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Uses of Statistical
Methodology in HIV/AIDS Projections
Ramesh S. Patil^{1}, Sajjan
C.G.^{2} and Nagaraja Rao C^{3}
^{
1}Lecturer/
Statistician, Department of Community Medicine, Navodaya Medical
College, Raichur  584101
^{
2}Lecturer,
Veershaiva College, Bellary 583104
^{
3}Professor,
Department of Statistics, Vijaya College, Bangalore560004
Corresponding
Adresses :
Email:
[email protected],
[email protected],
[email protected]
Research Article
Abstract:
Projections of AIDS incidence
are critical for assessing future health care needs. There is need
for more accurate forecasts for the future course of the epidemic.
Projections for the future of the epidemic have most often taken the
form of trying to estimate how many new AIDS cases will be diagnosed
(or reported) over some span of future years. Projections are very
central for planning interventions and managing the available
resources as they provide very valuable information on the number of
undiagnosed infections. Issues that are necessary to the
understanding and management of AIDS have generated several
statistical challenges such as the choice of infection density,
estimation of incubation period distribution, and dealing with
sensitivity and studying of incomplete data. There were various
mathematical and statistical approaches have been proposed to
predict the future AIDS cases. In studying of AIDS, our main
interest is in understanding the current situation and predicting
the future path.
Key words :
HIV/AIDS, Time Series, Delphi survey method.
1.
Introduction:
The first case of HIV infection
in India was diagnosed among commercial sex workers in Chennai,
Tamil Nadu, in 1986. Soon after, a number of screening centres were
established throughout the country. Initially the focus was on
screening foreigners, especially foreign students. Gradually, the
focus moved on to screening blood banks. By early 1987, efforts were
made up to set up a national network of HIV screening centres in
major urban areas.
A National AIDS Control
Programme was launched in 1987 with the program activities covering
surveillance, screening blood and blood products, and health
education. In 1992 the National AIDS Control Organization (NACO) was
established. NACO carries out India's National AIDS Programme, which
includes the formulation of policy, prevention and control
programmes.
Present Status
of HIV/AIDS in India
The estimates projected until
recently were that globally India is leading over South Africa in
terms of the overall number of people living with HIV. The United
Nations Report on HIV released on Tuesday the 30th of May'06 said
the World's second most populous nation has overtaken South Africa
as the country with the most people living with HIV virus. India is
home to about 5.7 million cases as against about 5.5 million cases
infected in South Africa.
�
NACO estimated that the number of
Indians living with HIV increased by 500,000 in 2003 to 5.7 million.
Around 38 percent of these people were women.
�
By the end of May 2005, the total number
of AIDS cases reported in India was 109,349 of whom 31,982 were
women. These data also indicated that 37% of reported AIDS cases
were diagnosed among people under 30. Many more AIDS cases go
unreported.
�
The UN Population Division projects that
India's adult HIV prevalence will peak at 1.9% in 2019. The UN
estimates there were 2.7 million AIDS deaths in India between 1980
and 2000. During 200015, the UN has projected 12.3 million AIDS
deaths and 49.5 million deaths during 201550.
�
A 2002 report by the CIA's (27)
predicted 20 million to 25 million AIDS cases in India by 2010, more
than any other country in the world.
The
future of HIV/AIDS in India :
There are many predictions
about the effect that AIDS will have on India in the future and a
lot of dispute about the accuracy of these estimates. Ruben del
Prado, deputy UNAIDS country coordinator for India, has predicted
that "there is going to be reversal of the epidemic by 2008 and
2009. This does not correlate with other UNrelated estimates,
however, which have suggested that:

India's adult HIV
prevalence will peak at 1.9% in 2019.

The number of AIDS
deaths in India (Which was estimated at 2.7 million for the period
19802000) will rise to 12.3 million during 200015, and to 49.5
million during 201550.

Economic growth in
India will slow by almost a percentage point per year as a result
of AIDS by 2019.
Routes of HIV transmission :
AIDS is the most dreaded human
misery with its impact felt in all over the world. It is a fatal
transmissible disorder of the immune system that is affected by HIV.
Slowly HIV attacks and destroys the immune system which is the body�s
main defense against disease. HIV infects the defense cells of the
immune system of the human body called CD4+ T lymphocytes and
gradually reduces the cell number, thus, making an infected person
defenseless and infections that eventually cause death. The end stage
of HIV infection person is AIDS. At the end of incubation period of
infected person, there is a rapid decrease in immune system which
leads to an increase in sickness until death occurs.
