The Modeling of the HIV Virus

1. The Modeling of the HIV Virus

2. Group Members Peter Phivilay Eric Siegel Seabass <|||>< With help from Joe Geddes

3. 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

4. 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)

5. 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)

6. 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))

7. Uninfected blood CD4+ cells over 10 years Before After

8. 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

9. Latently infected CD4+ cells in blood over 10 years Before After

10. Uninfected CD4+ cells in lymph over 10 years Before After

11. Latently (red), abortively (green), and actively (yellow) infected CD4+ cells in the lymph over 10 years Before After

12. Incorrect Model of Viral load dTp/dt = CLTL(t) – CPTP(t)

13. Incorrect Model of Viral load The effect without mutations

14. Viral Load over 1 year(in powers of 10)

15. Viral Load over 10 years (in powers of 10)

16. Number of Virus Strains over 10 years

17. Difficulties • Maple becomes slow and unreliable as the system increases in complexity

18. 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

19. More Difficulties • Finding values for parameters • First resource: • Internet • Papers • Journal Articles • Second resource: • Try different values and compare output to expected

20. 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

21. 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

22. References Kirschner, D. Webb, GF. Cloyd, M. Model of HIV-1 Disease Progression Based on Virus- Induced Lymph NodeHoming 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.

23. Acknowledgments • Joe Geddes for his help on the computers and strokes of brilliance • Prof. Najib Nandi for the account on the Linux machine