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Immune Response Models
The latently infected cell model (3.1)-(3.4) that was presented in section 3.1 does not explicitly model the immune response to the virus. Instead the effects of the said immune response are incorporated into relevant parameters. In particular, the rates at which CD8+ T cells kill infected cells and antibodies kill the virus are incorporated into the death rate constants δl and δa of the infected cells, and the clearance rate constant c of the virus. Parameters δl, δa and c therefore, “collectively reflect the immune system’s defensive strength against HIV infection” [80]. Similarly, parameters βT , and rT “collectively reflect HIVs offensive strength”[80] against the immune system.
The immune response to the virus can be explicitly modelled as illustrated by equations (3.22) – (3.26). The said immune response specifically focuses on the CD8+ cytotoxic T lymphocytes – CTL. The rationale behind this model is that HIV infects immune cells which are needed in the expansion of a CTL response against infections. As a result, the ability of these cells to deliver help is compromised. This model is a variation of the one presented in [87].
State variables I, P and E represent the helper-independent CTL response (CTLi), the helper-dependent CTL precursor response (CTLp) and the helper-dependent CTL effector response (CTLe), respectively. The help referred to is the uninfected CD4+ T cells, whose responsibility it is to coordinate the immune response. The CTLi proliferates with a rate constant ρI , while the CTLp proliferates with a rate constant ρP and differentiates to CTLe in the presence of infected cells, with a rate constant kE. Parameters δI , δP , δE are the respective death rate constants. The free virus particle dynamics are not explicitly modelled, as the assumption is that the said viral load correlates with the infected CD4+ T cells, T¤. Some authors explicitly model the virus but only model the effector CTLe response to the virus [70]
The Chronically Infected Cell Model
As pointed out in section 3.1, it has been observed from individuals on antiretroviral therapy that there are several distinct phases in the decay characteristics of the viral load [162]. There is a first initial rapid decline which has been associated with the clearance rate of the virus particles in plasma. The next phase, referred to as the first observable phase, has been attributed to the decay or decline of the actively infected cells. The later phase, the second observable phase, could however, be attributed to a variety of reasons: It could be due to the decline or decay of the latently infected cells or there could be a subset of the actively infected cells that has a much slower death rate. This subset of actively infected cells is believed to produce smaller amounts of virus particles over a longer period of time, and these cells are referred to as chronically infected. Equations (3.34)-(3.37) are a representation of the chronically infected cell model and similar to that as presented by [71, 162].
The Extended Model
It is apparent that there are other cells in the body besides CD4+ T cells that are as susceptible to the virus. The release of the virus from these cells and other infected compartments has been shown to affect the virus kinetics in plasma [113]. So while the chronically infected cell model fits the patient data, it is not the only reasonable biological model. The second observable phase of viral decay could also be linked to virus released from infected macrophages [162]. Macrophages live longer than the CD4+ T cells and are chronic virus producers. An upgrade of the latently infected model is the single compartment 6 dimensional (6D) model given by equations (3.38)-(3.43), and models the target cells co-circulating in plasma [85].
1 Introduction
1.1 Motivation
1.2 HIV/AIDS Therapy: A Control Engineering Problem
1.3 Thesis Objectives and Scope
1.4 Contribution
1.5 Organization of Thesis
2 Background
2.1 HIV and the Immune System
2.2 Drugs Used to Treat HIV Infection
2.3 Guidelines on the Use of Antiretroviral Agents
2.4 Treatment Interruption
2.5 Chapter Summary
3 HIV/AIDS Models
3.1 The Latently Infected Cell Model
3.2 Time Delay Models
3.3 Immune Response Models
3.4 The Chronically Infected Cell Model
3.5 The Extended Model
3.6 The External Virus Source Model
3.7 The Composite Long Lived Cell Model
3.8 Stochastic Models
3.9 Models Adopted in this Thesis
3.10 Model Parameters Affected by Therapy
3.11 Chapter Summary
4 Model Analysis
4.1 Steady State Analysis
4.2 Transient Response Analysis
4.3 Interruption of Highly Active Antiretroviral Therapy
4.4 Controllability Analysis
4.5 Identifiability Analysis
4.6 Model Reduction
4.7 Chapter Summary
5 Drug Dosage Design
5.1 The Dynamical System to be Controlled
5.2 Modelling Antiretroviral Drugs As Control Inputs
5.3 Prioritization of Objectives of Therapy
5.4 Model Predictive Control
5.5 Sampling
5.6 A Sequential Perturbation Approach to Dosage Design
5.7 Interruptible Drug Dosage Design
5.8 Chapter Summary
6 Conclusions and Future Work
6.1 Summary
6.2 Conclusions
6.3 Recommendations and Future Work
A Parameter Estimates
