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  dT p /dt = C L T L (t) – C P T P (t)  dT l p /dt = C L T l L (t) – C P T l P (t)  dT L /dt = C P T P (t) – C L T L (t) – kT L (t) + ų a T a L (t)  dT l L /dt = pkT L (t) – C L T l L (t) + C P T l P (t) – ų l T l L (t) – s l T l L (t) + s i T i L (t)  dT a L /dt = rkT L (t) – ų a T a L (t)  dT i L /dt = qkT L (t) – ų i T i L (t) + s l T l L (t) – s i T i L (t)

Modifications  dT p /dt = C L T L (t) – C P T P (t) + s*(1-( T p (t)+ T l p (t)+ T L (t)+ T l L (t)+ T a L (t)+ T i L (t))/Smax) - ų u *Tp(t)  dT L /dt = C P T P (t) – C L T L (t) – kT L (t) + ų a T a L (t) – ų u * T L (t)  dV/dt = bTil(t) - cV(t) - KR(t)  dS/dt = un*(q*k* T L (t) + S l * T l L (t))  dR/dt = [g* V(t) * R(t) * (1- R(t) / Rmax)]/ floor S(t)

Future Modifications  dT L /dt = C P T P (t) – C L T L (t) – kV(t)T L (t) + ų a T a L (t) – muU*Tp(t)  dT a L /dt = rkV(t)T L (t) – ų a T a L (t)  dT l L /dt = pkV(t)T L (t) – C L T l L (t) + C P T l P (t) – ų l T l L (t) – s l T l L (t) + s i T i L (t)  dT i L /dt = qkV(t)T L (t) – ų i T i L (t) + s l T l L (t) – s i T i L (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 dT p /dt = C L T L (t) – C P T P (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 Kirschner,D. Webb, GF. A Mathematical Model of Combined Drug Therapy of HIV Infection. Journal of Theoretical Medicine Perelson, A. Nelson, P. Mathematical Analysis of HIV-1 Dynamics in Vivo.. SIAM Review 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