Case Studies and Project Ideas: AIDS dynamics

Source:

Mathematical Biology, J.D. Murray, Springer-Verlag Publishers, London, 1989

NOTE: This model requires above average mathematical skills AND the maturity level to handle sensitive topics!

Goal:

to build a computer model to evaluate the development of an AIDS epidemic in a homosexual population over a period of 30 years.

Assumptions and guidelines:

Constant immigration of new susceptibles every year.
Susceptibles can become infectious or (eventually) die a natural death
Infectious people can either die a natural death, develop full-blown AIDS, or become non-infectious (but still test positive for the HIV virus -- this is known as being "seropositive")
People with AIDS can die of their disease or die a natural death.

Important factors:

There is a constant immigration that comes from recruitment of new homosexuals into the population.
Susceptibles can become infectious based on the probability of acquiring the disease from a sexual partner and from the number of partners per time.
All populations can die a natural death at a constant rate
Infectious persons develop AIDS based on the rate of conversion of the HIV virus to AIDS symptoms and the proportion of seropositives who are infectious
AIDS homosexuals die of AIDS at a constant rate

Building the Model:

The mathematical equations of this model are represented below in differential equation format. Each of the four populations: susceptibles, infectious, seropositives, and AIDS patients, are the populations we wish to track over the simulation period.

Keep in mind that when you are building your model, a positive value indicates a flow coming INTO a stock, and a negative flow indicates a flow going OUT of a stock. For example, in the differential equation for susceptibles, you will need one flow coming in and two flows going out! Think about this: how does the susceptible population increase? We're assuming that their is only ONE way. How does the susceptible population decrease? There are two ways: what are they?

The variables above are as follows:

X represents those who are susceptible to getting AIDS

Y represents those who are HIV-positive (infectious, capable of giving the HIV virus)

Z represents those who are seropositive non-infectious (sexually inactive due to their condition)

A represents those with full-blown AIDS

N re represents the total population (you do NOT need to use the differential equation for this population -- you can simply total all four of the populations to ensure that you have everyone accounted for!)

recruitment rate of susceptibles

the natural (non-AIDS) death rate

the probability of acquiring infection from a randomly chosen partner

, where

is the transmission probability

c is the number of sexual partners

d is the AIDS-related death rate

p is the proportion of seropositives who are infectious

v is the rate of conversion from infection

A large study of the AIDS epidemic was conducted in the early 1980's in England, with the following data points collected:

TOTAL number of initial population	100,000
Initial number of susceptibles	90,000
Initial number of AIDS-infected persons	0
Initial number of HIV-positive persons	10,000
Initial number of seropositive-noninfectious persons	0

Recruitment rate	13,333.3 persons per year
AIDS-related death rate	1.33 per year
Rate of conversion from infection	0.237
Natural death rate	1/32 (one in 32 persons per year)
Number of sexual partners	48 per year
Probability of acquiring AIDS	30% (0.30)
Rho	5.15

NOTE: the transmission probability

is calculated from:

where:

is the transmission probability

(rho) is a constant

v is the rate of conversion

c is the number of partners per year

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