MODELING TOBACCO SMOKING EFFECT ON HIV ANTIRETROVIRAL THERAPY

In this paper, we formulate a mathematical model to study how tobacco smoking affects antiretroviral therapy. The model is based on the fact that smoking induces metabolism of ARVs and HIV patient addiction to smoking affects adherence to drugs. Equilibrium states and effective reproduction number Re f are computed. Conditions for equilibria stability are derived. The model shows that in the absence of tobacco smoking and HIV, disease free equilibrium is stable when Re f < 1 and in the presence of tobacco smoking and HIV, endemic equilibrium is stable when Re f > 1. The analysis shows that tobacco smoking decreases the efficacy of antiretroviral therapy and this effect is critical when smoking induces metabolism of antiretroviral drugs by 30% to 70%. Even if a HIV infected smoker remains adherent to therapy, still the effect of tobacco smoking on drugs’ efficacy is inevitable. For management of HIV epidemic and its therapy, abstinence from smoking by HIV patients is recommended.


Introduction
The world has been suffering from HIV pandemic for over three decades now, with Sub Saharan Africa which has the majority of HIV infected individuals being a focal point for the pandemic [22,24,33]. Lack of vaccine and life prolonging drugs has led to significantly high AIDS mortality rate among HIV infected individuals. Antiretroviral drugs have dramatically reduced HIV morbidity and mortality from opportunistic infections, they reconstitute patients' immune system and improve their lives [16,26,27]. Under therapy CD4+ T-cells which are mostly devastated by HIV, regain their number by increasing production in the bone marrow [14].

Administration of antiretroviral drugs blocks a specific stage of HIV life cycle within a CD4+
T-cell. During the infection, a HIV virion attaches to the cell and inserts its genetic material in the form of RNA into the cell when fusion takes place [7]. The viral RNA in the host cell is reverse transcribed into viral DNA by protein reverse transcriptase and this DNA is then integrated into the DNA of host cell by the integrase [7,32]. Fusion inhibitors block fusion between the virus and the cell [5], Reverse Transcriptase Inhibitors (RTIs) block viral reverse transcription and Protease Inhibitors (PIs) block production of infectious virions.
Though antiretroviral drugs suppress the virus and improve quality of life [3], risk behaviour such as tobacco smoking negates and undermine the benefits of antiretroviral therapy among HIV infected individuals [10,15]. Apart from causing anergy and unresponsiveness to T-cells [30,31], tobacco smoking generation of poisonous, toxic and carcinogenic multi-components [4,28] interferes with metabolism of antiretroviral drugs. Nicotine which, in particular, is a main component of tobacco smoke induces the enzymes which are involved in metabolism of antiretroviral drugs (NNRTIs and PIs) [1,25]. Induction decreases the drugs' concentration and reduce their absorption [2].
Few cohort studies which have addressed the effect of tobacco smoking on the antiretroviral drugs concentrated on how tobacco smoking affects adherence to antiretroviral drugs [9,15,18,23,29] and ignore drug interaction. Though tobacco smoking is mentioned to affect adherence to antiretroviral drugs there have been inconsistent results as there are studies which claim the effect of tobacco smoking on drugs' adherence [9,10,18] and there are some which do not claim tobacco smoking effect on drugs' adherence [23,29].
A cohort study by Feldman et al. [10] considering a group of low-income women who are infected with HIV in highly active antiretroviral therapy era suggests that, smoking can negate antiretroviral therapy because it leads to poor viral responses, high risk of virologic rebound and immunologic failure. The study emphasizes that poor viral responses, virologic rebound and immunologic failure are due to non adherence to the therapy. According to the study, adherence to therapy is associated with high viral response and less viral and immunologic failure.
In this study mathematical modeling is used to explore the effect of tobacco smoking on antiretroviral therapy. For the best of our knowledge no study has ever used dynamical model to study the effect of tobacco smoking on antiretroviral therapy. However, mathematical models on antiretroviral therapy and its impact on HIV in-host dynamics have been developed. [14] developed a three components (Uninfected CD4+ T-cells, HIV infected CD4+ T-cells and Virus) model to determine threshold count of CD4+ T-cells for which treatment can be initiated without emergence of drug resistance. Srivastava et al. [32] extends a three components model to include pre and post-reverse transcription CD4+ T-cells. Li and Wang [17] included reverse transcriptase inhibitors in a three components model and show backward bifurcation due to viral rebound after ART stoppage. Consideration of tobacco smoking and HIV within the host differentiates our model from the rest and it the first attempt to model tobacco smoking effect on antiretroviral therapy. This work is organized into the following order; in the next Section we formulate the model and carry out analysis, numerical analysis is done in following Section and we conclude with discussions.

