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“The only thing necessary for these diseases to the triumph is for good people and governments to do nothing.”




R E Dorrington BA, BCom, BSc (Hons), MPhil, ASA, FIA



This addendum outlines changes made to the ASSA AIDS model since the abovementioned paper was published.  These changes were necessitated by the release of the 1996 Census results and an "AIDS Experts" workshop organised by the Department of Health, but the opportunity was also used to implement a number of improvements to the model.


AIDS, modelling, population, census, fertility, mortality, migration, South Africa.


Since the above paper was published before, inter alia, the release of the 1996 Census results and an "AIDS Experts" workshop organised by the Department of Health it does not describe the changes made to the model to incorporate insights that resulted from these sources of data.  It is the purpose of this addendum to document the changes to the model made since the publication of the original paper.


The starting population was derived in the same way as before.  However as the assumptions of mortality, fertility and migration have changed, as described below this resulted in a slightly lower starting population than in the original paper (32,8 million instead of 33,1 million).


HIV-negative fertility

Fertility rates up to 1992 were re-estimated in the light of the Census results.  In particular it was decided to apply the TFR projection model outlined in the paper only from 1993 onwards.  TFRs for the years before this were estimated as follows.  First Gompertz’s standard fertility curve was fitted to Sadie’s ASFRs in order to smooth some particularly high rates.  The TFRs were then set to the average of Sadie's adjusted TFRs and those implied by Udjo's projections (Udjo, 1998) on the basis of Sadie's adjusted ASFRs(1)  As TFRs were estimated only for five-yearly periods, TFRs for individual years were interpolated from polynomials of degree 3 fit to 1967.5-1977.5, 1972.5-1982.5 and 1982.5-1992.5 (2) and for individual ages using the Gompertz standard distribution

As the original method of determining ASFRs in the future produced negative rates at some ages when the TFR fell below 1,5 (i.e. some 60 years into the future) it was decided to alter the method.  Now the model produces ASFRs from a standard two-parameter Gompertz curve (with the two parameters changing with TFR to maintain a realistic shape).  In addition the user can set the limit below which the TFR will not fall.

HIV-positive fertility

As mentioned in the paper ideally it is necessary to model the ratio of HIV-positive fertility to HIV-negative fertility taking into account duration since becoming infected.  This was not done in the earlier version of the model resulting in an exaggeration of the impact of the epidemic on fertility.

This has been remedied in the final version by arbitrarily (3) setting the ratio of those who have been infected for t years to be equal to:


         is the ratio of fertility rates of those at age x who become infected to those who do not just before being infected (assumed to be close to the age specific ratios assumed by Zaba and Gregson (1998) for ages below 20, and one thereafter).

          is the initial impact on the fertility ratio of becoming infected (at age x)

      is a reduction factor which caters for the reduction in fertility over time since becoming infected.



HIV-negative mortality

As the staring population is as at 1 November 1985 and as the base mortality table used in the paper was centred around 1985 it was decided that it was more consistent to use as base mortality table one centred on 1 May 1996 (rather than mid-1985 as in the original version).  Thus the mortality was projected forward by 10 months from the original base.  In addition to this discussions with demographers and epidemiologists working in the area of mortality suggested that the infant mortality rate in the original table was probably a bit high and it was accordingly decreased by 6 per mille for males and 5 per mille for females.

Mortality of those born HIV-positive

The model reverts back to the original assumption of 30% per annum on the grounds that extra "sophistication" introduced by assuming a curve was unwarranted.

Neither of the above changes to mortality affects the results materially.


One of the most significant changes in terms of the effect on the size of the total population (and hence the magnitude of any absolute figures produced by the model) was the migration assumption.  The paper attempted to account for all possible migrants and thus arrive at a most likely estimate of the population including the undocumented migrants.  This exercise was attempted before the release of the 1996 Census results.

