# Coinfection Dynamics of HBV-HIV/AIDS with Mother-to-Child Transmission and Medical Interventions

Authors: Engida Endriyas Endashaw; Dawit Melese Gebru; Haileyesus Tessema Alemneh
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

In this study, we analyzed the effect of mother-to-child transmission (MTCT) of hepatitis B virus (HBV) and human immunodeficiency virus (HIV) on the transmission dynamics of their coinfection to make a recommendation based on reasons to public health sector, policy makers, and programme implementers. We proved that the solutions of the sub and full models are positive and bounded. The effective reproduction numbers of the models are derived using the next generation matrix method. The disease-free and endemic equilibria of the submodels and the coinfection model are computed, and the stability of those equilibria is analyzed using Routh-Hurwitz criteria after computing the associated effective reproduction numbers. We performed a sensitivity analysis to show the influence of different parameters on the effective reproduction number of HBV-HIV/AIDS coinfection model, and we identified the most sensitive parameters areτ2 and α1, which are the rate of MTCT of HIV and treatment rate for HBV infected class, respectively. The numerical simulation of the model is done using MATLAB and the findings from the simulations are discussed. From the results of numerical simulations, we observed that an increase in the rates of MTCT of HBV and HIV exacerbated HBV-HIV/AIDS coinfection, while a decrease in the rates of MTCT of these infections would decline the number of cases, minimize the spread, and help to eliminate HBV-HIV/AIDS coinfection from the society gradually.


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## Body

## 1. Introduction

Hepatitis B is a liver infection caused by HBV. It affects everyone including expectant women and their newly born children globally. The main routes of transmission are vertical transmission (MTCT), contact with infected blood, semen, and other bodily fluids [1, 2]. MTCT, by which HBV is transmitted from infected mothers to their infants, during pregnancy, at birth, or postnatally (during childcare or through breast milk), contributes significantly to the persistence of the high number of HBV carriers globally [3–6]. The chance of MTCT amongst infants born to HBV infected mothers is ranges from 10 to 40% in HBeAg negative mothers and greater than 90% in HBeAg positive mothers with HBV deoxyribonucleic acid (DNA) levels greater than 200,000 IU/ml [7]. Elimination of MTCT of HBV has been identified as a global public health priority. To prevent MTCT of HBV, World Health Organization advises universal immunization with at least three doses of HBV vaccine as first-line prevention against around birth infection for all newborns. Newborns should get their first dose of the vaccine along with hyperimmune hepatitis B immunoglobulin (HBIG) at birth [6]. However, in many resource-constrained countries, the initial dose is provided as a pentavalent vaccine in the EPI (Expanded Program on Immunization) program at 6 weeks of age, and therefore babies from birth to six weeks of age are not protected against vertical transmission [8].HIV also affects everyone including expectant women and their newly born children globally. It is a causative agent for acquired immunodeficiency syndrome (AIDS) which damages both humoral and cellular immunity resulting in increased susceptibility of the host to a wide range of [9, 10]. The main routes of transmission of HIV are the same as that of HBV [9, 11, 12]. MTCT, by which HIV is transmitted from infected mothers to their infants, contributes significantly to the persistence of the high numbers of HIV carriers globally [13, 14]. Without prevention, the chance of MTCT of HIV is 15–25% in developed countries and 25–35% in developing countries [15]. Primary HIV protection, the minimization of unintended pregnancies, good access to HIV diagnosis and counselling, the start of treatment, viral suppression for mothers living with HIV, safe delivery practices, good infant feeding practices, and access to postnatal antiretroviral (ARV) prophylaxis for infants are all important factors in the prevention of MTCT of HIV [16, 17]. Coinfection increases the morbidity and mortality beyond those caused by either infection alone. People coinfected with both infections 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 either infection alone [18, 19]. Thus, it is important to consider and study the effect of MTCT which contributes significantly to the persistence of the high number of HBV and HIV carriers globally.In the field of epidemiology, models are important tools in the study of extremely complex communicable diseases. Recently, numerous scholars are concentrated on the study of the transmission dynamics of a particular and coinfection of various diseases and how the diseases might be effectively managed and potentially eliminated. Wodajo and Mekonnen [20] proposed a mathematical model to study the dynamics of hepatitis B virus infection under the administration of vaccination and treatment, where HBV infection is transmitted in two ways through vertical and horizontal transmission. Their results show that the combined efforts of vaccination, effective treatment, and interruption of transmission make elimination of the infection plausible and may eventually lead to the eradication of the virus. Omondi et al. [21] developed a mathematical model describing the dynamics of HIV transmission by incorporating sexual orientation of individuals. They investigated the effect of the introduction of preexposure prophylaxis (PrEP) on the dynamics of the HIV. The results show that the introduction of PrEP has a positive effect on the limitation of the spread of HIV. Melese and Alemneh [22] developed a transmission dynamics model for VL–HIV coinfection by splitting the population into ten compartments. From the result they achieved, the authors concluded that increasing the rate of visceral leishmaniasis (VL) recovery (ϕ1), the recovery rate for VL–HIV Coinfection (ϕ2), removing reservoirs (c1), and minimizing the contact rate (βh) are important in controlling the transmission of individual and coinfection disease of VL and HIV. Teklu and Rao [23] proposed and analyzed a realistic compartmental mathematical model on the spread and control of HIV/AIDS-pneumonia coepidemic incorporating pneumonia vaccination and treatment for both infections at each infection stage in a population. The authors showed that pneumonia vaccination and treatment against disease have positive effect in decreasing pneumonia and coepidemic disease expansion and reducing the progression rate of HIV infection to the AIDS stage.However, mathematical models formulated to study the codynamics of HBV and HIV/AIDS are few in the literature, although the coexistence between the two infections exists. In our review of the literature, we found only three mathematical models of HBV-HIV/AIDS coinfection and we used them as the basis for our developed model as follows: Bowong et al. [24] developed a deterministic model for HBV and HIV coinfection. In their model, the authors did not consider treatment for all infections rather intended only on prevention against HBV infection. Nampala et al. [25] formulated an epidemiological model of hepatotoxicity and antiretroviral healing effects in HBV-HIV coinfection. The authors used numerical techniques to study the healing as well as toxic effect of the recently used HBV-HIV therapy, and as a result formulated a desirable combination for treating the effect of HBV and HIV infections. These authors concentrated on hepatotoxicity and treatment effects in their model rather than including protection strategies against all infections. The combined effect of vaccination and treatment on the transmission dynamics of HBV-HIV/AIDS coinfection has been studied by Endashaw and Mekonnen [26]. Their model subdivides the total population into nine mutually exclusive compartments depending on the disease status. The model has no recovery compartment for those who are recovered from HBV infection naturally. Moreover, the authors did not consider the effect of MTCT of both HBV and HIV infections on the transmission dynamics of the coinfection of these two viruses rather focusing on the horizontal transmission alone. They obtained results showing 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 are very essential to control the spread of HBV-HIV/AIDS coinfection as an important public health problem. We motivated by a study conducted by Endashaw and Mekonnen [26] and extended it by considering a recovery compartment for those individuals who are recovered from HBV infection naturally and MTCT of both HBV and HIV which are not considered in their study. Moreover, to the authors’ knowledge, none of the authors of these existing coinfection models have considered the effect of MTCT of hepatitis B virus and HIV on the coinfection dynamics of HBV-HIV/AIDS. As a result, we examine the effect of MTCT of HBV and HIV on the transmission dynamics of the coinfection of these two viruses with medical interventions. This extended model will be used to evaluate the effect of MTCT of both infections on their codynamics model with preventive (vaccination) and therapeutic (treatment) intervention strategies.


## 2. Baseline Model Description and Formulation

We grouped the entire populationNt into ten compartments as those who are susceptible to both diseases P1t, immune to HBV after vaccination P2t, only infected with HBV P3t, only infected with HIV P4t, infected with both HBV and HIV P5t, those who are receiving HBV treatment P6t, those who are receiving HIV treatment P7t, HBV-HIV/AIDS treated section P8t, individuals with suppressed viral load P9t, and HBV recovered class P10t. The entire population at time t, denoted by N(t), is given by
(1)Nt=P1t+P2t+P3t+P4t+P5t+P6t+P7t+P8t+P9t+P10t.(i)
The newly immunized individuals enter into the immunized compartmentP2 at a constant recruitment rate Λ(ii)
TheP4t compartment rises because of HIV infected newborns at a rate 1−d2τ2P4 and the transfer of individuals from P1t and P2t compartments by the infection rate λ2. It is supposed that an accurate contact of P4 compartment with both P1 and P2 classes lead to the birth of infected neonates with a rate of τ2. Of these newly born but infected neonates a fraction d2 dies during the birth due to the infection and the remaining complementary fraction 1−d2 enter into P4 class. Similarly, the P3t class increases due to HBV infected newborns at a rate 1−d1τ1P3 and the entering of individuals from P2t compartment by the force of infection λ1. It is assumed that the real contact among P3 and P2 classes lead to the birth of infected neonates with a rate of τ1. Of these newly born but infected neonates a fraction d1 dies during birth due to the infection and the remaining complementary fraction 1−d1 enters into P3 class(iii)
The combination among people is the same and become infected with HBV and HIV by the force of infectionλ1 and λ2, respectively(iv)
The efficacy of the vaccine wanes out at a rateω and the natural recovery rate of individuals from HBV infection in HBV-only infected class is r2(v)
The susceptible class rises due to the recruitment of unimmunized individuals, the transfer of naturally recovered individuals from recovered class, and the transfer of individuals whose vaccine efficacy wanes(vi)
Individuals inP1 and P2 compartments get HIV infection by the force of infection λ1, whereas individuals in P2 acquire HBV infection by the force of infection λ2. Individuals in HBV-only infected class get HIV infection by force of infection λ2 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 λ1 within a brief period of time before an infection with the first strain (HIV) has been established and an immune response has developed.(vii)
Individuals inP3 and P5 progress to P6 and P8 classes, respectively, after getting treatment. The proportion ζ of individuals in P4 progress to P7 due to the treatment rate α2 and the remaining proportion 1−ζ of individuals progress to P5 by force of infection λ1 before an infection with the first strain (HIV) has been established and an immune response has developed(viii)
Individuals who are recovered from HBV infection naturally in HBV-HIV/AIDS coinfected class enter in to HIV-only infected class at a rater1(ix)
Since the effective treatment reduces the viral load of the infected individuals inP6,P7, and P8 classes to the required undetectable level, individuals in these compartments progress to a suppressed viral load class at the progress rate θ1, θ2, and θ3, respectively(x)
If the vaccine efficacy not wanes, people who are immunized against HBV infection are susceptible to HIV-only(xi)
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(xii)
Recovered class increases due to a transfer of naturally recovered individuals from HBV infected class and decreases due to the progress of HBV recovered individuals to the susceptible class(xiii)
Those persons who get rid of HBV infection due to natural immunity inP3 class enter into the recovered class at rate of r2, but they do not susceptible to reacquiring the infection because they developed antibodies that protect them from HBV infection for the rest of their lives.Using Table1 and the model assumptions, the flow diagram for the transmission dynamics of the full model is given by:Table 1
Model Parameters and their descriptions.
ParametersDescriptionsd0Natural death rateΛRate of recruitment of individuals into immunized compartmentd1Probability of death of newborns infected with HBV at birthd2Probability of death of newborns infected with HIV at birthτ1Rate of MTCT of HBVτ2Rate of MTCT of HIVd3Mortality rate related to hepatitis B virusd4Death rate due to HIV/AIDSd5Mortality rate due to coinfectionα1Rate of treatment for hepatitis B virus infected individualsα2Rate of treatment forζ proportion of HIV/AIDS patientsα3Rate of treatment for those who are coinfectedλ1Infection related with hepatitis B virusλ2Infection related with HIV/AIDSr1A rate of return of HBV infected population fromP3 class naturally to former statusr2A rate of return of HBV infected population from coinfected class to former statusηFraction of vaccinated populationζFraction of HIV/AIDS patients entering to treatment compartment.θ1The transfer rate of HBV patients to suppressed viral load compartment after treatmentθ2The transfer rate of HIV/AIDS patients to suppressed viral load class after treatmentθ3The transfer rate of coinfected population to suppressed viral load class after treatmenth1Contact rate for HBVh2Contact rate for HIVωThe waning rate of vaccine efficacyr3The transfer rate of HB recovered individuals to susceptible classFrom the flow diagram of the model in Figure1, the dynamical system of the model is
(2)dP1dt=1−ηΛ+ωP2+r3P10−λ1+λ2+d0P1,dP2dt=ηΛ−λ2+ω+d0P2,dP3dt=1−d1τ1P3+λ1P1−r2+α1+λ2+d3+d0P3,dP4dt=1−d2τ2P4+λ2P1+P2+r1P5−ζα2+1−ζλ1+d4+d0P4,dP5dt=1−ζλ1P4+λ2P3−r1+α3+d5+d0P5,dP6dt=α1P3−θ1+d0P6,dP7dt=ζα2P4−θ2+d0P7,dP8dt=α3P5−θ3+d0P8,dP9dt=θ1P6+θ2P7+θ3P8−d0P9,dP10dt=r2P3−r3+d0P10,where the force of infection associated with HBV infection is given by
(3)λ1=h1P3+P5N.and the force of infection associated with HIV infection is given by
(4)λ2=h2+P4+P5N,with initial conditions,
(5)P10>0,P20>0,P30>0,P40>0,P50>0,P60>0,P70>0,P80>0,P90>0,andP100>0.Figure 1
Flow diagram of the full model.


