We assumed that individuals join the sexually active population at rate Π and spend an average length of time (1/μ) acquiring new sex partners; during this time period they acquire new sex partners at rate c. The model allowed individuals in the early stages of HIV infection, when they are eligible for the therapeutic vaccine (Y_E^U) to:

1. remain untreated
2. go on ARV only (move to Y_E^A)
3. go on ARV plus vaccine (move to Y_E^AV)

Individuals in this eligibility stage can go on and off ARV or on and off ARV plus vaccine. Individuals in late stage HIV-infection (Y_L^U) can go on and off treatment (ARV only). For individuals in the eligibility stage the per capita treatment rate of ARV is specified by σ_E^A, and the per capita treatment rate of ARV plus therapeutic vaccine is specified by σ_E^AV; the per capita rate of giving up ARV per year for these individuals is g_E^A and g_E^AV respectively. For individuals in the late stage of HIV-infection, when they are no longer eligible for the vaccine, the per capita treatment rate of ARV is specified by σ_L^A; the per capita rate of giving up ARV per year for these individuals is g_L^A. We assume that HIV-infected individuals will progress to disease and death at different rates:

• 1/v_E^U = average time spent in the untreated early stage of infection when individuals are eligible for both ARV and vaccine
• 1/v_L^U = average time spent in the late stage of infection if untreated
• 1/v_E^AV = average survival time for individuals when treated with ARV and vaccine in the early stage of infection
• 1/v_E^A = average survival time for individuals treated only with ARV in the early stage of infection
• 1/v_L^A = average survival time for individuals treated with ARV in the late stage of infection

We assume that HIV-infected individuals will transmit HIV with different probabilities:

• β_E^U = average per partnership transmission probability of HIV from untreated early-stage HIV infection
• β_E^A = average per partnership transmission probability of HIV from treated (ARV only) early-stage HIV infection
• β_E^AV = average per partnership transmission probability of HIV from treated (ARV plus vaccine) early-stage HIV infection
• β_L^U = average per partnership transmission probability of HIV from untreated late-stage HIV infection
• β_L^A = average per partnership transmission probability of HIV from treated (ARV only) late-stage HIV infection

The model is specified by the following equations: