Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Switched Positive Systems and Control of Mutation Rick Middleton and Esteban Hernandez richard.middleton@nuim.ie The Hamilton Institute The National University of Ireland, Maynooth In collaboration with: F. Blanchini, P. Colaneri, W. Huisinga, M. vonKleist August 25, 2011 Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Introduction & Motivating Problem HIV/AIDS: General Background Mathematical Model Switched Systems Theory Guaranteed Cost Control Optimal Control Computer Simulations Idealised Problem (4 state) A Less Idealised Problem Discussion & Conclusions Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions HIV/AIDS: General Background ◮ High profile disease Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions HIV/AIDS: General Background ◮ High profile disease ◮ Viral Infection that targets Immune System Cells: ◮ CD4+ T Lymphocytes: ‘T Cells’ (Blood & Tissue) ◮ Macrophages (Tissue) ◮ Dendritic Cells (Lymph) ◮ .... Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions HIV/AIDS: General Background ◮ High profile disease ◮ Viral Infection that targets Immune System Cells: ◮ CD4+ T Lymphocytes: ‘T Cells’ (Blood & Tissue) ◮ Macrophages (Tissue) ◮ Dendritic Cells (Lymph) ◮ .... ◮ Untreated, typically of the order of a decade to progress to AIDS (serious immune system malfunction) Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Integration, Transcription and Assembly Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Main drug classes and targets Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Basic Mathematical Model: Biochemical Reactions Reaction Rate Description ∅ → T s T Production of T cells T → ∅ d T T Death of T cells T + V → T ∗ r := β TV Infection of T Cells T ∗ → ∅ d T ∗ T ∗ Death of Infected Cells T ∗ → T ∗ + V pT ∗ Viral production V → ∅ d V V Viral death ˙ = s T − d T T − r T ˙ r − d T ∗ T ∗ T ∗ = pT ∗ − d V V ˙ V = Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Notes on simplified model ◮ With appropriate parameters, explains reasonably well observations of primary and asymptomatic phases of infection. Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Notes on simplified model ◮ With appropriate parameters, explains reasonably well observations of primary and asymptomatic phases of infection. ◮ Many different model extensions possible to include a variety of factors: ◮ Immune system response to infection (CTL etc.) ◮ Memory T Cells ◮ Alternate viral targets (e.g. Macrophages) ◮ Stochastic effects ◮ Different body compartments ◮ Effect of drugs - including Pharmacokinetics ◮ Viral Mutation Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Key extension 1: Macrophages Reaction Rate Description ∅ → M s M Production of Macrophages M → ∅ d M M Death of Macrophages M + V → M ∗ r := β M MV Infection of Macrophages M ∗ → ∅ d M ∗ M ∗ Death of Infected Cells M ∗ → M ∗ + V p M M ∗ Viral production Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Key extension 2: Viral Stimulation of Immune Cells T cell and Macrophage proliferation induced as body’s response to foreign object (virus). Reaction Rate Description ρ T TV V + T → V + 2 T Antigen stimulated proliferation C T + V ρ M MV V + M → V + 2 M Antigen stimulated proliferation C M + V Nonlinearity (Michelis-Menton) is important for appropriate model robustness. Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Problems with Anti Retroviral Therapy ◮ Cost, Side effects, Adherence ◮ Mutation and drug resistance: ◮ High mutation rate: probability of mutation = few % per reverse transcription Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Problems with Anti Retroviral Therapy ◮ Cost, Side effects, Adherence ◮ Mutation and drug resistance: ◮ High mutation rate: probability of mutation = few % per reverse transcription ◮ For mono-therapy, resistant mutations emerge and dominate within weeks (hence ART is always combination therapy: 3,4 or more drugs) Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Problems with Anti Retroviral Therapy ◮ Cost, Side effects, Adherence ◮ Mutation and drug resistance: ◮ High mutation rate: probability of mutation = few % per reverse transcription ◮ For mono-therapy, resistant mutations emerge and dominate within weeks (hence ART is always combination therapy: 3,4 or more drugs) ◮ Even with combination therapy, ART may fail. e.g. Sungkanuparph et al, HIV Medicine (2006): within 6 years or so, more than 40% of patients will have experienced ‘virological failure’ (Viral load returns to similar levels to that without ART). Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Mutation Model - extension of (Nowak & May 2000) m viral strains, V i , T ∗ i , and M ∗ i , i = 1 , 2 , . . . m . Reaction Rate Description T + V i → T ∗ r i := β i TV i Infection of T Cells i M + V i → M ∗ r Mi := β Mi MV i Infection of macrophages i T ∗ i → T ∗ p i T ∗ i + V i Viral production (T) i M ∗ i → M ∗ p MI M ∗ i + V i Viral production (M) i T + V i → T ∗ r ji := µ m ji β i TV i Viral mutation j M + V i → M ∗ r Mji := µ m ji β Mi MV i Viral mutation j Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating Problem Switched Systems Theory HIV/AIDS: General Background Computer Simulations Mathematical Model Discussion & Conclusions Simplified Mutation Model During therapy, pre-virological failure, assume: Constant T-cell, macrophage, CTL etc. counts. x ( t ) = A σ ( t ) x ( t ) ˙ ◮ x i : i = 1 ... m concentration of viral strain i where ◮ σ ( t ) ∈ { 1 , 2 , . . . , N } is drug therapy at time t ◮ A σ ( t ) = blockdiag { A i ,σ ( t ) } + µ M Hamilton Institute Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
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