There are four basic modes of HIV
transmission and they are:
1. Infected blood transfusion
2. Infected injecting
equipments
3. Unprotected sex
4. Infected mothertochild
transmission
HIV is transmitted through
penetrative (anal or vaginal) and oral sex; blood transfusion; the
sharing of contaminated needles in health care settings and through
drug injection; and, between mother and infant, during pregnancy,
childbirth and breastfeeding.
Treatment for HIV/AIDS :
Currently available drugs do not
cure HIV infection but they do prevent the development of AIDS. They
can stop the virus being made in the body and this stops the virus
from damaging the immune system, but these drugs cannot eliminate HIV
from the body. Infection with this virus results in the progressive
deterioration of the immune system, leading to 'immune deficiency'.
HIV is a very active virus that makes lots of copies of itself that
then damage the body�s immune cells (CD4 cells). Taking the medicines
everyday at the right time and in the right way keeps the right levels
of the medicines in the body which makes it very hard for the virus to
become resistant to the medicines. Current World Health Organization
(WHO) recommendations for HIV treatment state that three separate ARV
medicines need to be taken at all times. Some of these medicines can
produce side effects such as nausea and vomiting or headaches. Usually
most side effects are not serious and improve once the patient gets
used to the medicines.
Understanding HIV/AIDS numbers
:
In the beginning of public health
surveillance for HIV/AIDS in Asia Pacific countries, no distinction
was made between prevalent and cumulative numbers of HIV infection
and/or AIDS. However, with time and the progression of HIV infection
to AIDS and death, the constant widening difference between the
prevalent number of HIV infection and the cumulative number became
very obvious. As of the beginning of the new millennium, cumulative
numbers of HIV infection and/or AIDS cases are not commonly used,
except to put HIV/AIDS epidemics in this region into a historical
perspective. Public health programmes now almost exclusively use
prevalent and incident numbers. There is need a clear distinction to
be made for each of the following types of HIV/ AIDS numbers �
reported, official, estimated and actual. Official numbers of
HIV/AIDS may be reported cases or, in some instances, may be
officially estimated cases. Some care needs to be taken in evaluating
estimated numbers since, depending on the data, assumptions and
method(s) used to derive the estimate, the resultant figure can
represent a reliable working estimate of the actual HIV/AIDS numbers
or may represent gross overestimation or underestimation of these
numbers.
There were various mathematical
and statistical approaches have been proposed to predict the future
AIDS cases, numerous assumptions are required to account for the
intrinsic and extrinsic dynamics of disease spread [13], and detailed
models require specialized knowledge. This paper mainly describes the
general methods used for estimations / projections of AIDS cases and
reviews some statistical analysis of a few models which are developed
for the estimation of AIDS cases recently.
i.
Application of various statistical methods in context of projection of
AIDS cases.
ii.
Latest
development in projection of AIDS cases.
2. Some
Statistical Issues:
The statistical issues that arise
concerning the statistics of the AIDS epidemic illustrate the impact
of statistical forecasting in epidemiology. Analysis of studies of the
epidemiology and natural history of infection with the HIV and
subsequent onset of AIDS are complicated by many statistical issues.
Several such problems are associated with the nature of data
collection which is often unreliable and incomplete. In forecasting
health care needs, [15] the number of patients at various stages of
the illness and the rate of progression of AIDS will be a significant
factor in public health planning. Our interest is in understanding the
present state and predicting the future road. These are important
concern to health care system, administrators, policy makers,
epidemiologist and statisticians. Therefore, there is a need for
quality information to be collected and analysed in an objective
manner and presented in suitable format. Projections are very central
for planning interventions and managing the available resources as
they provide very valuable information on the number of undiagnosed
infections. Issues that are necessary to the understanding and
management of AIDS have generated several statistical challenges such
as the choice of infection density, estimation of incubation period
distribution, and dealing with sensitivity and studying of incomplete
data. To answer the crucial questions, there must be an effective
machinery to contribute to the available data among the researchers of
different disciplines for their study purpose viz.
1. What is the period
between infection and transmission?
2. How does the HIV
transmissibility vary with time after got infection and with disease
stage ?
3. What are the
cofactors affecting infectivity?
4. What is
responsible for the large variation in incubation time?
5. Will the current
trend in the spread of HIV virus continue?
6. Can we able to
explain the past growth of the epidemic?
7. Can we predict the
future size of the epidemic accurately?
To attempt the
above questions, researchers need different types of data. Identifying
the availability and sources of such data itself is the beginning.
Statisticians have to evaluate the source of data available for
prevalence and to develop estimation methods based on such data. The
issues which are essential to understanding and management of the
disease have generated numerous statistical challenges and which are
as follows:
i.
Estimation
of incubation period distribution
ii.
Projection
of the course of the epidemic
iii.
Selecting
proper infection density
iv.
Dealing
with confidentially and analysis of incomplete data
v.
Inadequacy
of data on HIV/AIDS for assessing the size and progression of
epidemic.
Today there is
challenging role to play in the field of research of AIDS. They have
responsibility about first to point out discrepancies in reported and
actual numbers of HIV/AIDS counts. The big difference between reported
and estimated numbers of HIV/AIDS counts in India, this is may be
because of following reasons:
i.
Unreliable
and inefficient reporting administration
ii.
Long and
variable incubation period
iii.
For
projection, use of improper methods
iv.
Allied
sensitivity and social disgrace
v.
Insufficiency in the present AIDS surveillance data
Importance of AIDS cases :
Projections for the future of the
epidemic have most often taken the form of trying to estimate how many
new AIDS cases will be diagnosed (or reported) over some span of
future years. In particular, we consider projection of the number of
future cases, and estimation and identification of two key
epidemiological unknowns, namely the properties of the incubation
distribution and those of the infectivity associated with
transmission. As data and projection methodologies improve, the
differences in projections may be reduced for subSaharan Africa. [16]
To provide a method for estimation and shortterm projection of AIDS
cases in areas where reporting of AIDS is unreliable. The method
relies on estimation of annual HIVinfected "cohorts" and on annual
progression rates from HIV infection to AIDS for each cohort.
Estimation of annual infections is based on observations as to when
HIV infections began to extensively spread and on the estimated shape
and intensity of the annual infection curve. Using published and
unpublished HIV serologic data, adult AIDS cases were estimated and
projected for selected countries or regions in areas where homosexual
men and IV drug users are the predominantly affected population
(Pattern I); where heterosexual transmission of HIV predominates
(Pattern II); and where HIV infection only began to spread extensively
after the mid1980s (Pattern III). This method is useful for
estimating the current and future AIDS case load, especially in areas
where the reporting of AIDS is unreliable. Such estimates are
critically needed for public health and health care planning.
Due to large discrepancies
between actual and estimated number of HIV/AIDS cases in India, there
is a need for reliable projections, which are to be base on standard
methodology which takes into account the transmission dynamics of the
HIV. [14] Stress the need of accurate projections of the number of
AIDS. If the projection methods are based not only on current
incidence but also on the past incidence of HIV then it will helps us
to know future path of the endemic better.
3. Methods for
estimation/projection of HIV infection and AIDS cases:
In this section, we describes the
limitations of the general methods used for estimating all the
important and needed HIV/AIDS numbers including: prevalence, incidence
and cumulative incidence of HIV infections, AIDS cases and AIDS
deaths; and HIVrelated diseases or conditions such as paediatric AIDS
and maternal AIDS orphans and HIVrelated tuberculosis cases. There
has been an increasing need for estimates and projections in recent
years for various purposes; monitoring and evaluating trends of
incidence, etc.
Estimating HIV Incidence
Incidence estimates are more
difficult to obtain than prevalence figures, but they are more
informative about the effects of prevention efforts and the future of
the epidemic. HIV incidence estimates can be obtained from:
1) observing
seroconversions in a longitudinal study;
2) inferring incidence
from serial crosssectional surveys;
3) using
capturerecapture methods in serial surveys;
4) backcalculation
from reported AIDS cases; and
5) identifying recent
seroconverters from a crosssectional sample
6) using two HIV
antibody tests of differing sensitivity for HIV antibodies.
The first method of estimating
incidence is to enroll an HIVnegative population in a longitudinal,
or cohort, study and to test the participants at regular intervals for
new HIV infections, thereby deriving an incidence rate (number of new
infections per total number of personyears of followup).
Longitudinal studies with incident infections have been a valuable
source of data.[6] Longitudinal studies are limited by the expense of
conducting such a study, by the characteristics of the population
enrolled, and the consideration that the longer the cohort is
followed, the less likely it is that they are still representative of
the population from which they were recruited.
The second method of estimating
incidence is by conducting serial crosssectional surveys in a
population. This method does not directly estimate incidence, but
incidence is indirectly estimated by the slope of the seroprevalence
against time if the population being surveyed remains representative
over time and if deaths and other losses to followup can be
considered negligible. This approach has been suggested for estimating
incidence from successive birth cohorts of recruits into the U.S.