### 2.1. Positivity of the Solutions and Boundedness of the Solution Region of the Full Model (1)

Theorem 1.
At the initial conditions (5), the solutions of the dynamical system (2) are nonnegative for time t>0 in the region
(6)Ω=P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+.Proof.
Let us take the first differential equation from the dynamical system (2)
(7)dP1dt=1−ηΛ+ωP2+r3P10−λ1+λ2+d0P1.
A separable first order ordinary differential equation of (7) for the variable P1 is
(8)dP1dt+λ1+λ2+d0P1=1−ηΛ+ωP2+r3P10.
Now taking the integrating factore∫λ1+λ2+d0dt, we get
(9)P1t=e−λ1+λ2+d0t1−ηΛ+ωP2+r3P10∫eλ1+λ2+d0tdt.
Integrating and simplifying (9) gives us
(10)P1t=C1e−λ1+λ2+d0t+1−ηΛ+ωP2+r3P10λ1+λ2+d0.where C1 is constant.After rearranging and computing (10) at t=0 we get
(11)C1=P10−1−ηΛ+ωP2+r3P10λ1+λ2+d0Substituting (11) in (10) we get
(12)P1t=P10e−λ1+λ2+d0t+1−ηΛ+ωP2+r3P10λ1+λ2+d01−e−λ1+λ2+d0t.which is the solution of (7).Since,P10>0,e−λ1+λ2+d0t>0 and 1−ηΛ+ωP2+r3P10/λ1+λ2+d0>0 for t≥0, equation is positive. Therefore, P1t=P10e−λ1+λ2+d0t+1−ηΛ+ωP2+r3P10/λ1+λ2+d01−e−λ1+λ2+d0t>0Following the same procedure,P2t>0, P3t>0, P4t>0, P5t>0, P6t>0, P7t>0, P8t>0, P9t>0, and P10t>0. Hence, from the above proof, we can conclude that whenever the initial values of the systems are all positive, then all the solutions of the dynamical system (2) are positive.Theorem 2.
The dynamical system (2) is positively invariant in the region
(13)Ω=P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+:N≤Λd0Proof.
The dynamics of total populationNt with respect to time t is computed as
(14)dNdt=Λ−d0N+τ1−d1τ1+d3P3+τ2−d2τ2+d4P4−d5P5
In the absence of mortality due to HBV infection, HIV/AIDS, and HBV-HIV/AIDS coinfection and comparing both sides of the equation (14) using standard comparison theorem,
(15)dNdt≤Λ−dN.
The solution for the inequality (15) after some steps becomes
(16)Nt≤N0e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size Nt⟶Λ/d0 whenever N0≤Λ/d0. This implies
(17)0≤Nt≤Λd0.
The inequality (17) shows that all the feasible solutions of the components of a dynamical system (2) with initial conditions enter the region Ω=P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+:N≤Λ/d0 for all t>0. Thus, the region Ω is positively invariant.


## 2.1. Positivity of the Solutions and Boundedness of the Solution Region of the Full Model (1)

Theorem 1.
At the initial conditions (5), the solutions of the dynamical system (2) are nonnegative for time t>0 in the region
(6)Ω=P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+.Proof.
Let us take the first differential equation from the dynamical system (2)
(7)dP1dt=1−ηΛ+ωP2+r3P10−λ1+λ2+d0P1.
A separable first order ordinary differential equation of (7) for the variable P1 is
(8)dP1dt+λ1+λ2+d0P1=1−ηΛ+ωP2+r3P10.
Now taking the integrating factore∫λ1+λ2+d0dt, we get
(9)P1t=e−λ1+λ2+d0t1−ηΛ+ωP2+r3P10∫eλ1+λ2+d0tdt.
Integrating and simplifying (9) gives us
(10)P1t=C1e−λ1+λ2+d0t+1−ηΛ+ωP2+r3P10λ1+λ2+d0.where C1 is constant.After rearranging and computing (10) at t=0 we get
(11)C1=P10−1−ηΛ+ωP2+r3P10λ1+λ2+d0Substituting (11) in (10) we get
(12)P1t=P10e−λ1+λ2+d0t+1−ηΛ+ωP2+r3P10λ1+λ2+d01−e−λ1+λ2+d0t.which is the solution of (7).Since,P10>0,e−λ1+λ2+d0t>0 and 1−ηΛ+ωP2+r3P10/λ1+λ2+d0>0 for t≥0, equation is positive. Therefore, P1t=P10e−λ1+λ2+d0t+1−ηΛ+ωP2+r3P10/λ1+λ2+d01−e−λ1+λ2+d0t>0Following the same procedure,P2t>0, P3t>0, P4t>0, P5t>0, P6t>0, P7t>0, P8t>0, P9t>0, and P10t>0. Hence, from the above proof, we can conclude that whenever the initial values of the systems are all positive, then all the solutions of the dynamical system (2) are positive.Theorem 2.
The dynamical system (2) is positively invariant in the region
(13)Ω=P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+:N≤Λd0Proof.
The dynamics of total populationNt with respect to time t is computed as
(14)dNdt=Λ−d0N+τ1−d1τ1+d3P3+τ2−d2τ2+d4P4−d5P5
In the absence of mortality due to HBV infection, HIV/AIDS, and HBV-HIV/AIDS coinfection and comparing both sides of the equation (14) using standard comparison theorem,
(15)dNdt≤Λ−dN.
The solution for the inequality (15) after some steps becomes
(16)Nt≤N0e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size Nt⟶Λ/d0 whenever N0≤Λ/d0. This implies
(17)0≤Nt≤Λd0.
The inequality (17) shows that all the feasible solutions of the components of a dynamical system (2) with initial conditions enter the region Ω=P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+:N≤Λ/d0 for all t>0. Thus, the region Ω is positively invariant.


## 3. Analysis of the HBV and HIV/AIDS Submodels

To lay down the foundation for the analysis of model (2), it is important to observe the dynamics of the submodels in advance.


### 3.1. HBV-Only Submodel

From the model (2), the HBV-only submodel is obtained by setting P4=P5=P7=P8=P9=P10=0. Thus, we have the following dynamical system
(18)dP1dt=1−ηΛ+ωP2+r3P10−λ1+d0P1,dP2dt=ηΛ−ω+d0P2,dP3dt=1−d1τ1P3+λ1P1−r2+α1+d3+d0P3,dP6dt=α1P3−d0P6,dP10dt=r2P3−r3+d0P10,where the force of infection is
(19)λ1=h1P3N1.with initial conditions,
(20)P10>0,P20>0,P30>0,P60>0,andP100>0.The total population of the dynamical system (18) is given by
(21)N1t=P1t+P2t+P3t+P6t+P10t.


#### 3.1.1. Positivity of the Solutions and Boundedness of the Solution Region of HBV-Only Submodel

As we did in subsection2.1, it is important to prove that all the solutions of the dynamical system (18) with positive initial conditions remain positive and the solution region is bounded. This fact is verified in theorem 3 and 4 as follows:Theorem 3.
The solutions of the dynamical system (18) with positive initial conditions are positive for time t≥0 in the region
(22)Ω1=P1,P2,P3,P6,P10∈ℝ5+.Proof.
TakingdP1/dt=1−ηΛ+ωP2+r2P3−λ1+d0P1 from the dynamical system (18) we get
(23)dP1dt+λ1+d0P1=1−ηΛ+ωP2+r2P3.
Equation (23) is a separable first order ordinary differential equation for the variable P1. After integrating and some simplifications, we get the solution
(24)P1t=P10e−λ1+d0t+1−ηΛ+ωP2+r2P3λ1+d01−e−λ1+d0t.
Since,P10>0,e−λ1+d0t>0 and 1−ηΛ+ωP2+r2P3/λ1+d0>0 for t≥0, the solution we found is positive. Following the same procedure, P2t>0, P3t>0, P6t>0, and P10t>0. Therefore, the solutions of the dynamical system are positive whenever the initial values positive.Theorem 4.
The dynamical system (18) is positively invariant in the region
(25)Ω1=P1,P2,P3,P6,P10∈ℝ5>+:N1≤Λd0.Proof.
The dynamics of total populationN1t with respect to time t is computed as
(26)dN1dt=Λ−d0N1−d3P3.
In the absence of mortality due to HBV infection and comparing both sides of equation (26) using standard comparison theorem, we obtain
(27)dN1dt≤Λ−d0N1.
After some steps the solution for equation (27) is
(28)N1t≤N10e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size N1t≤Λ/d0
This implies0≤N1t≤Λ/d0, which shows all the feasible solutions of the model (18) with initial conditions enter the region Ω1=P1,P2,P3,P6,P10∈ℝ5+:N1≤Λ/d0.
Thus, the regionΩ1 is bounded.


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

At DFE point it is assumed that there is no disease in the population. The DFE of the dynamical system (18) represented by E01 is obtained by setting the right-hand side of the dynamical system equal to zero, providing that P3=P6=0. After some simple calculation, E01 is equal to
(29)E01=Λω+d01−ηd0ω+d0,ηΛω+d0,0,0,0.


#### 3.1.3. Effective Reproduction Number of HBV-Only Submodel

The effective reproduction number of hepatitis B infected individuals, denoted byR1eff that of the dynamical system (18), 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 [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE01/∂xj∂viE01/∂xj−1 of the dynamical system (18) [28, 29], 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 E01 is the disease-free equilibrium point.The corresponding Jacobian matrices ofF and V computed at DFE point E01, respectively, are given as follows:
(30)F=1−d1τ1ω+d0+h1ω+d01−ηω+d0,and(31)V=r2+α1+d3+d0.From this, it follows that(32)R1eff=ρFV−1=1−d1τ1ω+d0+h1ω+d01−ηω+d0r2+α1+d3+d0.


#### 3.1.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel

Theorem 5.
The disease-free equilibrium pointE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is locally asymptotically stable if the effective reproduction number R1eff<1.Proof.
The Jacobean matrix of the dynamical system (18) at the DFE point E01 is
(33)J1E01=−d0ω00r30−ω+d000000−r2+α1+d3+d00000α1−d0000r20−r3+d0.
After some steps, the roots of (33) are
(34)λ1=λ2=−d0,λ3=−ω+d0,λ4=−r3+d0,λ5=−r2+α1+d3+d0.
This shows all the eigenvalues of the Jacobean matrix have negative real parts whenR1eff<1
Hence, the disease-free equilibriumE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is locally asymptotically stable if the effective reproduction number R1eff<1 and unstable otherwise.


#### 3.1.5. Global Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel

Theorem 6.
The disease-free equilibrium pointE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is globally asymptotically stable in the feasible region Ω1 if the effective reproduction number R1eff<1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(35)f1=u1P3,
whereu1=1/r2+α1+d3+d0.
The time derivative of (35) along the solution path yields
(36)f1̇=u11−d1τ1P3+h1P1P3N1−r2+α1+d3+d0P3=u11−d1τ1+h1P1N1−r2+α1+d3+d0P3
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 equilibriumE01=r2+α1+d3+d0/r2+α1+d3+d0,ηΛ/ω+d0,0,0,0. with R1eff=1−d1τ1ω+d0+h1ω+d01−η/ω+d0r2+α1+d3+d0,f1̇ can be simplified as follows:
(37)f1̇≤u11−d1τ1+h1P01N10−r2+α1+d3+d0P3.
Substitutingh1=R1effω+d0r2+α1+d3+d0−ω+d01−d1τ1/ω+d01−η in (37) and simplification gives us f1̇≤u1r2+α1+d3+d0R1eff−1P3. After substituting u1 we get f1̇≤R1eff−1P3.
This shows thatf1̇<0 when R1eff<1. Furthermore, f1̇=0if and only if P3=0 or R1eff=1 and Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 is the only singleton set in P1,P2,P3,P6,P10∈Ω1:f1̇=0. Thus, by LaSalle’s invariance principle [30], the DFE E01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 is globally asymptotically stable if R1eff<1.


#### 3.1.6. Existence of Endemic Equilibrium (EE) of HBV-Only Submodel

Suppose thatE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 be an arbitrary EE equilibrium point of HB-only submodel (18), which occurs when the disease persists in the society. After long steps we obtained the endemic equilibrium point E∗1=P1∗,P2∗,P∗3,P∗6,P∗10.where(38)P1∗=Λr3+d0ω+d01−ηr2+α1+d3+d0−1−d1τ1λ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,P2∗=ηΛω+d0,P3∗=λ1∗Λr3+d0ω+d01−ηλ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,P∗6=α1λ1∗Λr3+d0ω+d01−ηλ1∗d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d02ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,and
(39)P∗10=r2λ1∗Λr3+d0ω+d01−ηλ1∗r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d02r2+α1+d3+d0−1−d1τ1The associated force of infection of the dynamical system (18) is given by
(40)λ1∗=h1P3∗N1∗.where
(41)N1∗=λ1∗Λ/d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0+Λr3+d0ω+d0r2+α1+d3+d0−1−d1τ1λ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1.After substitutingP∗3 and N1∗ in (40) and simplifying it we got
(42)λ1∗=d0ω+d0r2+α1+d3+d0R1eff−1α1+d0ω+d01−η+d0ηα1+d3+d0−d0η1−d1τ1.Equation (42) shows that λ1∗>0 if R1eff>1. Hence, the endemic equilibrium point E∗1 of HBV-only submodel (18) exists whenever R1eff>1.


