Mathematical Analysis of the Transmission Dynamics of Hiv/tb Coinfection in the Presence of Treatment

This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two diseases. In the absence of TB infection, the model (HIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. The analysis of the respective reproduction thresholds shows that the use of a targeted HIV treatment (using anti-retroviral drugs) strategy can lead to effective control of HIV provided it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold. The full model, with both HIV and TB, is simulated to evaluate the impact of the various treatment strategies. It is shown that the HIV-only treatment strategy saves more cases of the mixed infection than the TB-only strategy. Further, for low treatment rates, the mixed-only strategy saves the least number of cases (of HIV, TB, and the mixed infection) in comparison to the other strategies. Thus, this study shows that if resources are limited, then targeting such resources to treating one of the diseases is more beneficial in reducing new cases of the mixed infection than targeting the mixed infection only diseases. Finally, the universal strategy saves more cases of the mixed infection than any of the other strategies. 1. Introduction. The inextricably linked pathogenesis and epidemiology of my-cobacterium tuberculosis (TB) and the human immuno-deficiency syndrome (HIV) are well known [18, 23, 24, 36]. The two diseases exhibit some sort of synergistic relationship, where each accelerates the progression of the other. For instance, since its emergence in the 1980s, the HIV/AIDS pandemic continues to play a major role in the resurgence of TB, resulting in increased morbidity and mortality worldwide.

1. Introduction.The inextricably linked pathogenesis and epidemiology of mycobacterium tuberculosis (TB) and the human immuno-deficiency syndrome (HIV) are well known [18,23,24,36].The two diseases exhibit some sort of synergistic relationship, where each accelerates the progression of the other.For instance, since its emergence in the 1980s, the HIV/AIDS pandemic continues to play a major role in the resurgence of TB, resulting in increased morbidity and mortality worldwide.
Additionally, HIV fuels progression to active disease in people infected with TB [23,40].Rates of recurrence of TB, both due to endogenous reactivation and exogenous re-infection, are increased among people infected with HIV [19,23,40].For instance, in many countries of eastern and southern Africa, the rates of TB notification have increased by five or more times as a result of HIV infection [37,38].Similarly, HIV infection increases the likelihood that a person will develop active TB [40].
The current statistics associated with the two diseases are staggering.While HIV accounts for 30-46 million infections (and resulted in over 20 million deaths) globally since its inception in the 1980s [39], TB affects at least 2 billion people (one-third of the world's population) and is the second greatest contributor of adult mortality amongst infectious diseases, causing approximately two million deaths a year worldwide [39].Currently, approximately 8% of global TB cases are attributable to HIV infection, but this proportion is expected to increase in the future.For instance, the number of HIV positives in India is estimated to be 3.97 million cases, and nearly 60% of the reported cases of AIDS had TB [19].
The largest number of TB cases occurs in Southeast Asia, which in 2004 accounted for an estimated 3 million new cases (one-third of the global total) for that year [39].However, the estimated incidence per capita in sub-Saharan Africa is nearly twice that of Southeast Asia, at 356 cases per 100,000 population in 2004.Also, the countries of Eastern Europe faced a serious epidemic in 2004, there were an estimated 166,000 new cases in Russia alone.
Thankfully, effective treatment does exist for each of these deadly diseases.For HIV, the use of anti-retroviral drugs (ARVs), notably the highly active antiretroviral therapy (HAART), has proven to be effective in curtailing its spread and AIDS-related mortality [14,21,28,29].However, these life-saving drugs are still not widely available in some resource-poor nations with high HIV incidence and prevalence.Tuberculosis, on the other hand, can be cured using drug therapy, such as DOTS (directly observed treatment short course).DOTS cures TB in 95% of cases, and a six-month supply of DOTS costs as little as $10 per person in some parts of the world [3].
The enormous public health burden inflicted by these two diseases necessitates the use of mathematical modelling to gain insights into their transmission dynamics and to determine effective control strategies.Unfortunately, not much has been done in terms of modelling the dynamics of HIV-TB coinfection at a population level.A few modelling studies, such as those in [27,30,35], have provided basic framework (using simplified models) for modelling the complex HIV-TB interaction in a community.The purpose of the current study is to complement the aforementioned studies, by designing and qualitatively analysing a new and more comprehensive deterministic model for gaining insights into the transmission dynamics and control of the two diseases in a population.The model allows for the assessment of treatment strategies for each disease (including the mixed infection).The robust model will be used to assess the public health (epidemiological) impact of four main treatment strategies, namely: (i) treating people infected with HIV only (HIV-only strategy), (ii) treating people infected with TB only (TB-only strategy), (iii) treating people infected with the mixed infection only (mixed-only strategy) and treating individuals infecetd with HIV, TB or the HIV-TB coinfection (universal strategy).
It is worth emphasizing that the two diseases differ in their modes of transmission.Whilst TB is an airborne disease (a susceptible individual may become infected with TB if he or she inhales bacilli, the causative agent of TB, in the air), HIV is transmitted predominantly via sexual contact or needle sharing (particularly among IV drug users).Thus, whilst HIV transmission almost exclusively involves sexually-active people (except for cases of vertical transmission), everyone (children and adults) is susceptible to TB infection.Children can acquire TB infection by having close contact with infected adults (usually family members).However, data from Health Canada [5] suggests that pediatric TB is on the decline.For instance, the number of reported TB cases in Canada in children under 15 years of age declined from 430 in 1970 to 109 in 2001 (the incidence of TB in children also decreased from 6.6 per 100,000 in 1970 to 1.9 per 100,000 in 2001).Although pediatric TB may be a factor in some nations, this study does not include children in the compartment of people susceptible to TB or HIV; rather, we consider sexually-active individuals only.
The paper is organized as follows.The model is formulated in Section 2. Two sub-models of the full model (HIV-only and TB-only) are analyzed in Section 3. The full model is analyzed (for the stability of the associated disease-free equilibrium) in Section 4, and numerical simulations are carried out in Section 5.

