Mathematical Analysis of the Role of HIV/HBV Latency in Hepatocytes

The biggest challenge of treating HIV is rampant liver-related morbidity and mortality. This is, to some extent, attributed to hepatocytes acting as viral reservoirs to both HIV and HBV. Viral reservoirs harbour latent provirus, rendering it inaccessible by combinational antiretroviral therapy (cART) that is specific to actively proliferating virus. Latency reversal agents (LRA) such as Shock and kill or lock and block, aiming at activating the latently infected cells, have been developed. However, they are CD4+ cell-specific only. There is evidence that the low replication level of HIV in hepatocytes is mainly due to the latency of the provirus in these cells. LRA are developed to reduce the number of latently infected cells; however, the impact of the period viral latency in hepatocytes especially, during HIV/HBV coinfection, needs to be investigated. Viral coinfection coupled with lifelong treatment of HIV/HBV necessitates investigation for the optimal control strategy. We propose a coinfection mathematical model with delay and use optimal control theory to analyse the effect of viral latency in hepatocytes on the dynamics of HIV/HBV coinfection. Analytical results indicate that HBV cannot take a competitive exclusion against HIV; thus, the coinfection endemic equilibrium implies chronic HBV in HIV-infected patients. Numerical and analytical results indicate that both HIV and HBV viral loads are higher with longer viral latency period in hepatocytes, which indicates the need to upgrade LRA to other non-CD4+ cell viral reservoirs. Higher viral load caused by viral latency coupled with the effects of cART partly explains why liver-related complications are the leading cause of mortality in HIV-infected persons.

1. Introduction 1.1. Background. Since the introduction of highly active antiretroviral therapy (HAART) in 1980s, there has been progressive improvement in pharmacologically managing HIV to the extent that to date, the life expectancy of HIVinfected people is very close to that of coinfected ones. HIV ceased to be a deadly disease but rather a chronic one. Worldwide, there has been commendable use of combinational antiretroviral therapy (cART), and to date, one could expect that there is a cure to this epidemic that has been on the scene for decades. The current cART is pharmacologically designed to hinder viral progression in the viral replication cycle. This cycle starts with binding onto the host cell and ends with budding or viral replication [1]. One of the key bottlenecks, which has slowed the therapeutic management of the disease, is its ability to establish silent reservoirs within the patient. Viral latency is the ability of a provirus that has successfully integrated into a cell, to remain transcriptionally silent and dormant within a host cell for some time, but capable of producing viral copies upon stimulation [2,3]. A cell that hosts a dormant virus is called a latently infected cell or a viral reservoir [4]. These reservoirs are predominantly CD4+ cells [1,4], but research has indicated that there are so many other types of cells that harbour HIV infection [5]. When a cell is latently infected, then it cannot be cleared, neither therapeutically nor through immune killing by cytotoxic T lymphocytes. It is thus asserted that HIV would have a complete cure, if it were not for these viral reservoirs [2]. The challenges of the viral reservoirs stem from the fact that the current cART responds only to active virally replicating host cells. Consequently, patients that are able to take their cART effectively are able to have undetectable viral load for some time, but due to latency, when the provirus rebound, the patient is again overwhelmed with an influx of viral copies. The problem of viral reservoirs has been addressed by numerous research. One major breakthrough was the introduction of a shock and kill strategy [5]. This strategy operates of the principle that a pharmacological agent, such as a cytokine or a small molecule, shocks the virus in the reservoir into transcription or activation [1]. A cell is reactivated the moment it is recognised as an antigen bearing cell that is eventually killed by cytotoxic T lymphocytes or therapeutically by cART. Since HIV reservoirs are predominantly memory CD+4 cells [1,2], all latency reversal agents are CD4+ specific, implying that only CD4+ cells are activated. Recent studies have indicated that due to the nature of memory CD4+ cells which are the majority HIV viral reservoirs, some type of patients would require 70 years of therapy to completely eradicate all the virus [1]. Viral reservoirs are significantly complicating management of HIV, because the current indicator of the therapeutic effect of cART is the measure of viral copies per some capacity of blood. However, in many cases, patients display very low viral copies and sometimes undetectable during routine checks. Unfortunately, the clinical results are not a sole indicator of the therapeutic effect of cART, because low viral replication may be due to high levels of latency. Most reservoirs are created during acute stage of infection, and they are predominantly CD4+ resting cells in the memory subsets [4]. These cells remain in this state until they are stimulated either through antigen recognition or by any other stimuli. In addition to the shock and kill strategy, pharmacologists have developed the block and lock therapy [1]. This involves using latencypromoting agents that would target all viral reservoirs to create an irreversible state, such that a cell that is latently infected will never have a chance to rebound and produce viral genomes [1]. It has been stated that, if it was not for these viral reservoirs, HIV would be cured to date [1,4]. A good example to justify this assertion is the "Berlin" patient, who recovered from HIV just because he had a bone marrow replacement, in which all resting CD4+ cells that were containing the latent provirus were removed [4]. A number of latency reversal agents such as histone deacetylase inhibitors, valproic acid, DNA methylation inhibitors, and protein kinase C agonists have been used [1,4]. However, in addition to their undesired effects such as blocking the activity of HIV-specific cytotoxic T lymphocytes, there has been little success in cell activation and total clearance of provirus reservoirs [4].
Given that HIV latency in host cells still prevails and varies with respect to individual patients, the effect of the period of latency on the dynamics of the infection needs to be investigated. It is worth mentioning that in addition to CD4+ cells, HIV infects other body cells. There is still a debate on HIV infection in hepatocytes, despite evidence from research that HIV productively infects hepatocytes and other hepatoma cells in a CD4-independent way [6]. HIV has also a direct cytopathic effect on hepatocytes, primarily triggering apoptosis via the HIV gp120 proteinreceptor signalling pathway [7]. These directly cytotoxic effects are enhanced in patients coinfected with HIV and viral hepatitis B (HBV), with each virus having significant effects on the others' replication [8]. There has been a lot of concern about high level of liver-related mortality in HIV-infected people, and coinfection with HBV has been mentioned out as one of the leading factors in addition to hepatotoxicity associated cART. The current cART meant to suppress viral replication is very ineffective in human hepatocytes [9], and this has been attributed to the fact that hepatocytes harbour both viruses. This study, therefore, uses mathematical models to investigate the impact of the latency period on the dynamics of HIV and HBV coinfection in the liver.