#### 3.1.7. Local Stability of the Endemic Equilibrium Point of HBV-Only Submodel

Theorem 7.
The endemic equilibrium pointE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 of the model (18) is locally asymptotically stable if R1eff>1.Proof.
The Jacobian matrix of (18) at the endemic equilibrium E∗1 is
(43)J1E∗1=−E1+d0ω−E20r30−ω+d0000E10−E3+E20000α1−d0000r20−r3+d0,where
(44)E1=h1P∗3N1∗=Λω+d0r3+d0r2+α1+d3+d0−1−d1τ1R1eff−1Λ/d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0,E2=h1P1∗N1∗=h1ω+d01−ηr2+α1+d3+d0−1−d1τ1ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0+Λr3+d0ω+d0r2+α1+d3+d0−1−d1τ1ω+d0r2+α1+d3+d0R1eff−1,E3=1−d1τ1−r2+α1+d3+d0.
Letλi be the eigenvalues of J1E∗1 where i=1, 2, 3, 4, 5.
The corresponding characteristic equation of (43) is
(45)λ+d0λ+ω+d0z3λ3+z2λ2+z1λ+z0=0,
This implies the first two eigenvalues of (45) are λ1=−d0<0 and λ2=−ω+d0<0. But, the algebraic sign of the remaining three eigenvalues is determined using Routh–Hurwitz stability criterion from the characteristic polynomial
(46)z3λ3+z2λ2+z1λ+z0=0,where
z3=1>0, z2=E1+E2+E3+r3+2d0
z1=E1+d0E2+E3+r3+d0+E2+E3r3+d0, and z0=E1+d0E2+E3r3+d0.
The sign of the coefficientsz0, z1, and z2 of the characteristic polynomial z3λ3+z2λ2+z1λ+z0=0 is positive if R1eff>1. In addition to that z1z2−z0z3>0. Thus, by Routh–Hurwitz stability criterion the endemic equilibrium E∗1 is stable if R1eff>1 and unstable if R1eff<1.


#### 3.1.8. Global Stability of the Endemic Equilibrium Point of HBV-Only Submodel

Theorem 8.
The endemic equilibrium pointE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 of the dynamical system (18) is globally asymptotically stable if R1eff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(47)L1P1,P3=b1P1−P1∗2+b2P3−P3∗2,b1>0,b2>0.
DifferentiatingL1P1,P3 with respect to time gives
(48)L̇1P1,P3=2b1P1−P1∗1−ηΛ+ωP2+r3P10−h1P3N1+d0P1+2b2P3−P3∗1−d1τ1P3+h1P3N1P1−r2+α1+d3+d0P3.
Substituting1−ηΛ+ωP2+r3P10=h3P∗3/N1∗+d0P1∗and r2+α1+d3+d0−1−d1τ1=h1P∗3/N1∗ with N1≈N1∗ in (48) yields:
(49)L̇1P1,P3=2b1P1−P1∗h1P∗3N1∗P1∗+d0P1∗−h1P3N1P1−d0P1+2b2P3−P3∗h1P3N1P1−h1P3N1∗P1∗=−2b1d0P1−P1∗2−2b1h1N1P3P1−P1∗2+2h1N1P3∗−P3P1−P1∗b1P1∗−b2P3≤0ifP3≥b1P1∗b2,P1≥P1∗,P3≤P∗3.
This impliesL̇1P1,P3≤0, and it vanishes if and only if P1∗=P1, P3∗=P3, and P1∗=b2P3/b1
This shows thatE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 is the largest compact invariant singleton set in P1∗,P2∗,P∗3,P∗6,P∗10∈Ω1:L̇1=0
Hence, by LaSalle’s invariance principle [30], the endemic equilibrium E∗1 is globally asymptotically stable in the invariant region whenever R1eff>1.


### 3.2. HIV/AIDS-Only Submodel

This submodel is obtained by settingP2=P3=P5=P6=P8=P9=P10=0. Thus, we have the following dynamical system
(50)dP1dt=Λ−λ2+d0P1,dP4dt=1−d2τ2P4+λ2P1−ζα2+d4+d0P4,dP7dt=ζα2P4−d0P7,where the force of infection is given by
(51)λ2=h2P4N2.with initial conditions, P10>0,P40>0, and P70>0The total population of the dynamical system (50) is given by:
(52)N2t=P1t+P4t+P7t.


#### 3.2.1. Positivity of the Solutions and Boundedness of the Solution Region of HIV/AIDS-Only Submodel

Here, we proved that all the solutions of the dynamical system (50) with positive initial conditions remain positive and the solution region is bounded as follows:Theorem 9.
The solutions of the dynamical system (50) with positive initial conditions are positive for time t≥0 in the region
(53)Ω2=P1,P4,P7∈ℝ3+Proof.
TakingdP1/dt=Λ−λ2+d0P1 from the dynamical system (50), we get
(54)dP1dt+λ2+d0P1=Λ.
(54) is a separable first order ordinary differential equation for the variable P1. After some steps we got the solution
(55)P1t=P10e−λ2+d0t+Λλ2+d01−e−λ2+d0t.
Since,P10>0,e−λ2+d0t>0 and Λ/λ2+d0>0 for t≥0, the solution we found is positive. Following the same procedure, P4t>0 and P10t>0. Therefore, the solutions of the model (50) are positive whenever the initial values positive.Theorem 10.
The dynamical system (50) is positively invariant in the region
(56)Ω2=P1,P4,P7∈ℝ3>+:N2≤Λd0.Proof.
The dynamics of total populationN2t with respect to time t is computed as
(57)dN2dt=Λ−d0N2−d4P4.
In the absence of mortality due to HIV infection and comparing both sides of equation (57) using standard comparison theorem, we obtain
(58)dN2dt≤Λ−d0N2.
After some steps the solution for the inequality (58) is
(59)N2t≤N0e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size N2t≤Λ/d0
This implies0≤N2t≤Λ/d0 which shows all the feasible solutions of the model (50) with initial conditions enter the region
(60)Ω2=P1,P4,P7∈ℝ3+:N2≤Λd0.
Thus, the regionΩ2 is positively invariant.


#### 3.2.2. Existence of the Disease - Free Equilibrium Point of HIV/AIDS-Only Submodel

The disease-free equilibrium point of HIV/AIDS-only submodel (50), represented by E02=P10,P40,P70, is obtained by setting the right-hand side of all the components of the model equal to zero, providing that P4=0. It is equal to
(61)E02=P10,P40,P70=Λd0,0,0.


#### 3.2.3. Effective Reproduction Numbers of HIV/AIDS-Only Submodel

In the same way that we have shown in sub section3.1.3, the effective reproduction number of HIV/AIDS-only infected individuals is
(62)R2eff=1−d2τ2+h2ζα2+d4+d0.


#### 3.2.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 11.
The disease-free equilibrium pointE02=Λ/d0,0,0 of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R2eff<1.Proof.
The Jacobean matrix of the dynamical system (50) at the DFE point E02=Λ/d0,0,0 is
(63)J2E02=−d0−h2001−d2τ2+h2−ζα2+d4+d000ζα2−d0.
The corresponding characteristic equation of (63) is
(64)−d0+λ−h200−ζα2+d4+d0−h2+1−d2τ2+λ00ζα2−d0+λ=0.
After some necessary steps, the roots of the characteristic equation (64) are λ1=λ2=−d0 and λ3=−ζα2+d4+d0+h2+1−d2τ2.
The simplified value ofλ3=ζα2+d4+d0R2eff−1.
Clearly,λ3 is negative if R2eff<1. This shows that all the eigenvalues of (64) have negative real parts when R2eff<1. Hence, the disease-free equilibrium E02=Λ/d0,0,0 of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R2eff<1 and unstable otherwise.


#### 3.2.5. Global Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 12.
The disease-free equilibrium pointE02=Λ/d0,0,0 of the dynamical system (50) is globally asymptotically stable in the feasible region Ω2 if the effective reproduction number R2eff<1.Proof.
To prove the global asymptotic stability of the disease-free equilibrium point, we used LaSalle-Lyapunov candidate function as follows:
LetW2 be a LaSalle-Lyapunov candidate function such that
(65)W2=i1P4.where
(66)i1=1ζα2+d4+d0.
The time derivative of (65) along the solution path yields
(67)W2̇=i1Ṗ4=i11−d2τ2+h2P1N2−ζα2+d4+d0P4.
Since the state variables of the model when the 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 equilibriumE02=Λ/d0,0,0 with R2eff<1, W2̇ can be simplified as
W2̇≤i11−d2τ2+h2P10/N20−ζα2+d4+d0P4. Substitution of i1 and simplification gives
(68)W2̇≤R2eff−1P4.
Inequality (68) shows that W2̇<0 when R2eff<1. Furthermore, W2̇=0 if and only if P4=0 or R2eff=1 and Λ/d0,0,0 is the only singleton set in P1,P4,P7∈ℝ3+∈Ω2:W2̇=0. Thus, by LaSalle’s invariance principle, the DFE E02=Λ/d0,0,0 is globally asymptotically stable if R2eff<1.


#### 3.2.6. Existence of Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Suppose thatE∗2=P∗1,P∗4,P∗7 be an arbitrary endemic equilibrium point of HIV/AIDS-only submodel (50), which occurs when the disease persists in the society. After long steps we obtained the endemic equilibrium point
(69)E∗2=Λζα2+d0d0ζα2+d4+d0R2eff−1+ζα2+d0,ΛR2eff−1ζα2+d4+d0R2eff−1+ζα2+d0,Λζα2R2eff−1d0ζα2+d4+d0R2eff−1+ζα2+d0.The associated force of infection of the dynamical system (50) is given by
(70)λ2∗=h2P4∗N2∗,where
(71)N2∗=h2Λζα2+d0d0ζα2+d4+d0ζα2+d4+d0R2eff−1+ζα2+d0.We can show the existence of the endemic equilibrium point of the model by substitutingP∗4 and N2∗ in (70). After substitution we got
(72)λ2∗=d0ζα2+d4+d0R2eff−1ζα2+d0.Clearly,λ2∗>0 if R2eff>1. Hence, an endemic equilibrium point E∗2=P∗1,P∗4,P∗7 of the dynamical system (50) exists whenever R2eff>1


#### 3.2.7. Local Stability of the Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 13.
The endemic equilibriumE∗2=P∗1,P∗4,P∗7 of the model (50) is locally asymptotically stable if R2eff>1.Proof.
The Jacobian matrix of the model (50) at the endemic equilibrium point E∗2 is
(73)J2E∗2=−F1+d0−F20F1F300ζα2−d0,
where
F1=d0ζα2+d4+d0R2eff−1/ζα2+d0, F2=ζα2+d4+d0, and F3=1−d2τ2
The corresponding characteristic equation of (73) is
(74)λ3+g2λ2+g1λ+g0=0,where
(75)g2=F1+2d0−F3,g1=F1F2+F1+d0d0−F3−F3d0,
and(76)g0=F1F2d0−F3d0F1+d0.
The sign of the coefficientsg0, g1, and g2 of the characteristic polynomial λ3+g2λ2+g1λ+g0=0 is positive if R2eff>1. In addition to that, g1g2>g0. Thus, by Routh–Hurwitz stability criterion the endemic equilibrium E∗2 is stable if R2eff>1 and unstable if R2eff<1


#### 3.2.8. Global Stability of the Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 14.
The endemic equilibriumE∗2=P∗1,P∗4,P∗7 of the dynamical system (50) is globally asymptotically stable if R2eff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(77)L2P1,P4=h3P1−P1∗2+h4P4−P4∗2,h3>0,h4>0.
DifferentiatingL2P2,P4 with respect to time gives us
(78)L̇2P1,P4=2h3P1−P1∗P1̇+2h4P4−P4∗Ṗ4=2h3P1−P1∗Λ−h2P4N2+d0P1+2h4P4−P4∗1−d2τ2P4+h2P4N2P1−ζα2+d4+d0P4.
SubstitutingΛ=h2P∗4/N2∗+d0P1∗ and 1−d2τ2+ζα2+d4+d0=h2P∗4/N2∗ with N2≈N2∗in (78) yields:
(79)L̇2P1,P4=2h3P1−P1∗h2P∗4N2∗P1∗+d0P1∗−h2P4N2P1−d0P1+2h4P4−P4∗h2P4N2P1−h2P4N2∗P1∗=−2h3d0P1−P1∗2−2h2h3N2P4P1−P1∗2+2h2N2P4∗−P4P1−P1∗h3P1∗−h4P4≤0ifP4≥h3P1∗h4,P1∗≤P1,P∗4≥P4.
This impliesL̇2P1,P4≤0, and it vanishes if and only if P1∗=P1, P4∗=P4, and P1∗=h4P4/h3
This shows thatE∗2=P∗1,P∗4,P∗7 is the largest compact invariant singleton set in P∗1,P∗4,P∗7∈Ω2:L̇2=0
Hence, by LaSalle’s invariance principle [30], the endemic equilibrium E∗2 is globally asymptotically stable in the invariant region whenever R2eff>1.


## 3.1. HBV-Only Submodel

From the model (2), the HBV-only submodel is obtained by setting P4=P5=P7=P8=P9=P10=0. Thus, we have the following dynamical system
(18)dP1dt=1−ηΛ+ωP2+r3P10−λ1+d0P1,dP2dt=ηΛ−ω+d0P2,dP3dt=1−d1τ1P3+λ1P1−r2+α1+d3+d0P3,dP6dt=α1P3−d0P6,dP10dt=r2P3−r3+d0P10,where the force of infection is
(19)λ1=h1P3N1.with initial conditions,
(20)P10>0,P20>0,P30>0,P60>0,andP100>0.The total population of the dynamical system (18) is given by
(21)N1t=P1t+P2t+P3t+P6t+P10t.