Model formulation
In this section we formulate the model by considering antiretroviral drugs: Reverse Transcription Inhibitors (RTIs) and Protease Inhibitors (PIs). The model divides T-cells into five classes: X represents density of uninfected T-cells, X 1 density of smoking partially impaired T-cells, X 2 density of HIV latently infected T-cells, X 3 density of smoking critically impaired T-cells and X 4 density of HIV productively infected T-cells. The density of free virus is repre- which is a decreasing function of free virus [6,14] is a recruitment rate for T-cells. The expression γX 1 +ηX 3 represents smoking impairment rate with γ < η being relative smoking impairment rates of X 1 and X 3 due to the fact that, smoking critically impaired T-cells have high concentration of tobacco smoke poisonous and carcinogenic compounds. Parameters β 1 and β 2 represent HIV infection rates for uninfected T-cells X and smoking partially impaired T-cells X 1 respectively. Since reverse transcription in smoking critically impaired Tcells X 3 is assumed to be spontaneous, in the presence of ARVs, HIV infects smoking critically impaired T-cells at a rate β 3 (1 − f 1 ε) where ε such that 0 ≤ ε ≤ 1 is the efficacy of RTIs in blocking reverse transcription in T-cells and f 1 (ϖ) = e −ϖ ϖ + 1 is the smoking effect in inducing metabolism of ARVs in critically impaired T-cells, ϖ ∈ (0, 1) is the rate at which smoking induces metabolism of ARVs. When ϖ = 0, smoking does not induce metabolism of ARVs and when ϖ = 1, smoking induces metabolism of ARVs at a highest rate.
Smoking partially impaired T-cells X 1 progress to smoking critically impaired T-cells X 3 at a rate ρ. HIV latently infected T-cells X 2 due to the presence of smoking partially impaired Tcells, progress to productively infected T-cells X 4 following successful reverse transcription at a rate σ (1 − f ε) where f = e −ϖ models smoking inducing effect in partially impaired T-cells.
Functions f and f 1 model relative smoking inducing effect in smoking partially and critically impaired T-cells as we assume smoking critically impaired T-cells experience high smoking inducing effect compared to smoking partially impaired T-cells. However, if ζ 1 denotes drugs' reverse transcription blocking rate in T-cells and ϑ adherence rate to treatment, then ε = ζ 1 ϑ .
Parameter α is smoking induced mortality in critically impaired T-cells and µ 1 is a HIV induced mortality in HIV productively infected T-cells which results into production of infectious virions at a rate Nµ 1 (1 − f 1 ξ ), ξ = κϑ such that 0 ≤ ξ ≤ 1 is the efficacy of PIs in blocking production of infectious virions in T-cells, κ is the rate at which PIs block production of infectious virions in T-cells and N is the number of infectious virions produced by each T-cell.
The model assumes that, metabolism of RTIs and PIs occurs at a cellular level. Tobacco smoking induces metabolism of RTIs and PIs uniformly. On HIV infection, smoking partially impaired T-cells join HIV latently infected compartment because their reverse transcription is not spontaneous and critically impaired T-cells join productively HIV infected T-cells as their reverse transcription is spontaneous. The model also assumes that unresponsiveness of T-cells increases with smoking rate. Interaction of variables is shown in Figure 1, state variables and model parameters are described in Tables 1 and 2 X β    Considering the relationship between variables, parameters and assumptions, the model is governed by the following system of differential equations;

Boundedness of solutions
Summing equations for T-cells in system (1), we obtain which results into Applying integration factor e µt , the general solution of equation (4) is given by We consider two cases to determine limiting solution in equation (5), when T t (0) ≥ Λ µ and when This implies that when T t (0) ≥ Λ µ , the RHS of equation (5) is positive and decreasing implying that T t (0) is the maximum value.
This shows that when T t (0) ≤ Λ µ , the RHS of equation (5) is negative and increasing implying Λ µ is the maximum value. Hence Generally we see that This result implies that both uninfected and infected T-cells are bounded. It follows that From equation (1f), it can be shown that V is also bounded. We consider the equation Using equation (10), it is seen that which works out to be The upper bound for free virus depends on the efficacy of PIs ξ and the magnitude of the effect of tobacco smoking f 1 . We find that, the system (1) is positive invariant in the set The solutions of the model (1)

Disease Free Equilibrium and Effective reproduction
Number R e f The model system (1) has disease free equilibrium given by In this steady state, the immune system is strong and responds accordingly to infections. CD4+ T-cells perform their role as expected. To show the potentials of ARVs on HIV in the presence of tobacco smoking, we compute effective reproduction number R e f in next section.