Mathematical
Modelling. Mathematical models of ordinary or partial differential equations have been used for decades to help understand the within-host dynamics of viral infections. Nowak et al. [10] introduced a basic within-host viral infection model with three variables, namely, target healthy cells, infectious cells, and viral population. This model has been widely adopted and improved to model various aspects and dynamics of viral infections, with and without treatment [11,12]. However, this basic model failed to capture some vital aspect of immunopathogenesis. It assumed that upon infection, cells instantly begin producing virus. Biologically, there is a time delay between viral entry into a host cell and the time the cell begins to release viral copies (intracellular delay) [13,14].
The first intracellular delay model was introduced by Herz et al. [13], to characterize the time between the initial viral entry into a target cell and subsequent viral production. Their study reveals that combining the intracellular delay with less than 100% effective drug therapy results in increased infected cell death as compared to the case of perfect drug therapy. They further report that including a delay changes the estimated value of the viral clearance rate but does not change the productively infected T cell loss rate.
Since Herz et al. [13] studied a number of viral dynamics models, some have included one type of delay to cater for time between viral entry and actual production of virus [15,16], while others have included more than one delay to cater for the time between viral infection and the actual time when cytotoxic T lymphocytes reach out to kill the infectious cells [14,16,17]. Pharmacological delay in the viral treatment has also been studied using mathematical models with medication [18,19]. Intracellular delay of HIV infection dynamics in CD4+ cells [20][21][22], as well as HBV in hepatocytes [23,24], has been studied. While some researchers used the basic model of Nowak et al. [10] and incorporated a delay similar to that of Herz et al. [13], others included latency and two delays, on the assumption that once a virus gains access to a host cell, the cell either becomes productive or remains latent until activation [25,26].
Both discrete time delay [22,27] and continuous delay described by gamma distribution [15,28] have been studied. There are some studies on HIV dynamics in macrophages [26,29,30], but majority of within-host mathematical 2 Journal of Applied Mathematics models on HIV consider viral infection in CD4+ cells. To date, only our previous work in [30][31][32][33] considers the mathematical approach of HIV dynamics in hepatocytes. In this study, we look at the dynamics of HIV/HBV coinfection in human hepatocytes incorporating intracellular delay and antiretroviral therapy.

