Modeling African AIDS

Tom Keske tkeske at mediaone.net
Fri May 12 23:56:06 EST 2000


MODELING AFRICAN AIDS


Africa has a total population of about 761 million [1].

Sub-Saharan Africa accounts for almost 70% of global HIV infections
and 83% of cumulative AIDS deaths.  Yet, it contains only 10% of
the world's population [2].

Currently, 23.3 million people are living with HIV/AIDS in
sub-Saharan Africa, 12.2 million of whom are women and 10.1
million men [2].   The majority of AIDS cases in Africa are believed
to be the product of heterosexual transmission.

I have been attempting to do computer modeling of the AIDS epidemic,
using my own software (copies available on request).

One goal is to attempt to reconcile the numbers of HIV cases in
Africa against the timeframe in which HIV has allegedly existed in
humans, and against the per-contact infectivity rates of HIV.

Another goal is to examine the ratio of male to female HIV cases.
HIV infection occurs more easily in the male-to-female direction,
than in the female-to-male direction [3].  It is therefore curious
that Africa shows a near-equal number of male/female infections.

The model is using infectivity data from a 10-year study of heterosexual
couples, where one partner was known to be HIV+.   The study analyzed
how many of the uninfected partners also become HIV+ over the 10-year
period [3].   The study lists per-contact infectivity rates as:

   male-to-female: .0009
   female-to-male: .0001125


The program allows the user to define various information such as
infectious activities, risk subgroups, average frequency of
contacts per subgroup, and number of years to model.   It also
allows for definition of time-varying infection rates.   The
program collects this information using an interactive dialog.

The program starts with a specified number of infected people.
Sexual partners are selected randomly for each contact (assuming
worst case of 100% promiscuity).  The program tracks each member
of the modeled population, and whether or not they are currently
infected.  When an uninfected person has contact with an infected
person, the program makes a decision as to whether the uninfected
person will now become infected.  This decision is made randomly,
but strictly according to the probability of infection for the
type of contact.

I attempted modeling a sample population of 100,000 people, half
men and half women, using the above infectivity rates, over a 70
year period.   The 70 year period is chosen to match the 70-year
period in which HIV is alleged to have existed in human beings,
according to the recent Los Alamos computer model that suggested
1930 as the most likely date at which HIV infected the first
human being.

The program started with a one infected male, to be consistent
with the theory that 1930 was the very first time that HIV
entered any human being.

The first peculiar result was that this yielded only about 21
infections, in the entire 70-year period (Appendix A).

At first, I simply assumed that the software had to be in error,
and started to investigate what the bugs might be.

I thought that surely the results must be wrong, because the
California study of heterosexuals [3] had produced 68 infections
in 360 women, in only a 10-year period.

When I could not see anything wrong in the software, I realized
that there was something significantly different about the
Northern California study.  This started with 360 infected men,
not a single infected man.

I attempted to use the program to duplicate the results of the
California study, and found that it did this very well (Appendix
B).  Starting with 360 infected men, the program reported 73 new
infections among the women- very close to the real results, even
with a conservative assumption for numbers of sexual contacts.

Thus, I realized that the program was making perfect sense.  It
was the epidemic itself that wasn't making sense.  Either the
figures from the 10-year study was seriously wrong, or something
else happened in Africa to accelerate the spread of HIV, in a
drastic way.  For example, something such as a sudden, mass
infection by way of contaminated vaccines.

Those 21 infected Africans would have a lot of explaining to do
to the rest of 23.3 million HIV-infected people in sub-Saharan Africa
(statistician gallows-humor for grim situations).

I was also trying to look at the female-male ratio of infections.
If the male-to-female infectivity is 8 times higher, why should
Africa be showing near-equal numbers of infections, again except
for possibilities of other, equal-opportunity mass infections?

The program was showing a female-to-male ratio of about 3:1 after
70 years.   Again, this seemed odd, and I started looking for
program bugs.  Why wasn't the ratio more like 8:1?

I thought that the sample of 21 infections wasn't enough, so I
ran more simulations.

Appendix C shows excerpts of a 450-year simulation, which
yielded about 22,000 total infections.

Over a number of such trials, I found an average female-male ratio
of about 2.8, which seemed to make no sense.   Yet, this value
seemed very consistent, with only a small amount of variability.

