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Technical Note - BThe full list of countries is provided in thenotes to Table T10. The data indicate thesubstantial variation in HIV prevalenceamong adults, from negligible levels insome countries, to a high prevalence ofnearly 34 per cent in Botswana. The meanHIV prevalence in the full cross-sectionof countries, in contrast to Botswana’scase, is three per cent. As anotherexample of the variation in the spread ofthe HIV/AIDS epidemic across countries,the average annual increase in AIDS cases(taken as a proportion of per 100population aged 15–64 years in 1998)during the period from 1980 to 1998 wasabout 0.18 for the full sample, but rangedfrom 0 in some, to 1.93 in Botswana. Oursample of countries includes those withextremely low levels of humandevelopment in 1999—0.258 for SierraLeone –as also countries such as Norway,with an HDI of 0.939 in 1999. Similarvariation is apparent from data oneducation variables such as adult literacy,mean years of schooling and schoolingenrolment, life expectancy at birth andreal GDP per capita.a) The econometric model forHD Impact on HIV transmissionThe starting point of our discussion is thefollowing simple equation that can beused to describe the dynamics of HIVtransmission in an adult population,whose HIV prevalence rate is denotedby H. 18(3) dH/dt = (1 – H)*H p*ß *T - θ*HHere dH/dt is the derivative of H withrespect to time “t” and equation (3)describes the movement of HIVprevalence over time. H pis the infectionrate among the “sexual” partners of theabove population, ß is the proportion ofunprotected sexual interactions in anygiven interval and T is the rate oftransmission of HIV during unprotectedsex. θ denotes the proportion exiting fromthe infected pool in any time period.The HIV prevalence rate among partners,the proportion of people in thepopulation of interest who haveunprotected sex and the rate oftransmission of HIV will depend on anumber of variables, such as the level ofknowledge people possess about risk ofinfection, income levels and hence theopportunity cost of infection, concernsabout health (including one’s owninfection status, especially if there isassortative matching), opportunity,inequalities that allow some people to payoff others to incur increased risk as well asindicative of poverty, cultural norms thatgovern the practice of unprotected sexand the like. Specifically, let us assumethat the impact of these other forces is feltin such a way that(4) H p*ß *T = (α + π*x)*ρ*HHere x indicates all variables other thanan average individual’s HIV-status, H.H is taken to enter multiplicatively in thisformulation. Thus, we can write(5) dH/dt =(1 - H) * (α + π*x) *ρ*H - θ*Hand(6) dH/dt = (α*ρ + ρ*π*x - θ)*H -(α + π*x)*ρ*H 2This formulation, under differentassumptions, leads to three relatedeconometric specifications, as isdemonstrated below. The differentialequation in (6) has a readily obtainablesolution given by,(7) Z(t) = exp(-k*t) * z(0) + (m/k)*(1 - exp(-k*t))Regional Human Development Report174 HIV/AIDS and Development in South Asia 2003