Model Formulation
We define five variables in the model: healthy hepatocytes T, HIV-infected hepatocytes I h , HBV-infected hepatocytes I b , HIV viral load V h , and HBV viral load V b . Healthy hepatocytes proliferate at a constant rate λ, are infected by HIV and HBV at rates β 1 and β 2 , respectively, and are cleared naturally at rate d 1 . In order to incorporate the intracellular phase of the virus life cycle, we assume that HIV production delays by τ 1 , and for HBV, it is τ 2 , behind the infection of a hepatocyte. This implies that recruitment of HIV and HBV virus-producing hepatocytes at time t is not given by the density TðtÞI h ðtÞ and TðtÞI b ðtÞ of newly infected cells. They are rather given by the density of cells that were infected at given that they are still alive at time t [13]. The probability that a healthy hepatocyte will survive HIV and HBV latency and produce virions after activation is e −d 1 τ 1 and e −d 1 τ 2 , respectively. HIV-and HBV-infected hepatocytes are cleared at rates d 2 and d 3 , whereas the number of HIV and HBV virions produced by one infectious hepatocytes are n 1 and n 2 . Due to low HIV infection in hepatocytes [34], we suppose that some of the infectious HIV copies that come from other cells such as CD4+ cells are moving freely in the liver. HIV viral copies are produced from other cells and macrophages at a constant rate m. HIV and HBV viral copies are cleared from the liver at rates c 1 and c 2 , respectively. The system of equations describing HIV/HBV dynamics in hepatocytes given the above considerations can be stated as   Proof. By the fundamental theory of functional differential equations [14], we suppose that there is a unique local solution TðtÞ, I h ðtÞ, I b ðtÞ, V h ðtÞ, and V b ðtÞ, for the given initial conditions in (3), to system (1) in ½0, T f , where T f is a finite number. Using the constant of variation formula, we get the following solutions to system (1): System (4) shows that the solutions of (1) are positive for all t ≥ 0.
We assume a functional Differentiating Equation (5) and incorporation system (1) result in It is indicated that where Therefore Thus, FðtÞ is bounded and so are the functions TðtÞ, I h ðtÞ, I b ðtÞ, V h ðtÞ and V b ðtÞ.

Steady
States of the System. For steady states, it is assumed that there is no delay dependence, that is 2.2.1. Local Stability and Disease-Free Equilibrium. System (1) has a disease-free steady state defined as D f = ðλ/d 1 , 0, 0, 0, 0Þ. The local stability of D f is governed by the basic reproduction number R 0 , which is the number of secondary viral infections resulting from one virally infected cell in a wholly susceptible cell population. R 0 is established using the next generation operator method as in [35]. It can be shown that the spectral radius of the next generation matrix, which defines the R 0 , is given by where R h and R b are the numbers of secondary infections resulting from one HIV and HBV infectious hepatocyte, respectively. Each HIV and HBV productive hepatocyte infects β 1 λ/d 1 and β 2 λ/d 1 target hepatocytes for a viral lifespan of n 1 /c 1 and n 2 /c 2 . This is conditioned on that when a cell is infected by HIV and HBV, it will survive for the time period of τ 1 and τ 2 , with a probability e −d 1 τ 1 and e −d 1 τ 2 .
From Theorem 3 of Van den Driessche and Watmough [35], we deduce the following theorem.
Theorem 3. The disease-free equilibrium D f is locally asymptotically stable when R 0 < 1, and unstable otherwise.
In order to establish other conditions that determine the stability of D f , we use the Jacobian matrix of system (1).
from which it is seen that traceðJ D f Þ < 0 and We establish that det ðJ D f Þ < 0 only if We thus deduce the following theorem.
Theorem 4. The disease-free equilibrium D f is locally asymptotically stable when R 0 < 1 given that R h < 1 and R b < 1. Figure 1 shows higher level of R b and compared to R h , which is an indication that HBV is more aggressive in hepatocytes than HIV is. Both reproduction numbers grow with respective increasing infection rates and are inversely proportional to delay in viral production.
To further ascertain the influence of one virus on the other, we express R h in terms of R b to have For strictly positive parameters, ∂R b /∂R h > 0 and ∂R h /∂R b > 0, implying that the presence of HIV in the liver influences the increase of HBV and vice versa [36,37].