military.[7]
The third method is a variant on
the crosssectional survey approach that uses "capturerecapture," a
methodology long used by biologists to study wildlife populations. It
requires some sort of unique identifier, but not necessarily names, of
individuals included in repeated surveys, so that the seroconverters
among those repeatedly tested can be identified. This method was used
to estimate incidence rates among injecting drug users in San
Francisco by repeated testing in both clinic and street settings over
a 5year period while asking participants to receive their test
results under a unique identifier constructed from the day of the
month of their birth and their parents' first names.[8]
The fourth method uses "back
calculation," which combines the available data on the numbers of
reported AIDS cases and the incubation period distribution of AIDS
(the mathematical function that estimates the probability of
developing AIDS for each year following HIV infection) to derive how
many HIV infections occurred during years past.[9] With information on
past infections and AIDS cases, current HIV prevalence can be
estimated. This technique requires fairly complete surveillance of
AIDS cases and an accurate estimate of the incubation period
distribution. It is limited by its inability to estimate HIV
infections in recent years with any precision. More significantly, the
large, and as yet largely unmodeled, effect of antiretroviral therapy
on the incubation period has rendered backcalculation currently
ineffective in estimating prevalence. The complexity of treatment
regimens and their effects appear unlikely to be captured by an
adjustment to the incubation distribution. For this reason, back
calculation may no longer be a useful method of estimating HIV
prevalence.
The fifth method is relatively
new. It uses two HIV enzyme immunoassays: one is a current, highly
sensitive test and the other has been made insensitive ("detuned"), in
order to identify recent seroconverters from a single crosssectional
sample. As the quantity and avidity of antibody in peripheral blood
increases progressively in the first weeks and months after HIV
infection, a newly infected person will test positive on the sensitive
assay and negative on the "detuned," as it is often called, or less
sensitive assay. [10] One source of variation with this method is the
viral subtype (clade) of HIV being tested. The average window of time
captured by the two assays also needs to be determined and validated
separately for assays of different manufacture. False positive
seroconversions can occur in individuals with latestage HIV
infection, in which antibody levels decline, and in persons receiving
antiretroviral treatment. Despite these limitations, this method has
grown in use because it is the only method that allows an incidence
estimate from a single crosssectional sample. It is described by CDC
as the serological testing algorithm for recent HIV seroconversion or
STARHS.[11]
A sixth approach does not
estimate HIV incidence per se but uses the number of reported AIDS
cases in the youngest age range of adult cases, ages 1325, as a
surrogate for recent trends in incidence.[12] The justification for
this approach is that onset of sexual and drugusing risk behavior in
the teenage years (or later) leads to the inference that AIDS cases in
this age group will be predominately those with a short incubation
time from infection to AIDS and that therefore most of the cases
reflect relatively recent infections (less than, say, 5 years on
average).
Methods for
estimating/projecting HIV prevalence:
1. Before the advent
of effective drug therapy to prevent or delay the relentless
progression from HIV infection to the development of AIDS, most of the
developed countries considered that reported AIDS cases are to be
sufficiently reliable for estimating/projecting HIV prevalence by
using a backcalculation method. The backcalculation method used
annual progression rates from HIV infection to AIDS and reported
annual AIDS cases (usually after adjustments for delayed and
incomplete reports) to calculate how many annual HIV infections would
have been needed to generate the estimated/projected annual AIDS
cases.
2. In the late 1980s
and early 1990s to estimate HIV prevalence, there was use of �ratio�
method that used an estimated ratio of prevalent HIV infections to
prevalent AIDS cases. As the backcalculation method required
reliable estimates of AIDS cases, in the same way, the ratio method
also required reliable estimates of AIDS cases, which were usually not
available. Apart from this, most users of the ratio method did not
realize that in all HIV epidemics the ratio of prevalent HIV infection
to prevalent AIDS cases changes rapidly over time. This HIV/AIDS
ratio falls from many thousands to one during the first few years of
an HIV epidemic, to less than ten to one after the first decade. This
decline occurs whether HIV incidence is increasing or decreasing
because, in the absence of effective treatment, virtually all
HIVinfected individuals progress to AIDS. The HIV to AIDS case ratio
is, therefore, almost all HIV and no or few AIDS cases.
3. An easy and useful
method to estimate/project the current HIV prevalence in a �mature�
HIV epidemic (one that has been in progress for about 10 years or
longer) is to multiply by the estimated annual AIDS cases by 20. If
the median period for HIV infection to the development of AIDS is
assumed to be 10 years, then about 10 years after the start of an HIV
epidemic, about 5% of prevalent HIV infections will develop AIDS on an
annual basis. For example, if the estimated annual number of AIDS
cases is 1000, then the estimated HIV prevalence would be about 20 000
(1000 multiplied by 20). Conversely, if HIV prevalence is estimated
to be 20 000, then, by taking 5% of the HIV prevalence, one can
calculate rapidly the expected annual number of AIDS cases to be about
1000. This is a �quick check and balance� method to see if the
national estimate of HIV prevalence is compatible with the estimated
annual number of AIDS cases or the reverse � if the estimated annual
number of AIDS cases �matches� with the estimated national HIV
prevalence.
