The AIDS Epidemic
Presented by
Jay Wopperer
HIV/AIDS-- Public Enemy #1?
What causes AIDS?
• Intravenous Drug Use.
• Homosexual activity.
• Heterosexual activity.
• Blood Transfusions
• Work related fluid exchange contact, (i.e. doctors, nurses, etc..)
Statistic on AIDS
• 40 million adults and 2.7 million children were living
with HIV at the end of 2001. • 3 million people had died from AIDS or AIDS related
diseases in 2001• 1.2% of the overall world population has HIV or
AIDS (8.6% in Africa, .6% in US)• In the US of the infected population 79% are men,
21% are women.• Average age range 30-34.• NYC has the most people suffering from AIDS (over
120,000).
The Cell Structure of AIDS
AIDS and Ethnicity
HIV/AIDS -The Graphical Model
The Infection of HIV in the Body
0
500
1000
1500
Time in Months
Cells
HIV antibodies
HIV/AIDS
CD4 T-Cells
AIDS-- The Mathematical Model
• We can think of AIDS as an S => I Model. In other words once one is infected, a person remains so until death.
INaN
IIN
SI
IN
SSaNS
Nc
N
0
Where the contact rate depends on the population size:
What does our N’ model mean?
INaN
As always N(t) = S(t) + I(t)
What does our S’ model mean?
IN
SSaNS
S is the number of people susceptible to AIDSN is our total populationI is the number of people infected
is our death rate, a is the birth rate
What does our I’ model mean?
IIN
SI
Disease carried death.
In other words we have the infected rate minus those that are going to die off.
Death rate for infectives so measures theincrease in the death rate attributed to disease.
Mean Life Span of an Infective
• If there is no disease (I = 0), so N` = (a - )N
• Mean Life Span of an Infective (MLI) =
• A single infective in an infinite population of susceptibles creates:
1
00R
Analysis of R0:• If R0>1
– This implies that very few introduces infectives will grow by a factor of R0 every MLI period.
– If this is our case, then the population, N, will dilute our infectives, I. In other words: I/N -> 0.
– So our I class can be rejected, therefore populations grows exponentially.
– In fact, it grows by a factor of:
aa
ea
1
Continued
1
a
We can make the claim provided:
Thus, we expect I/N to grow or decline by a factor:
00
1 Ra
R
over an MLI period.
Another important parameter combination is the number ofoffspring an infective has over its MLI:
aP0
Results
• Case 1: R0 < 1– Disease is weakly contagious or highly pathogenic.
– N (t) => infinity, I (t) => 0
– Disease has no effect.
• Case 2: R1 < 1 < R0
– N (t) => infinity, I (t) => infinity, but I (t) / N (t) => 0
Results (Continued)
• Case 3: 1 < R1
– P0 > 1, P0 = 1
– P0 < 1 and R1 <
– Disease contagiousness dominates both pathogenicity and host birth rate.
– N (t) => infinity and I (t) => infinity. We see that N (t) grows exponentially but at a much slower rate than cases 1 and 2.
– I (t) / N (t) =>
aa )(
0
0
Results (Continued)
• Case 4: P0 < 1 and R0 > – Disease is highly contagious but births from
infectives are insufficient for exponential growth.– Here xe = ye = – The disease ‘stabilizes the population size and
becomes endemic.
• Results we obtained by papers written by H. Thieme, O Diekmann and M. Kretzschmar
),1
())(),((e
e
e x
y
xtItN
)(
))(( 00
c
xe a
Highly Contagious Diseases can “Control” a Population
Case 1 taeN
I
~
0
Case 2 0
,
N
I
IN
CN
I
IN
,
Case 3 Case 4
NN
II
a
ao
o
)()(0
a
0
Virus Dynamics
• The effect of AIDS on CD4+ T cells
• HIV ‘docks’ on the CD4 receptor of the T cells.
• The new model is very similar to the SEIR model.
ratedeathTd
sourcesotherandThymuss
TdT
TpTsTf
cVTNV
TKVTT
kVTTfT
p
pMAX
)1()(
)(
*
**
Steady States
• We obtain two steady states for our model:
)~
,~
,~
(
)0,0,(
* VTT
T Our trivial steady state.
Our nontrivial steady state.
Analysis of Our Steady States
To determine the stability of our steady states, we use the Jacobian matrix:
cN
kTkV
kTkVTf
0
0)(
Our Trivial Steady State
cN
Tk
TkTf
0
0
0)(
We evaluate our steady state at (T,0,0):
We see the trace < 0
We see the det =
)1( 0RcTNkc
If R0 < 1, det > 0 therefore stable
If R0 > 1, det < 0 therefore unstable
Our Nontrivial Steady State
cN
TkVk
TkVkTf
0
~~
~0
~)
~(
We evaluate our steady state at )~
,~
,~
( * VTT
Through this analysis it is difficult to determine stability.
Perilson
Mathematician for claims the stability of our model will look like this
R0
S
S
U
Conclusion
• These models are all very new, some as recent as last year. Therefore AIDS research is still in its primitive stages.
• The complexity of AIDS makes it difficult to hone in on one specific problem, thus making it very difficult to model and predict.