Global Stability of Disease-Free Equilibrium.
The global stability is of system (1) is derived using a theorem by Castillo-Chavez [38] as shown in Appendix A.
Theorem 5. Castillo-Chavez et al. [38]. We write system (1) in the form 0Þ called the Metzler matrix, whose off-diagonal elements are nonnegative, as , SinceĜ ≥ 0 for all ðX, ZÞ in the region where the model makes biological meaning, then D f is globally asymptotically stable.

Existence of Boundary and Interior Equilibria with
Possible Competitive Exclusion. System (1) is expected to have three nontrivial equilibria as follows: Solving system (1) by equating the left-hand side to zero results in two equilibria, that is, HIV-only and coinfection equilibria, indicating that E B ðT + , 0, The absence of HBV-only endemic equilibrium is an indication that, given the proliferation of HIV from other cells other than hepatocytes, it is not likely that HBV will take competitive exclusion over HIV in the liver. The HIV-only endemic equilibrium E H ðT * , I * h , 0, V * h , 0Þ is given by where It is analytically cumbersome to deduce the conditions under which this steady state exists. Assessing the endemicity of either infection under the circumstance that HIV viral copies that infect hepatocytes are only proliferated in hepatic cells, it is assumed that m = 0. In this case, we have both HIV-only (E h ) and HBVonly (E b ) endemic equilibria.
For the case of HIV-only, the endemic equilibrium, the point under the assumption that m = 0, is given by We can then derive the following result.  Journal of Applied Mathematics Lemma 6. If HIV that infects liver cells proliferates only in hepatic cells, then HIV-only endemic equilibrium will exist when R h > 1.

Local Stability of E h .
Assuming that E h exists when ðV b , I b Þ ⟶ ð0, 0Þ as t ⟶ ∞, the Jacobian matrix of system (1) is derived as The determinant of J E h is given by Given that traceðJ E h Þ < 0, we deduce the following. Considering the HBV-only endemic equilibrium when m = 0, E b exists when ðV h , I h Þ ⟶ ð0, 0Þ as t ⟶ ∞ and is given by Thus, E b exists when R b > 1. Using the same method as above, the conditions for local stability of of E b can be established.

Existence of the Interior Equilibrium
It is not likely that both virus exist in same magnitude over the infection period. As t ⟶ ∞, either HIV or HBV will dominate the liver. HIV dominates when R h > R b . In this case, if R b > 1, then I b and V b will exist (Equations (26) and (28)), but I h and V h will not lie within a biologically feasible solution set, as seen in Equations (25) and (27). Thus, the coinfection endemic equilibrium E HB will not exist when the liver is dominated by HIV. On the other hand, when hepatitis dominates the liver at endemicity of the coinfection, that is, R b > R h together with R b > 1 (of course, HBV can only become endemic when R b > 1), then only I h and V h will surely exist with respect to Equations (25) and (27). We investigate the parametric conditions that allow for the feasibility of I b and V b .
Using Equation (26) Using Equation (28) Rearranging (29) gives and rearranging (30) gives Thus, for a biologically feasible solution, we have We therefore deduce the following result.
Result 2. Whenever R b > 1, the coinfection endemic equilibrium state E HB exists anywhere in the biologically feasible region 6 Journal of Applied Mathematics

Numerical Results of the Model.
In this section, we present the numerical results of system (1), with parameter values as indicated in Table 1. Due to lack of previous studies on the dynamics of HIV in hepatocytes, we assume that HIV and HBV have the same period of latency in hepatocytes. This indicates the probability that an HIV or HBV producing hepatocyte surviving through the time period τ 1 = τ 2 is theoretically equal. Biologically, according to Alshorman et al. [25], the initial value of healthy cells ðTð0ÞÞ, proliferation rate λ, and clearance rate d 1 are related by λ = d 1 Tð0Þ; thus, initial values are as follows: T 0 ð Þ = 3:0769 × 10 6 , I h 0 ð Þ = 10, Numerical results of system (1) are shown in Figure 2. Variables approach an infected steady state, because parameter values as shown in Table 1 satisfy the condition in Result 2. Comparing the dynamics with and without delay, Figure 3 shows that the peak times of both virus as well as the decay time of healthy hepatocytes are longer with delay than without. We summarise the peak values and times as presented in Table 2.
With the same initial values for infectious classes and viral copies, despite the influx of HIV from cells other than hepatocytes, HBV peak values are a hundred-fold higher than HIV. Additionally, HBV peaks later than HIV, which also seems to support another study [3] that indicates that the presence of HIV changes the natural his-tory of HBV, with the progression of liver-related diseases 17 times more in the coinfected than in HBV monoinfected individuals. With varying delay period, Figure 4 shows that the longer it takes for HIV and HBV to replicate in hepatocytes, the more the HIV and the less the HBV virus multiplies.