4. In the absence of
reliable AIDS case estimates or data, epidemiologists have estimated
HIV prevalence by using the results of serological surveys and
extrapolating these data to the total population of the age group
1549 year. This has been and continues to be the primary method used
in developing countries to estimate HIV prevalence. In this method,
major problems are, the limited number of HIV seroprevalence studies
that may be representative of specific populations or subgroups, and
the wide variability in estimates of the size(s) of important HIVrisk
behaviour groups or cohorts, viz. FSW, IDU and patients seen in STI
clinics.
Estimation of HIV prevalence
by using HIV serological data :
Using the available HIV
serological data to derive a seroprevalence estimate, many
epidemiologists have developed their own methods, assumptions and
biases. Although HSS systems are not designed to provide data for
making HIV prevalence estimates, they are widely used for this
purpose, simply because there are usually no better serological data
available. HIV prevalence in the 1549 yearold population has been
calculated according to the following general formulae:
(1) The number of HIV infections
in each of the major highrisk groups = the estimated number of the
highrisk group (estimated for a specific population or a province)
multiplied by estimated HIV seroprevalence rate (from HSS data); and
(2) The number of HIV infections
in the 1549 yearold population = estimated HIV seroprevalence rate
in antenatal women in the province (from HSS data) multiplied by the
estimated number of 1549 yearolds in the province (from census
estimates).
Major sources of error:
1. Obviously error
will occur while estimating HIV prevalence. The data quality and
representativeness of the usual grab samples collected for most HSS
systems can be seriously questioned. However, there have not been any
systematic ways to quantify the probable range of error(s) related to
such data quality issues. There has also been little effort to use the
full range of data available, e,g. HIV prevalence from existing
surveys, HIV prevalence in groups outside HSS, other data sources,
etc.
2. Errors in
estimating the size(s) of specific RBG can be quite large (up to
several times higher or lower).
3. The probable
heterogeneity of HIV risk within any specific RBG is well known, but
frequently findings from sentinel HIV sites that tend to capture
persons from those RBG with the highest or very highrisk behaviours
are then extrapolated to the total RBG. This lead obviously will tend
to higher HIV prevalence estimates.
4. In this method, a
major assumption used is that HIV prevalence found in ANC can, with
adjustment for the estimated male to female ratio, be used as a
surrogate for HIV prevalence in the total 1549 yearold population.
However, this assumption has not been validated for other populations.
5. Measurement and/or
estimation of the male to female (M:F) ratio of HIV infections has
been carried out using a variety of methods and assumptions. In most
of the epidemiological settings outside Africa (where there is a
slight excess of infected females, compared with males) there has been
a consistent and fairly large preponderance of infected males compared
with females.
6. In heterosexual
HIV epidemics in Africa, a marked urbantorural HIV differential, of
up to 10fold or more, was noted in the early phase of HIV spread.
This differential narrowed markedly with time and after 10 years or
more had been reduced to about 12fold. One current assumption is
that changes in the urbantorural HIV prevalence differentials in
other developing country populations follow the same general course as
that which has been observed in Africa. It is quite possible (and
indeed probable) that, in other regions, heterosexual transmission of
HIV may remain more localized in the highest RBG in urban centres and
may penetrate or diffuse much more slowly (if at all) into most rural
populations.
History of methods for
projecting HIV Cases :
There is great uncertainty in
projecting the future, especially for a complex problem such as HIV
transmission. Even so, attempts to predict future trends and
prevalence of HIV have been carried out with a very wide range of
errors, using the following methods.
Delphi survey method :
The Delphi survey method was
developed in an attempt to improve the reliability of the judgments
needed in relatively uncertain situations, as well as to provide a
means of quantifying such judgments. Essentially, the Delphi method
obtains educated guesses from selected experts in a reiterative
fashion, and then uses the average and range of the Delphi responses
as projections. The main advantages of the Delphi method are speed
and low cost. Though, it is difficult to select truly knowledgeable
experts (i.e., experienced quantitative epidemiologists who are
familiar with the epidemiology of HIV and general demographics of a
specific country or population) to develop reliable estimates or
projections of the number of HIV infections. This method should be
used only for populations where no data are available.
Mathematical and
computer/simulation models :
Mathematical and
computer/simulation models have been used to develop short and
longrange projections of HIV prevalence. Yet, such models should be
used primarily for hypothesis testing � not for making estimates and
projections of the annual incidence/prevalence of HIV infection for a
specific country or population(s). That was the conclusion of a
United Kingdom expert committee that reviewed the situation in the
United Kingdom in 1994. The committee concluded that the general
uncertainty of many of the needed input parameters, such as the size
of the risk groups, as well as reliable data on their current sex
partner exchange rates, made estimation and projection of HIV/AIDS
incidence and prevalence in the UK extremely uncertain. As a result,
they stated clearly that model outputs should not be used for specific
programme or policy development.