### 3.1.1. Positivity of the Solutions and Boundedness of the Solution Region of HBV-Only Submodel

As we did in subsection2.1, it is important to prove that all the solutions of the dynamical system (18) with positive initial conditions remain positive and the solution region is bounded. This fact is verified in theorem 3 and 4 as follows:Theorem 3.
The solutions of the dynamical system (18) with positive initial conditions are positive for time t≥0 in the region
(22)Ω1=P1,P2,P3,P6,P10∈ℝ5+.Proof.
TakingdP1/dt=1−ηΛ+ωP2+r2P3−λ1+d0P1 from the dynamical system (18) we get
(23)dP1dt+λ1+d0P1=1−ηΛ+ωP2+r2P3.
Equation (23) is a separable first order ordinary differential equation for the variable P1. After integrating and some simplifications, we get the solution
(24)P1t=P10e−λ1+d0t+1−ηΛ+ωP2+r2P3λ1+d01−e−λ1+d0t.
Since,P10>0,e−λ1+d0t>0 and 1−ηΛ+ωP2+r2P3/λ1+d0>0 for t≥0, the solution we found is positive. Following the same procedure, P2t>0, P3t>0, P6t>0, and P10t>0. Therefore, the solutions of the dynamical system are positive whenever the initial values positive.Theorem 4.
The dynamical system (18) is positively invariant in the region
(25)Ω1=P1,P2,P3,P6,P10∈ℝ5>+:N1≤Λd0.Proof.
The dynamics of total populationN1t with respect to time t is computed as
(26)dN1dt=Λ−d0N1−d3P3.
In the absence of mortality due to HBV infection and comparing both sides of equation (26) using standard comparison theorem, we obtain
(27)dN1dt≤Λ−d0N1.
After some steps the solution for equation (27) is
(28)N1t≤N10e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size N1t≤Λ/d0
This implies0≤N1t≤Λ/d0, which shows all the feasible solutions of the model (18) with initial conditions enter the region Ω1=P1,P2,P3,P6,P10∈ℝ5+:N1≤Λ/d0.
Thus, the regionΩ1 is bounded.


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

At DFE point it is assumed that there is no disease in the population. The DFE of the dynamical system (18) represented by E01 is obtained by setting the right-hand side of the dynamical system equal to zero, providing that P3=P6=0. After some simple calculation, E01 is equal to
(29)E01=Λω+d01−ηd0ω+d0,ηΛω+d0,0,0,0.


### 3.1.3. Effective Reproduction Number of HBV-Only Submodel

The effective reproduction number of hepatitis B infected individuals, denoted byR1eff that of the dynamical system (18), 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 [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE01/∂xj∂viE01/∂xj−1 of the dynamical system (18) [28, 29], 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 E01 is the disease-free equilibrium point.The corresponding Jacobian matrices ofF and V computed at DFE point E01, respectively, are given as follows:
(30)F=1−d1τ1ω+d0+h1ω+d01−ηω+d0,and(31)V=r2+α1+d3+d0.From this, it follows that(32)R1eff=ρFV−1=1−d1τ1ω+d0+h1ω+d01−ηω+d0r2+α1+d3+d0.


### 3.1.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel

Theorem 5.
The disease-free equilibrium pointE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is locally asymptotically stable if the effective reproduction number R1eff<1.Proof.
The Jacobean matrix of the dynamical system (18) at the DFE point E01 is
(33)J1E01=−d0ω00r30−ω+d000000−r2+α1+d3+d00000α1−d0000r20−r3+d0.
After some steps, the roots of (33) are
(34)λ1=λ2=−d0,λ3=−ω+d0,λ4=−r3+d0,λ5=−r2+α1+d3+d0.
This shows all the eigenvalues of the Jacobean matrix have negative real parts whenR1eff<1
Hence, the disease-free equilibriumE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is locally asymptotically stable if the effective reproduction number R1eff<1 and unstable otherwise.


### 3.1.5. Global Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel

Theorem 6.
The disease-free equilibrium pointE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is globally asymptotically stable in the feasible region Ω1 if the effective reproduction number R1eff<1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(35)f1=u1P3,
whereu1=1/r2+α1+d3+d0.
The time derivative of (35) along the solution path yields
(36)f1̇=u11−d1τ1P3+h1P1P3N1−r2+α1+d3+d0P3=u11−d1τ1+h1P1N1−r2+α1+d3+d0P3
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 equilibriumE01=r2+α1+d3+d0/r2+α1+d3+d0,ηΛ/ω+d0,0,0,0. with R1eff=1−d1τ1ω+d0+h1ω+d01−η/ω+d0r2+α1+d3+d0,f1̇ can be simplified as follows:
(37)f1̇≤u11−d1τ1+h1P01N10−r2+α1+d3+d0P3.
Substitutingh1=R1effω+d0r2+α1+d3+d0−ω+d01−d1τ1/ω+d01−η in (37) and simplification gives us f1̇≤u1r2+α1+d3+d0R1eff−1P3. After substituting u1 we get f1̇≤R1eff−1P3.
This shows thatf1̇<0 when R1eff<1. Furthermore, f1̇=0if and only if P3=0 or R1eff=1 and Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 is the only singleton set in P1,P2,P3,P6,P10∈Ω1:f1̇=0. Thus, by LaSalle’s invariance principle [30], the DFE E01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 is globally asymptotically stable if R1eff<1.


### 3.1.6. Existence of Endemic Equilibrium (EE) of HBV-Only Submodel

Suppose thatE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 be an arbitrary EE equilibrium point of HB-only submodel (18), which occurs when the disease persists in the society. After long steps we obtained the endemic equilibrium point E∗1=P1∗,P2∗,P∗3,P∗6,P∗10.where(38)P1∗=Λr3+d0ω+d01−ηr2+α1+d3+d0−1−d1τ1λ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,P2∗=ηΛω+d0,P3∗=λ1∗Λr3+d0ω+d01−ηλ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,P∗6=α1λ1∗Λr3+d0ω+d01−ηλ1∗d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d02ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,and
(39)P∗10=r2λ1∗Λr3+d0ω+d01−ηλ1∗r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d02r2+α1+d3+d0−1−d1τ1The associated force of infection of the dynamical system (18) is given by
(40)λ1∗=h1P3∗N1∗.where
(41)N1∗=λ1∗Λ/d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0+Λr3+d0ω+d0r2+α1+d3+d0−1−d1τ1λ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1.After substitutingP∗3 and N1∗ in (40) and simplifying it we got
(42)λ1∗=d0ω+d0r2+α1+d3+d0R1eff−1α1+d0ω+d01−η+d0ηα1+d3+d0−d0η1−d1τ1.Equation (42) shows that λ1∗>0 if R1eff>1. Hence, the endemic equilibrium point E∗1 of HBV-only submodel (18) exists whenever R1eff>1.


### 3.1.7. Local Stability of the Endemic Equilibrium Point of HBV-Only Submodel

Theorem 7.
The endemic equilibrium pointE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 of the model (18) is locally asymptotically stable if R1eff>1.Proof.
The Jacobian matrix of (18) at the endemic equilibrium E∗1 is
(43)J1E∗1=−E1+d0ω−E20r30−ω+d0000E10−E3+E20000α1−d0000r20−r3+d0,where
(44)E1=h1P∗3N1∗=Λω+d0r3+d0r2+α1+d3+d0−1−d1τ1R1eff−1Λ/d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0,E2=h1P1∗N1∗=h1ω+d01−ηr2+α1+d3+d0−1−d1τ1ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0+Λr3+d0ω+d0r2+α1+d3+d0−1−d1τ1ω+d0r2+α1+d3+d0R1eff−1,E3=1−d1τ1−r2+α1+d3+d0.
Letλi be the eigenvalues of J1E∗1 where i=1, 2, 3, 4, 5.
The corresponding characteristic equation of (43) is
(45)λ+d0λ+ω+d0z3λ3+z2λ2+z1λ+z0=0,
This implies the first two eigenvalues of (45) are λ1=−d0<0 and λ2=−ω+d0<0. But, the algebraic sign of the remaining three eigenvalues is determined using Routh–Hurwitz stability criterion from the characteristic polynomial
(46)z3λ3+z2λ2+z1λ+z0=0,where
z3=1>0, z2=E1+E2+E3+r3+2d0
z1=E1+d0E2+E3+r3+d0+E2+E3r3+d0, and z0=E1+d0E2+E3r3+d0.
The sign of the coefficientsz0, z1, and z2 of the characteristic polynomial z3λ3+z2λ2+z1λ+z0=0 is positive if R1eff>1. In addition to that z1z2−z0z3>0. Thus, by Routh–Hurwitz stability criterion the endemic equilibrium E∗1 is stable if R1eff>1 and unstable if R1eff<1.


### 3.1.8. Global Stability of the Endemic Equilibrium Point of HBV-Only Submodel

Theorem 8.
The endemic equilibrium pointE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 of the dynamical system (18) is globally asymptotically stable if R1eff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(47)L1P1,P3=b1P1−P1∗2+b2P3−P3∗2,b1>0,b2>0.
DifferentiatingL1P1,P3 with respect to time gives
(48)L̇1P1,P3=2b1P1−P1∗1−ηΛ+ωP2+r3P10−h1P3N1+d0P1+2b2P3−P3∗1−d1τ1P3+h1P3N1P1−r2+α1+d3+d0P3.
Substituting1−ηΛ+ωP2+r3P10=h3P∗3/N1∗+d0P1∗and r2+α1+d3+d0−1−d1τ1=h1P∗3/N1∗ with N1≈N1∗ in (48) yields:
(49)L̇1P1,P3=2b1P1−P1∗h1P∗3N1∗P1∗+d0P1∗−h1P3N1P1−d0P1+2b2P3−P3∗h1P3N1P1−h1P3N1∗P1∗=−2b1d0P1−P1∗2−2b1h1N1P3P1−P1∗2+2h1N1P3∗−P3P1−P1∗b1P1∗−b2P3≤0ifP3≥b1P1∗b2,P1≥P1∗,P3≤P∗3.
This impliesL̇1P1,P3≤0, and it vanishes if and only if P1∗=P1, P3∗=P3, and P1∗=b2P3/b1
This shows thatE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 is the largest compact invariant singleton set in P1∗,P2∗,P∗3,P∗6,P∗10∈Ω1:L̇1=0
Hence, by LaSalle’s invariance principle [30], the endemic equilibrium E∗1 is globally asymptotically stable in the invariant region whenever R1eff>1.


## 3.1.1. Positivity of the Solutions and Boundedness of the Solution Region of HBV-Only Submodel

As we did in subsection2.1, it is important to prove that all the solutions of the dynamical system (18) with positive initial conditions remain positive and the solution region is bounded. This fact is verified in theorem 3 and 4 as follows:Theorem 3.
The solutions of the dynamical system (18) with positive initial conditions are positive for time t≥0 in the region
(22)Ω1=P1,P2,P3,P6,P10∈ℝ5+.Proof.
TakingdP1/dt=1−ηΛ+ωP2+r2P3−λ1+d0P1 from the dynamical system (18) we get
(23)dP1dt+λ1+d0P1=1−ηΛ+ωP2+r2P3.
Equation (23) is a separable first order ordinary differential equation for the variable P1. After integrating and some simplifications, we get the solution
(24)P1t=P10e−λ1+d0t+1−ηΛ+ωP2+r2P3λ1+d01−e−λ1+d0t.
Since,P10>0,e−λ1+d0t>0 and 1−ηΛ+ωP2+r2P3/λ1+d0>0 for t≥0, the solution we found is positive. Following the same procedure, P2t>0, P3t>0, P6t>0, and P10t>0. Therefore, the solutions of the dynamical system are positive whenever the initial values positive.Theorem 4.
The dynamical system (18) is positively invariant in the region
(25)Ω1=P1,P2,P3,P6,P10∈ℝ5>+:N1≤Λd0.Proof.
The dynamics of total populationN1t with respect to time t is computed as
(26)dN1dt=Λ−d0N1−d3P3.
In the absence of mortality due to HBV infection and comparing both sides of equation (26) using standard comparison theorem, we obtain
(27)dN1dt≤Λ−d0N1.
After some steps the solution for equation (27) is
(28)N1t≤N10e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size N1t≤Λ/d0
This implies0≤N1t≤Λ/d0, which shows all the feasible solutions of the model (18) with initial conditions enter the region Ω1=P1,P2,P3,P6,P10∈ℝ5+:N1≤Λ/d0.
Thus, the regionΩ1 is bounded.


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

At DFE point it is assumed that there is no disease in the population. The DFE of the dynamical system (18) represented by E01 is obtained by setting the right-hand side of the dynamical system equal to zero, providing that P3=P6=0. After some simple calculation, E01 is equal to
(29)E01=Λω+d01−ηd0ω+d0,ηΛω+d0,0,0,0.


## 3.1.3. Effective Reproduction Number of HBV-Only Submodel

The effective reproduction number of hepatitis B infected individuals, denoted byR1eff that of the dynamical system (18), 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 [27, 28]. It is obtained by taking the spectral radius of the matrix FV−1=ρFV−1=∂FiE01/∂xj∂viE01/∂xj−1 of the dynamical system (18) [28, 29], 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 E01 is the disease-free equilibrium point.The corresponding Jacobian matrices ofF and V computed at DFE point E01, respectively, are given as follows:
(30)F=1−d1τ1ω+d0+h1ω+d01−ηω+d0,and(31)V=r2+α1+d3+d0.From this, it follows that(32)R1eff=ρFV−1=1−d1τ1ω+d0+h1ω+d01−ηω+d0r2+α1+d3+d0.


## 3.1.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel

Theorem 5.
The disease-free equilibrium pointE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is locally asymptotically stable if the effective reproduction number R1eff<1.Proof.
The Jacobean matrix of the dynamical system (18) at the DFE point E01 is
(33)J1E01=−d0ω00r30−ω+d000000−r2+α1+d3+d00000α1−d0000r20−r3+d0.
After some steps, the roots of (33) are
(34)λ1=λ2=−d0,λ3=−ω+d0,λ4=−r3+d0,λ5=−r2+α1+d3+d0.
This shows all the eigenvalues of the Jacobean matrix have negative real parts whenR1eff<1
Hence, the disease-free equilibriumE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is locally asymptotically stable if the effective reproduction number R1eff<1 and unstable otherwise.


