# 1 Modelling the interactions between HIV and the immune system in hmans R. Ouifki and D. Mbabazi 10/21/2015AIMS.

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1 Modelling the interactions between HIV and the immune system in hmans R. Ouifki and D. Mbabazi 10/21/2015AIMS

2 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 model Basic model with RTI Basic model with PI Basic model with HAART 10/21/2015 AIMS

3 I.1 HIV-1 Disease Progression The pattern of disease progression in HIV infection is divided into three stages: 1. Primary Infection HIV moves to lymphoid tissue and viral reservoirs 2.Asymptomatic Stage Virus continues to replicate and CD4 + cell numbers decline. 3.AIDS CD4 + cells fall below200 micro litre and opportunistic infections begin to appear 1. 2. 3. I. HIV dynamics without treatment 10/21/2015 AIMS

4 Infection rate k T V c d clearance death VirusTarget cellInfected cell proliferation from other sources virions/day 10/21/2015 AIMS I.2 Model of viral infection

5 T : T-cells, T* : Infected T-cells, V : Virus 10/21/2015 AIMS I.3. Basic Model

10/21/2015 AIMS 6 I.4 Dynamics of T-cells without HIV

10/21/2015 AIMS 7 What happens after infection with HIV? In the absence of HIV, the population of T-cells stabilises at the value

10/21/2015 AIMS 8 The equilibrium points are obtained by determining constant solutions of the system. That is finding I.5 Equilibrium points and Stability

10/21/2015 AIMS 9 This implies that From the last equation, we obtain

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12 Bifurcation Diagram 1 Viral free steady state Unstable. Infected steady state stable Viral free steady state stable 10/21/2015 AIMS

10/21/2015 AIMS 13 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?

14 II. Basic Model With treatment 10/21/2015 AIMS HIV Proteins synthesis And packaging T Cell New virus Mature Virus Reverse transcription RNADNA Protease Inhibiors work here Reverse transcriptase Inhibiors work here A graphic of HIV life cycle

15 Model with RTI Treatment : is the efficacy of RTI 10/21/2015 AIMS

16 Steady states: The viral free steady state The infected steady state The basic reproductive rate: Basic model with lower infection rate 10/21/2015 AIMS

17 Model with PI Treatment: is the efficacy of PI 10/21/2015 AIMS

18 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 10/21/2015 AIMS

19 RTI and PI Treatment 10/21/2015 AIMS

20 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, 10/21/2015 AIMS

21 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 10/21/2015 AIMS

22 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 10/21/2015 AIMS

The infected cells remain at their steady state value The treatment is 100% effective. We obtain The following expression for V 10/21/201523 AIMS

24 Using nonlinear regression analysis the parameters were estimated by fitting the formula for V to the plasma HIV-1 RNA measurements. 10/21/2015 AIMS

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