Method for shortterm (less
than 5 years) projection of AIDS cases/deaths:
A simple scenario/modelling
approach for estimation and projection of AIDS cases was developed
during the late 1980s by the Surveillance, Forecasting, and Impact
Assessment (SFI) unit of the former WHO Global Programme on AIDS
(GPA). This scenario/modelling approach or method can be used to
provide working estimates and shortterm projections of AIDS cases and
deaths for policy development and public health planning. HIV/AIDS
scenarios can be made up or constructed with or without models to
�fit� the observed HIV/AIDS data and trends. The following is an
outline of the general methods used in this scenario/modelling
approach to develop working estimates and projections of HIV
infections and AIDS cases and deaths.
(1) Assemble and analyse
available HIV seroprevalence data to estimate the most recent
pattern(s), prevalence and trends of HIV infection for a specific
population.
(2) Based on these data and other
epidemiological observations, different HIV patterns and prevalence
levels (i.e., scenarios) can be constructed with some confidence to
the year 2005 for specific countries/populations.
(3) An AIDS model can be used to
derive annual and cumulative estimates and projection of AIDS
cases/deaths and other HIVrelated conditions, based on the general
HIV scenario(s) constructed.
EPIMODEL :
EPIMODEL is a simple
microcomputer programme developed by WHO in the late 1980s to estimate
past and current prevalence, and to make shortterm projections of
AIDS cases and deaths in areas where AIDS case reporting was largely
incomplete and unreliable. Most the problems encountered by users of
EPIMODEL are associated with the quality of input parameters supplied
by users. The basic module of EPIMODEL uses estimates of HIV
prevalence and distributes this prevalence by annual HIVinfected
cohorts back to the estimated start of the HIV epidemic along a
selected epidemic curve. EPIMODEL then applies annual progression
rates from HIV infection to the development of AIDS to each of the
annual HIV cohorts to calculate annual numbers of adult AIDS cases and
deaths. EPIMODEL provides default values for several input parameters
that may be considered appropriate for modelling HIV/AIDS in a
subSaharan African population, but all input parameters for EPIMODEL
can be easily changed to better �fit� the specific population that is
being modelled. It must be recognized that, in any large population,
the spread of HIV infection and the subsequent appearance of AIDS
cases is usually the consequence of several epidemics, i.e., in
different �risk groups� or different geographical areas.
EPIMODEL was not designed to
provide projection of HIV infection. The basic module of EPIMODEL was
designed to estimate and project adult AIDS cases and deaths. This
module can, with the additional input of a population denominator,
calculate annual incidence and prevalence rates for HIV infection.
Other modules of EPIMODEL include a Child module and Tuberculosis
module.
Aside from the potential errors
described above, additional sources of potential error in using
EPIMODEL include the following:
(1) One problem of EPIMODEL is in
making only a single point of prevalence, with a starting year then
generating a curve. Also, the greatest error could occur in estimating
HIV point prevalence. Usually only subsets of data are used,
representativeness of populations tested is not considered.
(2) The �stage� of the HIV
epidemic will have a significant impact on the estimates of annual HIV
incidence and on estimates of annual deaths due to severe immune
deficiency related to HIV infection. The stage and duration of the
modelled HIV epidemic will also have a major impact on the estimated
cumulative incidence of HIV infections and AIDS deaths.
(3) Another possible source of
error in producing estimates and projections of AIDS cases and deaths
with EPIMODEL is the selection of the median interval period from HIV
infection to death due to severe immunodeficiency related to HIV
infection. The median interval from HIV infection to the development
of severe immune deficiency appears to be similar in all populations
(i.e., in developed and developing countries) and is estimated to be
about 78 years. However, there is a consensus that the survival
period from the development of severe immune deficiency to death is
much shorter in most developing countries than in developed countries,
where the advent of HAART therapy has significantly increased survival
of patients with moderate immune deficiency related to their HIV
infection.
The default median progression
period from infection to AIDS in EPIMODEL is 10 years and the default
median interval from AIDS to death for developing countries is less
than 1 year. This has resulted in a median interval from HIV
infection to death of 11 years. The change from this 11year median
survival period to the 9year median progression period from infection
to death results in much higher (up to 30% higher) cumulative numbers
of HIV infections. In addition, use of a 9year median survival
period results in a higher (up to 60% higher) annual number of AIDS
deaths.