I also found that the ratio of 2.8 did not change for
different absolute values of infectivity, so long as the
infectivity ratio was the same.  E.e. infectivity values of
.8 and .1 yielded the same female/male ratio as .0009/.0001125.

At first I imagined that the number of female infections should
grow indefinitely, compared to the males.  Then, I realized that
this wasn't the case.   As the pool infected females grows, their
collective ability to infect a male grows, until it reaches a
point of balance.

Then, I imagined that an equilibrium should be reached around a
ratio of 8:1, where females would have an equal chance of
infecting a new male, as the males would have chance for infecting
a new female.

However, this doesn't represent true equilibrium, either.  If you
have a female/male ratio of 8/1, and infect one more man and one
more woman, your ratio becomes 9/2 = 4.5, which is significantly
different from the old ratio of 8.

Then, I imagined that the ratio should keep oscillating, between
a lower bound of about 4.5, and an upper bound of 8.

However, this situation does not represent a true equilibrium,
either.   A true equilibrium would be a female/male ratio such
that the next iteration of infections would maintain the same
ratio.  Is this possible?

If "F/M" is the ratio of infected females/males in the current
round, then an equilibrium would have to satisfy the equality of

     (F/M) = ((F +.8M)/ (M + .1F))

     <or>

     F + (.1 * (F^2)) = (M * F) + (.8 * (M^2)

We are only concerned about the ratio of "F/M", not the value of
F and M, so if we let "M" be 1, then it becomes

     F + (.1 * (F^2)) = F + .8

This boils down to F = the square root of 8, or 2.83.

Working this out was a "Eureka" moment, when I realized that
the program results made sense after all, and could reveal
new things about the epidemic that were not immediately
obvious.

An 8-to-1 infectivity ratio should give us about a 3-to-1
ratio of infected females and males, far different than what
we see in Africa.

Another important thing that the software modeling shows:
This equilibrium is reached almost immediately, and shows
little deviation throughout the course of the entire epidemic.
Therefore, the significant deviation in Africa from a 3:1
ratio would not be expected as simply a random fluctuation.

There is a great deal more modeling work to do, because I see
different studies that show absurd levels of discrepancies
concerning infectivity.   I am relatively convinced that no
model of HIV is likely to make appreciable sense.    For example,
if we assume much higher heterosexual infectivity rates, then we
will have a more difficult time explaining the relative lack of
heterosexual AIDS in the West.

This doesn't mean that we can learn nothing from modeling.
The bizarre confusion yielded by sensible models is telling us
that we are are missing something basic and profound in our
understanding of AIDS.

One of the greatest likelihoods why there can be no sensible
model is that most models must assume that there is no totally
unnatural, significant factor at work, such as contaminated
vaccines.  The failure of nearly all models will serve to point
ever more to such a conclusion.


Tom Keske
Boston, Mass.


References

[1] http://www.stats.demon.nl/africa.htm
[2] http://HIVInSite.ucsf.edu/international/africa/


[3] "Heterosexual transmission of human immunodeficiency virus
     (HIV) in northern California: results from a ten-year study"

     .Am J Epidemiol 1997 Aug 15;146(4):350-7

     Padian NS, Shiboski SC, Glass SO, Vittinghoff E

     Department of Obstetrics, Gynecology and Reproductive
     Sciences, University of California, San Francisco, USA.

     To examine rates of and risk factors for heterosexual
     transmission of human immunodeficiency virus (HIV),
     the authors conducted a prospective study of infected
     individuals and their heterosexual partners who have
     been recruited since 1985. Participants were recruited
     from health care providers, research studies, and
     health departments throughout Northern California, and
     they were interviewed and examined at various study
     clinic sites. A total of 82 infected women and their
     male partners and 360 infected men and their female
     partners were enrolled. Over 90% of the couples were
     monogamous for the year prior to entry into the study; <
     3% had a current sexually transmitted disease (STD).
     The median age of participants was 34 years, and the
     majority were white. Over 3,000 couple-months of data
     were available for the follow-up study. Overall, 68
     (19%) of the 360 female partners of HIV-infected men
     (95% confidence interval (CI) 15.0-23.3%) and two (2.
     4%) of the 82 male partners of HIV-infected women (95%
     CI 0.3-8.5%) were infected. History of sexually
     transmitted diseases was most strongly associated with
     transmission. Male-to- female transmission was
     approximately eight-times more efficient than female-
     to-male transmission and male-to-female per contact
     infectivity was estimated to be 0.0009 (95% CI 0.0005-
     0.001). Over time, the authors observed increased
     condom use (p < 0.001) and no new infections.
     Infectivity for HIV through heterosexual transmission
     is low, and STDs may be the most important cofactor
     for transmission. Significant behavior change over
     time in serodiscordant couples was observed.