Optimal Control Problem
The optimal control method has been studied widely in a number of settings including analysis of control strategies in infectious diseases. In within-host viral dynamics particularly, controls are antiretroviral drugs used to suppress the progression of a pathogen. In some studies, controls are instantaneous [41,42], while in others, a pharmacodynamics delay is incorporated [18,43]. HIV and HBV combinational antiretroviral therapy (cART) consists of infusion, integrase, reverse transcriptase, and protease inhibitors, all aiming at blocking viral replication in vivo [16]. In HIV/HBV-coinfected persons, cART is supposed to suppress replication rate of both virus in all body cells. It has been reported that, generally, cART is not as effective in hepatocytes as compared in CD4+ cells, possibly because of high level of HIV latency in this type of cells [3].
Different classes of antiretroviral drugs serve different purposes and act at different stages of viral progression. Most of the drugs are enzyme inhibitors, and they incapacitate different enzymes that aid viral multiplication within a host cell. In order to monitor effectiveness of cART in HIV-or HIV/HBV-coinfected patients, frequent viral load testing, regulated by WHO [44], is carried out.
Time-dependent resultant efficacy ϕðtÞ, for 0 ≤ ϕðtÞ ≤ 1, is considered. It is assumed that in the case of two enzyme inhibitors, whose efficacies are ϕ 1 ðtÞ and ϕ 2 ðtÞ, we have ϕðtÞ = ϕ 1 ðtÞ + ϕ 2 ðtÞ − ϕ 1 ðtÞϕ 2 ðtÞ. Thus, 1 − ϕðtÞ = ð1 − ϕ 1 ðtÞÞð1 − ϕ 2 ðtÞÞ, which indicates that the two drugs act independently with additive efficacy. The same argument can be used even   Table 1. Horizontal axes represent time in days.  Table 1. Horizontal axes represent time in days. 8 Journal of Applied Mathematics when three types of enzyme inhibitors are used. Considering an instantaneous control efficacy, the optimal control problem is given as It is worth noting that Equation (35) is similar to Equation (1), with all parameters taking the same descriptions as indicated in Section 2. However, the resultant efficacy ϕðtÞ is expected to reduce the number of viral copies in a coinfected person. Thus, the number of HIV (n 1 d 2 I h ðtÞ + m) and HBV (n 2 d 3 I b ðtÞ) copies generated are reduced to ð1 − ϕðtÞÞn 1 d 2 I h ðtÞ + ð1 − ϕðtÞÞm and ð1 − ϕðtÞÞn 2 d 3 I b ðtÞ, respectively. This indicated that the higher the value of ϕðtÞ, the fewer the viral copies, and when the cART is 100% efficacious (ϕðtÞ = 1), then no viral copies will be produced.
Treatment strategy ϕðtÞ that minimises the number of HIV and HBV viral load in the liver and consequently maximises healthy hepatocytes for the entire treatment duration T f is investigated. The strategy ϕðtÞ should minimise the viral load, as well as the cost of treatment and the corresponding side effects of the medication that result in reduced density of the pathogen. The objective function is defined as Parameter A is established by combining the benefits and costs of the treatment such that it equalises the size of the term ϕðtÞ and reflects the severity of the side effects of the drugs. The cost function is quadratic because it should be nonlinear at the optimal controls ϕ * ðtÞ. However, there is no linear relationship between the effect of the therapy on hepatocytes and viral load [45,46]. We seek an optimal control strategy ϕ * ðtÞ such that where U is the control set defined by 3.1. Existence of an Optimal Control. The existence of an optimal control ϕ * ðtÞ can be proved by the results of Fleming and Rishe in [47].