Effective reproduction Number R e f
The average new impairments due to tobacco smoking and the average new infections due to HIV in the presence of antiretroviral therapy are measured by effective reproduction number R e f which is computed by next generation approach [34]. The effective reproduction number infections H i and transfer terms Z i are: The matrices H and Z work out to be If we denote the effective reproduction number for T-cells by R e f such that we find that The threshold quantities R eS and R eA are partial effective reproduction numbers due to tobacco smoking and HIV respectively. The partial effective reproduction number due to HIV R eA depends on efficacies of antiretroviral therapy in blocking reverse transcription ε and production of infectious virus ξ , the rate at which smoking affects drugs' efficacy in blocking reverse transcription f and the rate at which smoking affects drugs' efficacy in blocking production of infectious virions f 1 .

Analysis of partial effective reproduction number R eA
To analyze partial effective reproduction number due to HIV R eA , we write it as a function of efficacies of antiretroviral drugs in blocking reverse transcription and production of infectious virions, and the rate at which smoking affects the efficacy of antiretroviral drugs. The partial effective reproduction number is therefore: Variation of the effective reproduction number R eA with respect to the efficacy of antiretroviral therapy in blocking reverse transcription in partially impaired T-cells ε gives and variation of R eA with respect to the efficacy of antiretroviral therapy in blocking production of infectious virions ξ , gives Equations (22) and (23) show that if tobacco smoking does not induce metabolism of antiretroviral drugs, they can avert HIV new infections within a HIV smoker successfully. Since tobacco smoking affects antiretroviral therapy by inducing their metabolism [25] and by reducing drugs' adherence [10], we express the efficacy of the antiretroviral drugs as a product of drugs' blocking rate and adherence rate. We assess the effect of tobacco smoking on the efficacies of antiretroviral drugs by differentiating partial effective reproduction number R eA with respect to the rate at which smoking induces metabolism of antiretroviral drugs ϖ and adherence rate ϑ to obtain: We find that: If tobacco smoking continues and equation (26) holds, the efficacy of antiretroviral drugs will decrease as smoking inducing effect increases, this will result into increased HIV new infections and decreasing the number of CD4+ T-cells. However, if equation (27) holds, adherence to drugs will reduce HIV new infections significantly but as adherence drops, new infections will increase.

Stability Analysis of Disease Free Equilibrium
In this section local and global stabilities of disease free equilibrium are investigated using linearization method, Hurwitz criterion and comparison theorem.

Local stability of Disease free equilibrium
The system (1) is linearized at disease free equilibrium to obtain a matrix Local stability of disease free equilibrium holds if the matrix J A has negative eigenvalues. From the first column of the matrix J A , we find the eigenvalue −µ < 0. The matrix then reduces to The system (1) has a stable disease free equilibrium if the coefficients of the polynomial (30) obey the following conditions: From the polynomial (30), we see that This implies that, the coefficients of a polynomial (30) do not change sign when there is low smoking rate and low HIV infection rate. Using coefficients of polynomial (30) The equations (34) and (35) hold if (36) R eS < 1 and R eA < 1.
Local stability of disease free equilibrium is concluded in the following theorem: Theorem 2.2.

Disease free equilibrium of the model (1) is locally asymptotically stable if equations (34) and
(35) hold and R eS < 1 and R eA < 1, it is unstable otherwise.

Global stability of Disease free equilibrium
The global stability of disease free equilibrium is investigated by using comparison theorem as applied by Diekmann et al. [8]. We state and prove the Theorem . Theorem 2.2. Disease free equilibrium is globally asymptotically stable if R e f < 1 when (X 1 , X 2 , X 3 , X 4 ,V ) → (0, 0, 0, 0, 0) and X → Λ µ .

Proof
Using comparison theorem the infected classes of the model (1) are written as The matrix H − Z has negative eigenvalues, equation (38) will hold if and only if X → Λ µ , 0, 0, 0, 0) and R e f < 1 and this completes the proof.