Asian Epidemic Model (AEM) :
This model uses behavioral inputs
to model HIV prevalence trends over time. This model has been able to
fit 10 years of epidemiological and behavioral data in Thailand. The
model contains six major population subgroups: general population
males and females, male clients of sex workers, direct and indirect
sex workers, and injecting drug users. The size of each population and
behavioural time trends (condom use, frequency of intercourse, etc.)
will be determined from analysis of existing behavioural studies in
the country. The transmission parameters (e.g. HIV transmission
probabilities, STD cofactors, circumcision cofactors) will then be
adjusted to fit to time trends in epidemiological HIV data in the
country. This model will then produce estimates of new infections
that would be more consistent with observed behavioural trends.
Time series analysis
:
A time series is a chronological
sequence of observations on a particular variable. The data points
may be plotted to create a model, enabling one to quickly see trends,
cycles, seasonal variations, or irregular fluctuations that occur over
time. Once a pattern has been identified, it may be extrapolated into
the future and used in forecasting [20].
In forecasting the AIDS endemic
with time series analysis, then we have to pay attention to the
following some questions:
1.
How
regular are the past HIV/AIDS trends? What are the chances that these
patterns change?
2.
Is future
HIV/AIDS counts dependent at least partially on the present observable
counts?
3.
How
reliable and accurate are the past data on HIV infection?
Future numbers of HIV/AIDS counts
are necessarily based on the present incidence. However, question 1
and 2 are satisfactory answered, but question 3 is not. This happens
due to sensivity factor associated with AIDS and lack of satisfactory
diagnostic facilities which have led to large under reporting of AIDS
cases. If the available data are corrected using a suitable method
then they can be subjected to Time series analysis.
Extrapolation :
In the western area registration
of AIDS cases are fairly complete and get reliable estimates of AIDS
prevalence and incidence. Due to long asymptomatic period of infection
and the fact that spread is mainly limited to specific exposure groups
that are often difficult to contact, estimates of the prevalence and
incidence of HIV infections cannot easily be obtained from registers
of cases of HIV infection. However, [17] in some countries where AIDS
registration is incomplete, HIV prevalence can only estimated by
extrapolation from surveys. In this method the value of dependent
variable x say, the number of AIDS cases is estimated for a given
value of y, the independent variable, say the number of HIV
seropositive or the number of persons exposed to risk of infection,
which lie outside the existing range of y values. NewtonGregory
Forward interpolation formula is wellknown one. In graphical
approach, the fitted curve is extrapolated to a future time point.
This classical method of extrapolation, which recognized a polynomial
relation between x and y, is unlikely to be apply in the AIDS course
in view of behaviour of AIDS data. So is the case with the multiple
regression models.
To choose the time period to
model and the type of model to fit, it is important to identify
changes in trends of AIDS incidence, especially in projecting AIDS
cases by extrapolation. Because variations in the numbers of cases
diagnosed from period to period can obscure changing trends, adjusted
data on incidence were plotted with smoothed curves obtained from the
lowness procedure [21]. Adjusted data on incidence, not the smoothed
data, were used in backcalculation and extrapolation analyses.
A method to correct AIDS
counts :
The method is briefly explained
below,
Define:
n_{ij }: Number of New
AIDS cases reported during the period I for the period of diagnosis j.
where 1< j < I < t, where t being the number of periods under
observation ( A period may be of 6 months duration ). It is considered
that (i) n_{ij }are independently Poisson distributed with θ_{ij}
and (ii) all the reported casers are diagnosis correctly.
P_{k }: Proportion of
cases reported where k is the number of periods between diagnosis and
report ; ∑ P_{k } = 1.
n._{j }: Cumulative
reported incidence for each period of diagnosis.
N._{j }: Actual number of
diagnosed AIDS cases which are not observable due
to reporting delays. Here
it is considered that every diagnosed case will
be reported later or
sooner.
The parameter θ_{ij
}are then defined by,
θ_{ij } =
N._{j }P_{k} with k = I � j, P_{k
}> 0
3.921
Maximization of log likelihood of
equation (3.921) results in the following equations:
N._{j } =
∑^{t}_{i=j }n_{ij }/ ∑^{tj}_{0
}P_{k, } j= 1,2,3,�.t
3.922
P_{k} _{ } = ∑^{tk}_{i=1
}n_{i+k.j }/ ∑^{tk}_{i=1 }N._{i}
_{ } k= 0,1,2,3,�.( t � 1 ) 3.923
Solutions of equations (3.922)
and (3.923) can obtain using an iterative proportional fitting
algorithm of [18]. It can be shown that equations (3.922) and (3.923)
are conditional solutions to the problem of estimating the size of the
multinomial population [19] and [5] have used similar estimators for
the actual number of AIDS cases and subsequently fitted a line or
curve by regression technique to the estimate.
