# A Pearson Type III Curve to Approximate the Age Distribution of

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A Pearson Type III Curve to Approximate the Age Distribution of
HIV/AIDS Infection in Botswana
Parameswara Krishan
University of Botswana and University of Alberta
E-mail: krishnanp@mopipi.ub.bw

Introduction
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The Sub-Saharan countries in Africa suffer from the scourge of HIV epidemic. Botswana has the highest
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prevalence rate of the disease. The Central Statistical Office (CSO) of Botswana has conducted two surveys
to assess the extent of the epidemic. The field work for the third has just been completed. The current
Botswana Aids Impact Survey (BAIS III) is equipped to collect incidence data as well.

Incidence and Prevalence
The incidence rate (I) describes the rate of development of the disease over a period of time (usually one
year) standardized for 100 members of the population at risk. Prevalence rate (PR) shows the number of
existing cases of the disease during a period standardized for 100 members of the risk population.
P and I are related by:

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P= I*D,                                                (1)
where D is the duration of the disease. HIV has a long incubation period, resulting in an infected person
unaware of knowing it till the symptoms appear or a proper blood test is carried out. Given the nature of the
epidemic, collection of incidence data is rather involved.

Estimation of Incidence from Prevalence
BAIS II of 2004 has shown the age distribution of males and females positive and affected by the disease.
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We approximate the distribution by a simple probability curve. The prevalence data by age were subject to
distribution fitting by using Karl Pearson’s criterion (Kendall and Stuart, 1977). Pearson type III gamma
will provide a good approximation to the data.
The type III distribution, with parameters ‘m’ and ‘p’ is given by :
dF(x,m,p)= mp * xp-1 exp(-mx)dx / Γ(p)
where 0<x ≤∞: m>0, p>0.
If p is ≤ 1, the Pearson curve is J- shaped; otherwise it is unimodal.
The prevalence data are unimodal. The population mean of this distribution is p/m and variance
p/m2                                            (2)

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The 2004 BIAS data (CSO, 2005) yielded a mean of 36 years for Males (variance of 4.58). By Pearson’s
method of moments m^ = 1.72 and p^ = 61.9. For Botswana Females, the mean age was 33.4(variance of
10.7). Then m^ = 3.12 and p^ = 104.1. In view of non-availability of incomplete gamma function tables with
us, the test of goodness of fit was not carried out. Instead, we fit the simplest version of Type III – Negative
Exponential. The advantage here is that only one parameter has to be estimated.

The Negative Exponential
The density function of a negative exponential is given by:
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DF(x,θ) = θ exp(-θx)dx ; θ>0, 0<x<∞                                                 (3)

The estimate of θ is: θ^ =1/mean(x)                                                (4)

2004 HIV/AIDS Data
For Botswana males, θ^ was estimated as 0.0258,                                                      (5)
^
and for females θ 0.0290                                                                     (6)
We may note that the values are close.
The incidence rate at age X is given by: ∂/∂x {(θ exp(-θx)} = - θ2 *exp[-θx].                        (7)
The negative sign may be ignored as incidence rate is always positive. The rates per 10,000 have been
computed and are shown in Table 1.
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Table 1: Incidence rate per 10000 in 2004.
Age group                 Males                   Females
15-19                     4.2                     5.1
20-24                     3.7                     4.4
25-29                     3.2                     3.8
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30-34                     2.8                     3.3
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35-39                     2.5                     2.8
40-44                     2.2                     2.5

The rates decline with age:
•       Females have higher incidence compared to males
•       The rates are high at the prime sexually active ages
•       The fit underestimates the prevalence rates

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REFERENCES
[1]   Central Statistical Office (2005), Botswana Aids Impact Survey II – Statistical report, Gaborone:
Central Statistical Office.
[2]   Kendall, M.G. and A. Stuart (1977), Advanced Theory of Statistics, Vol 1, London: Griffin, (4th
Edition).
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