2.
Model formulation and basic properties.The total sexually-active population at time t, denoted by N (t), is subdivided into mutually-exclusive compartments, namely susceptible (S(t)), newly-and asymptomatically-infected individuals with HIV (H 1 (t)), HIV-infected individuals with clinical symptoms of AIDS (H 2 (t)), individuals infected with TB in latent (asymptomatic) stage (L(t)), individuals infected with TB in the active stage (T (t)), untreated dually-infected individuals (with both diseases) having latent TB and in the asymptomatic stage of HIV infection (I 1 HL (t)), untreated dually-infected individuals with active TB and in the asymptomatic stage of HIV infection (I 1 HT (t)), untreated dually-infected individuals with latent TB and showing symptoms of AIDS (I 2 HL (t)), untreated dually-infected individuals with active TB and showing symptoms of AIDS (I 2 HT (t)), treated individuals infected with HIV only (W H (t)), treated individuals infected with TB only (W T (t)), dually-infected individuals with latent TB treated of HIV (W H HL (t)), dually-infected individuals with active TB treated of HIV only (W H HT (t)), individuals infected with both diseases treated of TB (W T HT (t)) and those treated of both HIV and TB only (W M HT (t)), so that . The susceptible population is increased by the recruitment of individuals (assumed susceptible) into the population, at a rate Π.Both singly-and dually-infected individuals transmit either HIV or TB infection as follows (note that we split the disease transmission process into those generated by singly-infected and duallyinfected individuals to make the formulation easier to follow).

2.1.
Transmission by singly-infected individuals.Susceptible individuals acquire HIV infection, following effective contact with people infected with HIV only (i.e., those in the H 1 , H 2 and W H classes) at a rate λ H , given by where, β H is the effective contact rate for HIV transmission.Further, the modification parameters η 2 ≥ 1 and η H < 1 account for the relative infectiousness of individuals in the H 2 (AIDS) and W H (treated HIV-infected individuals) classes in comparison to those in the H 1 (asymptomatic HIV) class.That is, individuals in the H 2 class are more infectious than those in the H 1 class (because of their higher viral load); and, likewise, treated HIV-infected individuals are less infectious than those in the H 1 class (because the use of treatment significantly reduces the viral load in those treated).
Similarly, susceptible individuals acquire TB infection from individuals with active TB only at a rate λ T , given by where, β T is the effective contact rate for TB infection, and the parameter η T < 1 accounts for the reduction in infectiousness among individuals with active TB who are treated (in comparison to those who are not treated).A fraction, l, of susceptible individuals who acquire TB infection moves to the latent TB class (L) at the rate λ T , and the remaining fraction, 1 − l, moves to the active TB class T .It is assumed that individuals in the latent TB class do not transmit infection.

