Far East Journal of Theoretical Statistics
Volume 16, Issue 2, Pages 235 - 268
(July 2005)
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THRESHOLD INVESTIGATIONS FOR A NON-LINEAR STOCHASTIC PARTNERSHIP MODEL FOR THE HETEROSEXUAL SPREAD OF HIV/AIDS AMONG SENIORS
Robert J. Gallop (U. S. A.) and Charles J. Mode (U. S. A.)
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Abstract: While
HIV/AIDS has received much attention the past
decade, most educational ads and campaigns are
directed at the younger generations, whereas
HIV/AIDS among older adults is commonly called
the overlooked epidemic. Thus, little is known
about the evolution of the disease in this
population. To determine whether or not the
disease will spread rapidly with the
introduction of a few infectives, inference
from an accurate model is needed. With the
lack of substantial investigations on this
population, little data has been accrued;
therefore, simulations based on an accurate
model are required. Threshold conditions, by
definition, specify conditions in terms of the
parameter such that the disease will spread if
a small number of infected individuals are
introduced into a large susceptible
population. To acquire threshold conditions
for the spread of HIV/AIDS in the senior
sector, a discrete time stochastic framework
where semi-Markovian life cycle models for
single males, single females, and couples is
outlined based on theory of competing risks
with parameters sufficiently describing the
disease impact and population dynamics over
time. By taking conditional expectations of
the present given the past in the stochastic
process and forcing the time increment in the
stochastic process to become small, the
stochastic process will give rise to a
deterministic formulation in the form of a
system of differential equations. This
deterministic formulation is a reflection of
the population-average or central tendency of
the process. By exploiting the stability
properties of the deterministic formulation at
the infection-free equilibrium gives rise to a
threshold condition. If the system is
unstable, then the introduction of a few
infectives will result in a severe epidemic.
If the system is stable, then the tendency is
for the population to be resistant to new
infections. The semi-Markovian framework will
incorporate stochasticity in the simulations;
therefore, to provide a synopsis of the
behavior of the system, the deterministic
solution and stochastic simulations will be
displayed graphically, where the stochastic
simulations capture the intrinsic variability
in disease progression. Proximity to the
threshold condition will serve as an indicator
of need for further educational efforts among
seniors. In addition, statistical comparisons
of cumulative growth curves of the epidemic
under various parameter settings will be
conducted. |
Keywords and phrases: heterosexual, partnerships, stochastic model, embedded deterministic model, Monte Carlo simulations, threshold conditions. |
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