Transcript Document

WE’RE
BACK...
Altering HIV
Treatment to Maximize
Life Span
Denise Tabasco
Emma Benn
Rachel Epstein
Audrey Hucks
The Big Question
When is the best time to
treat HIV so that the
infected person will have
a maximum lifespan?
Two Key Issues of
Treatment Strategy
Time of Initiation of
Treatment
Time at which drug
resistant virus appears
Methods
 We need to learn about and research the various
treatments for HIV
– how each works
– effects of each
– known information about resistance to each
– studies done on each
 We’ll use the model to experiment with different values
and times of initiation of We need to learn the basics of
HIV/AIDS
 treatment
 We’ll find studies/experiments done which illustrate
different treatment initiation
Equipment
 Maple/Java
• run programs
• graph
• experiment
• use as visual displays of information
 Netscape ( to search for information)
general information on HIV
 treatments
studies
Medscape



MODEL WITHOUT
TREATMENT

1. T ' (t )  S (t )  TT (t )  pT (t )V (t )  k V (t )T (t )
s s

2. T ' (t )  k V (t )T (t )   T (t )  piTs (t )Vs (t )
s
s s
T s
S
C  Vs (t )
i

3. V ' s(t )  NpiTs(t )
Ci  Vs (t ) 
Vs (t )
GsVs (t )
 kvT (t )Vs (t ) 
Ci  Vs(t )
B  Vs(t )
Model With Treatment


4. T ' (t )  SO (t )  T T (t )  pT (t )V (t )  kSV (t )  krVr (t )T (t )
C  V (t )
5.
piTs (t )V (t )
T 's (t )  k sVs (t )T (t )  Ti Ts (t ) 
Ci  V (t )
p
T
(
t
)
V
(
t
)
i
r
 6. T 'r (t )  k rVr (t )T (t )  T Tr (t ) 
i
Ci  V (t )

7.
pq (t ) Npi Ts (t )V (t ) 1  q (t ) Npi Tr (t )V (t )
V ' s (t ) 

C i  V (t )
C i  V (t )
G sVs (t )
 k v T (t )Vs (t )  
B  V (t )

8. V '
q(t ) Npi Tr (t )V (t ) 1  q(t )Npi Ts (t )V (t )
GrVr (t )

 k vT (t ) 
r (t ) 
Ci  V (t )
Ci  V (t )
B  V (t )
OH NO!!!
There’s a problem with
the Kirschner/Webb
model!!!
Doh!
When we reproduced the
graphs. . .
Without Treatment
Without Treatment
15
1000
virus
T ce lls
1500
500
0
10
5
0
0
50
100
weeks
150
0
50
100
w eeks
150
0.01 m_T
0.5 m_Ti
0.01 k_V
0.00025 k_s
0.00025 k_r
0.5 p
0.5 pi
10 N
188 C
8B
330 Gs
328.35 Gr
55 B_s
0.1 delta t
0.999 q
0.9 mu
0.15 eta
0.05 rho
1/day
1/day
mm^3/day
mm^3/day
mm^3/day
mm^3/day
mm^3/day
rate that tcells kill virus
rate that susceptible virus infects tcells
rate that resistant virus infects tcells
proliferation rate of Tcells
proliferation rate of infected Tcells
number of virus on lysis
1/mm^3 half saturation constant for tcells
mm^3
half saturation constant for virus input
mm^3/day rate of input of external sensitive virus to serum
mm^3/day rate of input of external resistant virus to serum
mm^3
half saturation constant for Tcell source
probability of mutation after resistant strain appears
treatment parameter
treatment parameter
treatment parameter
Here’s what worked:
Apparently, there was a typo in one of
the most sensitive parameters, p (the
proliferation rate of uninfected CD4+
T cells) so rather than p equaling 0.5,
it should be 0.05
6.
piTr (t )V (t )
T 'r (t )  k rVr (t )T (t )  Ti Tr (t ) 
Ci  V (t )
7.
pq(t ) NpiTs (t )V (t ) 1  q(t ) NpiTr (t )V (t )
V 's (t ) 

Ci  V (t )
Ci  V (t )
GsVs (t )
 kvT (t )Vs (t )  
B  V (t )
Nowak Model
dVi
 Vi  A r  sz  pX i 
dt
dX i
X

Z


( kVi  uVX i ) 1 

dt
L 

dZ
( kV  uVZ) 1  X  Z 
dt
L 

References
Herod, James V., Shonkwiler, Ronald W., and

Yeargers, Edward K. An Introduction to the
Mathematics of Biology: with computer algebra models
Birkhauser Boston, Boston, 1996.
HIV and AIDS: Martin A. Nowak and Andrew J.
McMichael, “How HIV Defeats the Immune System,” in
Scientific American, Vol. 273, No. 2, page 58, August
1995.
HIV and T cells: A. S. Perelson, D. E. Kirschner and R.
J. De Boer, “The Dynamics of HIV infection of CD4+ T
Cells,” Math. Biosci. 114, pp. 81-125, 1993.
Kirschner, D. E. and Perelson, A. S. “A Model for the
Immune System Response to HIV: AZT Treatment
Studies.”
Kirschner, D. E. and Webb, G. F. “A Mathematical
Model of Combined Drug Therapy of HIV Infection.”
Journal of Theoretical Medicine, Vol. 1, pp. 25-34,
Thanks for you attention
and...
Have a nice
day!