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.