Changing Risk Behaviours and the HIV Epidemic: A Mathematical Analysis in the Context of Treatment as Prevention

Background Expanding access to highly active antiretroviral therapy (HAART) has become an important approach to HIV prevention in recent years. Previous studies suggest that concomitant changes in risk behaviours may either help or hinder programs that use a Treatment as Prevention strategy. Analysis We consider HIV-related risk behaviour as a social contagion in a deterministic compartmental model, which treats risk behaviour and HIV infection as linked processes, where acquiring risk behaviour is a prerequisite for contracting HIV. The equilibrium behaviour of the model is analysed to determine epidemic outcomes under conditions of expanding HAART coverage along with risk behaviours that change with HAART coverage. We determined the potential impact of changes in risk behaviour on the outcomes of Treatment as Prevention strategies. Model results show that HIV incidence and prevalence decline only above threshold levels of HAART coverage, which depends strongly on risk behaviour parameter values. Expanding HAART coverage with simultaneous reduction in risk behaviour act synergistically to accelerate the drop in HIV incidence and prevalence. Above the thresholds, additional HAART coverage is always sufficient to reverse the impact of HAART optimism on incidence and prevalence. Applying the model to an HIV epidemic in Vancouver, Canada, showed no evidence of HAART optimism in that setting. Conclusions Our results suggest that Treatment as Prevention has significant potential for controlling the HIV epidemic once HAART coverage reaches a threshold. Furthermore, expanding HAART coverage combined with interventions targeting risk behaviours amplify the preventive impact, potentially driving the HIV epidemic to elimination.


The Model
The model describes a two disease system in which becoming infected with the first disease is a prerequisite for becoming susceptible to the second disease. The first disease represents risk behaviour, which spreads in the population through social influence. The second disease is HIV, which requires existence of risk behaviour to propagate. HIV is modelled as a two stage, with the first called the acute phase and the second the chronic phase.
We define the model so that the total population in the system is constant. The fraction of the general population not engaged in risk behaviour is denoted by g. The fraction of the population engaged in risk behaviour and susceptible to HIV is denoted by s. The fractions of the population in the acute and chronic HIV phases are denoted by a and c, respectively. Individuals in the s, a, and c groups influence members in the general population to engage in risk behaviour at a rate per person β sac . The subpopulations g, s, and a have the same death rate δ gsa . The death rate of the HIV chronic phase is δ c . The rate per person that HIV positive individuals move from the acute phase to the chronic phase is ρ a . The infectivity of HIV positive individuals in the acute and chronic phases are λ a and λ c , respectively. The model is described by the following system of four differential equations, which correspond to the system of equations (6) in the main paper, with δ gsah replaced by δ gsa and equation (7) in the main paper used to define λ c and δ c : (S1.1) We rescale to dimensionless time by multiplying t by δ gsa . Relabelling the parameters to reflect these rescalings, as well as writing φ = λc λa gives the system of equations, (S1. 2) The parameters β, λ, and δ are all assumed to be strictly positive. The parameter φ satisfies 0 < φ < 1.
The physically relevant region for this system of equations is the simplex The variables g, s, a, and c satisfy the constraint g + s + a + c = 1, which allows us to reduce the system of four equation (S1.2) to the following three equations: (S1.4) The physically relevant region for this reduced system of equations is the convex hull To show that the system of equations (S1.4) is a well-defined model, it is necessary to verify that R 1 is a trapping region. To prove this note that the vector field defined by (S1.4) is inward pointing at every point on the boundary of R 1 , except at (s, a, c) = (0, 0, 0), which is an equilibrium point.

