The AIDS Epidemic

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The AIDS Epidemic

• Presented by
• Jay Wopperer

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HIV/AIDS-- Public Enemy 1?
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What causes AIDS?

• Intravenous Drug Use.
• Homosexual activity.
• Heterosexual activity.

• Blood Transfusions
• Work related fluid exchange contact, (i.e.
  doctors, nurses, etc..)

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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).

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The Cell Structure of AIDS
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AIDS and Ethnicity
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HIV/AIDS -The Graphical Model
HIV antibodies
HIV/AIDS
CD4 T-Cells
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AIDS-- The Mathematical Model

• We can think of AIDS as an S gt I Model. In other
  words once one is infected, a person remains so
  until death.

Where the contact rate depends on the
population size
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What does our N model mean?
As always N(t) S(t) I(t)
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What does our S model mean?
S is the number of people susceptible to AIDS N
is our total population I is the number of
people infected
is our death rate, a is the birth rate
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What does our I model mean?
Disease carried death.
Death rate for infectives so a measures
the increase in the death rate attributed to
disease.
In other words we have the infected rate minus
those that are going to die off.
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Mean Life Span of an Infective

• If there is no disease (I 0), so N (a - m)N
• Mean Life Span of an Infective (MLI)
• A single infective in an infinite population of
  susceptibles creates

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Analysis of R0

• If R0gt1
• 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 -gt
  0.
• So our I class can be rejected, therefore
  populations grows exponentially.
• In fact, it grows by a factor of

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Continued
We can make the claim provided
Thus, we expect I/N to grow or decline by a
factor
over an MLI period.
Another important parameter combination is the
number of offspring an infective has over its MLI
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Results

• Case 1 R0 lt 1
• Disease is weakly contagious or highly
  pathogenic.
• N (t) gt infinity, I (t) gt 0
• Disease has no effect.
• Case 2 R1 lt 1 lt R0
• N (t) gt infinity, I (t) gt infinity, but I (t) /
  N (t) gt 0

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Results (Continued)

• Case 3 1 lt R1
• P0 gt 1, P0 1
• P0 lt 1 and R1 lt
• Disease contagiousness dominates both
  pathogenicity and host birth rate.
• N (t) gt infinity and I (t) gt infinity. We see
  that N (t) grows exponentially but at a much
  slower rate than cases 1 and 2.
• I (t) / N (t) gt

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Results (Continued)

• Case 4 P0 lt 1 and R0 gt
• 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

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Highly Contagious Diseases can Control a
Population
a a-m
Case 3
Case 4
a abo
bo a m
Case 2
Case 1
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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.

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Steady States

• We obtain two steady states for our model

Our trivial steady state. Our nontrivial
steady state.
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Analysis of Our Steady States
To determine the stability of our steady states,
we use the Jacobian matrix
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Our Trivial Steady State
We evaluate our steady state at (T,0,0)
We see the trace lt 0
We see the det
If R0 lt 1, det gt 0 therefore stable
If R0 gt 1, det lt 0 therefore unstable
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Our Nontrivial Steady State
We evaluate our steady state at
Through this analysis it is difficult to
determine stability.
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Perilson
Mathematician for claims the stability of our
model will look like this
S
R0
S
U
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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.