The Modeling of the HIV Virus
Group Members
Peter Phivilay
Eric Siegel
Seabass <|||><
With help from Joe Geddes
Goals
Accurately implement the current models
Modify existing equations to make them
more mathematically accurate and
biologically realistic
Create equations to model the viral load,
number of HIV strains, and the immune
response
Model the effects of the number of viral
strains on the progression of the virus
Original System of Equations
 dTp/dt = CLTL(t) – CPTP(t)
 dTlp/dt = CLTlL(t) – CPTlP(t)
 dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t)
 dTlL/dt = pkTL(t) – CLTlL(t) + CPTlP(t) – ųlTlL(t) – slTlL(t) + siTiL(t)
 dTaL/dt = rkTL(t) – ųaTaL(t)
 dTiL/dt = qkTL(t) – ųiTiL(t) + slTlL(t) – siTiL(t)
Modifications
 dTp/dt = CLTL(t) – CPTP(t) +
s*(1-(Tp(t)+Tlp (t)+TL (t)+TlL (t)+TaL (t)+TiL (t))/Smax) ųu*Tp(t)
 dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t) – ųu* TL(t)
 dV/dt = bTil(t) - cV(t) - KR(t)
 dS/dt = un*(q*k* TL(t) + Sl * TlL(t))
 dR/dt = [g* V(t) * R(t) * (1- R(t) / Rmax)]/ floor S(t)
Future Modifications
 dTL/dt = CPTP(t) – CLTL(t) – kV(t)TL(t) + ųaTaL(t) –
muU*Tp(t)
 dTaL/dt = rkV(t)TL(t) – ųaTaL(t)
 dTlL/dt = pkV(t)TL(t) – CLTlL(t) + CPTlP(t) – ųlTlL(t) – slTlL(t) + siTiL(t)
 dTiL/dt = qkV(t)TL(t) – ųiTiL(t) + slTlL(t) – siTiL(t)
 dS/dt = un*(q*k*V(t)*Tl(t) + Sl * Tll(t))
Uninfected blood CD4+ cells over 10
years
Before
After
Incorrect display of uninfected
T cells
The cell count does not get low enough to
induce AIDS
Uninfected CD4+ cells
in blood
Uninfected CD4+
cells in lymph
Latently infected CD4+ cells in blood
over 10 years
Before
After
Uninfected CD4+ cells in lymph over
10 years
Before
After
Latently (red), abortively (green), and
actively (yellow) infected CD4+ cells in the
lymph over 10 years
Before
After
Incorrect Model of Viral load
dTp/dt = CLTL(t) – CPTP(t)
Incorrect Model of Viral load
The effect without mutations
Viral Load over 1 year
(in powers of 10)
Viral Load over 10 years
(in powers of 10)
Number of Virus Strains over
10 years
Difficulties
Maple becomes slow and unreliable as the
system increases in complexity
Solution?
Don’t use Maple!
Switched the project to Python
Simpler
Faster
Lacks built-in plotting routines
Wrote data to file and opened in Excel
Switched project to a faster computer
Dual-processor machine running Linux
More Difficulties
Finding values for parameters
First resource:
Internet
Papers
Journal Articles
Second resource:
Try different values and compare output to
expected
Analysis
Written a biologically accurate equation for
the viral load
Modeled the effects of mutations and the
number of strains
Added terms to the model while
maintaining its purpose
Failed to display the delay before the viral
explosion
Future Goals
Correct viral load equation to delay viral
explosion
Add V(t) for infection terms rather than just
a constant
Possibly add equations to represent the
cytotoxic T-cells and macrophages.
Adjusting the parameters and equations to
explore the various treatment options
References
Kirschner, D. Webb, GF. Cloyd, M. Model of HIV-1 Disease Progression Based on
Virus- Induced Lymph Node Homing and Homing-Induced Apoptosis of CD4+
Lymphocytes. JAIDS Journal. 20000.
Kirschner,D. Webb, GF. A Mathematical Model of Combined Drug Therapy of HIV
Infection. Journal of Theoretical Medicine. 1997
Perelson, A. Nelson, P. Mathematical Analysis of HIV-1 Dynamics in Vivo.. SIAM
Review. 1999
Nowak, MA. May, MR. Anderson, RM. The Evolutionary Dynamics of HIV-1
Quasispecies and the development of immunodeficiency disease.
Acknowledgments
Joe Geddes for his help on the computers
and strokes of brilliance
Prof. Najib Nandi for the account on the
Linux machine