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

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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][4][5][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 resourceconstrained 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 (ϕ 1 ), the recovery rate for VL-HIV Coinfection (ϕ 2 ), removing reservoirs (c 1 ), and minimizing the contact rate (β h ) 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.

Baseline Model Description and Formulation
We grouped the entire population NðtÞ into ten compartments as those who are susceptible to both diseases ðP 1 ðtÞÞ , immune to HBV after vaccination ðP 2 ðtÞÞ, only infected with HBV ðP 3 ðtÞÞ, only infected with HIV ðP 4 ðtÞÞ, infected with both HBV and HIV ðP 5 ðtÞÞ, those who are receiving HBV treatment ðP 6 ðtÞÞ, those who are receiving HIV treatment ðP 7 ðtÞÞ, HBV-HIV/AIDS treated section ðP 8 ðtÞÞ, individuals with suppressed viral load ðP 9 ðtÞÞ, and HBV recovered class ðP 10 ðtÞÞ. The entire population at time t, denoted by N(t), is given by (vii) Individuals in P 3 and P 5 progress to P 6 and P 8 classes, respectively, after getting treatment. The proportion ζ of individuals in P 4 progress to P 7 due to the treatment rate α 2 and the remaining proportion ð1 − ζÞ of individuals progress to P 5 by force of infection λ 1 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 rate r 1 (ix) Since the effective treatment reduces the viral load of the infected individuals in P 6 ,P 7 , and P 8 classes to the required undetectable level, individuals in these compartments progress to a suppressed viral load class at the progress rate θ 1 , θ 2 , and θ 3 , 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 in P 3 class enter into the recovered class at rate of r 2 , 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 Table 1 and the model assumptions, the flow diagram for the transmission dynamics of the full model is given by: From the flow diagram of the model in Figure 1, the dynamical system of the model is

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.

Theorem 4. The dynamical system (18) is positively invariant in the region
Proof. The dynamics of total population N 1 ðtÞ with respect to time t is computed as In the absence of mortality due to HBV infection and comparing both sides of equation (26) using standard comparison theorem, we obtain After some steps the solution for equation (27) is As t ⟶ ∞, the population size which shows all the feasible solutions of the model (18) with initial conditions enter the region Ω 1 = fðP 1 , P 2 , P 3 , P 6 , P 10 Þ ∈ ℝ 5 + : N 1 ≤ Λ/d 0 g. Thus, the region Ω 1 is bounded.

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 E 0 1 is obtained by setting the right-hand side of the dynamical system equal to zero, providing that fP 3 = P 6 = 0g. After some simple calculation, E 0 1 is equal to 6 Computational and Mathematical Methods in Medicine

Effective Reproduction Number of HBV-Only Submodel.
The effective reproduction number of hepatitis B infected individuals, denoted by R 1 eff 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 [28,29], where F i is the rate of appearance of a new infection in the compartment i, v i is the transfer of infection from one compartment i to another, and E 0 1 is the disease-free equilibrium point. The corresponding Jacobian matrices of F and V computed at DFE point E 0 1 , respectively, are given as follows: and From this, it follows that 3.1.4. Local Stability Analysis of the Disease-Free Equilibrium Point of HBV-Only Submodel (18) is locally asymptotically stable if the effective reproduction number R 1 eff < 1.
Proof. The Jacobean matrix of the dynamical system (18) at the DFE point E 0 1 is After some steps, the roots of (33) are This shows all the eigenvalues of the Jacobean matrix have negative real parts when R 1 eff < 1 Hence, the disease-free equilibrium E 0 (18) is locally asymptotically stable if the effective reproduction number R 1 eff < 1 and unstable otherwise.

Global Stability Analysis of the Disease-Free
Equilibrium Point of HBV-Only Submodel (18) is globally asymptotically stable in the feasible region Ω 1 if the effective reproduction number R 1 eff < 1.
Proof. Consider the following LaSalle-Lyapunov candidate function: where The time derivative of (35) along the solution path yields 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 diseasefree equilibrium E 0 f 1 can be simplified as follows: Substituting (37) and simplification gives us is the only singleton set in fð P 1 , P 2 , P 3 , P 6 , P 10 Þ ∈ Ω 1 : _ f 1 = 0g. Thus, by LaSalle's invariance principle [30], the DFE E 0

Existence of Endemic Equilibrium (EE) of HBV-Only
Submodel. Suppose that E * 1 = ðP 1 * , P 2 * , P * 3 , P * 6 , P * 10 Þ 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 The associated force of infection of the dynamical system (18) is given by where After substituting P * 3 and N 1 * in (40) and simplifying it we got Equation (42) shows that λ 1 * > 0 if R 1 eff > 1. Hence, the endemic equilibrium point E * 1 of HBV-only submodel (18) exists whenever R 1 eff > 1.