Untreated individuals.
Dually-infected individuals are assumed capable of transmiting either HIV or TB, but not the mixed infection.Untreated duallyinfected individuals (i.e., those in the I 1 HL , I 1 HT , I 2 HL , and I 2 HT classes) transmit HIV at a rate λ 1 HT , where In (3), c 2 η 2 (with c 2 ≥ 1) accounts for the assumed increase in infectiousness for dually-infected individuals in the AIDS stage compared to dually-infected individuals in the asymptomatic HIV stage; while the modification parameter η D > 1 accounts for the assumption that dually-infected individuals with active TB transmit HIV at a higher rate than the corresponding dually-infected individuals with latent TB (in other words, it is assumed that dually-infected people with active TB transmit HIV at a rate higher than that of dually-infected individuals (W T HT ) with latent TB, since it is known that active TB accelerates HIV progression in people infected with both diseases).
Similarly, untreated dually-infected individuals with active TB (i.e., those in the I 1 HT and I 2 HT classes) transmit TB at a rate λ 2 HT , with In other words, equation (5) shows that dually-infected individuals treated of TB (W T HT ) transmit HIV at the same rate (β H ) as the corresponding (untreated) HIV-infected individuals with HIV infection only (in the H 1 class), while duallyinfected individuals treated of HIV with latent and active TB (W H HL and W H HT ), and dually-infected individuals treated of both diseases (W M HT ) transmit HIV at the reduced rate, η H β H (in comparison to the untreated HIV-infected individuals in the H 1 class).Thus, treating dually-infected individuals of one disease (only) does not limit (or reduce) their ability to transmit the other disease.
Similarly, treated individuals with mixed infection (involving active TB) transmit TB at a rate λ T M , where ), at a rate σ.The parameters ψ 1 and ψ 2 (with ψ 2 > ψ 1 > 1) account for the assumed increase in probability of acquiring TB infection for HIV-infected individuals; the parameter ψ 2 is associated with those with clinical symptoms of AIDS (H 2 ) while ψ 1 is associated with those without symptoms (H 1 ).That is, it is assumed that individuals with HIV infection are more prone to TB infection than wholly-susceptible individuals.Further, those with AIDS acquire TB infection at a higher rate than those in the asymptomatic stage of HIV infection owing to the weaker immune status of the former.The population of individuals infected with TB only (in the L or T class) is reduced following acquisition of HIV-infection, which can result following effective contact with individuals infected with HIV (in the H 1 , H 2 , W H , I 1 HL , I 1 HT , I 2 HL , I 2 HT , W H HL , W H HT , W T HT and W M HT classes).Further, individuals in the latent TB class (L) progress to the active TB class (T ) at a rate α, and become re-infected (exogenously) after effective contact with individuals in the active TB class (at a rate λ R ) or with individuals having mixed infection involving active TB (at a rate λ R1 ).The rates λ R and λ R1 are, respectively, given by where β T η r (with η r > 0 being the modification parameter for exogenous reinfection) is the contact rate associated with the exogenous re-infection.Individuals in the I 1 HL class undergo exogenous re-infection at a rate λ R2 , where A fraction, l, of susceptible individuals who acquire TB infection from untreated dually-infected individuals (at the rate λ 2 HT ) move to the latent TB class, while the remaining fraction, 1 − l, move to the active TB class.Susceptible individuals who had effective contact with dually-infected individuals treated of TB can acquire either TB or HIV infection.Those who acquire HIV move to the H 1 class (at the rate β H ), and, on the other hand a fraction, l, of those who acquire TB infection move to the latent TB class (L), while the remaining fraction, 1 − l, move to the active TB class (T ).Similarly, susceptible individuals who acquire infection from dually-infected individuals treated of HIV alone can become infected with TB at the rate β T (where a fraction, l, move to the latent class; and the remaining fraction, 1 − l, move to the active TB class), and those infected with HIV move to H 1 class at the reduced rate η H β H . Dually-infected individuals treated of both diseases transmit HIV infection at the reduced rate η H β H , and TB infection at the reduced rate η T β T (where a fraction, l, of the TB cases move to the latent class and the remaining fraction, 1 − l, move to the active TB class).Note that, since we have no data to show that the aforementioned fractions (that move to the latent TB class) are distinct, we assume that they are all equal (to l).Further, individuals in the I 1 HL class progress to the I 1 HT class at an increased rate θ 1 α, where θ 1 ≥ 1 (this is to account for the fact that HIV infection accelerates TB progression in dually-infected individuals); and are treated for HIV at a rate τ 1 .Finally, a fraction, ξ, of individuals in the I 1 HL class progress to active TB and AIDS stage (I 2 HT ) at a rate γ HT , and the remaining fraction, 1 − ξ, moves to the I 2 HL at the same rate γ HT .