     PMID: 9270414, UI: 97416464



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APPENDIX A:  MODELING AFRICAN HETEROSEXUAL AIDS OVER 70 YEARS

Below is the output of the vepid.c epidemic modeling program, for
a sample population of 100000 heterosexuals, half male and half
female.  For average contacts, I am using a generous estimate of
67 per year (between 1-2 contacts per week with purely random
partners).  This is intended to reflect a "worst-case" scenario,
and is not attempting to characterize African sex patterns.  The
figure of 67 contacts is similar to published data about another
high-risk group- gay men in San Francisco.  This value is chosen so
that the program results could not be accused of simply
underestimating sexual frequencies or numbers of partners.

% vepid
  Enter Total Population Size ( <= 100000): 100000
  Enter no. of activities to model:  1
  DO ANY INFECTIVITY RATES VARY WITH TIME (y or n)? n
  Does activity #1 involve exactly 2 partners (y or n)? y

  Enter av probability of infection,
         activity #1, ACTIVE  role: .0001125
  Enter av probability of infection,
         activity #1, PASSIVE role: .0009

  DO YOU WISH TO DEFINE POPULATION RISK SUBGROUPS (y or n)? y
  ENTER NUMBER OF POPULATION RISK SUBGROUPS: 2
  Enter size of subgroup #1: 50000

  Enter activity #1, ACTIVE , average no. contacts per year
         for subgroup #1:  67
  Enter activity #1, PASSIVE, average no. contacts per year
         for subgroup #1:  0

  Enter number initially infected for subgroup #1:  1
  Enter size of subgroup #2: 50000

  Enter activity #1, ACTIVE , average no. contacts per year
         for subgroup #2:  0
  Enter activity #1, PASSIVE, average no. contacts per year
         for subgroup #2:  67

  Enter number initially infected for subgroup #2:  0
  Enter number of years to model: 70
  Enter random seed (any number between 1 and 4294967295): 2345345

  NUM ACTIVITIES: 1
  ACTIVITY #1, ONE_PARTNER
        AV prob infection, ACTIVE , 0.0001125    num_adjust,  = 0

        AV prob infection, PASSIVE, 0.0009       num_adjust,  = 0

  POP SIZE: 100000
  NUM SUBGROUPS = 2

     TOTAL FOR SUBGROUP 0 = 50000
     TOTAL INFECTED IN SUBGROUP: 1

     Contacts/yr for activity #1, ACTIVE : 67

     TOTAL FOR SUBGROUP 1 = 50000
     TOTAL INFECTED IN SUBGROUP: 0

     Contacts/yr for activity #1, PASSIVE: 67

  New infections in year #1 = 0, GRAND TOTAL = 1
  New infections in year #2 = 2, GRAND TOTAL = 3
  New infections in year #3 = 0, GRAND TOTAL = 3
  New infections in year #4 = 0, GRAND TOTAL = 3
  New infections in year #5 = 0, GRAND TOTAL = 3
  New infections in year #6 = 0, GRAND TOTAL = 3
  New infections in year #7 = 0, GRAND TOTAL = 3
  New infections in year #8 = 0, GRAND TOTAL = 3
  New infections in year #9 = 1, GRAND TOTAL = 4
  New infections in year #10 = 1, GRAND TOTAL = 5

              . . .

  New infections in year #60 = 1, GRAND TOTAL = 16
  New infections in year #61 = 1, GRAND TOTAL = 17
  New infections in year #62 = 1, GRAND TOTAL = 18
  New infections in year #63 = 0, GRAND TOTAL = 18
  New infections in year #64 = 0, GRAND TOTAL = 18
  New infections in year #65 = 0, GRAND TOTAL = 18
  New infections in year #66 = 1, GRAND TOTAL = 19
  New infections in year #67 = 0, GRAND TOTAL = 19
  New infections in year #68 = 0, GRAND TOTAL = 19
  New infections in year #69 = 2, GRAND TOTAL = 21
  New infections in year #70 = 0, GRAND TOTAL = 21

  TOTAL CONTACTS: 469000000
  TOTAL DUMMY CONTACTS, NO PARTNER: 0
  REDUNDANT INFECTIONS: 0

  Subgroup #1 infections: initial = 1, new = 4, total = 5
  Subgroup #2 infections: initial = 0, new = 16, total = 16

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APPENDIX B - DUPLICATION OF NORTHERN CALIFORNIA HETEROSEXUAL
             TRANSMISSION STUDY RESULTS [3]
             PMID: 9270414, UI: 97416464

Below, is the output of the "vepid.c" epidemic modeling software, when
trying to duplicate the results of the Padian, et al 10-year study
of heterosexual transmission.