Subsequently it was decided that it would be more reasonable not to try to guess the extent of the "uncountable", undocumented, migrants but rather to set migration to be consistent with a best estimate of the population taking into account the 1996 Census results.


1.                    The population was projected to 1996 and the number of females compared to the census post-enumeration adjusted count.  From this comparison it appeared that, after allowing for what appeared to be an undercount in the census 0-4 year olds (since some of this could be due to age exaggeration the shortfall was assumed to be that on the 0-14 year olds in total) and apportioning those with no stated age, there were some 500 000 female migrants between 1985 and 1996.

2.                    This together with the 200 000 up to 1985 means that the total number of foreign-born females in the population was about 700 000 (of which the census identified only 300 000).

3.                    If we assume that the ratio of male foreigners to female foreigners is the same as reflected in the count (i.e. approximately 1.5 males to 1 female after including those with no stated country of birth) then there must be about 1 00 000 foreign males, only 500 000 of whom were identified as foreigners in the census.  Some 500 000 of these have been included up to 1985 leaving some 500 000 to be accounted for as migrants since 1985.

4.                    After taking into account the "documented" migrants implied by Sadie's projections and allowing for deaths during the period the "undocumented" migrants were set to 398 000 for females and 535 000 for males. This resulted in the number of undocumented migrants starting at about 4 000 and rising to around 180 000 in 1996. These were apportioned to ages to produce (fairly crudely) the numbers of females distributed by age according to the 1996 census.

5.                    NOTE.  This computation merely ensures some consistency of males to the number of females counted in the census (which were assumed to be counted for the adult ages).  The adjustments in no way account for the uncounted/uncountable undocumented migrants.  (My guess is that one could add a further 1 million to the 1996 figure for these migrants).


1.                    The contagion matrix was altered to ensure that the forces of transmission are symmetrical around the diagonal, which appears to be an intuitively reasonable assumption to make.

2.                    The model was adjusted to incorporate a 10% chance of infants becoming infected via breast-feeding (in addition to the assumption of a 25% vertical (mother to child) rate of transmission).

3.                    The percentages in the various risk groups were altered to correspond to the latest Metropolitan model (essentially increasing the proportion STD from 17% to 20% and reducing the RSK group accordingly).


Calibration is now to the rates reported for ANC attendees rather than those estimated for all pregnant women (this in itself makes no difference to the calibration of the model).

Since the paper a further year’s ANC prevalence survey results has been released.  The estimate of overall prevalence at 22,8% is substantially higher than the 20,9% projected by the model.  Since, according to the dynamics of the model, it is difficult to understand how the recorded figure could have jumped from 17% to nearly 23% in one year, it was decided that a greater understanding of the nature of the data was needed before embarking on further calibration.  Thus the model was calibrated excluding the latest data, although the rate has been included on the graph for comparative purposes.



Obviously as a result of the changes listed above the output from the model changes.  The most important differences are:

1.                    The total population now peaks at slightly over 49 million in 2010 but decreases very much more slowly.

2.                    The number of AIDS deaths expected for the calendar year 1998 (as opposed to the 1 November 1998 to 31 October 1999 reported on in the paper) is a little over 100 000.

3.                    The tables of results in the original paper have been updated and are published, amongst other output, at the ASSA web site.


Udjo E (1998) Additional Evidence Regarding Fertility and Mortality Trends in South Africa and Implications for Population Projections.  Statistics SA, Pretoria.

(1) Athough the TFRs published by the two researchers appear to be quite different this is a function of differences in the ASFRs rather than the number of births each would project (which turn out to be very similar).

(2) Y= 0,002x2 – 0,155x + 6,917

Y = -0,006x2 + 0,074x + 5,166

Y = 0,003x2 – 0,201x + 7,094

(3) This function was chosen as being one that both resulted in the pattern of ratios after 30 years that was comparable with that in Zaba and Gregson (1998) as well and a steady decrease in the TFR as the epidemic progresses of 4-5% for every 10% in prevalence (which is also consistent with Zaba and Gregson).