## 3.1.5. Global Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel

Theorem 6.
The disease-free equilibrium pointE01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 of the dynamical system (18) is globally asymptotically stable in the feasible region Ω1 if the effective reproduction number R1eff<1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(35)f1=u1P3,
whereu1=1/r2+α1+d3+d0.
The time derivative of (35) along the solution path yields
(36)f1̇=u11−d1τ1P3+h1P1P3N1−r2+α1+d3+d0P3=u11−d1τ1+h1P1N1−r2+α1+d3+d0P3
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 equilibriumE01=r2+α1+d3+d0/r2+α1+d3+d0,ηΛ/ω+d0,0,0,0. with R1eff=1−d1τ1ω+d0+h1ω+d01−η/ω+d0r2+α1+d3+d0,f1̇ can be simplified as follows:
(37)f1̇≤u11−d1τ1+h1P01N10−r2+α1+d3+d0P3.
Substitutingh1=R1effω+d0r2+α1+d3+d0−ω+d01−d1τ1/ω+d01−η in (37) and simplification gives us f1̇≤u1r2+α1+d3+d0R1eff−1P3. After substituting u1 we get f1̇≤R1eff−1P3.
This shows thatf1̇<0 when R1eff<1. Furthermore, f1̇=0if and only if P3=0 or R1eff=1 and Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 is the only singleton set in P1,P2,P3,P6,P10∈Ω1:f1̇=0. Thus, by LaSalle’s invariance principle [30], the DFE E01=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0 is globally asymptotically stable if R1eff<1.


## 3.1.6. Existence of Endemic Equilibrium (EE) of HBV-Only Submodel

Suppose thatE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 be an arbitrary EE equilibrium point of HB-only submodel (18), which occurs when the disease persists in the society. After long steps we obtained the endemic equilibrium point E∗1=P1∗,P2∗,P∗3,P∗6,P∗10.where(38)P1∗=Λr3+d0ω+d01−ηr2+α1+d3+d0−1−d1τ1λ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,P2∗=ηΛω+d0,P3∗=λ1∗Λr3+d0ω+d01−ηλ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,P∗6=α1λ1∗Λr3+d0ω+d01−ηλ1∗d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d02ω+d0r3+d0r2+α1+d3+d0−1−d1τ1,and
(39)P∗10=r2λ1∗Λr3+d0ω+d01−ηλ1∗r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d02r2+α1+d3+d0−1−d1τ1The associated force of infection of the dynamical system (18) is given by
(40)λ1∗=h1P3∗N1∗.where
(41)N1∗=λ1∗Λ/d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0+Λr3+d0ω+d0r2+α1+d3+d0−1−d1τ1λ1∗ω+d0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3+d0ω+d0r3+d0r2+α1+d3+d0−1−d1τ1.After substitutingP∗3 and N1∗ in (40) and simplifying it we got
(42)λ1∗=d0ω+d0r2+α1+d3+d0R1eff−1α1+d0ω+d01−η+d0ηα1+d3+d0−d0η1−d1τ1.Equation (42) shows that λ1∗>0 if R1eff>1. Hence, the endemic equilibrium point E∗1 of HBV-only submodel (18) exists whenever R1eff>1.


## 3.1.7. Local Stability of the Endemic Equilibrium Point of HBV-Only Submodel

Theorem 7.
The endemic equilibrium pointE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 of the model (18) is locally asymptotically stable if R1eff>1.Proof.
The Jacobian matrix of (18) at the endemic equilibrium E∗1 is
(43)J1E∗1=−E1+d0ω−E20r30−ω+d0000E10−E3+E20000α1−d0000r20−r3+d0,where
(44)E1=h1P∗3N1∗=Λω+d0r3+d0r2+α1+d3+d0−1−d1τ1R1eff−1Λ/d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0,E2=h1P1∗N1∗=h1ω+d01−ηr2+α1+d3+d0−1−d1τ1ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0ω+d01−ηr3+d0α1+d0+d0r2+ηd0r3+d0r2+α1+d3+d0−1−d1τ1−r2r3d0+Λr3+d0ω+d0r2+α1+d3+d0−1−d1τ1ω+d0r2+α1+d3+d0R1eff−1,E3=1−d1τ1−r2+α1+d3+d0.
Letλi be the eigenvalues of J1E∗1 where i=1, 2, 3, 4, 5.
The corresponding characteristic equation of (43) is
(45)λ+d0λ+ω+d0z3λ3+z2λ2+z1λ+z0=0,
This implies the first two eigenvalues of (45) are λ1=−d0<0 and λ2=−ω+d0<0. But, the algebraic sign of the remaining three eigenvalues is determined using Routh–Hurwitz stability criterion from the characteristic polynomial
(46)z3λ3+z2λ2+z1λ+z0=0,where
z3=1>0, z2=E1+E2+E3+r3+2d0
z1=E1+d0E2+E3+r3+d0+E2+E3r3+d0, and z0=E1+d0E2+E3r3+d0.
The sign of the coefficientsz0, z1, and z2 of the characteristic polynomial z3λ3+z2λ2+z1λ+z0=0 is positive if R1eff>1. In addition to that z1z2−z0z3>0. Thus, by Routh–Hurwitz stability criterion the endemic equilibrium E∗1 is stable if R1eff>1 and unstable if R1eff<1.


## 3.1.8. Global Stability of the Endemic Equilibrium Point of HBV-Only Submodel

Theorem 8.
The endemic equilibrium pointE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 of the dynamical system (18) is globally asymptotically stable if R1eff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(47)L1P1,P3=b1P1−P1∗2+b2P3−P3∗2,b1>0,b2>0.
DifferentiatingL1P1,P3 with respect to time gives
(48)L̇1P1,P3=2b1P1−P1∗1−ηΛ+ωP2+r3P10−h1P3N1+d0P1+2b2P3−P3∗1−d1τ1P3+h1P3N1P1−r2+α1+d3+d0P3.
Substituting1−ηΛ+ωP2+r3P10=h3P∗3/N1∗+d0P1∗and r2+α1+d3+d0−1−d1τ1=h1P∗3/N1∗ with N1≈N1∗ in (48) yields:
(49)L̇1P1,P3=2b1P1−P1∗h1P∗3N1∗P1∗+d0P1∗−h1P3N1P1−d0P1+2b2P3−P3∗h1P3N1P1−h1P3N1∗P1∗=−2b1d0P1−P1∗2−2b1h1N1P3P1−P1∗2+2h1N1P3∗−P3P1−P1∗b1P1∗−b2P3≤0ifP3≥b1P1∗b2,P1≥P1∗,P3≤P∗3.
This impliesL̇1P1,P3≤0, and it vanishes if and only if P1∗=P1, P3∗=P3, and P1∗=b2P3/b1
This shows thatE∗1=P1∗,P2∗,P∗3,P∗6,P∗10 is the largest compact invariant singleton set in P1∗,P2∗,P∗3,P∗6,P∗10∈Ω1:L̇1=0
Hence, by LaSalle’s invariance principle [30], the endemic equilibrium E∗1 is globally asymptotically stable in the invariant region whenever R1eff>1.


## 3.2. HIV/AIDS-Only Submodel

This submodel is obtained by settingP2=P3=P5=P6=P8=P9=P10=0. Thus, we have the following dynamical system
(50)dP1dt=Λ−λ2+d0P1,dP4dt=1−d2τ2P4+λ2P1−ζα2+d4+d0P4,dP7dt=ζα2P4−d0P7,where the force of infection is given by
(51)λ2=h2P4N2.with initial conditions, P10>0,P40>0, and P70>0The total population of the dynamical system (50) is given by:
(52)N2t=P1t+P4t+P7t.


### 3.2.1. Positivity of the Solutions and Boundedness of the Solution Region of HIV/AIDS-Only Submodel

Here, we proved that all the solutions of the dynamical system (50) with positive initial conditions remain positive and the solution region is bounded as follows:Theorem 9.
The solutions of the dynamical system (50) with positive initial conditions are positive for time t≥0 in the region
(53)Ω2=P1,P4,P7∈ℝ3+Proof.
TakingdP1/dt=Λ−λ2+d0P1 from the dynamical system (50), we get
(54)dP1dt+λ2+d0P1=Λ.
(54) is a separable first order ordinary differential equation for the variable P1. After some steps we got the solution
(55)P1t=P10e−λ2+d0t+Λλ2+d01−e−λ2+d0t.
Since,P10>0,e−λ2+d0t>0 and Λ/λ2+d0>0 for t≥0, the solution we found is positive. Following the same procedure, P4t>0 and P10t>0. Therefore, the solutions of the model (50) are positive whenever the initial values positive.Theorem 10.
The dynamical system (50) is positively invariant in the region
(56)Ω2=P1,P4,P7∈ℝ3>+:N2≤Λd0.Proof.
The dynamics of total populationN2t with respect to time t is computed as
(57)dN2dt=Λ−d0N2−d4P4.
In the absence of mortality due to HIV infection and comparing both sides of equation (57) using standard comparison theorem, we obtain
(58)dN2dt≤Λ−d0N2.
After some steps the solution for the inequality (58) is
(59)N2t≤N0e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size N2t≤Λ/d0
This implies0≤N2t≤Λ/d0 which shows all the feasible solutions of the model (50) with initial conditions enter the region
(60)Ω2=P1,P4,P7∈ℝ3+:N2≤Λd0.
Thus, the regionΩ2 is positively invariant.


### 3.2.2. Existence of the Disease - Free Equilibrium Point of HIV/AIDS-Only Submodel

The disease-free equilibrium point of HIV/AIDS-only submodel (50), represented by E02=P10,P40,P70, is obtained by setting the right-hand side of all the components of the model equal to zero, providing that P4=0. It is equal to
(61)E02=P10,P40,P70=Λd0,0,0.


### 3.2.3. Effective Reproduction Numbers of HIV/AIDS-Only Submodel

In the same way that we have shown in sub section3.1.3, the effective reproduction number of HIV/AIDS-only infected individuals is
(62)R2eff=1−d2τ2+h2ζα2+d4+d0.


### 3.2.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 11.
The disease-free equilibrium pointE02=Λ/d0,0,0 of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R2eff<1.Proof.
The Jacobean matrix of the dynamical system (50) at the DFE point E02=Λ/d0,0,0 is
(63)J2E02=−d0−h2001−d2τ2+h2−ζα2+d4+d000ζα2−d0.
The corresponding characteristic equation of (63) is
(64)−d0+λ−h200−ζα2+d4+d0−h2+1−d2τ2+λ00ζα2−d0+λ=0.
After some necessary steps, the roots of the characteristic equation (64) are λ1=λ2=−d0 and λ3=−ζα2+d4+d0+h2+1−d2τ2.
The simplified value ofλ3=ζα2+d4+d0R2eff−1.
Clearly,λ3 is negative if R2eff<1. This shows that all the eigenvalues of (64) have negative real parts when R2eff<1. Hence, the disease-free equilibrium E02=Λ/d0,0,0 of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R2eff<1 and unstable otherwise.


### 3.2.5. Global Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 12.
The disease-free equilibrium pointE02=Λ/d0,0,0 of the dynamical system (50) is globally asymptotically stable in the feasible region Ω2 if the effective reproduction number R2eff<1.Proof.
To prove the global asymptotic stability of the disease-free equilibrium point, we used LaSalle-Lyapunov candidate function as follows:
LetW2 be a LaSalle-Lyapunov candidate function such that
(65)W2=i1P4.where
(66)i1=1ζα2+d4+d0.
The time derivative of (65) along the solution path yields
(67)W2̇=i1Ṗ4=i11−d2τ2+h2P1N2−ζα2+d4+d0P4.
Since the state variables of the model when the 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 equilibriumE02=Λ/d0,0,0 with R2eff<1, W2̇ can be simplified as
W2̇≤i11−d2τ2+h2P10/N20−ζα2+d4+d0P4. Substitution of i1 and simplification gives
(68)W2̇≤R2eff−1P4.
Inequality (68) shows that W2̇<0 when R2eff<1. Furthermore, W2̇=0 if and only if P4=0 or R2eff=1 and Λ/d0,0,0 is the only singleton set in P1,P4,P7∈ℝ3+∈Ω2:W2̇=0. Thus, by LaSalle’s invariance principle, the DFE E02=Λ/d0,0,0 is globally asymptotically stable if R2eff<1.


### 3.2.6. Existence of Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Suppose thatE∗2=P∗1,P∗4,P∗7 be an arbitrary endemic equilibrium point of HIV/AIDS-only submodel (50), which occurs when the disease persists in the society. After long steps we obtained the endemic equilibrium point
(69)E∗2=Λζα2+d0d0ζα2+d4+d0R2eff−1+ζα2+d0,ΛR2eff−1ζα2+d4+d0R2eff−1+ζα2+d0,Λζα2R2eff−1d0ζα2+d4+d0R2eff−1+ζα2+d0.The associated force of infection of the dynamical system (50) is given by
(70)λ2∗=h2P4∗N2∗,where
(71)N2∗=h2Λζα2+d0d0ζα2+d4+d0ζα2+d4+d0R2eff−1+ζα2+d0.We can show the existence of the endemic equilibrium point of the model by substitutingP∗4 and N2∗ in (70). After substitution we got
(72)λ2∗=d0ζα2+d4+d0R2eff−1ζα2+d0.Clearly,λ2∗>0 if R2eff>1. Hence, an endemic equilibrium point E∗2=P∗1,P∗4,P∗7 of the dynamical system (50) exists whenever R2eff>1


### 3.2.7. Local Stability of the Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 13.
The endemic equilibriumE∗2=P∗1,P∗4,P∗7 of the model (50) is locally asymptotically stable if R2eff>1.Proof.
The Jacobian matrix of the model (50) at the endemic equilibrium point E∗2 is
(73)J2E∗2=−F1+d0−F20F1F300ζα2−d0,
where
F1=d0ζα2+d4+d0R2eff−1/ζα2+d0, F2=ζα2+d4+d0, and F3=1−d2τ2
The corresponding characteristic equation of (73) is
(74)λ3+g2λ2+g1λ+g0=0,where
(75)g2=F1+2d0−F3,g1=F1F2+F1+d0d0−F3−F3d0,
and(76)g0=F1F2d0−F3d0F1+d0.
The sign of the coefficientsg0, g1, and g2 of the characteristic polynomial λ3+g2λ2+g1λ+g0=0 is positive if R2eff>1. In addition to that, g1g2>g0. Thus, by Routh–Hurwitz stability criterion the endemic equilibrium E∗2 is stable if R2eff>1 and unstable if R2eff<1


### 3.2.8. Global Stability of the Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 14.
The endemic equilibriumE∗2=P∗1,P∗4,P∗7 of the dynamical system (50) is globally asymptotically stable if R2eff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(77)L2P1,P4=h3P1−P1∗2+h4P4−P4∗2,h3>0,h4>0.
DifferentiatingL2P2,P4 with respect to time gives us
(78)L̇2P1,P4=2h3P1−P1∗P1̇+2h4P4−P4∗Ṗ4=2h3P1−P1∗Λ−h2P4N2+d0P1+2h4P4−P4∗1−d2τ2P4+h2P4N2P1−ζα2+d4+d0P4.
SubstitutingΛ=h2P∗4/N2∗+d0P1∗ and 1−d2τ2+ζα2+d4+d0=h2P∗4/N2∗ with N2≈N2∗in (78) yields:
(79)L̇2P1,P4=2h3P1−P1∗h2P∗4N2∗P1∗+d0P1∗−h2P4N2P1−d0P1+2h4P4−P4∗h2P4N2P1−h2P4N2∗P1∗=−2h3d0P1−P1∗2−2h2h3N2P4P1−P1∗2+2h2N2P4∗−P4P1−P1∗h3P1∗−h4P4≤0ifP4≥h3P1∗h4,P1∗≤P1,P∗4≥P4.
This impliesL̇2P1,P4≤0, and it vanishes if and only if P1∗=P1, P4∗=P4, and P1∗=h4P4/h3
This shows thatE∗2=P∗1,P∗4,P∗7 is the largest compact invariant singleton set in P∗1,P∗4,P∗7∈Ω2:L̇2=0
Hence, by LaSalle’s invariance principle [30], the endemic equilibrium E∗2 is globally asymptotically stable in the invariant region whenever R2eff>1.


## 3.2.1. Positivity of the Solutions and Boundedness of the Solution Region of HIV/AIDS-Only Submodel

Here, we proved that all the solutions of the dynamical system (50) with positive initial conditions remain positive and the solution region is bounded as follows:Theorem 9.
The solutions of the dynamical system (50) with positive initial conditions are positive for time t≥0 in the region
(53)Ω2=P1,P4,P7∈ℝ3+Proof.
TakingdP1/dt=Λ−λ2+d0P1 from the dynamical system (50), we get
(54)dP1dt+λ2+d0P1=Λ.
(54) is a separable first order ordinary differential equation for the variable P1. After some steps we got the solution
(55)P1t=P10e−λ2+d0t+Λλ2+d01−e−λ2+d0t.
Since,P10>0,e−λ2+d0t>0 and Λ/λ2+d0>0 for t≥0, the solution we found is positive. Following the same procedure, P4t>0 and P10t>0. Therefore, the solutions of the model (50) are positive whenever the initial values positive.Theorem 10.
The dynamical system (50) is positively invariant in the region
(56)Ω2=P1,P4,P7∈ℝ3>+:N2≤Λd0.Proof.
The dynamics of total populationN2t with respect to time t is computed as
(57)dN2dt=Λ−d0N2−d4P4.
In the absence of mortality due to HIV infection and comparing both sides of equation (57) using standard comparison theorem, we obtain
(58)dN2dt≤Λ−d0N2.
After some steps the solution for the inequality (58) is
(59)N2t≤N0e−d0t+Λd01−e−d0t.
Ast⟶∞, the population size N2t≤Λ/d0
This implies0≤N2t≤Λ/d0 which shows all the feasible solutions of the model (50) with initial conditions enter the region
(60)Ω2=P1,P4,P7∈ℝ3+:N2≤Λd0.
Thus, the regionΩ2 is positively invariant.


## 3.2.2. Existence of the Disease - Free Equilibrium Point of HIV/AIDS-Only Submodel

The disease-free equilibrium point of HIV/AIDS-only submodel (50), represented by E02=P10,P40,P70, is obtained by setting the right-hand side of all the components of the model equal to zero, providing that P4=0. It is equal to
(61)E02=P10,P40,P70=Λd0,0,0.


## 3.2.3. Effective Reproduction Numbers of HIV/AIDS-Only Submodel

In the same way that we have shown in sub section3.1.3, the effective reproduction number of HIV/AIDS-only infected individuals is
(62)R2eff=1−d2τ2+h2ζα2+d4+d0.


## 3.2.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 11.
The disease-free equilibrium pointE02=Λ/d0,0,0 of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R2eff<1.Proof.
The Jacobean matrix of the dynamical system (50) at the DFE point E02=Λ/d0,0,0 is
(63)J2E02=−d0−h2001−d2τ2+h2−ζα2+d4+d000ζα2−d0.
The corresponding characteristic equation of (63) is
(64)−d0+λ−h200−ζα2+d4+d0−h2+1−d2τ2+λ00ζα2−d0+λ=0.
After some necessary steps, the roots of the characteristic equation (64) are λ1=λ2=−d0 and λ3=−ζα2+d4+d0+h2+1−d2τ2.
The simplified value ofλ3=ζα2+d4+d0R2eff−1.
Clearly,λ3 is negative if R2eff<1. This shows that all the eigenvalues of (64) have negative real parts when R2eff<1. Hence, the disease-free equilibrium E02=Λ/d0,0,0 of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R2eff<1 and unstable otherwise.


## 3.2.5. Global Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 12.
The disease-free equilibrium pointE02=Λ/d0,0,0 of the dynamical system (50) is globally asymptotically stable in the feasible region Ω2 if the effective reproduction number R2eff<1.Proof.
To prove the global asymptotic stability of the disease-free equilibrium point, we used LaSalle-Lyapunov candidate function as follows:
LetW2 be a LaSalle-Lyapunov candidate function such that
(65)W2=i1P4.where
(66)i1=1ζα2+d4+d0.
The time derivative of (65) along the solution path yields
(67)W2̇=i1Ṗ4=i11−d2τ2+h2P1N2−ζα2+d4+d0P4.
Since the state variables of the model when the 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 equilibriumE02=Λ/d0,0,0 with R2eff<1, W2̇ can be simplified as
W2̇≤i11−d2τ2+h2P10/N20−ζα2+d4+d0P4. Substitution of i1 and simplification gives
(68)W2̇≤R2eff−1P4.
Inequality (68) shows that W2̇<0 when R2eff<1. Furthermore, W2̇=0 if and only if P4=0 or R2eff=1 and Λ/d0,0,0 is the only singleton set in P1,P4,P7∈ℝ3+∈Ω2:W2̇=0. Thus, by LaSalle’s invariance principle, the DFE E02=Λ/d0,0,0 is globally asymptotically stable if R2eff<1.


## 3.2.6. Existence of Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Suppose thatE∗2=P∗1,P∗4,P∗7 be an arbitrary endemic equilibrium point of HIV/AIDS-only submodel (50), which occurs when the disease persists in the society. After long steps we obtained the endemic equilibrium point
(69)E∗2=Λζα2+d0d0ζα2+d4+d0R2eff−1+ζα2+d0,ΛR2eff−1ζα2+d4+d0R2eff−1+ζα2+d0,Λζα2R2eff−1d0ζα2+d4+d0R2eff−1+ζα2+d0.The associated force of infection of the dynamical system (50) is given by
(70)λ2∗=h2P4∗N2∗,where
(71)N2∗=h2Λζα2+d0d0ζα2+d4+d0ζα2+d4+d0R2eff−1+ζα2+d0.We can show the existence of the endemic equilibrium point of the model by substitutingP∗4 and N2∗ in (70). After substitution we got
(72)λ2∗=d0ζα2+d4+d0R2eff−1ζα2+d0.Clearly,λ2∗>0 if R2eff>1. Hence, an endemic equilibrium point E∗2=P∗1,P∗4,P∗7 of the dynamical system (50) exists whenever R2eff>1


## 3.2.7. Local Stability of the Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 13.
The endemic equilibriumE∗2=P∗1,P∗4,P∗7 of the model (50) is locally asymptotically stable if R2eff>1.Proof.
The Jacobian matrix of the model (50) at the endemic equilibrium point E∗2 is
(73)J2E∗2=−F1+d0−F20F1F300ζα2−d0,
where
F1=d0ζα2+d4+d0R2eff−1/ζα2+d0, F2=ζα2+d4+d0, and F3=1−d2τ2
The corresponding characteristic equation of (73) is
(74)λ3+g2λ2+g1λ+g0=0,where
(75)g2=F1+2d0−F3,g1=F1F2+F1+d0d0−F3−F3d0,
and(76)g0=F1F2d0−F3d0F1+d0.
The sign of the coefficientsg0, g1, and g2 of the characteristic polynomial λ3+g2λ2+g1λ+g0=0 is positive if R2eff>1. In addition to that, g1g2>g0. Thus, by Routh–Hurwitz stability criterion the endemic equilibrium E∗2 is stable if R2eff>1 and unstable if R2eff<1


## 3.2.8. Global Stability of the Endemic Equilibrium Point of HIV/AIDS-Only Submodel

Theorem 14.
The endemic equilibriumE∗2=P∗1,P∗4,P∗7 of the dynamical system (50) is globally asymptotically stable if R2eff>1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(77)L2P1,P4=h3P1−P1∗2+h4P4−P4∗2,h3>0,h4>0.
DifferentiatingL2P2,P4 with respect to time gives us
(78)L̇2P1,P4=2h3P1−P1∗P1̇+2h4P4−P4∗Ṗ4=2h3P1−P1∗Λ−h2P4N2+d0P1+2h4P4−P4∗1−d2τ2P4+h2P4N2P1−ζα2+d4+d0P4.
SubstitutingΛ=h2P∗4/N2∗+d0P1∗ and 1−d2τ2+ζα2+d4+d0=h2P∗4/N2∗ with N2≈N2∗in (78) yields:
(79)L̇2P1,P4=2h3P1−P1∗h2P∗4N2∗P1∗+d0P1∗−h2P4N2P1−d0P1+2h4P4−P4∗h2P4N2P1−h2P4N2∗P1∗=−2h3d0P1−P1∗2−2h2h3N2P4P1−P1∗2+2h2N2P4∗−P4P1−P1∗h3P1∗−h4P4≤0ifP4≥h3P1∗h4,P1∗≤P1,P∗4≥P4.
This impliesL̇2P1,P4≤0, and it vanishes if and only if P1∗=P1, P4∗=P4, and P1∗=h4P4/h3
This shows thatE∗2=P∗1,P∗4,P∗7 is the largest compact invariant singleton set in P∗1,P∗4,P∗7∈Ω2:L̇2=0
Hence, by LaSalle’s invariance principle [30], the endemic equilibrium E∗2 is globally asymptotically stable in the invariant region whenever R2eff>1.


## 4. Analysis of the HBV-HIV/AIDS Model at Equilibrium Points

### 4.1. Disease-Free Equilibrium Point of HBV-HIV/AIDS Coinfection Model

To find the DFE point, we equated the right-hand side of the full mode (2) to zero and evaluated it at P3=P4=P5=0. Therefore, the DFE point represented by E03 is equal to:
(80)E03=Λω+d01−ηd0ω+d0,ηΛω+d0,0,0,0,0,0,0,0,0.


### 4.2. Effective Reproduction Number of the Full Model

We represented the effective reproduction number of the coinfected model byR3eff. To find R3eff, we used the next generation matrix method that was formulated in [28, 29]. In the same way that we have shown in sub section 3.1.3, R3eff can be manipulated as follows:
(81)F=1−d1τ1+h1ω+d01−ηω+d00h1ω+d01−ηω+d001−d2τ2+h2h2000,V=r2+α1+d3+d0000ζα2+d4+d0−r100r1+α3+d5+d0.This implies the next-generation matrixFV−1 becomes
(82)FV−1=1−d1τ1+ω+d01−ηh1/ω+d0r2+α1+d3+d00ω+d01−ηh1ω+d0r1+α3+d5+d001−d2τ2+h2ζα2+d4+d0h2r1+α3+d5+d0+r1−d2τ2+h2r1+α3+d5+d0ζα2+d4+d0000.The spectral radius (the largest eigenvalue) of (82) is the effective reproduction number of the full model. Hence, after some steps, the spectral radius of equation (82) becomes
(83)R3eff=maxh1ω+d01−η+ω+d01−d1τ1ω+d0r2+α1+d3+d0,h2+1−d2τ2ζα2+d4+d0.