Subgroup #1 represents 360 males (all initially infected).
Subgroup #2 represents the infected men's 360 female partners
(0 initial infections).

The abstract of the study did not list a number of sexual contacts
per year, so I made a conservative estimate of one intercourse every
other week (26 contacts per year).

The final result of the modeling after 10 years was 73 female infections
(versus 68 actual).


  % vepid
  Enter Total Population Size ( <= 100000): 720
  Enter no. of activities to model:  1
  DO ANY INFECTIVITY RATES VARY WITH TIME (y or n)? n
  Does activity #1 involve exactly 2 partners (y or n)? y

  Enter av probability of infection,
         activity #1, ACTIVE  role: .0001125
  Enter av probability of infection,
         activity #1, PASSIVE role: .0009

  DO YOU WISH TO DEFINE POPULATION RISK SUBGROUPS (y or n)? y
  ENTER NUMBER OF POPULATION RISK SUBGROUPS: 2
  Enter size of subgroup #1: 360

  Enter activity #1, ACTIVE , average no. contacts per year
         for subgroup #1:  26
  Enter activity #1, PASSIVE, average no. contacts per year
         for subgroup #1:  0

  Enter number initially infected for subgroup #1:  360
  Enter size of subgroup #2: 360

  Enter activity #1, ACTIVE , average no. contacts per year
         for subgroup #2:  0
  Enter activity #1, PASSIVE, average no. contacts per year
         for subgroup #2:  26

  Enter number initially infected for subgroup #2:  0
  Enter number of years to model: 10
  Enter random seed (any number between 1 and 4294967295): 987987987

  NUM ACTIVITIES: 1
  ACTIVITY #1, ONE_PARTNER
        AV prob infection, ACTIVE , 0.0001125    num_adjust,  = 0

        AV prob infection, PASSIVE, 0.0009       num_adjust,  = 0

  POP SIZE: 720
  NUM SUBGROUPS = 2

     TOTAL FOR SUBGROUP 0 = 360
     TOTAL INFECTED IN SUBGROUP: 360

     Contacts/yr for activity #1, ACTIVE : 26

     TOTAL FOR SUBGROUP 1 = 360
     TOTAL INFECTED IN SUBGROUP: 0

     Contacts/yr for activity #1, PASSIVE: 26

  New infections in year #1 = 13, GRAND TOTAL = 373
  New infections in year #2 = 9, GRAND TOTAL = 382
  New infections in year #3 = 8, GRAND TOTAL = 390
  New infections in year #4 = 9, GRAND TOTAL = 399
  New infections in year #5 = 7, GRAND TOTAL = 406
  New infections in year #6 = 6, GRAND TOTAL = 412
  New infections in year #7 = 6, GRAND TOTAL = 418
  New infections in year #8 = 3, GRAND TOTAL = 421
  New infections in year #9 = 7, GRAND TOTAL = 428
  New infections in year #10 = 5, GRAND TOTAL = 433
  TOTAL CONTACTS: 187200
  TOTAL DUMMY CONTACTS, NO PARTNER: 0
  REDUNDANT INFECTIONS: 12

  Subgroup #1 infections: initial = 360, new = 0, total = 360
  Subgroup #2 infections: initial = 0, new = 73, total = 73


--------------------------------------------------------------

APPENDIX C:   Female/Male HIV Infection Ratios

   Below is a table of female/male HIV infection ratios for various
   numbers of years of a modeled epidemic.   The modeled epidemic
   started with one male infection, and uses infectivity rates of
   .0009 (male-to-female), .0001125 (female-to-male)


Year   Male       Female    Ratio (female/male)
----    -------       -------   ------------------
0           1                0         0
50         3                7         2.3
100       8              19         2.3
200     46            117         2.5
400   2537         6975         2.7
450   6487       16668         2.6
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