Theorem 7.
Consider the control problem with system (35). There exists an optimal control ϕ * ∈ U such that Proof. To use an existence result in Fleming and Rishe in [47], we must check the following properties.
(1) The set of controls and corresponding state variables is nonempty (2) The control set U is convex and closed The control ϕðtÞ from (38) and state variables TðtÞ, I h ðtÞ, I b ðtÞ, V h ðtÞ, and V b ðtÞ from (4) are nonnegative and nonempty which satisfy condition (1). The control set is convex and closed by definition in (38). Since the state system (35) is linear in ϕðtÞ, the right-hand side satisfies condition (3). Using the boundedness of solution as indicated in Section 2.1, the integrand in the objective function (36) is convex on U (because its quadratic in ϕðtÞ); thus, condition (4) is satisfied. Finally, we can easily see that there exists a constant β > 1 and positive numbers c 1 and c 2 satisfying Equation (40).

Characterization of the Optimal Control.
In order to derive the necessary condition for the optimal control, Pontryagin's maximum principle with delay given by Gollmann et al. [18] is used. First, we define the Lagrangian for the optimal control problem (35) as   Table 1. Horizontal axes represent time in days.

Journal of Applied Mathematics
The principle converts the system (35), the objective function (36), and the optimal control (39) into a problem of minimising the Hamiltonian given by where λ i for i = 1, ⋯, 5 are the adjoint functions and f i are the right hand side of the system of state variables in (35). Using the principle in [18], we arrive at the following theorem.

Theorem 8.
Given an optimal control ϕ * ∈ U and solutions T * , I * h , I * b , V * h , and V * b of the corresponding state system (35), there exist adjoint functions λ 1 , ⋯, λ 5 which satisfy with transversality conditions λ i ðT f Þ = 0, for i = 1, ⋯, 5, and the corresponding optimal control given by and χ ½0,T f −τ i is the characteristic function on interval ½0, T f − τ i defined by Proof. Using Pontryagin's maximum principle with delay in state [18], we can obtain the adjoint equations and transversality conditions as For i = 1, 2, the optimal control ϕ * can be solved using the optimality condition and by the bounds of ϕðtÞ defined in (38), it is easy to show that Equation (44) holds.
Combining the state variables in Equation (35), the system of adjoint variables (44), and the optimal control (44), the optimality system is derived as where ϕ * ðtÞ is given by Equation (44). Numerical results of the optimality system (48) are obtained by finite difference approximation method [40], between t = 0 and T f = 400 days. Parameter values are as shown in Table 1, and the initial values are as shown in (34). Figures 5 and 6 show that medication is able to control either infection. This is in support to the objective of the control strategy, which is aimed at minimising the viral load. However, the reduction in viral load seems not to imply significant growth in healthy hepatocytes as also shown in Figure 5. Nonetheless, it is the opposite when compared to Figure 6.
With initial viral copies of both infections set as 10, numerical results indicate that at the end of control period, there are 3141 copies of HIV and 1918 copies of HBV. Comparing with 45820 and 3:094 × 10 6 copies of HIV and HBV, respectively, without control, is a clear indication that therapy is more effective in reducing HBV load. The control profile in Figure 7 shows the treatment administration schedule over the period of 300 days. The control should be applied at maximum effectiveness during the first two weeks because of the high viral load in the patient. There is periodicity in control application for the next 4 months before a steady upper bound, but the control does not go below 80%. This indicates that considering all the assumptions taken in this study, if efficacy is maintained beyond 80%, it is possible to reduce the HIV/HBV load in coinfected patients and the systemic cost. Figure 8 shows that there is an increase in both HIV and HBV copies with increase in delay period. We have seen that without the control presented in Section 2.4, the delay only increases HIV copies and not HBV. This suggests that the longer the virus hides in a host cell, the less effective the therapy and consequently the more viral load in the patient  Table 1. Horizontal axes represent time in days.  Table 1. Horizontal axes represent time in days.

12
Journal of Applied Mathematics is. We can also observe from unlike in Figure 6, where HBV is tens of thousands more than HIV, that when control is applied, HBV load becomes less than HIV load, which is an indication that therapy is more effective in controlling HBV.

