as follows:
(64)JE0BH=−μqσ−ωBd−ωHd−ωB+ωHd00000−q+μ0−ωHμρq+μ−ωHμρq+μ000000ωBd−σ+α+δB+μ0ωB0000000ωH−πγ+δH+μωH+r00000000−r+β+δBH+μ000000α00−ϕ+μ000000πγ00−θ+μ000000β00−ϵ+μ000000ϕθϵ−μ,
where(65)k1=q+μ,k2=σ+α+δB+μ,k3=πγ+δH+μ,k4=r+β+δBH+μ,k5=ϕ+μ,k6=θ+μ,k7=ϵ+μ,d=q+μ1−ρq+μ.
Letλi be eigenvalues of JE0BH where i=1,⋯,9.
Then, the eigenvalues ofJE0BH are λ1=λ2=−μ<0 or λ3=−q+μ<0 or λ4=−ϵ+μ<0 or λ5=−θ+μ<0 or λ6=−μ+ϕ<0 or λ7=−r+β+μ+δBH<0 or l2λ2+l1λ+l0=0 where l2=1>0,(66)l1=σ+α+δB+μ1−RBeff+πγ+δH+μ1−RHeff,andl0=σ+α+δB+μ1−RBeffπγ+δH+μ1−RHeff.
To check the algebraic sign of the remaining two eigenvalues for the quadratic equationl2λ2+l1λ+l0=0, we can apply Routh-Hurwitz stability criteria. Since l2=1>0, both l1 and l0 should be positive.
Clearly,l1 and l0 are positive if RBeff<1 and RHeff<1.
This impliesl1 and l0 are positive, if RBHeff=maxRBeff,RHeff<1.
This impliesλ8<0 and λ9<0.
This implies all eigenvalues are negative.
Hence, the disease-free equilibrium pointE0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 of (1) is locally asymptotically stable if RBHeff=maxRBeff,RHeff<1.Theorem 12.
The disease-free equilibrium pointE0BH=bq+μ1−ρ/μq+μ,ρb/q+μ,0,0,0,0,0,0,0 of (1) is globally asymptotically stable in the feasible region Ω if the effective reproduction number RBHeff<1 and is unstable otherwise.Proof.
Consider the following LaSalle-Lyapunov candidate function:(67)L=α1IB+α2IH+IBH,where
(68)α1=1/2r+β+δBH+μσ+α+δB+μ,α2=1/2r+β+δBH+μπγ+δH+μ.
L is a continuous function for all S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0, and it has first-order partial derivative and has a minimum at
(69)E0BH=bq+μ1−ρμq+μ,ρbq+μ,0,0,0,0,0,0,0.
Its time derivative along the solution path yields the following:(70)L̇=α1İB+α2İH+İBH=α1ωBIB+IBHSN−ωHIH+IBHIBN−σ+α+δB+μIB+α2ωHIH+IBHS+VN+rIBH−ωB1−πIB+IBHIHN−πγ+δH+μIH+ωB1−πIB+IBHIHN+ωHIH+IBHIBN−r+β+δBH+μIBH.
Since the state variables of the model when the HBV-HIV/AIDS is endemic in the population do not exceed the state variables of the model in a population free of HBV-HIV/AIDS, at the disease-free equilibrium(71)E0BH=bq+μ1−ρμq+μ,ρbq+μ,0,0,0,0,0,0,0,
withRBHeff=maxωBq+μ−μρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH, L̇ can be simplified as follows:
(72)L≤̇α1ωBIBS0S0+V0+ωBIBHS0S0+V0—σ+α+δB+μIB+α2ωHIHS0+V0N+ωHIBHS0+V0N−πγ+δH+μIH−r+β+δBH+μIBH=α1ωBS0S0+V0−σ+α+δB+μIB+α2ωH−πγ+δH+μIH+α1ωBS0S0+V0+α2ωH−r+β+δBH+μIBH=α1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ−σ+α+δB+μIB+α2RH0πγ+δH+μ−πγ+δH+μIH+α1RBeffq+μσ+α+δB+μq+μ1−ρq+μ1−ρq+μ+α2RH0πγ+δH+μ−r+β+δBH+μIBH=α1σ+α+δB+μRBeff−1IB+α2πγ+δH+μRH0−1IH+α1RBeffσ+α+δB+μ+α2RH0πγ+δH+μ−r+β+δBH+μIBH.
Substitutingα1 and α2, we get
(73)12r+β+δBH+μRBeff−1IB+12r+β+δBH+μRH0−1IH+r+β+δBH+μRBeff2+RH02−1IBH≤r+β+δBH+μRBHeff2+RBHeff2−1IBH=12r+β+δBH+μRBHeff−1IB+12r+β+δBH+μRBHeff−1IH+r+β+δBH+μRBHeff2+RBHeff2−1IBH=12r+β+δBH+μRBHeff−1IB+12r+β+δBH+μRBHeff−1IH+r+β+δBH+μRBHeff−1IBH
This impliesL̇≤0 when RBHeff<1. Furthermore,
L̇=0 if and only if IB=IH=IBH=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in Ω contained in S,V,IB,IH,IBH,TB,TH,TBH,SV∈ℝ9>0 is reduced to the DFE. This proves the global asymptotic stability of the DFE E0BH on Ω if the effective reproduction number RBHeff<1 and is unstable otherwise.


