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ADDENDUM TO ASSA600: AN AIDS MODEL OF THE THIRD KIND?
R E Dorrington BA, BCom,
BSc (Hons), MPhil, ASA, FIA
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.
modelling, population, census, fertility, mortality, migration, South
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
2. STARTING POPULATION
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).
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
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
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.
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
setting the ratio of those who have been infected for t years to be
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.
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
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.
5. INTERNATIONAL MIGRATION
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).
6. ASSUMPTION PAGE
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
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
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.
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).