Existence of Endemic Equilibrium
The two epidemics tobacco smoking and HIV are endemic when there is high rate of tobacco smoking and HIV infection. The endemic equilibrium of the model system (1) is defined by The expressions of X * , X * 1 , X * 2 , X * 3 and X * 4 are in terms of free virus V * . We obtain free virus V * which determines the number of endemic equilibria by substituting X * , X * 1 , X * 2 , X * 3 in the equation dV dt at equilibrium. There are multiple endemic equilibria.

Global stability of endemic equilibrium
Local stability of disease free equilibrium in Π 2 imply local stability of endemic equilibrium by reverse condition [34], we therefore proceed with global stability of endemic equilibrium.
Lyapunov function and LaSalle's invariance principle have been useful in analysis of global stability of endemic equilibrium. In this work, logarithmic Lyapunov function as applied by Mpeshe et al. [20] and Mpeshe et al. [21] is used to investigate the stability of endemic equilibrium given in equation (40).
Logarithmic Lyapunov function U is defined by where a i are positive constants, X * i is an equilibrium point and X i is a density of cells or free virus in compartment i. Using logarithmic Lyapunov function (41) while T-cells' compartments are represented by T , a system (1) is written as Differentiation of equation (42) with respect to time gives At endemic equilibrium Π * 2 , we have Simplifications and rearrangement give Following equations (43) and (44), F is non-positive [19,22]. Therefore it follows that F ≤ 0 for X, X 1 , X 2 , X 3 , X 4 ,V > 0. This implies that dU dt ≤ 0 for X, X 1 , X 2 , X 3 , X 4 ,V > 0 and it is zero This indicates that the largest compact invariant set Π 2 in which dU dt = 0 is the singleton Π * 2 which is the endemic equilibrium. By LaSalle invariant principle, Π * 2 is globally asymptotically stable in the interior of Π 2 when R e f > 1. We summarize this result in the following theorem Theorem 2.1. If the effective reproduction number for T-cells R e f > 1, then the model system (1) has a unique endemic equilibrium Π * 2 which is globally asymptotically stable in the interior of Π 2 .

Numerical Analysis and Discussions
In this section numerical results for the model system (1) are presented and discussed by considering two scenarios; the case when tobacco smoking induces metabolism of ARVs without affecting drugs' adherence and the case when tobacco smoking induces metabolism of ARVs and affects drug adherence. However, we start with the general dynamics when smoking induces metabolism of ARVs and when it does not induce metabolism of ARVs. Using parameter values in Table 3      As smoking rate increases smokers become depressed and stressed hence fail to take ARVs as required, thus drug adherence drops. When smoking inducing effect increases and adherence to ARVs decreases, free virus and HIV actively infected T-cells increase. This is depicted in

Conclusion
A deterministic model for tobacco smoking effect on antiretroviral therapy is presented and analyzed to provide insights how antiretroviral therapy is affected by tobacco smoking. The  model is formulated basing on the fact that, tobacco smoking induces metabolism of ARVs and has an influence on drugs' adherence. To measure and assess the effect of drugs' adherence we expressed drugs' efficacy as the product of adherence and the rate at which RTIs block reverse transcription and the product of adherence and the rate at which PIs block production of infectious virions.
The equilibrium states and effective reproduction number R e f which is given as the maximum of the partial effective reproduction numbers due to tobacco smoking R eS and HIV R eA is computed and analyzed. Analysis of HIV partial effective reproduction number R eA indicates that tobacco smoking decreases the efficacy of ARVs because as tobacco smoking induces metabolism of ARVs, HIV new infections increases. Additionally, when drugs' adherence drops as a result of stress and depression due to high smoking rate, HIV new infections increases.
Numerical analysis guarantees a stable endemic equilibrium when R e f > 1. The effect of tobacco smoking on ARVs is studied numerically by considering two cases; the case when smoking induces metabolism of ARVs and does not affect drugs' adherence, and the case when smoking induces metabolism of ARVs and affects drugs' adherence. For the two cases it has been found that, tobacco smoking decreases the efficacy of ARVs. This effect is significant when tobacco induces metabolism of ARVs by 30% to 70%. We consistently found that even if HIV smoker remains adherent to therapy, tobacco smoking effect on ARVs will still persist.
Since tobacco smoking complicates HIV epidemic and its therapy, the study calls for HIV patients to abstain from smoking to ease management of the epidemic and its therapy.

Conflict of Interests
The authors declare that there is no conflict of interests.