Professor Conducts AIDS Research
For the past six years, Professor Fabio Milner has been
conducting AIDS related research. Milner's research on AIDS
modeling has been oriented in two disjoint directions. First, in
collaboration with an Italian team of mathematicians, computer
scientists, physicians, and epidemiologists, deterministic
mathematical models were designed to simulate the HIV/AIDS
epidemic among intravenous drug users in Latium, a region of
Rome, Italy. Secondly, in collaboration with a psychologist,
(formerly a Purdue professor) from the Centers for Disease
Control and Prevention (CDC) in Atlanta, research was done to
help understand the psychological motivations college students
may have that lead them from various social situations to sexual
behavior that is at high risk for the dissemination of sexually
transmitted diseases (STDs). A description of the first project
follows.
The first model for propagation of HIV/AIDS among
intravenous drug users was based on a fairly standard model of
epidemics of SIR type due to Kermack and McKendrick, which
structures the population into three classes: susceptibles, who
may contract the disease, infecteds who may transmit the disease
to susceptibles by direct contact, and removeds who do not
transmit or contract the disease and, consequently, do not
participate in the dynamics of the epidemic. For the HIV/AIDS
model, susceptibles are individuals who test negative to the
human immunodeficiency virus HIV, infecteds are those who
test positive, and removeds are those who have developed
full-blown AIDS. The model consists of an initial-boundary
value problem for a system of three differential equations, one
modeling the dynamics of each of the three epidemic classes.
The equations that describe the dynamics of susceptibles and
removeds are ordinary, since there does not seem to be a great
distinction in behavior among those individuals within each of
these two groups. The rate of change of the number of
susceptibles is assumed to consist of three terms: first, a mortality and
removal term assumed to be proportional to the size of the group
(this is the rate at which susceptibles leave the population);
second, a recruitment or immigration term assumed to be equal to
the product of the initial size of the population and the mortality
rate, so that the total population size will stay constant; third, a
removal term due to contagion which represents the rate of passage
from the class of susceptibles to that of infecteds. The rate
of change of the number of removeds is assumed to consist of
two terms: first, a mortality term assumed proportional to the
size of the group and, secondly, a recruitment term representing
the rate of passage from the class of infecteds to that of
removeds. Infecteds are structured by age of infection, that is,
the time elapsed since they became infected with HIV. This is
an essential feature in the dynamics of this particular epidemic,
since infectiousness as well as transition rate from seropositivity
to AIDS are strongly dependent on this variable. Therefore, the
equation modeling the dynamics of infecteds is a partial
differential equation for which the "boundary" condition
corresponding to age of infection equal to zero represents the
transition from the class of susceptibles to that of infecteds; and
finally, a removal term representing the transition from the class
of infecteds to that of removeds.
The dependence of infectiousness and transition rate of
passage into the AIDS class were taken from the literature. The
former has been measured mostly in terms of the level of blood
antibodies in infected individuals, the latter from some groups of
individuals for whom regular blood samples were available over
many years, thus making it possible to determine with some
accuracy when they became infected and to follow them up to
see when they developed AIDS. The mortality rate was assumed
to be the same in all three epidemic classes. This was done to
make the mathematical model tractable but is clearly unrealistic.
However, the mortality rate in the two classes which are
responsible for the dynamics of the epidemic, susceptibles and
infecteds, is indeed almost the same, while that for individuals
with full-blown AIDS is much larger but does not affect the
propagation of the disease as these individuals do not interact
with susceptibles since they are mostly confined to hospital
beds. The assumption that the population size is constant is a
severe one. In fact, little is known concerning this population,
since intravenous drug users usually do not offer information
about their drug use habits, so most of the knowledge about
them is tentative at best and frequently nonexistent. The contact
rate between susceptibles and infecteds, which is a very
important parameter in the model, can only be guessed. The
same is true of the number of infected individuals present
"initially" and of what is meant by this word. That is, when the
first HIV infected individuals appeared in this population and
how many there were are unknown quantities needed in the
model.
These three quantities were estimated as an inverse problem
designed to give the best fit of the few reliable data available,
which were the total numbers of new AIDS cases for several
years. The values thus obtained were one infected individual
initially present in 1979, a mean contact rate estimated at 53,
which amounts to about one new needle sharing partner per
week. This value may seem high but the reason is that the
model only uses the product of this number and the
infectiousness, and so the contact rate may actually be 5.3, for
example, with an infectiousness equal to ten times the level of
blood antibodies. This said, the resulting parameters used seem
quite plausible and allowed the use of the model for the two
main purposes of sensitivity analysis and prediction. The
predicted numbers of new AIDS cases for the two years
following the first study were almost perfect. The sensitivity
analysis showed little sensitivity to some characteristics of the
infectiousness and transition rates to AIDS, but a strong one to
contact rate.
A more complex model structuring susceptibles and
infecteds by activity level was also described, in which
individuals with a high contact rate were separated from those
with a low contact rate. One of the main differences with the
simpler model was that the complex one has multiple non-trivial
steady states, which would allow the epidemic to stabilize at
different endemic levels depending on initial conditions. This
underlines the importance of being able to better establish what
the initial conditions and parameters of the model are in reality,
which will necessitate further demographic and sociological
studies of the population involved.
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