Local Stability of the Endemic Equilibrium Point of HBV-Only Submodel
Proof. The Jacobian matrix of (18) at the endemic equilibrium E * 1 is This implies the first two eigenvalues of (45) are λ 1 = − d 0 < 0 and λ 2 = −ðω + d 0 Þ < 0. But, the algebraic sign of the remaining three eigenvalues is determined using Routh-Hurwitz stability criterion from the characteristic polynomial where The sign of the coefficients z 0 , z 1 , and z 2 of the characteristic polynomial z 3 λ 3 + z 2 λ 2 + z 1 λ + z 0 = 0 is positive if R 1 eff > 1. In addition to that z 1 z 2 − z 0 z 3 > 0. Thus, by Routh-Hurwitz stability criterion the endemic equilibrium E * 1 is stable if R 1 eff > 1 and unstable if R 1 eff < 1.

HIV/AIDS-Only
Submodel. This submodel is obtained by setting P 2 = P 3 = P 5 = P 6 = P 8 = P 9 = P 10 = 0. Thus, we have the following dynamical system where the force of infection is given by with initial conditions, P 1 ð0Þ > 0, P 4 ð0Þ > 0, and P 7 ð0Þ > 0 The total population of the dynamical system (50) is given by: Proof. Taking dP 1 /dt = Λ − ðλ 2 + d 0 ÞP 1 from the dynamical system (50), we get (54) is a separable first order ordinary differential equation for the variable P 1 . After some steps we got the solution Since, P 1 ð0Þ > 0, e −ðλ 2 +d 0 Þt > 0 and Λ/ðλ 2 + d 0 Þ > 0 for t ≥ 0, the solution we found is positive. Following the same procedure, P 4 ðtÞ > 0 and P 10 ðtÞ > 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
Proof. The dynamics of total population N 2 ðtÞ with respect to time t is computed as In the absence of mortality due to HIV infection and comparing both sides of equation (57) using standard comparison theorem, we obtain After some steps the solution for the inequality (58) is As t ⟶ ∞, the population size N 2 ðtÞ ≤ Λ/d 0 This implies 0 ≤ N 2 ðtÞ ≤ Λ/d 0 which shows all the feasible solutions of the model (50) with initial conditions enter the region Thus, the region Ω 2 is positively invariant.

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 E 0 2 = ðP 1 0 , P 4 0 , P 7 0 Þ, is obtained by setting the right-hand side of all the components of the model equal to zero, providing that P 4 = 0. It is equal to

Effective Reproduction Numbers of HIV/AIDS-Only
Submodel. In the same way that we have shown in sub section 3.1.3, the effective reproduction number of HIV/ AIDS-only infected individuals is

Local Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel
Theorem 11. The disease-free equilibrium point E 0 2 = ðΛ/ d 0 , 0, 0Þ of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R 2 eff < 1.
Proof. The Jacobean matrix of the dynamical system (50) at the DFE point E 0 2 = ðΛ/d 0 , 0, 0Þ is The corresponding characteristic equation of (63) is After some necessary steps, the roots of the characteristic equation (64) are λ 1 = λ 2 = −d 0 and λ 3 The simplified value of λ 3 This shows that all the eigenvalues of (64) have negative real parts when R 2 eff < 1. Hence, the disease-free equilibrium E 0 2 = ðΛ/d 0 , 0, 0Þ of the dynamical system (50) is locally asymptotically stable if the effective reproduction number R 2 eff < 1 and unstable otherwise.

Global Stability Analysis of the Disease-Free Equilibrium Point of HIV/AIDS-Only Submodel
Theorem 12. The disease-free equilibrium point E 0 2 = ðΛ/ d 0 , 0, 0Þ of the dynamical system (50) is globally asymptotically stable in the feasible region Ω 2 if the effective reproduction number R 2 eff < 1.
Proof. To prove the global asymptotic stability of the diseasefree equilibrium point, we used LaSalle-Lyapunov candidate function as follows: Let W 2 be a LaSalle-Lyapunov candidate function such that where The time derivative of (65) along the solution path yields 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 equilibrium E 0 2 = ðΛ/d 0 , 0, 0Þ with R 2 eff < 1, _ W 2 can be simplified as

Existence of Endemic Equilibrium Point of HIV/AIDS-Only Submodel.
Suppose that E * 2 = ðP * 1 , P * 4 , P * 7 Þ 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 The associated force of infection of the dynamical system (50) is given by where We can show the existence of the endemic equilibrium point of the model by substituting P * 4 and N 2 * in (70). After substitution we got Clearly, λ 2 * > 0 if R 2 ef f > 1. Hence, an endemic equilibrium point E * 2 = ðP * 1 , P * 4 , P * 7 Þ of the dynamical system (50) exists whenever R 2 eff > 1
Proof. The Jacobian matrix of the model (50) at the endemic equilibrium point E * 2 is where 11 Computational and Mathematical Methods in Medicine The corresponding characteristic equation of (73) is where and The sign of the coefficients g 0 , g 1 , and g 2 of the characteristic polynomial λ 3 + g 2 λ 2 + g 1 λ + g 0 = 0 is positive if R 2 ef f > 1. In addition to that, g 1 g 2 > g 0 . Thus, by Routh-Hurwitz stability criterion the endemic equilibrium E * 2 is stable if R 2 eff > 1 and unstable if R 2 eff < 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 P 3 = P 4 = P 5 = 0. Therefore, the DFE point represented by E 0 3 is equal to:

Effective Reproduction Number of the Full Model.
We represented the effective reproduction number of the coinfected model by R 3 ef f . To find R 3 ef f , 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, R 3 ef f can be manipulated as follows: This implies the next-generation matrix FV −1 becomes 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  (2) is locally asymptotically stable if R 3 eff < 1.
Hence, by some algebraic manipulations together with Routh-Hurwitz stability criteria, the disease-free equilibrium point E 0 3 of the dynamical system (2) is locally asymptotically stable if R 3 eff = max fR 1 eff , R 2 eff g < 1

Global Stability Analysis of the Disease-Free Equilibrium Point of the HBV-HIV/AIDS Coinfection Model
Theorem 16. The disease-free equilibrium point E 0 0, 0, 0, 0, 0, 0, 0, 0Þ of the model (2) is globally asymptotically stable in the feasible region Ω if the effective reproduction number R 3 ef f < 1.
Proof. Consider the following LaSalle-Lyapunov candidate function: where and The time derivative of (87) along the solution path yields Since the state variables of the model when the HBV-HIV/AIDS is endemic in the population do not exceed the state variables of the model in a population free of HBV-HIV/AIDS, at the disease-free equilibrium E 0 L can be simplified as follows:

Computational and Mathematical Methods in Medicine
After simplifying (91) we got Substituting m 1 and m 2 (92) gives us This implies _ L ≤ 0 when R 3 eff < 1. Furthermore, _ L = 0 if and only if P 3 = P 4 = P 5 = 0. Thus, by LaSalle's invariance principle [30], the largest invariant set in Ω contained in f ðP 1 , P 2 , P 3 , P 4 , P 5 , P 6 , P 7 , P 8 , P 9 , P 10 Þ ∈ ℝ 10 + g is reduced to the DFE. This proves the global asymptotic stability of the DFE E 0 3 on Ω if the effective reproduction number R 3 eff < 1and unstable otherwise.
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.

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 τ 2 and α 1 , which are the rate of vertical transmission of HIV and treatment rate for HBV infected class, respectively. The graphical representation of sensitivity indices of R 3 eff is given as follows: The sign of the sensitivity index of each parameter value in Figure 2 shows that what will happen to R 3 eff if the parameter is increased or decreased. R 3 eff increases when sensitivity indices with positive signs increase, while R 3 eff decreases when sensitivity indices with negative signs increase and vice versa. The indices in the figure also shows that the percentage change of R 3 eff for each increase value of parameter for 1%. As the figure shows, the sensitivity indices of τ 2 and α 1 are 0.88 and −0:759, respectively. These parameters deserve the most numerical attention. For instance, increasing τ 2 for 10% will increases R 3 ef f for approximately 8

Numerical Simulation
We performed the numerical simulation using the parameter values in Table 2 and the following initial values.
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.

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  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. Figures 4(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).  Figure 6(a) show that if the rate of MTCT of HIV (τ 2 ) 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 (τ 2 ) 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 5(c) shows the profile of HB-only infected individuals for different values of τ 1 . Three different values were considered with increasing τ 1 whose values were 0.04, 0.6, and 0.9, respectively. It was observed that when τ 1 =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 τ 1 =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 τ 1 =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. Figure 5(d) is demonstrating the effect of τ 1 on HBV-HIV/AIDS coinfected individuals. In this figure, for τ 1 = 0.4, τ 1 = 0.7, and τ 1 = 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 τ 1 , 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 preven-tion and control measures applied to forestall the spread of the infection and disease-induced deaths.

18
Computational and Mathematical Methods in Medicine increase but decreased due to the effective control measure applied to forestall the spread of HIV infection. But there was a slight increase when τ 2 =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.

Conclusion
The importance of epidemiological models lies in their ability to provide meaningful biological interpretations and pos-sible 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 (R 3 eff ) of the improved model and compared it with the effective reproduction number (R BH eff ) in [26]. The effective reproduction number of the improved model is R 3 eff = max fðh 1 ðω + d 0 ð1 − ηÞÞ + ðω + d 0 Þð1 − d 1 Þτ 1 /ðω + d 0 Þðr 2 + α 1 + d 3 + d 0 ÞÞ, ð h 2 + ð1 − d 2 Þτ 2 /ζα 2 + d 4 + d 0 Þg. Based on the data given in Table 2, we evaluated the numerical value of R 3 eff . As the  19 Computational and Mathematical Methods in Medicine numerical value indicates, R 3 ef f = maxf2, 0:5g = 2 which is greater than R BH eff =maxf0:94, 0:28g = 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 R 3 eff , the most sensitive parameters are τ 2 and α 1 , 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 (τ 1 ) and HIV (τ 2 ) 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.

Data Availability
Data used to support the findings of this study are included in the article.