Smoothing of Exponential
trend :
Exponential smoothing was applied
to several of the models in order to obtain clearer graphs for
analysis. Exponential smoothing is a forecasting method that weights
recent observations more heavily than remote observations. The
equation for exponential smoothing is
S_{t} = αy_{t} +
(1  a)S_{t1}
In this equation, S_{t}
is the smoothed curve and α is the smoothing constant, which is always
between zero and one. The trend inherent in HIV/AIDS data by moving
average method can be improved by assigning the weights to each year
reported AIDS case in geometric progression. Here greater weights are
assigned to latest observations. Number of HIV positives can be taken
as weights since the endemic is relatively young and there will be
very few HIV positives in the very beginning of the spread than in
recent years.
If the weights assigned are math
version normal { 1, (1ω), (1ω)^{2}, �..(1ω)^{n1} }
to �n� observations ( 0 < ω < 1 ), then the weighted averages till
the current year t and the succeeding year.
Taking n to be large, and higher
powers of w and ( 1 � w ) and doing certain algebraic manipulations,
the following relationship between x_{t+1} , the forecasting
value for the next period and x_{t }, the forecast value for
the current period can be established.
X _{t+1 } = W x_{t1 }+ (1ω) x_{t }
(3.931)
i.e. the new forecast = [ω X
observed value + (1ω) X old forecast ]. Here x_{t+1 }is the
smoothed forecast, _{ }w is the smoothing coefficient and ( T
w/w ) is the trend factor. The forecast for the first period is
generally taken from some old forecast if available or is often
considered.
To minimizing forecast error:
To attempt the
situation in which the trend is upward but the forecast is low or
conversely, a factor is added to make forecast value closer to the
actual value. Equation (5.91) may be written as
X_{t+1 }= ω ( x_{t1 }� x_{t }) + x_{t}
By induction,
X_{t } = ωx_{t }+ (1ω) x_{t1 }= ω ( x_{t
}� x_{t1} ) + x_{t1}
Where
the quantity ( x_{t }� x_{t1} ) is the error. The
trend coefficient which is required for preparing the forecast is
calculated by the formula
Θ_{t }=
[ w X change in smooth value ] + [ ( 1w ) X proceeding trend
coefficient ]
Then the forecast
F_{t } is obtained by the relation,
F_{t
} = smoothed value + ( trend factor X trend coefficient )
And error of the
forecast, E_{t }=X_{t } F_{t
}
The forecasts
resulting from the single parameter exponential smoothing is
consistently low_{ }because of there is an upward trend in the
actual number of AIDS cases. To rise above of this, a second smoothing
constant say the HIV seropositivity rate may be selected for trend
itself.
The
Multiple Regression Model :
While time series analysis was
useful in gathering information about the population of HIV/AIDS
patients as a whole, a second method, the multiple regression model,
was used to examine HIV/AIDS mortality on an individual basis. We
used the multiple regression method to build statistical models
describing the dependence of the incubation period on a person�s age
at HIV diagnosis and the chronological time since the start of the
study. The Multiple Regression model uses more than one independent
or predictor variable (denoted x_{1}, x_{2}, etc.) to
explain dependent, or response, variable, y. The equation is shown
here:
y = �_{0} + �_{1}
x_{1} + ߭_{2} x_{2} + ε
The two predictor variables, x_{1}
and x_{2} , are a person�s age at the time of HIV contraction
and the chronological time counting from the start of the study.
These were used to explain the response variable, y, which is the
length of time between a person�s contraction of HIV and their
diagnosis of AIDS (the incubation period). �_{0} is the
yintercept. The error term, ε, explains the variation in the
response variable that could occur given our combination of predictor
variables.
BackCalculation :
For long incubation period, this
method is designed specially for AIDS cases. It is a method in which
the number of AIDS cases can be projected from those already infected
with the AIDS virus i.e. it reconstruct the past pattern of HIV
infections and it is used widely to predict the number of AIDS cases
apart from knowing the present situation [4][5]. Further, this
projected can consider as the lower bound as this number will be
expected even if there are no future infections. It preassumes the
knowledge of incubation distribution among the infected that can
develop AIDS. There is no need of assumption about the number of
infected individuals or the probability of an infected individual
eventually developing AIDS. Because of the long incubation period,
this method does not account for further infection cases but can
produce accurate short term projection. In this procedure, convolution
equation namely,
Z = X + T
Where T is the variable denoting
the length of incubation period, is the basis of the Backcalculation
method. And X & Z are the random variables denoting the chorological
times of infection and diagnosis for AIDS respectively.
Let N denote the total number
diagnosed upto the year L+1. Then,