As noted earlier, individuals with latent TB (only) are re-infected (exogenously) at a rate λ R or λ R1 .A fraction, φ, of those individuals re-infected at the rate λ R1 progress to active TB, and the remaining fraction, 1 − φ, acquire HIV infection and move to the I 1 HT class.Individuals in this I 1 HT class are treated for HIV at the rate τ 1 and for active TB at a rate τ 3 .Finally, I 1 HT individuals progress to I 2 HT at an increased rate η 1 σ, with η 1 ≥ 1.In other words, this study assumes that the presence of mixed infection accelerates progression of both diseases (to either active TB or AIDS stage).
Similarly, individuals with AIDS and latent TB (I 2 HL ) are re-infected (exogenously) at the rate λ R2 .Individuals in this class are treated for HIV at the rate τ 2 , progress to I 2 HT class at the rate θ 2 α (θ 2 ≥ 1) and die due to the two diseases at a rate δ HT .Finally, individuals in the I 2 HT class are treated for HIV at the rate τ 2 and active TB at the rate τ 3 ; and they die at an increased rate ωδ HT , with ω > 1. Treatment for HIV, using ARVs, is administered to individuals in both H 1 and H 2 classes, at the rate τ 1 and τ 2 , respectively; while individuals with active TB are treated at the rate τ 3 .Individuals successfully treated for HIV are assumed to eventually (after long period of time, lasting decades) succumb to the disease (due to the failure of treatment or resistance development) and progress to AIDS at a reduced rate θ t σ 1 , where 0 < θ t < 1.Individuals successfully treated of TB return to the latent TB stage at a rate ρ.Finally, dually-infected individuals treated of HIV can further be treated for TB; while dually-infected individuals treated of TB can similarly be treated for HIV.Dually-infected individuals who have received treatment for both diseases (W M HT class) and dually-infected individuals treated of TB (W T HT class) eventually progress to the final stage of HIV disease and latent TB (I 2 HL ) at rates σ HT and σ T , respectively.On the other hand, dually-infected individuals treated of HIV with latent TB (W H HL class) progress to the class of dually-infected individuals treated of HIV with active TB (I 1 HT ) at the rate θ 1 α and/or to the untreated dually-infected individuals with AIDS and latent TB at a rate θ t σ.Dually-infected individuals treated of HIV with active TB (W H HT ) progress to the class of individuals with AIDS and active TB (I 2 HT ) at a rate σ H . Further, natural mortality occurs in all classes at a rate µ, while individuals in the AIDS (H 2 ) and active TB (T ) classes suffer an additional disease-induced death at rates δ H and δ T , respectively.
Combining all the aforementioned assumptions and definitions, the model for the transmission dynamics of HIV and TB in a sexually-active population is given by the following system of differential equations: where, Since the model ( 7) monitors human populations, all the variables and parameters of the model are non-negative.Consider the biologically-feasible region The following steps are followed to establish the positive invariance of D (i.e., all solutions in D remain in D for all time).The rate of change of the total population, obtained by adding all the equations in model (7), is given by It is easy to see that whenever N > Π/µ, then dN/dt < 0. Since dN/dt is bounded by Π − µN , a standard comparison theorem [25] can be used to show that . Thus, every solution of the model ( 7) with initial conditions in D remains there for t > 0 (the ω-limit sets of the system (7) are contained in D).Thus, D is positive-invariant and attracting.Hence, it is sufficient to consider the dynamics of the flow generated by (7) in D. In this region, the model can be considered as been epidemiologically and mathematically well-posed [17].
3. Analysis of the sub-models.Before analyzing the full model (7), it is instructive to gain insights into the dynamics of the models for HIV only (HIV-only model ) and TB only (TB-only model ). where, For this model, it can be shown that the region, is positively-invariant and attracting.Thus, the dynamics of the HIV-only model will be considered in D 1 .

