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Modelling the interactions between HIVand the immune system in hmans

R. Ouifki and D. Mbabazi

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Introduction

Models including drug therapy and intracellular delays have been developed to understand the dynamics of HIV-1 infection and estimate the kinetic parameters.

We present three types of models of HIV dynamics: Basic modelBasic model with RTIBasic model with PIBasic model with HAART

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I.1 HIV-1 Disease Progression

The pattern of disease progression in HIV infection is divided into three stages:

1. Primary InfectionHIV moves to lymphoid tissue andviral reservoirs

2. Asymptomatic StageVirus continues to replicate and

CD4+ cell numbers decline.

3. AIDSCD4+ cells fall below 200micro litre and opportunistic

infections begin to appear

1. 2. 3.

I. HIV dynamics without treatment

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Infection rate

kTV

c d

clearance death death

T*

Virus Target cell Infected cell

proliferation from other sources

virions/day

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I.2 Model of viral infection

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T: T-cells, T*: Infected T-cells, V : Virus

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I.3. Basic Model

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I.4 Dynamics of T-cells without HIV

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What happens after infection with HIV?

In the absence of HIV, the population of T-cells stabilises at the value

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The equilibrium points are obtained by determining constant

solutions of the system. That is finding

I.5 Equilibrium points and Stability

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This implies that

From the last equation, we obtain

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Bifurcation Diagram

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Viral free steady state Unstable.

Infected steady state stableViral free steady state

stable

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1. The model fits well the first two stages of the disease progression,

BUT not the AIDS stage. This is because the model always has a stable

equilibrium point (Disease free or infected).

2. To eradicate HIV from the body all we need to do is to bring

bellow one . For this one can decrease either

• k (Treatment with RTI)

• N (Treatment with PI)

• Or both (HAART).

What did we learn from our analysis of the basic model?

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II. Basic Model With treatment

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HIVProteins synthesis

And packaging

T Cell

New virus Mature VirusReverse

transcription

RNA DNA

Protease Inhibiors work here

Reverse transcriptaseInhibiors work here

A graphic of HIV life cycle

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Model with RTI Treatment:

is the efficacy of RTI

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Steady states:

The viral free steady state

The infected steady state

The basic reproductive rate:

Basic model with lower infection rate

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Model with PI Treatment:

is the efficacy of PI

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Steady states:

The viral free steady state

The infected steady state

The basic reproductive rate:

The first three equations correspond to a basic basic model with lower viral production number

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RTI and PI Treatment

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Steady states:

The viral free steady state

The infected steady state

The basic reproductive rate:

The first threes equations correspond to a basic model with lower infection rate and lower viral production number

Where is the combined efficacy,

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The viral free steady state is locally asymptotically stable.

The viral free steady state becomes unstable and the infected steady exists and is locally asymptotically stable.

Stability (RTI, PI or Combined therapy)

where X can be RTI, PI or c

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Parameter estimations (Perelson et al. (1996))

Experimental data were collected from five infected patients whose base- line values of measurements taken at days -7, -4, -1 and 0. Ritonavir was administered (600mg twice a day). After treatment HIV-1 RNA concentrations in plasma was measured (every 2 hours until the sixth hour, every 6 hours until day 2 and every day until day 7).

The basic model with PI treatment was used to estimate the kinetic parameters.To simplify it was supposed that, before the treatment, the system was at the infected steady state equilibrium, then

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• The infected cells remain at their steady state value

• The treatment is 100% effective.

We obtain The following expression for V

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Using nonlinear regression analysis the parameters were estimated by fitting the formula for V to the plasma HIV-1 RNA measurements.

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