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A Model for the Study of HIV/AIDS
Brandy L. Rapatski
Juan Tolosa
Richard Stockton
College of NJ
Classic Model – Compartment Model
S
Susceptible
I
Infected
R
Recovered
Removed/Recovered Class
•Immunity (can’t infect or be infected)
•Death
•Isolation
Assumptions
 An average infective makes contact sufficient to transmit
infection with bN others per unit time. (N = total
population).
 Probability a random contact by an infective is with a
susceptible is S/N
 The number of new infections in unit time is
bN (S/N)I = bSI
 There is no entry into or departure from the population
except possibly through death from the disease.
S
dS
  bSI
dt
dI
 bSI  I
dt
dR
 I
dt
I
R
Qualitative Approach
dS
  bSI
dt
dI
 ( bS   ) I
dt
dS
 0 S(t) decreases for all time
dt
dI

 0 if S  , I(t) decreases to 0.
dt
b
dI

 0 if S  , I(t) increases to maximum then decreases.
dt
b
bS (0)
 threshold value

Basic Reproduction Number
bS (0)
R0 

The number of secondary infections caused by a
single infective introduced into a wholly susceptible
population over the course of the infection of this
single infective.
R0  1, epidemic dies out
R0  1, there is an epidemic
Progression of Infection by Stage
We investigate how infectivity varies with stage of infection.
1/6
Year
First
Stage
7 Years
Second
Stage
3 Years
Third Stage
including AIDS
Before modern medical intervention
AIDS is roughly the last year of the third stage
Death
Progression of Infection by Stage
an alternative set of durations
Infectivity varies by stage of the disease
1/6
Year
First
Stage
7 Years
Second
Stage
2 Years
AIDS & Death
Third
Stage*
Before modern medical intervention
*Here last year of AIDS is omitted due to sexual inactivity
Untreated Viral Loads
[------------------2nd Stage-------------------]
[1st Stage]
[-----3rd Stage----]
Anderson, R. M. “The spread of HIV and sexual mixing patterns,” pp. 71-86 in Mann, J. and D. Tarantola (eds.), AIDS in
the World II: Global Dimensions, Social Roots, and Responses. The Global AIDS Policy Coalition. (New York: Oxford
University Press, 1996).
Modes of HIV Transmission




Sexual Contact
IV Drug Use
Vertical Transmission (Mother to Child)
Blood Transfusion
Modeling SF Gay Population: the
Data
 HIV exploded in San Francisco between the years 1978
and 1984.
 San Francisco obtained high quality data on 6875 gay
men: Infection rates and number of sexual contacts.
 Blood samples were taken and frozen during the years
HIV was quietly breaking out – as part of a hepatitis
vaccine study.
San Francisco
San Francisco Transmission Dynamics
 Analysis of 1978-1984
 Six Activity Levels (from survey data)
 Infectiousness depends on stage (3 stages)
 Bathhouse Assumption
 Men vary in how often they visit the
bathhouses but once inside choose partners at
random.
Model Compartments
Individuals can be in any of four stages (including susceptible)
and in any of six activity groups. So the model keeps track of
fraction of people in each of the 4 x 6 subpopulations.
Simple Transmission Dynamics
dS

 bIS  new case rate
dt
S  fraction of susceptibl es, between 0 and 1
I  fraction infected  1  S
b  effective contact rate
Multiple Group Transmission Dynamics:
from Group j to group i
dSi

 bijIjSi 
dt
 rate of new group i cases caused by group j
Si  fraction of susceptibl es, between 0 and 1
Ij  fraction infected  1  Sj
bij  effective contact rate, which equals
numbers of contacts times infectious ness
Go to Six-Group Models
Six-Group Models
80
70
60
50
model
40
data
30
20
10
0
0
1
2
3
4
5
6
7
8
Current Work
 Test and Treat US MSM HIV/AIDS population
Go to Test and Treat Model
Juan’s Stuff 
Contact Information
[email protected]
[email protected]
Current MSM Epidemic in US
 MSM accounted for 71% of all HIV infections among male adults and adolescents in
2005 even though only about 5% to 7% of male adults and adolescents in the United
States identify themselves as MSM
 Through its National HIV Behavioral Surveillance system, CDC found that 25% of
the MSM surveyed in 5 large cities were infected with HIV and 48% of those infected
were unaware of their infections.
 Young black MSM in this study were more likely to be unaware of their infection –
approximately 9 of 10 young black MSM compared to 6 of 10 young white men.
 Of the men who tested positive, most (74%) had previously tested negative for HIV
infection and 59% believed that they were at low or very low risk.