3.1.1.
Local stability of disease-free equilibrium (DFE).The model ( 9) has a DFE, obtained by setting the right-hand sides of the equations in the model to zero, given by The linear stability of E 0 can be established using the next-generation operator method on the system (9).Using the notation in [33], the matrices F and V , for the new infection terms and the remaining transfer terms respectively, are, respectively, given by (noting that S * = N * at the DFE E 0 ) and, Thus, The following result follows from Theorem 2 of [33].
The threshold quantity R H is the reproduction number for HIV [1].It measures the average number of new HIV infections generated by a single HIV-infected individual in a population where a certain fraction of infected individuals are treated.
3.1.2.Analysis of R H .The objective here is to determine, using the threshold quantity R H , whether or not treating individuals with HIV, either those in the asymptomatic stage (modelled by the rate τ 1 ) or AIDS stage (modelled by the rate τ 2 ), can lead to HIV elimination in the community.It is evident from (11) that and, Thus, a sufficiently effective HIV treatment program that focusses on treating infected individuals in the asymptomatic stage (at a high rate, τ 1 → ∞) or those with AIDS symptoms (at a rate τ 2 → ∞) can lead to effective disease control if it results in making the respective right-hand side of ( 12) or ( 13) less than unity.The profiles of R H , as a function of treatment rates τ 1 and τ 2 , are depicted in Figure 1A.For the set of parameters used in these simulations, it is evident from this figure that while a strategy that focuses on treating asymptomatic individuals alone can dramatically reduce R H from around R H = 17 to a value of R H less than unity (R H = 0.523), the strategy that focuses on treating AIDS individuals only reduces R H from R H = 17 to R H = 10 (thus, HIV cannot be eliminated in the latter case, but will be in the former).It was further shown, in Figure 1A, that the combined treatment of HIV-infected individuals with or without AIDS symptoms reduces R H to values less than unity faster than a strategy that targets individuals without AIDS symptoms.Further sensitivity analysis on the treatment parameters is carried out by computing the partial derivatives of R H with respect to the treatment parameters (τ 1 and τ 2 ), giving, , Consider the case τ 2 = 0 (that is, only HIV-infected individuals with no AIDS symptoms being treated).It follows from ( 14) that ∂RH ∂τ1 < 0 if Thus, the targeted treatment of HIV-infected individuals in the asymptomatic stage will have positive impact in reducing HIV burden only if η H < ∆ I .Such a treatment will fail to reduce the HIV burden if η H = ∆ I , and will have a detrimental impact in the community (increase R H ) if η H > ∆ I .This result is summarized below.
Lemma 3.2.The targeted treatment of HIV-infected individuals in the asymptomatic stage will have positive impact if η H < ∆ I , no impact if η H = ∆ I and will have detrimental impact if η H > ∆ I .
Similarly, it follows from (15) that ∂RH ∂τ2 < 0 if giving the following result.
Lemma 3.3.The targeted treatment of HIV-infected individuals with AIDS symptoms will have positive impact if It is worth emphasizing that if Condition (16) or (17) does not hold, then the use of the corresponding targeted treatment strategy would increase HIV burden in the community (since it increases R H ), although such treatment may be beneficial to those HIV-infected (individuals) treated.That is, the use of ARVs will increase disease burden if it fails to reduce the infectiousness of those treated below a certain threshold (η H < ∆ I if asymptomatic HIV-infected individuals are targeted; or η H < ∆ A if individuals with AIDS symptoms are targeted).
Returning back to Lemma 1. Biologically speaking, this lemma implies that HIV can be eliminated from the community (when R H < 1) if the initial sizes of the sub-populations of the model are in the basin of attraction of E 0 .To ensure that elimination of the virus is independent of the initial sizes of the sub-populations, it is necessary to show that the DFE is globally asymptotically stable.This is established below.
Proof.Consider the following Lyapunov function: where, with Lyapunov derivative (where a dot represents differentiation with respect to t), Since all the model parameters are nonnegative, it follows that Ḟ ≤ 0 for R H ≤ 1 with Ḟ = 0 if and only if is the singleton {E 0 }.Therefore, by the LaSalle's Invariance Principle [26], every solution to the equations of the model ( 9), with initial conditions in D 1 , approaches E 0 as t → ∞, whenever R H ≤ 1 (note that substituting H 1 = H 2 = W H = 0 in the first equation of (9) shows that S → S * as t → ∞).
The above result shows that HIV will be eliminated from the community if the epidemiological threshold, R H , can be brought to a value less than unity.
3.1.4.Existence of endemic equilibria.To find conditions for the existence of an equilibrium for which HIV is endemic in the population (i.e., at least one of H * * 1 , H * * 2 and W * * H is non-zero), denoted by E 1 = (S * * , H * * 1 , H * * 2 , W * * H ), the equations in (9) are solved in terms of the force of infection at steady-state (λ * * H ), given by Setting the right hand sides of the model to zero (and noting that λ H = λ * * H at equilibrium) gives Using (19) in the expression for λ * * H in (18) shows that the nonzero (endemic) equilibria of the model satisfy where, It is clear that a 11 > 0, and a 12 > 0 for R H > 1.Thus, the linear system (20) has a unique positive solution, given by λ * * H = a 12 /a 11 , whenever R H > 1.The components of the endemic equilibrium, E 1 , are then determined by substituting λ * * H = a 12 /a 11 into (19).Noting that R H < 1 implies that a 12 < 0. Thus, for R H < 1, the force of infection at steady-state (λ * * H ) is negative (which is biologically meaningless).Hence, the model has no positive equilibria in this case.These results are summarized below.
Lemma 3.5.The HIV-only model (9) has a unique endemic equilibrium whenever R H > 1, and no endemic equilibrium otherwise.