Equilibrium Points
To find the equilibrium points of the system of equations (S1.4), set ds dt = 0, da dt = 0, and dc dt = 0 and find all real roots of the resulting algebraic system.
The risk-free equilibrium point s 0 = 0 , a 0 = 0 , c 0 = 0 (S1.6) of the system (S1.4) corresponds to no risk behaviour, nor any disease in the population. The second equilibrium point corresponds to endemic risk behaviour, but no disease in the system. The value of this equilibrium point, which we call the risk-endemic equilibrium point, is Observe that the risk-endemic equilibrium point is in the physical region R 1 if β ≥ 1.
In terms of the original model equations (S1.1) this implies that if δ c ≥ δ gas , then there is at most one endemic HIV equilibrium point. Deaths caused by HIV occur after the acute phase, so this assumption corresponds to assuming that HIV infection increases the death rate. The first step in proving that δ ≥ 1 implies that there can be at most one disease-endemic equilibrium point is to simplify equation (S1.11) through the change in parameters (S1.12) Note that the map (S1.14) to The inverse is In terms of the new parameters, the quadratic equation (S1.11) becomes and the condition δ ≥ 1 is equivalent to ψ 2 ψ 3 ≥ ψ 1 . The result that there can be at most one diseaseendemic equilibrium point follows from the following lemma.
Proof. For the equation (S1.17) to have two positive roots, the coefficient of a must be strictly negative, which is equivalent to The inequality (S1.19) simplifies to (S1.20) For equation (S1.17) to have two roots greater than or equal to zero, the constant term must satisfy Substituting from the inequality (S1.20) into the inequality (S1.21) implies that 1 − β ≥ 0. However, this result and the inequality (S1.20) contradict ψ 3 > 0.
The only equilibrium point of the system (S1.4) which can correspond to endemic disease is given by (S1.22) The requirement that s 2 ≤ 1 for this equilibrium point to be in R 1 implies that λ ≥ ψ 3 . A necessary and sufficient condition for this disease-endemic equilibrium point to be in R are given in the following proposition.
Proof. Observe that the constant term in equation (S1.17) changes sign at ψ 3 = λ(β−1) β . Therefore, one of the two roots of equation (S1.17) changes sign at this value of ψ 3 . Lemma 1 implies that it must be the largest root. It follows that equation (S1.17) has a single positive root if and only if the inequality (S1.23) holds.
It remains to show that if there is a positive root, then it must always correspond to an equilibrium point in R 1 by verifying that which is equivalent to (S1.25) Equations (S1.22) and the inequality (S1.23) imply that (S1.26)

Stability of the Equilibrium Points
Conditions for each of the three equilibrium points -the risk-free equilibrium point, the risk-endemic equilibrium point, and the disease-endemic equilibrium point -to be stable in the physical region R are determined in this section. We shall prove local asymptotic stability for each of the three fixed points and global asymptotic stability for the risk-free equilibrium point. The Jacobian matrix of the system of equation (S1.4) is (S1.27) An equilibrium point is locally asymptotically stable if the real parts of all three eigenvalues of the Jacobian matrix evaluated at the equilibrium point are strictly negative. The equilibrium point is unstable if the real part of any eigenvalue is strictly positive.

Risk-Free Equilibrium Point
The risk-free equilibrium point is the only equilibrium point in the the physical region R when β < 1. In this case, it is shown in the following theorem that the risk-free equilibrium point is locally asymptotically stable.
In the following theorem, the LaSalle Invariance Principal [2, Chapt. 2: Corollary 6.5] is used to show global asymptotic stability of the risk-free equilibrium point in the physical region R. This implies that any solution (g(t), s(t), a(t), c(t)) of the system of equation (S1.2) approaches (1, 0, 0, 0) as t goes to infinity, for all initial values in R.
Proof. First, we use the constraint g + s + a + c = 1 to eliminate the equation for c in the system (S1.2) and obtain: (S1.30) We define the Lyapunov-LaSalle function . Lyapunov functions of this form are somtimes referred to as Volterra-style Lyapunov functions [3]. A similar Lyapunov-LaSalle function was used to prove global stability for SIR, SIRS, and SIS models in [4]. The function L satisfies (S1.32) and dL dt = 0 holds only at (g 0 , s 0 , a 0 ). Since R 2 is a compact positively invariant set, it follows from the LaSalle Invariance Principal [2, Chapt. 2: Corollary 6.5] that (1, 0, 0) is globally asymptotically stable in R 2 and that (1, 0, 0, 0) is globally asymptotically stable in R.

Risk-Endemic Equilibrium Point
As β increases through one, the risk-endemic equilibrium point (s 1 , a 1 , c 1 ) passes through the risk-free equilibrium point into the physical region R. The following theorem implies that a transcritical bifurcation occurs and the risk-endemic equilibrium point becomes locally asymptotically stable.

Disease-Endemic Equilibrium Point
The disease-endemic equilibrium point passes through the risk-endemic equilibrium point and into the physical region R when λ increases through βψ3 β−1 = βδ(1+ρ) (β−1)(δ+ψρ) for β > 1. The following theorem implies that at this point a second transcritical bifurcation occurs and the disease-endemic equilibrium point becomes locally asymptotically stable.