## 4.5. Existence and Stability of Endemic Equilibrium (EE) of HBV-HIV/AIDS Coinfection Model

Let the unique endemic equilibrium point of a dynamical system in (1) which exists when the disease persists in the community is denoted by E∗BH=S∗,V∗,I∗B,I∗H,I∗BH,T∗B,T∗H,T∗BH,S∗V. From the analysis of HBV-only submodel (22) and HIV/AIDS-only submodel (41), there exists an endemic equilibrium point for HBV-only submodel and HIV/AIDS-only submodel if RB0>1 and RHeff>1, respectively.This implies the endemic equilibriumE∗BH=S∗,V∗,I∗B,I∗H,I∗BH,T∗B,T∗H,T∗BH,S∗V for HBV-HIV/AIDS coinfection exists because the effective reproduction number for HBV- HIV/AIDS coinfection is the maximum of RBeff,RH0, which is greater than one where
(74)S∗=bq+1−ρλ∗H+μσ+α+λ∗H+δB+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,V∗=ρbλ∗H+q+μ,I∗B=λ∗Bb1−ρλ∗H+μ+qλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,I∗H=λ∗Hbλ∗H+q+μq+1−ρλ∗H+μσ+α+λ∗H+δB+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μλ∗B1−πβ+δBH+μ+πγ+δH+μr+β+δBH+μ+λ∗Hρbλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μλ∗B1−πβ+δBH+μ+πγ+δH+μr+β+δBH+μ+λ∗Bλ∗Hrbq+1−ρλ∗H+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μλ∗B1−πβ+δBH+μ+πγ+δH+μr+β+δBH+μ,I∗BH=λ∗B1−πλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μI∗H+bλ∗H1−ρλ∗H+μ+qr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,T∗B=λ∗Bαb1−ρλ∗H+μ+qϕ+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,T∗H=πγI∗Hθ+μ,T∗BH=βλ∗B1−πλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μI∗H+bλ∗H1−ρλ∗H+μ+qε+μr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ,S∗V=εβλ∗B1−πλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μI∗H+bλ∗H1−ρλ∗H+μ+qμε+μr+β+δBH+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ+λ∗Bαbϕ1−ρλ∗H+μ+qμϕ+μλ∗H+q+μλ∗B+λ∗H+μα+λ∗H+δB+μ+σλ∗H+μ+θπγI∗Hμθ+μ.The stability analysis of the endemic equilibrium of the full model (1) in terms of the model parameters analytically is difficult. Hence, we will give an explanation of the stability analysis of the endemic equilibrium E∗BH of the full model (1) under the section numerical simulation.