3.1.5.
Local stability of endemic equilibrium.Using standard linearization of the HIV-only model around the endemic equilibrium is laborious and not really tractable mathematically.Here, the center manifold theory [6], as described in [9] (Theorem 4.1), will be used to establish the local asymptotic stability of the endemic equilibrium (see also [10,33]).To apply this method, the following simplification and change of variables are made first.Let S = x 1 , H 1 = x 2 , H 2 = x 3 , and , W H = x 4 , so that N = x 1 + x 2 + x 3 + x 4 .Further, by using vector notation X = (x 1 , x 2 , x 3 , x 4 ) T , the HIV-only model ( 9) can be written in the form dX dt = (f 1 , f 2 , f 3 , f 4 ) T , as follows: The Jacobian of the system (21), at E 0 , is given by from which it can be shown that Consider the case when R H = 1.Suppose, further, that β H is chosen as a bifurcation parameter.Solving (22) for β H gives R H = 1 when Note that the above linearized system, of the transformed system (21) with β H = β * , has a zero eigenvalue which is simple.Hence, the center manifold theory [6] can be used to analyze the dynamics of ( 21) near β H = β * .In particular, Theorem 4.1 in [9] will be used to show the LAS of the endemic equilibrium point of (21) (which is the same as the endemic equilibrium point of the original system ( 9)), for It can be shown that the Jacobian of ( 21) at β H = β * (denoted by J β * ) has a right eigenvector (associated with the zero eigenvalue) given by w = [w 1 , w 2 , w 3 , w 4 ] T , where The denominator ] (associated with the zero eigenvalue), where For convenience, the theorem in [9] (see also [6,10,33]) is reproduced below.
Theorem 3.6 (Castillo-Chavez & Song [9]).Consider the following general system of ordinary differential equations with a parameter φ where 0 is an equilibrium point of the system (that is, f (0, φ) ≡ 0 for all φ) and assume is the linearization matrix of the system (24) around the equilibrium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts; A2: Matrix A has a right eigenvector w and a left eigenvector v (each corresponding to the zero eigenvalue).
Let f k be the kth component of f and The local dynamics of the system around 0 is totally determined by the signs of a and b.
Computations of a and b : For the system ( 21), the associated non-zero partial derivatives of F (at the DFE) are given by It follows from the above expressions that For the sign of b, it can be shown that the associated non-vanishing partial derivatives of F are