### 4.3. Local Stability Analysis of the Disease-Free Equilibrium Point of the HBV-HIV/AIDS Coinfection Model

Theorem 15.
The disease-free equilibrium pointE03=h2P∗4/N2∗/d0ω+d0,ηΛ/ω+d0,0,0,0,0,0,0,0,0 of the dynamical system (2) is locally asymptotically stable if R3eff<1.Proof.
The Jacobean matrix of the dynamical system (2) at the DFE point E03 is
(84)JE03=−d0ω−h1k8−h2k8−h1+h2k80000r30−k10−h2dηω+d−h2dηω+d0000000h1k8+k20h1k800000000h2+k3h2+r1000000000−k40000000α100−k50000000ζα200−k60000000α300−k70000000θ1θ2θ3−d0000r2000000−r3+d0.where k1=ω+d0,k2=1−d1τ1−r2+α1+d3+d0,k3=1−d2τ2−ζα2+d4+d0,k4=r+α3+d5+d0,k5=θ1+d0,k6=θ2+d0,k7=θ3+d0,k8=ω+d01−η/ω+d0, k9=r3+d0
The characteristic equation of (84) is
(85)d0+λ2k1+λk4+λk5+λk6+λk7+λk9+λt2λ2+t1λ+t0=0
where(86)t2=1,t1=r2+α1+d3+d01−R1eff+ζα2+d4+d01−R2eff,t0=r2+α1+d3+d01−R1effζα2+d4+d01−R2eff.
Thus, the first seven eigenvalues from the characteristic equation (85) are λ1=λ2=−d0,λ3=−ω+d0,λ4=−θ3+d0,λ5=−θ2+d0,λ6=−d0+θ1,λ7=−r1+α3+d0+d5, and λ8=−r3+d0. But we determined the remaining two eigenvalues from the characteristic polynomial t2λ2+t1λ+t0=0using the Routh-Hurwitz stability criteria. Clearly, t2=1>0 and both t1 and t0 are positive if R1eff<1 and R2eff<1. This implies t1 and t0 are positive, if R3eff=maxR1eff,R2eff<1. This shows λ8<0 and λ9<0. From these all, we see that all eigenvalues are negative.
Hence, by some algebraic manipulations together with Routh-Hurwitz stability criteria, the disease-free equilibrium pointE03 of the dynamical system (2) is locally asymptotically stable if R3eff=maxR1eff,R2eff<1


### 4.4. Global Stability Analysis of the Disease-Free Equilibrium Point of the HBV-HIV/AIDS Coinfection Model

Theorem 16.
The disease-free equilibrium pointE03=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0,0,0,0,0,0 of the model (2) is globally asymptotically stable in the feasible region Ω if the effective reproduction number R3eff<1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(87)L=m1P3+m2P4+P5,where
(88)m1=1/2r1+α3+d5+d0r2+α1+d3+d0,and
(89)m2=1/2r1+α3+d5+d0ζα2+d4+d0.
The time derivative of (87) along the solution path yields
(90)L̇=m11−d1τ1P3+h1P3+P5P1N−h2P4+P5P3N−r2+α1+d3+d0P3+m21−d2τ2P4+h2P4+P5P1+P2N+rP5−h11−πP3+P5P4N−ζα2+d4+d0P4+h11−πP3+P5P4N+h2P4+P5P3N−r1+α3+d5+d0P5.
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 equilibriumE03=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0,0,0,0,0,0 with R3eff=maxh1ω+d01−η+ω+d01−d1τ1/ω+d0r2+α1+d3+d0,ω+d0r2+α1+d3+d0/ζα2+d4+d0, L̇ can be simplified as follows:
(91)L̇≤m11−d1τ1P3+h1P3P10P10+P20+h1P5P10P10+P20—r2+α1+d0+d3P3+m21−d2τ2P4+h2P4P10+P20N+h2P5P10+P20N−ζα2+d4+d0P4−r1+α3+d5+d0P5.
After simplifying (91) we got
(92)L̇≤m1r2+α1+d0+d3R1eff−1P3+m2ζα2+d4+d0R2eff−1P4+m1R1effr2+α1+d0+d3+m2R2effζα2+d4+d0−r1+α3+d5+d0P5.
Substitutingm1 and m2 (92) gives us
(93)L̇≤12r1+α3+d5+d0R3eff−1P3+12r1+α3+d5+d0R3eff−1P4+r1+α3+d5+d0R3eff−1P5.
This impliesL̇≤0 when R3eff<1. Furthermore, L̇=0 if and only if P3=P4=P5=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in Ω contained in P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+ is reduced to the DFE. This proves the global asymptotic stability of the DFE E03 on Ω if the effective reproduction number R3eff<1and unstable otherwise.


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

The endemic equilibrium point of a dynamical system (2) which exists when the disease persists in the community is denoted by E∗3=P1∗,P2∗,P∗3,P∗4,P∗5,P∗6,P∗7,P∗8,P∗9,P∗10. From the analysis of HBV-only submodel (18) and HIV/AIDS-only submodel (50), there exists an endemic equilibrium point for HBV-only submodel and HIV/AIDS-only submodel if R1eff>1 and R2eff>1, respectively.This implies that the endemic equilibrium pointE∗3 for HBV- HIV/AIDS coinfection exists because the effective reproduction number for HBV- HIV/AIDS coinfection is the maximum of R1eff and R2eff, which is greater than one. After some steps we got E∗3=P1∗,P2∗,P∗3,P∗4,P∗5,P∗6,P∗7,P∗8,P∗9,P∗10. Where
(94)P1∗=Λω+1−ηλ∗2+d0r2+α1+λ∗2+d3+d0−1−d1τ1λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗2=ηΛλ∗2+ω+d0,P∗3=λ∗1Λω+1−ηλ∗2+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗4=λ∗2Λλ∗2+ω+d0ω+1−ηλ∗2+d0r2+α1+λ∗2+d3+d0−1−d1τ1λ∗2+ω+d0λ∗1+λ∗2+d0λ∗1+λ∗2+d0+r2λ∗2+d0λ∗11−ζα3+d5+d0+ζα2+d4+d0r1+α3+d5+d0+λ∗2ηΛλ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0λ∗11−ζα3+d5+d0+ζα2+d4+d0r1+α3+d5+d0+λ∗1λ∗2r1Λω+1−ηλ∗2+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0λ∗11−ζα3+d5+d0+ζα2+d4+d0r1+α3+d5+d0,P∗5=1−ζλ∗1P∗4λ∗2+ω+d0λ∗1+λ∗2+d0r2+α1+λ∗2+d3+d0−r2λ∗1+λ∗1λ∗2Λω+1−ηλ∗2+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗6=λ∗1α1Λω+1−ηλ∗2+d0ϕ+μλ∗H+q+μλ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗7=ζα2P∗4θ2+d0,P∗8=α3λ∗11−ζλ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0p∗4+Λλ∗21−ηλ∗2+d0+ωθ3+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗1+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗9=α3θ3λ∗11−ζλ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0P∗4+Λλ∗21−ηλ∗2+d0+ωd0θ3+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0+λ∗1α1Λθ11−ηλ∗2+d0+ωd0θ1+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0+θ2ζα2P∗4d0θ2+d0,P∗10=r2λ∗1Λω+1−ηλ∗2+d0r3+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0.The local and global stability analysis of the endemic equilibrium of the full model (2) in terms of the model parameters analytically is difficult. Hence, we will give an explanation of the stability analysis of E∗3 of this model in Section 8.


## 4.1. Disease-Free Equilibrium Point of HBV-HIV/AIDS Coinfection Model

To find the DFE point, we equated the right-hand side of the full mode (2) to zero and evaluated it at P3=P4=P5=0. Therefore, the DFE point represented by E03 is equal to:
(80)E03=Λω+d01−ηd0ω+d0,ηΛω+d0,0,0,0,0,0,0,0,0.


## 4.2. Effective Reproduction Number of the Full Model

We represented the effective reproduction number of the coinfected model byR3eff. To find R3eff, we used the next generation matrix method that was formulated in [28, 29]. In the same way that we have shown in sub section 3.1.3, R3eff can be manipulated as follows:
(81)F=1−d1τ1+h1ω+d01−ηω+d00h1ω+d01−ηω+d001−d2τ2+h2h2000,V=r2+α1+d3+d0000ζα2+d4+d0−r100r1+α3+d5+d0.This implies the next-generation matrixFV−1 becomes
(82)FV−1=1−d1τ1+ω+d01−ηh1/ω+d0r2+α1+d3+d00ω+d01−ηh1ω+d0r1+α3+d5+d001−d2τ2+h2ζα2+d4+d0h2r1+α3+d5+d0+r1−d2τ2+h2r1+α3+d5+d0ζα2+d4+d0000.The spectral radius (the largest eigenvalue) of (82) is the effective reproduction number of the full model. Hence, after some steps, the spectral radius of equation (82) becomes
(83)R3eff=maxh1ω+d01−η+ω+d01−d1τ1ω+d0r2+α1+d3+d0,h2+1−d2τ2ζα2+d4+d0.


## 4.3. Local Stability Analysis of the Disease-Free Equilibrium Point of the HBV-HIV/AIDS Coinfection Model

Theorem 15.
The disease-free equilibrium pointE03=h2P∗4/N2∗/d0ω+d0,ηΛ/ω+d0,0,0,0,0,0,0,0,0 of the dynamical system (2) is locally asymptotically stable if R3eff<1.Proof.
The Jacobean matrix of the dynamical system (2) at the DFE point E03 is
(84)JE03=−d0ω−h1k8−h2k8−h1+h2k80000r30−k10−h2dηω+d−h2dηω+d0000000h1k8+k20h1k800000000h2+k3h2+r1000000000−k40000000α100−k50000000ζα200−k60000000α300−k70000000θ1θ2θ3−d0000r2000000−r3+d0.where k1=ω+d0,k2=1−d1τ1−r2+α1+d3+d0,k3=1−d2τ2−ζα2+d4+d0,k4=r+α3+d5+d0,k5=θ1+d0,k6=θ2+d0,k7=θ3+d0,k8=ω+d01−η/ω+d0, k9=r3+d0
The characteristic equation of (84) is
(85)d0+λ2k1+λk4+λk5+λk6+λk7+λk9+λt2λ2+t1λ+t0=0
where(86)t2=1,t1=r2+α1+d3+d01−R1eff+ζα2+d4+d01−R2eff,t0=r2+α1+d3+d01−R1effζα2+d4+d01−R2eff.
Thus, the first seven eigenvalues from the characteristic equation (85) are λ1=λ2=−d0,λ3=−ω+d0,λ4=−θ3+d0,λ5=−θ2+d0,λ6=−d0+θ1,λ7=−r1+α3+d0+d5, and λ8=−r3+d0. But we determined the remaining two eigenvalues from the characteristic polynomial t2λ2+t1λ+t0=0using the Routh-Hurwitz stability criteria. Clearly, t2=1>0 and both t1 and t0 are positive if R1eff<1 and R2eff<1. This implies t1 and t0 are positive, if R3eff=maxR1eff,R2eff<1. This shows λ8<0 and λ9<0. From these all, we see that all eigenvalues are negative.
Hence, by some algebraic manipulations together with Routh-Hurwitz stability criteria, the disease-free equilibrium pointE03 of the dynamical system (2) is locally asymptotically stable if R3eff=maxR1eff,R2eff<1


## 4.4. Global Stability Analysis of the Disease-Free Equilibrium Point of the HBV-HIV/AIDS Coinfection Model

Theorem 16.
The disease-free equilibrium pointE03=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0,0,0,0,0,0 of the model (2) is globally asymptotically stable in the feasible region Ω if the effective reproduction number R3eff<1.Proof.
Consider the following LaSalle-Lyapunov candidate function:(87)L=m1P3+m2P4+P5,where
(88)m1=1/2r1+α3+d5+d0r2+α1+d3+d0,and
(89)m2=1/2r1+α3+d5+d0ζα2+d4+d0.
The time derivative of (87) along the solution path yields
(90)L̇=m11−d1τ1P3+h1P3+P5P1N−h2P4+P5P3N−r2+α1+d3+d0P3+m21−d2τ2P4+h2P4+P5P1+P2N+rP5−h11−πP3+P5P4N−ζα2+d4+d0P4+h11−πP3+P5P4N+h2P4+P5P3N−r1+α3+d5+d0P5.
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 equilibriumE03=Λω+d01−η/d0ω+d0,ηΛ/ω+d0,0,0,0,0,0,0,0,0 with R3eff=maxh1ω+d01−η+ω+d01−d1τ1/ω+d0r2+α1+d3+d0,ω+d0r2+α1+d3+d0/ζα2+d4+d0, L̇ can be simplified as follows:
(91)L̇≤m11−d1τ1P3+h1P3P10P10+P20+h1P5P10P10+P20—r2+α1+d0+d3P3+m21−d2τ2P4+h2P4P10+P20N+h2P5P10+P20N−ζα2+d4+d0P4−r1+α3+d5+d0P5.
After simplifying (91) we got
(92)L̇≤m1r2+α1+d0+d3R1eff−1P3+m2ζα2+d4+d0R2eff−1P4+m1R1effr2+α1+d0+d3+m2R2effζα2+d4+d0−r1+α3+d5+d0P5.
Substitutingm1 and m2 (92) gives us
(93)L̇≤12r1+α3+d5+d0R3eff−1P3+12r1+α3+d5+d0R3eff−1P4+r1+α3+d5+d0R3eff−1P5.
This impliesL̇≤0 when R3eff<1. Furthermore, L̇=0 if and only if P3=P4=P5=0. Thus, by LaSalle’s invariance principle [30], the largest invariant set in Ω contained in P1,P2,P3,P4,P5,P6,P7,P8,P9,P10∈ℝ10+ is reduced to the DFE. This proves the global asymptotic stability of the DFE E03 on Ω if the effective reproduction number R3eff<1and unstable otherwise.