## 5. Sensitivity Analysis

In this subsection, sensitivity analysis is performed to identify the most influential parameters for the spread as well as for control of infection in the community. To perform this, we use the techniques described in [36]. The results of sensitivity analysis and the set of parameters used are given in Figure 2 and Table 2, respectively.Figure 2
Sensitivity index of parameters.Table 2
Parameter values used in numerical simulations.
ParametersValueSourceμ0.01[31]b0.04∗N0AssumptionδH0.333[31]δB0.01[32]α0.3Assumptionγ0.6[33]δBH,β,r0.001, 0.015, and 0.001, respectivelyAssumptionσ0.075Assumptionρ0.65[34]π0.396[33]ωB0.4[34]ωH0.03Assumptionq0.1[34, 35]ϕ,θ,ε0.014, 0.013, and 0.012, respectivelyAssumptionFigure2 is about the sensitivity index of parameters. It shows that the most sensitive parameters are ωB and ωH, which are the effective contact rate for HBV and HIV/AIDS transmission, respectively. The least sensitive parameters are δB and q which are HBV-induced mortality and vaccine efficacy waning rate, respectively.


## 6. Numerical Simulation

We performed the numerical simulation using the parameter values in Table2 and the initial values given below.Initially,S0=100,000, V0=1000, IB0=10,000, IH0=3000, IBH0=1000, TB0=500, TH0=500, TBH0=100, and SV0=200. Initially, S0=100,000, V0=1000, IB0=10,000, IH0=3000, IBH0=1000, TB0=500, TH0=500, TBH0=100, and SV0=200.