It also follows from the above expressions that
Thus, a < 0 and b > 0. So (by Theorem 2, Item (iv)), we have established the following result (note that this result holds for R H > 1 but close to 1): Theorem 3.7.The unique endemic equilibrium guaranteed by Theorem 2 is LAS for R H near 1.
In summary, the HIV-only model ( 9) has a globally-asymptotically stable DFE whenever R H ≤ 1, and a unique endemic equilibrium point whenever R H > 1.The unique endemic equilibrium point is LAS at least near R H = 1.The dynamics of TB-only is also explored as below.

TB-only model. The model for the transmission dynamics of TB only (obtained by setting
with, The TB-only model ( 25) is formulated along the lines of the model in Feng et al. [12], but with the additional features of (i) newly-infected individuals can have either latent or active TB (0 < l < 1) and (ii) treated (TB-infected) individuals can transmit TB (η T = 0).

Local stability of DFE.
The TB-only model ( 25) has a DFE given by Here, the F and V matrices are given, respectively, by where, Thus, the following result is established (from Theorem 2 of [33]).
Lemma 3.8.The DFE of the model (25), given by (26), The threshold quantity, R T , is the reproduction number for TB.

3.2.2.
Analysis of R T .Here, the reproduction threshold, R T , will be analyzed to determine whether or not treating people with active TB (modelled by the rate τ 3 ) can lead to the effective control or elimination of TB in the population.It follows from (27), that from which it is evident that the parameters β T and η T play an important role in determining the value of R T .A plot of R T as a function of τ 3 is depicted in Figure 1B.This figure shows that, for the set of parameters used in the simulations, an effective strategy for treating people with active TB may not be adequate to eliminate TB in the community, since it only brings R T down to about R T = 5 at steady state (and the condition R T < 1 is needed for effective control of TB in a population).Biologically speaking, Lemma (3.8) implies that TB can be eliminated from the community (when R T < 1) if the initial sizes of the sub-populations of the model are in the basin of attraction of E 0t .Since TB models are often shown to exhibit the phenomenon of backward bifurcation [9,12], where the stable DFE co-exists with a stable endemic equilibrium when the associated reproduction threshold (R T ) is less than unity, it is instructive to determine whether or not the TB-only model ( 25) exhibits this feature.This is done below.
3.2.3.Bifurcation analysis.Models of TB dynamics with exogenous re-infection are known to exhibit the phenomenon of backward bifurcation [9,12], where the stable DFE co-exists with a stable endemic equilibrium.Here, the model ( 25) will be analysed to see whether or not the new features (l = 1 and η T = 0) added to some of the earlier TB models (e.g., the model in [12]) would have any effect on the expected reinfection-induced backward bifurcation property of TB disease.Here, too, the Centre Manifold theory will be used on the model system (25).Let S = x 1 , L = x 2 , T = x 3 , and W T = x 4 , so that N = x 1 + x 2 + x 3 + x 4 , so that the model ( 25) is re-written in the form: The Jacobian of the system (28), at the DFE (26), is given by from which it can also be shown that Suppose β T is chosen as a bifurcation parameter.Solving (29) for R T = 1, gives .
It can be shown that the Jacobian of (28) at β T = β * (denoted by J(E 0t ) βT =β * = J β * ) has a right eigenvector (corresponding to the zero eigenvalue) given by w = [w 1 , w 2 , w 3 , w 4 ] T , where Further, the Jacobian J β * has a left eigenvector (associated with the zero eigenvalue) given by v where Theorem 2 will be used to establish the presence of backward bifurcation in the TB-only model (25).
Computations of a and b : For the system (28), the associated non-zero partial derivatives of F (at the DFE) are given by It follows from the above expressions that from which it can be shown that a > 0 iff where, Hence, a > 0 whenever (note that For the sign of b, it can be shown that the associated non-vanishing partial derivatives of F are Thus, we have established the following result: It should be noted that the inequality in Theorem 4 does not hold if η r = 0 (since the right hand side of the inequality is positive).Thus, the backward bifurcation phenomenon of the TB-only model (28) will not occur if η r = 0 (i.e., the TBonly model will not undergo backward bifurcation in the absence of exogenous re-infection).
Numerical simulations are carried out, using an appropriate set of parameter values (satisfying the inequality in Theorem 4), to illustrate the backward bifurcation phenomenon of model (28) (see Figure 2).
It should be emphasized that these parameter values are chosen only to illustrate the backward bifurcation phenomenon of model (28), and may not all be realistic epidemiologically.With the chosen set of parameter values, the ratio = 2. Thus, the inequality in Theorem 4 will hold by choosing a value of η r > 2, such as η r = 3.Thus, it follows from Theorem 4 (and Figure 2) that the additional features l = 1 (i.e., all newly-infected individuals have latent TB) and η T = 0 (i.e., treated people do not transmit the disease) do not affect the backward bifurcation property of the TB disease (since, the inequality in Theorem 4 still holds if l = 1 and η T = 0 with η r = 3).In summary, unlike the HIV-only model (9), the TB-only model (28) undergoes backward bifurcation, where multiple stable equilibria co-exist when R T < 1. Backward bifurcations have been observed in a number of epidemiological settings, such as those associated with behavioral responses to perceived risks [10,15], multigroup models [7,8,20,31], vaccination models [2,4,11,22,32], disease treatment [34] and models of the transmission of TB with exogenous re-infection [9,12] and HTLV-I [13].