Discussion and Conclusion
This study proposed a mathematical model representing coinfection of HBV and HIV in human hepatocytes. The model included a time delay to represent the time between viral entry into a host cell and the actual time when the cell is able to replicate viral copies. When the virus gain entry into a host cell, the integrated provirus may remain latent for a number of years without replication. The cell hosting a provirus is regarded as a viral reservoir. The reservoir is not able to attract cytotoxic killing from the virus-specific lymphocyte, neither can it be cleared using cART. The effect of latency period on the dynamics of HIV/HBV infection was investigated. Analytically, model solutions were found to be positive and bounded. The disease-free equilibrium D f was computed and the basic reproduction number R 0 deduced. It was established that D f is locally asymptotically stable, when R 0 < 1 and unstable when R 0 > 1. Results indicate that HBV cannot gain competitive exclusiveness over HIV given the influx of HIV from other cells; thus, there is no HBV-only endemic equilibrium point. Coinfection endemic exists with some parametric conditions indicated in Result 2. Failure to have a possible HBV-only endemic equilibrium, but rather an HIV-only and coinfection endemic equilibria, is an indication that HBV becomes chronic in HIV-infected patients [3]. Assuming equal intracellular delay for both HIV and HBV of 5 days, numerical results show that there is higher HIV load with viral latency than without. However, the effect of latency period on the number of viral copies is not seen at the start of the infection. This is an indication that the influx of HIV viremia is the one that has an impact on the HBV viral load that consequently shoot up. This could signify that hepatocytes acting as HIV reservoirs have a greater impact on HIV dynamics, which consequently influences HBV. It has been previously reported that HIV alters the dynamics of HBV in a manner that is not yet known [3].
With parameter values used in simulations, it was found that R 0 = R b , where R b is the number of secondary infections resulting from one HBV infectious hepatocyte. This is evidenced by higher HBV viral load as compared to HIV in numerical simulations and is in line with study by Parvez [3], who reports that HIV coinfection significantly alters the natural history of hepatitis B and therefore complicates the disease management. They further assert that, in HIVcoinfected individuals, liver-related mortality is over 17 times higher than those with HBV monoinfection. Even though the use of cART has changed the overall outcome of HIV in infected patients, from a fatal to a chronic one, pharmacologists are still trying to find an absolute cure to  Table 1. Horizontal axes represent time in days.
13 Journal of Applied Mathematics the disease. Their efforts have been hindered by viral reservoirs that harbour the provirus for a long time making cART inefficient, since it targets only actively replicating virus. Given both challenges of viral latency and lifelong treatment, optimal therapy that minimises active viral load and cost of treatment, combined with side effects, needed to be investigated. By using Pontryagin's maximum principle of delayed models, the optimal control problem was characterized and the expression of the optimal control was computed. Using the finite difference approximation method, simulation results show that the resultant control strategy is effective in reducing the active viral load of either infection. The optimal efficacy that minimises the viral load for the entire control period of 400 days is over 80%. It is important to note that as latency reversal agents are manufactured, there is also a need to ensure that the current cART is highly efficacious in eliminating all actively replicating virus.
Numerical results from the optimal control model with varying delay periods indicate that the longer the delay is, the more HIV and HBV viral load builds up. An increase in HIV and HBV copies with increase in latency period, in the presence of cART, is an indication that viral latency hinders drug management and control [3,9]. Generally, latent reversal agents should not only be CD4+ cell-specific since there is evidence that CD4+ cells are not the only viral reservoirs. For the liver specifically and in HIV/HBV-coinfected patients, HIV latency does not only lead to higher HIV load and hinder drug management, but it also implicates higher HBV viral load in the liver. where the components of the column-vector X ∈ ℝ m denotes the number of uninfected individuals and the components of vector Z ∈ ℝ n denotes the number of infected individuals. Let U 0 = ðX * , 0Þ denote the disease-free equilibrium of this system. The fixed point U 0 = ðX * , 0Þ is a globally asymptotically stable equilibrium for this system provided that R 0 < 1 (locally asymptotically stable) and the following two conditions satisfied:

Data Availability
The data supporting this coinfection model is from previously published research articles as cited in Table 1.

Conflicts of Interest
The authors declare that there are no conflicts of interest concerning the publication of this paper.