## 7. Results and Discussions

Numerical simulations are carried out using MATLAB numerical solver (ode45) to visualize the dynamics of HBV-only submodel (22), HIV/AIDS-only submodel (41), and HBV-HIV/AIDS coinfection model (1). Figures 3–8 are about the stability of disease-free and endemic equilibria of HB-only submodel, HIV/AIDS-only submodel, and HBV-HIV/AIDS coinfection model at different reproduction numbers and the associated parameters given in Table 2. Figures 3 and 7 were plotted using the effective reproduction number RBHeff=0.046, which is less than unity, ωB=0.09, and μ=0.5 and keeping all other associated parameters as listed in Table 2. As the figures clearly show, the behavior of the infectious classes of HBV-only submodel (22) and HBV-HIV/AIDS coinfection model (1), each infectious class converges to the disease-free equilibrium point of the models. The convergence of all infectious classes to their disease-free equilibrium point of the models shows that the disease-free equilibria of the models are globally asymptotically stable, which indicates the absence of HBV and HBV-HIV/AIDS coinfection in the society. The simulation result of the stability of the DFE of HIV/AIDS-only submodel using the basic reproduction number RH0=0.003, ωH=0.003, and μ=0.5 and keeping all other associated parameters as listed in Table 2 is represented by Figure 5. The figure shows the convergence of the infection class to the disease-free equilibrium point of the model, which shows that the disease-free equilibrium point of the model is globally asymptotically stable. The global stability of the DFE of the model further indicates the infection is not spreading in the society or the disease dies out in the community. On the other hand, the simulations given in Figures 4, 6, and 8 are about the stability of endemic equilibrium of HB-only submodel, HIV/AIDS-only submodel, and HBV-HIV/AIDS coinfection model, respectively. The stability of endemic equilibria of the models was determined using RBeff=2.37, ωB=0.95, and q=0.4 and keeping all other associated parameters as listed in Table 2 for HB-only submodel, RH0=3.5, ωH=0.9, and δH=0.01 and keeping all other associated parameters as listed in Table 2 for HIV/AIDS-only submodel, and RBHeff=6.8, ωB=0.9, ωH=0.5, and α=0.03 and keeping all other associated parameters as listed in Table 2 for HBV-HIV/AIDS coinfection model. In all cases, the basic and effective reproduction numbers are greater than one, and the simulation results show the convergence of the solutions of the models to their endemic equilibrium points. The convergence of the solutions of the models to their endemic equilibrium point indicates that the endemic equilibrium points of the models are locally asymptotically stable (i.e., the diseases are spreading in the society). The effect of vaccination on the susceptible individuals for HBV is shown in Figure 9. When the proportion of immunized individuals increases from 0.91 to 0.99, the number of susceptible individuals decreases due to enough immunization coverage. In the first year, the trend of decreasing of the susceptible individuals for three different proportions seems the same, but after a year, the number of susceptible individuals decreases better as the proportion of vaccinated individuals increases. Since it is difficult to immunize all susceptible individuals, the number of susceptible individuals cannot be zero over the period of administration of vaccine. The simulations given in Figures 10–12 show the effects of variation of treatment rates on the infectious population. Figure 10 shows that the number of HB infectious individuals decreases, and almost all become near to zero after four years if the treatmentratealpha=0.525. It also shows using treatment at the highest possible rate decreases the number of HBV infectious individuals faster compared to using treatment at the lowest rate. The decreasing of the number of HB infectious individuals indicates that the effective treatment reduces the viral load of HBV infectious individuals to the required undetectable level, and hence, there will be no risk of transmitting the infection to others in the society. Similarly, Figure 11 shows that the number of HIV/AIDS infectious individuals decreases and becomes near to zero after seven years by increasing the treatment rate gamma from 0.6 to 0.9. This implies the effective treatment reduced the viral load of HIV infectious individuals to the required undetectable level. The effect of varying treatment rates on HBV and HIV/AIDS coinfectious individuals simultaneously is examined in Figure 12. As the figure shows, the number of HBV-HIV/AIDS coinfectious individuals initially 500 was decreased to about less than 20 individuals at the end of five years when the treatmentratebeta=0.18, which shows using treatment at the highest possible rate decreases the number of HBV-HIV/AIDS coinfectious individuals better compared to using treatment at the lowest rates. The decreasing of the number of HBV-HIV/AIDS coinfectious individuals indicates that the effective treatment reduces the viral load of HBV-HIV/AIDS coinfectious individuals to the required undetectable level and the risk of transmitting the infection to others in the society decreases. Figures 13–16 are the results of the numerical simulations of the effective reproduction number of HBV-HIV/AIDS coinfection model with variable parameter values. The simulation results in Figures 13 and 16 show that if the effective contact rates ωB and ωH are less than 0.42 and 0.58, respectively, the reproduction number of HBV-HIV/AIDS coinfection is less than one, which indicates HBV-HIV/AIDS coinfection dies out, whereas if the effective contact rates ωB and ωH are greater than 0.42 and 0.58, respectively, the reproduction number of HBV-HIV/AIDS coinfection is greater than one, which shows the infection is spreading in the community. On the other hand, Figures 14 and 15, respectively, show the absence of disease in the society if the proportion of vaccinated individuals ρ and the treatment rate α are greater than 0.14 and 0.28, respectively, and the existence of HBV-HIV/AIDS coinfection in the society if the proportion of vaccinated individuals ρ and the treatment rate α are less than 0.14 and 0.28, respectively.Figure 3
Stability of DFE of HB-only submodel atRBeff=0.046, ωB=0.09, and μ=0.5.Figure 4
Stability of EE of HB-only submodel atRBeff=2.37, ωB=0.95, and q=0.4.Figure 5
Stability of DFE of HIV/AIDS-only submodel atRH0=0.003, ωH=0.003, and μ=0.5.Figure 6
Stability of EE of HIV/AIDS-only submodel atRH0=3.5, ωH=0.9, and δH=0.01.Figure 7
Stability of DFE of HBV-HIV/AIDS coinfection model atRBHeff=0.046, ωB=0.09, and μ=0.5.Figure 8
Stability of EE of HBV-HIV/AIDS coinfection model atRBHeff=6.8, ωB=0.9, ωH=0.5, and α=0.03.Figure 9
The effect of vaccination on HB susceptible individuals.Figure 10
The effect of treatment on HB infectious individuals.Figure 11
Effect of treatment on HIV/AIDS infectious individuals.Figure 12
Effect of varying treatment rate on HBV-HIV/AIDS coinfectious individuals simultaneously.Figure 13
Simulation of the reproduction number of HBV-HIV/AIDS coinfection at variable contact rateωB.Figure 14
Simulation of the reproduction number of HBV-HIV/AIDS coinfection at a variable proportion of vaccinated individualsρ.Figure 15
Simulation of the reproduction number of HBV-HIV/AIDS coinfection at variable treatment rateα.Figure 16
Simulation of the reproduction number of HBV-HIV/AIDS coinfection at variable contact rate for HIV transmissionωH.