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

The endemic equilibrium point of a dynamical system (2) which exists when the disease persists in the community is denoted by E∗3=P1∗,P2∗,P∗3,P∗4,P∗5,P∗6,P∗7,P∗8,P∗9,P∗10. From the analysis of HBV-only submodel (18) and HIV/AIDS-only submodel (50), there exists an endemic equilibrium point for HBV-only submodel and HIV/AIDS-only submodel if R1eff>1 and R2eff>1, respectively.This implies that the endemic equilibrium pointE∗3 for HBV- HIV/AIDS coinfection exists because the effective reproduction number for HBV- HIV/AIDS coinfection is the maximum of R1eff and R2eff, which is greater than one. After some steps we got E∗3=P1∗,P2∗,P∗3,P∗4,P∗5,P∗6,P∗7,P∗8,P∗9,P∗10. Where
(94)P1∗=Λω+1−ηλ∗2+d0r2+α1+λ∗2+d3+d0−1−d1τ1λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗2=ηΛλ∗2+ω+d0,P∗3=λ∗1Λω+1−ηλ∗2+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗4=λ∗2Λλ∗2+ω+d0ω+1−ηλ∗2+d0r2+α1+λ∗2+d3+d0−1−d1τ1λ∗2+ω+d0λ∗1+λ∗2+d0λ∗1+λ∗2+d0+r2λ∗2+d0λ∗11−ζα3+d5+d0+ζα2+d4+d0r1+α3+d5+d0+λ∗2ηΛλ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0λ∗11−ζα3+d5+d0+ζα2+d4+d0r1+α3+d5+d0+λ∗1λ∗2r1Λω+1−ηλ∗2+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0λ∗11−ζα3+d5+d0+ζα2+d4+d0r1+α3+d5+d0,P∗5=1−ζλ∗1P∗4λ∗2+ω+d0λ∗1+λ∗2+d0r2+α1+λ∗2+d3+d0−r2λ∗1+λ∗1λ∗2Λω+1−ηλ∗2+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗6=λ∗1α1Λω+1−ηλ∗2+d0ϕ+μλ∗H+q+μλ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗7=ζα2P∗4θ2+d0,P∗8=α3λ∗11−ζλ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0p∗4+Λλ∗21−ηλ∗2+d0+ωθ3+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗1+d0α1+λ∗2+d3+d0+r2λ∗2+d0,P∗9=α3θ3λ∗11−ζλ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0P∗4+Λλ∗21−ηλ∗2+d0+ωd0θ3+d0r1+α3+d5+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0+λ∗1α1Λθ11−ηλ∗2+d0+ωd0θ1+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0+θ2ζα2P∗4d0θ2+d0,P∗10=r2λ∗1Λω+1−ηλ∗2+d0r3+d0λ∗2+ω+d0λ∗1+λ∗2+d0α1+λ∗2+d3+d0+r2λ∗2+d0.The local and global stability analysis of the endemic equilibrium of the full model (2) in terms of the model parameters analytically is difficult. Hence, we will give an explanation of the stability analysis of E∗3 of this model in Section 8.


## 5. Sensitivity Analysis

The reason why this section is important is that it tells us which parameters deserve the most numerical attention. That is, it highlights which parameters should be prioritized in prevention and controlling strategies. On the basic parameters, we carried out sensitivity analysis using the techniques outlined by [36, 37]. As the magnitude and direction of the sensitivity analysis result of the model parameters in Figure 2 indicates, the most sensitive parameters are τ2 and α1, which are the rate of vertical transmission of HIV and treatment rate for HBV infected class, respectively. The graphical representation of sensitivity indices of R3eff is given as follows:Figure 2
Magnitude and direction of sensitivity analysis result of parameters for the full model.The sign of the sensitivity index of each parameter value in Figure2 shows that what will happen to R3eff if the parameter is increased or decreased. R3eff increases when sensitivity indices with positive signs increase, while R3eff decreases when sensitivity indices with negative signs increase and vice versa. The indices in the figure also shows that the percentage change of R3eff for each increase value of parameter for 1%. As the figure shows, the sensitivity indices of τ2 and α1 are 0.88 and −0.759, respectively. These parameters deserve the most numerical attention. For instance, increasing τ2 for 10% will increases R3eff for approximately 8.8%. On the other hand, increasing α1 for 10% will decreases R3eff for 7.59%. Biologically, this means that, increasing treatment rate by 10% for HBV infected individuals decreases a patient’s chance of infecting others by approximately 7.59% during his/her infectious period. On the other hand, a single HIV/AIDS infectious individual gets approximately 8.8% chance to infect healthy individuals during his/her infectious period if the rate of MTCT of HIV increased by 10%. Hence, increasing treatment rate for HBV infected individuals and decreasing the rate of MTCT of HIV are very important to eliminate HBV- HIV/AIDS coinfection.


## 6. Numerical Simulation

We performed the numerical simulation using the parameter values in Table2 and the following initial values.Table 2
Parameter values used in numerical simulations.
ParametersValueSourceΛ0.04∗N0[26]α10.3[26]d5,α3,r10.001, 0.015, 0.001 respectively[26]r20.075[26]h20.03[26]θ1,θ2,θ30.014, 0.013, and 0.012, respectively[26]d00.01[31]d40.333[31]τ1,d1,d3,r30.4, 0.01, 0.1, 0.2, respectivelyAssumptionτ20.3[32]d20.2[32]α20.6[33]ζ0.396[33]η0.65[34]h10.4[34]ω0.1[34, 35]Initially,P10=100,000, P20=1000, P30=10,000, P40=3000, P50=1000, P60=500, P70=500, P80=100, P90=200, and P100=750.Numerical simulations for all models here are manipulated using MATLAB numerical solver (ode45). We choose ODE45 for the reason that the state of being exact and the speed at which the result of numerical process of calculating complicated system faster.


## 7. Results and Discussions

In this section, numerical results are manipulated for the submodels and the coinfection model using MATLAB numerical solver (ode45). We chose ODE45 for the reason that the state of being exact and the speed at which the result of numerical process of calculating complicated system faster. Figure3 has been plotted under consideration of the effective reproduction number of the full model less than unity. It is about the behavior of infected classes of the coinfection model at R3eff=0.49, h1=0.09, d0=0.5, and all other associated parameters as listed in Table 2. As the figure clearly shows, each infectious class of HBV- HIV/AIDS coinfection model (2) converges to the disease-free equilibrium point of the model. The convergence of all infectious classes to the disease-free equilibrium point of the model shows that the disease-free equilibrium of the model is globally asymptotically stable, which indicates the absence of HBV- HIV/AIDS coinfection in the society.Figure 3
Population Profile of infected classes of the coinfection model atR3eff=0.49, h1=0.09, and d0=0.5.Figures4(a), 4(b), and 4(c) are demonstrate the stability of endemic equilibrium of HBV- HIV/AIDS coinfection model for three different initial conditions keeping all other associated parameters as listed in Table 2. In each case, the effective reproduction number is greater than one and the simulation results show the convergence of the solutions of the model to the endemic equilibrium point. The convergence of the solutions of the model to the endemic equilibrium point for different initial conditions indicates that the endemic equilibrium point of the model is locally asymptotically stable (i.e., the disease is spreading in the society). Effect of the rate of MTCT of HBV on the effective reproduction number of HB-only submodel and HBV-HIV/AIDS coinfection model, respectively, are shown in Figures 5(a) and 5(b). The figures show that if the rate of MTCT of HBV (τ1) is less than 0.02, the reproduction number of HB-only submodel and HBV-HIV/AIDS coinfection model are less than one, which indicates both infections die out in the community. Whereas, if the rate of MTCT of HBV (τ1) is greater than 0.02, the reproduction number of HB-only submodel and HBV-HIV/AIDS coinfection model are greater than one, which shows the infections are spreading in the community. On the other hand, Figure 6(a) is demonstrating the effect of the rate of MTCT of HIV on the effective reproduction number of HIV/AIDS-only submodel. The simulation results in Figure 6(a) show that if the rate of MTCT of HIV (τ2) is less than 0.68, the reproduction number of HIV-only submodel is less than one, which indicates HIV infection dies out in the community. Whereas, if the rate of MTCT of HIV (τ2) is greater than 0.68, the effective reproduction number of HIV/AIDS-only submodel is greater than one, which shows the infection is spreading in the community.Figure 4
Stability of EE of HBV-HIV/AIDS coinfection model for three initial conditions.
(a)
Stability of EE of HBV-HIV/AIDS coinfection model atR3eff=3.14, h1=0.9, and h2=0.5(b)
Stability of EE of HBV-HIV/AIDS coinfection model atR3eff=3.42, h1=0.7, and h2=0.6(c)
Stability of EE of HBV-HIV/AIDS coinfection model atR3eff=4.0, h1=0.8, and h2=0.65Figure 5
Effect of MTCT of HBV onR1eff, R3eff, HB-only infected individuals, and HBV-HIV/AIDS coinfected individuals.
(a)
Effect ofτ1 on R1eff(b)
Effect ofτ1 on R3eff(c)
Effect ofτ1 on HB-only infected individuals(d)
Effect ofτ1 on HBV-HIV/AIDS coinfected individualsFigure 6
Effect of MTCT of HIV onR2eff, HIV/AIDS-only infected individuals, and HBV-HIV/AIDS coinfected individuals.
(a)
Effect ofτ2 on R2eff(b)
Effect ofτ2 on HIV/AIDS-only infected individuals(c)
Effect ofτ2 on HBV-HIV/AIDS coinfected individualsFigure5(c) shows the profile of HB-only infected individuals for different values of τ1. Three different values were considered with increasing τ1 whose values were 0.04, 0.6, and 0.9, respectively. It was observed that when τ1=0.9, there was a rapid growth in HB-only infected population from 10000 to 13225 up to one and a half years. Thereafter, there was a gradual decrease in the population. The gradual decrease in the population was due to enough immunization coverage and effective treatment applied to prevent the spread of the infection. That means with the current prevention and control lowering the number of patients is impossible within one and a half years when the transmission rate is high. Similarly, when τ1=0.6, the number of HB-only infected individuals rises rapidly from 10000 to 10836 within a year. Thereafter, there was a decline in HB-only infected individuals as time progresses as a result of effective prevention and control measures. On the other hand, we did not observe a gradual or rapid increase in HB-only infected individuals when τ1=0.04, but a decrease. This is due to the lowest vertical transmission rate of HBV and effective prevention and control measures applied to forestall the spread of the infection. That is when the transmission rate is low, it is possible to manage the number of HB infected individuals with the current prevention and control.Figure5(d) is demonstrating the effect of τ1 on HBV-HIV/AIDS coinfected individuals. In this figure, for τ1=0.4, τ1= 0.7, and τ1= 0.9, there was a slow increase in HBV-HIV/AIDS coinfected individuals up to thirty years due to enough immunization coverage and effective treatment applied to prevent the spread of the infection. However, for the same values of τ1, there was a rapid growth of coinfected individuals to different peaks from thirty to eighty years. This indicates that the current prevention and control should be modified after thirty years. After eighty years, there was a rapid decrease in the population. This is due to the effective prevention and control measures applied to forestall the spread of the infection and disease-induced deaths.Figure6(b) is about the profile of HIV/AIDS-only infected individuals for three different values of τ2. The figure shows that when τ2=0.3 and 0.6, the number of HIV/AIDS-only infected individuals did not increase but decreased due to the effective control measure applied to forestall the spread of HIV infection. But there was a slight increase when τ2=0.9 within one and a half years. Thereafter, there was a decline in HIV/AIDS-only infected individuals as time progresses as a result of treatment and becomes near to zero after twenty-four years. The effect of τ2 on HBV-HIV/AIDS coinfected individuals is examined in Figure 6(c). As the figure shows when τ2=0.3 and 0.6, the number of HIV/AIDS-only infected individuals did not increase but decreased due to the effective control measure applied to forestall the spread of HIV infection. But there was a slight increase when τ2=0.9 within one and a half years. Thereafter, there was a decline in HIV/AIDS-only infected individuals as time progresses as a result of treatment and becomes near to zero after twenty-four years.


## 8. Conclusion

The importance of epidemiological models lies in their ability to provide meaningful biological interpretations and possible disease prevention and control measures. In this study, we improved the model in [26] to show that the combined effect of MTCT of hepatitis B virus and HIV on their codynamics model, which were not considered in [26]. We derived the effective reproduction number (R3eff) of the improved model and compared it with the effective reproduction number (RBHeff) in [26]. The effective reproduction number of the improved model is R3eff=maxh1ω+d01−η+ω+d01−d1τ1/ω+d0r2+α1+d3+d0,h2+1−d2τ2/ζα2+d4+d0. Based on the data given in Table 2, we evaluated the numerical value of R3eff. As the numerical value indicates, R3eff= max2,0.5=2 which is greater than RBHeff=max0.94,0.28=0.94 in [26]. This tells us that the transmission possibility of the infection in the improved model is high due to MTCT of both infections, which was not observed in the previous work. This is because, in the improved model, MTCT of hepatitis B virus and HIV causes HBV-HIV/AIDS coinfection to spread and may cause an epidemic, whereas, this infection will not spread, there will be a decline in the number of cases, and will eventually die out because one infectious case will infect less than one person on average in his/her infectious period in [26]. We proved that the disease-free equilibrium points of the models are locally and globally asymptotically stable if the associated reproduction numbers 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 are greater than one. From the sensitivity analysis calculated to show the impact of different parameters on R3eff, the most sensitive parameters are τ2 and α1, which are the rate of MTCT of HIV and treatment rate for HBV infected class, respectively. As shown in Section 8, if the rate of MTCT of HBV (τ1) and HIV (τ2) are less than 0.02 and 0.3, respectively, the coinfection of the two viruses will not spread in the community, there will be a decline in the number of cases, and will eventually dies out in the community. Hence, an increase in the rates of MTCT of HBV and HIV exacerbated HBV-HIV/AIDS coinfection, while a decrease in the rates of MTCT of these infections would decline the number of cases, minimize the spread, and help to eliminate the coinfection of HBV and HIV from the community gradually. From the numerical results, we recommend that public policymakers and other concerned bodies must focus on decreasing the rates of MTCT of HBV and HIV in addition to the recommendation given in the study [26] to control the spread of HBV-HIV/AIDS coinfection. Last but not least, it should be noted that this study did not take into account the importance of screening in the dynamics of HBV-HIV/AIDS coinfection. It may affect the transmission dynamics of HBV-HIV/AIDS coinfection in a population. We leave this for future consideration.



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