The Modeling of the HIV Virus. Group Members Peter Phivilay Eric Siegel Seabass With help from Joe Geddes.

The Modeling of the HIV VirusThe Modeling of the HIV Virus

Group MembersGroup Members

Peter PhivilayEric Siegel

Seabass <|||><With help from Joe Geddes

GoalsGoalsAccurately implement the current modelsModify 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 EquationsOriginal System of Equations dTp/dt = CLTL(t) – CPTP(t)

dTlp/dt = CLTl

L(t) – CPTlP(t)

dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t)

dTlL/dt = pkTL(t) – CLTl

L(t) + CPTlP(t) – ųlTl

L(t) – slTlL(t) + siTi

L(t)

dTaL/dt = rkTL(t) – ųaTa

L(t)

dTiL/dt = qkTL(t) – ųiTi

L(t) + slTlL(t) – siTi

L(t)

ModificationsModifications dTp/dt = CLTL(t) – CPTP(t) +

s*(1-(Tp(t)+Tlp (t)+TL (t)+Tl

L (t)+TaL (t)+Ti

L (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 ModificationsFuture Modifications

dTL/dt = CPTP(t) – CLTL(t) – kV(t)TL(t) + ųaTaL(t) –

muU*Tp(t)

dTaL/dt = rkV(t)TL(t) – ųaTa

L(t)

dTlL/dt = pkV(t)TL(t) – CLTl

L(t) + CPTlP(t) – ųlTl

L(t) – slTlL(t) + siTi

L(t)

dTiL/dt = qkV(t)TL(t) – ųiTi

L(t) + slTlL(t) – siTi

L(t)

dS/dt = un*(q*k*V(t)*Tl(t) + Sl * Tll(t))

Uninfected blood CD4+ cells over 10 Uninfected blood CD4+ cells over 10 yearsyears

Before After

Incorrect display of uninfected Incorrect display of uninfected T cellsT 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 Latently infected CD4+ cells in blood over 10 yearsover 10 years

Before After

Uninfected CD4+ cells in lymph over Uninfected CD4+ cells in lymph over 10 years10 years

Before After

Latently (red), abortively (green), and Latently (red), abortively (green), and actively (yellow) infected CD4+ cells in the actively (yellow) infected CD4+ cells in the

lymph over 10 yearslymph over 10 years

Before After

Incorrect Model of Viral loadIncorrect Model of Viral loaddTp/dt = CLTL(t) – CPTP(t)

Incorrect Model of Viral loadIncorrect Model of Viral loadThe effect without mutations

Viral Load over 1 yearViral Load over 1 year(in powers of 10)(in powers of 10)

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

Number of Virus Strains over Number of Virus Strains over 10 years10 years

DifficultiesDifficulties

Maple becomes slow and unreliable as the system increases in complexity

Solution?Solution?

Don’t use Maple!

Switched the project to PythonSimplerFasterLacks built-in plotting routines

Wrote data to file and opened in ExcelSwitched project to a faster computer

Dual-processor machine running Linux

More DifficultiesMore Difficulties

Finding values for parametersFirst resource:

InternetPapersJournal Articles

Second resource:Try different values and compare output to

expected

AnalysisAnalysis

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 GoalsFuture 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

ReferencesReferencesKirschner, 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.

AcknowledgmentsAcknowledgments

Joe Geddes for his help on the computers and strokes of brilliance

Prof. Najib Nandi for the account on the Linux machine