## 8. Conclusion

In this study, we formulated and analyzed a deterministic model to see the effect of vaccination and treatment on the transmission dynamics of HBV-HIV/AIDS coinfection. The basic reproduction number of HBV-HIV/AIDS coinfection model derived before implementing any intervention wasRBH0=maxωB/μ+σ+δB,ωH/μ+δH and the effective reproduction number derived after implementing both vaccination and treatment was RBHeff=maxωBq+μ1−ρ/q+μα+μ+σ+δB,ωH/μ+πγ+δH. Based on the data given in Table 2, we evaluated the numerical values of the basic and effective reproduction numbers to evaluate the effectiveness of our intervention. As the numerical values indicate, the basic reproduction number RBH0=max4.2,0.09=4.2 and the effective reproduction number RBHeff=max0.94,0.28=0.94. This tells us one infectious individual infects at least 4 healthy individuals in his/her infectious time if we do not implement any intervention mechanism. However, after incorporating vaccination and treatment efforts together in our model to reduce the burden caused by HBV-HIV/AIDS coinfection, the effective reproduction number becomes RBHeff=max0.94,0.28=0.94, which is less than the threshold value one. This implies the disease dies out in the community, and there will be no risk of transmitting the infection to others due to the effective vaccination and treatment efforts. We proved that the disease-free equilibrium points of the sub and full models are locally and globally asymptotically stable if the associated reproduction numbers RBeff, RHeff, and RBHeff are less than one and the endemic equilibrium points of the sub and full models are locally and globally asymptotically stable whenever the associated reproduction numbers RBeff, RHeff, and RBHeff are greater than one. From the sensitivity analysis carried out to illustrate the influence of different parameters on the effective reproduction number of HBV-HIV coinfection model, ωB and ωH, which are the effective contact rate for HBV and HIV transmission, respectively, are the most sensitive parameters. As the simulation results show, if the effective contact rates ωB and ωH are less than 0.42 and 0.58 for HBV and HIV transmission, respectively, the proportion of immune individuals following vaccination to HBV infection is greater than 0.14, the treatment rate α for HB chronic carriers is greater than 0.28, and the treatment rate β is greater than or equal 0.18 for HBV-HIV/AIDS-coinfected individuals, then the number of HBV infectious individuals, the number of HIV infectious individuals, and the number of HBV and HIV/AIDS coinfectious individuals decrease significantly and even fall to zero over time. Hence, it is concluded that decreasing the effective contact rates for HBV and HIV transmission and the use of vaccination and treatment at the highest possible rate are essential to control the spread of HBV-HIV/AIDS coinfection. From the numerical results, we recommend that public policymakers and other concerned bodies must focus on increasing vaccination coverage against HBV infection, treatment of hepatitis B and HIV/AIDS infections, and HBV-HIV/AIDS coinfection to control the spread of HBV-HIV/AIDS coinfection. Finally, we should mention that, for the sake of simplicity, the role of screening in the transmission dynamics of HBV-HIV/AIDS coinfection in a population was not considered in this study. It may affect the transmission dynamics of HBV-HIV/AIDS coinfection in a population. We leave this for future consideration.



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