5.
Simulations.The full model ( 7) is now simulated, using the parameter estimates in Table 2 (unless otherwise stated), to assess the potential impact of treatment strategies against HIV and TB, as follows.7) for various values of the associated reproduction thresholds (R H and R T ).For the case when R H < 1 and R T < 1, (that is, R c < 1), the solution profiles can converge to the DFE or the EEP owing to the phenomenon of backward bifurcation in the full model (7). Figure 3 shows convergence of the solutions to the DFE for R c < 1 (in line with Lemma 6), whereas Figure 4 illustrates the backward bifurcation phenomenon of the full model (7), with some solutions converging to an EEP and others to a DFE when the threshold quantity R c is less than unity.Note that, for the set of parameter values used, the simulations have to be run for long time periods (in hundreds of years) to generate the backward bifurcation pictures depicted in Figure 4.

Evaluation of treatment strategies.
As stated earlier, the paper offers four main treatment strategies namely the, (i) HIV-only strategy, (ii) TB-only strategy, (iii) mixed-only strategy, and (iv) universal strategy.The full model ( 7) is now simulated (for a four-year period) to assess these strategies as follows:  (7).Total infectives as a function of time using different initial conditions, with R H = 0.29, R T = 0.50; so that R c = 0.50.(A) HIV cases, (B) TB cases, and (C) HIV-TB cases.All other parameters as in Table 2.   (7).Backward bifurcation diagrams using different initial conditions and parameter values such that R H = 0.87, R T = 0.96; so that R c = 0.96.(A) HIV cases, (B) TB cases, and (C) HIV-TB cases.All other parameters as in Table 2. 5.2.1.HIV-only treatment strategy.Here, simulations are carried out to monitor the impact of treating for HIV only.That is, singly-infected individuals in the H 1 and H 2 classes are treated at the rates τ 1 and τ 2 , respectively; while dually-infected individuals are treated for HIV at the rate τ 2 (i.e., individuals infected with TB, either singly or dually, are not treated for TB, so that τ 3 = 0).Using a modest rate of τ 1 = τ 2 = 0.5, the results, depicted in Figure 5A, show a significant reduction of the number of new cases of HIV as well as those of the mixed HIV-TB infection (although more new cases of HIV are prevented than those of the mixed infection).Similar trends were observed when the treatment rate was increased by 10-fold (Figure 5B).(7).Cumulative new cases averted using HIV-only treatment strategy with (A) τ 1 = τ 2 = 0.5 and τ 3 = 0 and (B) τ 1 = τ 2 = 5 and τ 3 = 0.All other parameters as in Table 2. 5.2.2.TB-only treatment strategy.In these simulations, only individuals infected with TB are treated for TB.That is, only individuals in the active TB classes (i.e., those in T, I 1 HT , I 2 HT , and W H HT ) are treated (at the rate τ 3 ) and those with HIV or latent TB are not treated (i.e., τ 1 = τ 2 = 0).Figure 6A shows a significant reduction of the cumulative number of new TB cases followed by that of the mixed infection.It should be stated that the number of TB cases prevented (Figure 6A) exceeds the corresponding number of HIV cases prevented under the HIV-only treatment strategy (Figure 5A).However, more cases of the mixed infection were prevented in the HIV-only treatment strategy than in this (TB-only) treatment scenario.Similar trends were observed when the treatment rate is increased by 10-fold (Figure 6B).5.2.3.Mixed-only treatment strategy.In these simulations, only individuals with the mixed infection are treated for both diseases (HIV infected individuals are treated at the rates τ 1 and τ 2 ; and those with active TB are treated at the rate τ 3 ).That is, we set τ 1 = τ 2 = τ 3 = 0 in the H 1 , H 2 , T, W H and W T classes but use these  (7).Cumulative new cases averted using mixed-only treatment strategy with (A) τ 1 = τ 2 = τ 3 = 0.5 and (B) τ 1 = τ 2 = τ 3 = 5.All other parameters as in Table 2.
cases of HIV and TB as the HIV-only and TB-only strategies, the universal strategy saves more mixed infections than any of the other three strategies.
Conclusions.A realistic deterministic model for the transmission dynamics of HIV and TB in a population is designed and rigorously analysed.The HIV-only model is shown to have a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity; and has a unique and locally-asymptotically stable endemic equilibrium when the number exceeds unity.On the hand, it was shown (using Centre Manifold theory) that the model with TB infection only undergoes the phenomenon of backward bifurcation, when the associated reproduction number is less than unity.The full model has a diseasefree equilibrium which is locally-asymptotically stable whenever the maximum of the reproduction numbers of the two sub-models described above is less than unity.Numerical simulations show that the full model, like the TB-only sub-model, also undergoes backward bifurcation.These results have important public health implication, as they govern the elimination and/or persistence of the two diseases in a community.By analyzing the various associated reproduction numbers, it was shown that the targeted use of ARVs for individuals with or without AIDS symptoms can lead to HIV elimination in the community if the (average) relative infectiousness of the individuals treated, in comparison to untreated HIV-infected individuals, does not exceed a certain critical value.
Numerical simulations of the full model were carried out to assess the impact of the associated (four) treatment strategies.Some of main epidemiological findings of this study include:  (7).Cumulative new cases averted using the universal treatment strategy with (A) τ 1 = τ 2 = τ 3 = 0.5 and (B) τ 1 = τ 2 = τ 3 = 5.All other parameters as in Table 2.
(i) Treating any of the two diseases alone prevents significant number of cumulative new cases of the disease being treated as well as that of the mixed infection (and more cumulative new cases are prevented for higher treatment rates); (ii) The HIV-only strategy prevents more cumulative new cases of the mixed infection than the TB-only strategy; (iii) For low treatment rates, the mixed-only strategy saves the least cumulative new cases of HIV, TB and the mixed infection in comparison to the other strategies.That is, if resources are low and the objective is to minimize cases of mixed infection, than such resources should be targeted to treating either HIV or TB but not the mixed HIV-TB infection; (iv) For high treatment rates, the mixed-only strategy compares reasonably well (in terms of cumulative new cases averted) with the other strategies; (v) The universal strategy saves more cumulative new cases of the mixed infection than any of the other strategies.
Overall, this study shows that the prospects of effectively controlling the spread of HIV and TB in a community, using effective treatment for both diseases, is bright.

3. 1 .
HIV-only model.The model with HIV only (obtained by setting L = T = I 1 HL = I 1 HT = I 2 HL = I 2 HT = W T = W H HL = W H HT = W T HT = W M HT = 0 in (7)) is given by

Figure 1 .
Figure 1.Reproduction numbers as a function of treatment rates.(A) R H , (B) R T then the TB-only model (28) undergoes a backward bifurcation at R T = 1.

Lemma 4 . 1 .
The DFE of the full HIV-TB model(7), given by E 1 , is LAS if R c < 1, and unstable if R c > 1.

Figure 3 .
Figure3.Simulations of the full model(7).Total infectives as a function of time using different initial conditions, with R H = 0.29, R T = 0.50; so that R c = 0.50.(A) HIV cases, (B) TB cases, and (C) HIV-TB cases.All other parameters as in Table2.

Figure 4 .
Figure 4. Simulations of the full model(7).Backward bifurcation diagrams using different initial conditions and parameter values such that R H = 0.87, R T = 0.96; so that R c = 0.96.(A) HIV cases, (B) TB cases, and (C) HIV-TB cases.All other parameters as in Table2.
. Derivation of model equations.The populations of individuals in the H 1 and H 2 classes are reduced due to TB infection (following effective contact with indi-