The AIDS Epidemic - Arizona State Universitykuang/class/AIDS.ppt · PPT file · Web viewThe AIDS...
Transcript of The AIDS Epidemic - Arizona State Universitykuang/class/AIDS.ppt · PPT file · Web viewThe AIDS...
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).
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
IINS
I
INS
SaNS
NcN
0
Where the contact rate depends on the population size:
What does our S’ model mean?
INS
SaNS
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?
IINS
I
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 xy
xtItN
)())(( 00
c
xe a
Highly Contagious Diseases can “Control” a Population
Case 1 taeN
I
~
0Case 2 0
,
NI
IN
CNI
IN
,
Case 3 Case 4
NNII
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.
ratedeathTdsourcesotherandThymuss
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 StatesTo determine the stability of our steady states, we use the Jacobian matrix:
cNkTkVkTkVTf
0
0)(
Our Trivial Steady State
cNTkTkTf
00
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
cNTkVkTkVkTf
0
~~~0~)~(
We evaluate our steady state at )~,~,~( * VTT
Through this analysis it is difficult to determine stability.