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Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Switched Positive Systems and Control ofMutation
Rick Middleton and Esteban Hernandez
richard.middleton@nuim.ie
The Hamilton InstituteThe National University of Ireland, Maynooth
In collaboration with: F. Blanchini, P. Colaneri, W. Huisinga, M. vonKleist
August 25, 2011
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Introduction & Motivating ProblemHIV/AIDS: General BackgroundMathematical Model
Switched Systems TheoryGuaranteed Cost ControlOptimal Control
Computer SimulationsIdealised Problem (4 state)A Less Idealised Problem
Discussion & Conclusions
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
HIV/AIDS: General Background
I High profile disease
I Viral Infection that targets Immune System Cells:I CD4+ T Lymphocytes: ‘T Cells’ (Blood & Tissue)I Macrophages (Tissue)I Dendritic Cells (Lymph)I ....
I Untreated, typically of the order of a decade to progress toAIDS (serious immune system malfunction)
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
HIV/AIDS: General Background
I High profile diseaseI Viral Infection that targets Immune System Cells:
I CD4+ T Lymphocytes: ‘T Cells’ (Blood & Tissue)I Macrophages (Tissue)I Dendritic Cells (Lymph)I ....
I Untreated, typically of the order of a decade to progress toAIDS (serious immune system malfunction)
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
HIV/AIDS: General Background
I High profile diseaseI Viral Infection that targets Immune System Cells:
I CD4+ T Lymphocytes: ‘T Cells’ (Blood & Tissue)I Macrophages (Tissue)I Dendritic Cells (Lymph)I ....
I Untreated, typically of the order of a decade to progress toAIDS (serious immune system malfunction)
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Integration, Transcription and Assembly
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Main drug classes and targets
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Basic Mathematical Model: Biochemical Reactions
Reaction Rate Description
∅ → T sT Production of T cellsT → ∅ dTT Death of T cells
T + V → T ∗ r := βTV Infection of T CellsT ∗ → ∅ dT∗T ∗ Death of Infected Cells
T ∗ → T ∗ + V pT ∗ Viral productionV → ∅ dV V Viral death
Ṫ = sT − dTT − rṪ ∗ = r − dT∗T ∗
V̇ = pT ∗ − dV V
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Notes on simplified model
I With appropriate parameters, explains reasonably wellobservations of primary and asymptomatic phases of infection.
I Many different model extensions possible to include a varietyof factors:
I Immune system response to infection (CTL etc.)I Memory T CellsI Alternate viral targets (e.g. Macrophages)I Stochastic effectsI Different body compartmentsI Effect of drugs - including PharmacokineticsI Viral Mutation
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Notes on simplified model
I With appropriate parameters, explains reasonably wellobservations of primary and asymptomatic phases of infection.
I Many different model extensions possible to include a varietyof factors:
I Immune system response to infection (CTL etc.)I Memory T CellsI Alternate viral targets (e.g. Macrophages)I Stochastic effectsI Different body compartmentsI Effect of drugs - including PharmacokineticsI Viral Mutation
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Key extension 1: Macrophages
Reaction Rate Description
∅ → M sM Production of MacrophagesM → ∅ dMM Death of Macrophages
M + V → M∗ r := βMMV Infection of MacrophagesM∗ → ∅ dM∗M∗ Death of Infected Cells
M∗ → M∗ + V pMM∗ Viral production
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Key extension 2: Viral Stimulation of Immune Cells
T cell and Macrophage proliferation induced as body’s response toforeign object (virus).
Reaction Rate Description
V + T → V + 2T ρTTVCT+V Antigen stimulated proliferationV + M → V + 2M ρMMVCM+V Antigen stimulated proliferation
Nonlinearity (Michelis-Menton) is important for appropriate modelrobustness.
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Problems with Anti Retroviral Therapy
I Cost, Side effects, AdherenceI Mutation and drug resistance:
I High mutation rate: probability of mutation = few % perreverse transcription
I For mono-therapy, resistant mutations emerge and dominatewithin weeks (hence ART is always combination therapy: 3,4or more drugs)
I 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 haveexperienced ‘virological failure’ (Viral load returns to similarlevels to that without ART).
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Problems with Anti Retroviral Therapy
I Cost, Side effects, AdherenceI Mutation and drug resistance:
I High mutation rate: probability of mutation = few % perreverse transcription
I For mono-therapy, resistant mutations emerge and dominatewithin weeks (hence ART is always combination therapy: 3,4or more drugs)
I 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 haveexperienced ‘virological failure’ (Viral load returns to similarlevels to that without ART).
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Problems with Anti Retroviral Therapy
I Cost, Side effects, AdherenceI Mutation and drug resistance:
I High mutation rate: probability of mutation = few % perreverse transcription
I For mono-therapy, resistant mutations emerge and dominatewithin weeks (hence ART is always combination therapy: 3,4or more drugs)
I 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 haveexperienced ‘virological failure’ (Viral load returns to similarlevels to that without ART).
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Mutation Model - extension of (Nowak & May 2000)
m viral strains, Vi , T∗i , and M
∗i , i = 1, 2, . . .m.
Reaction Rate Description
T + Vi → T ∗i ri := βiTVi Infection of T CellsM + Vi → M∗i rMi := βMiMVi Infection of macrophagesT ∗i → T ∗i + Vi piT ∗i Viral production (T)M∗i → M∗i + Vi pMIM∗i Viral production (M)T + Vi → T ∗j rji := µmjiβiTVi Viral mutationM + Vi → M∗j rMji := µmjiβMiMVi Viral mutation
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
HIV/AIDS: General BackgroundMathematical Model
Simplified Mutation Model
During therapy, pre-virological failure, assume:Constant T-cell, macrophage, CTL etc. counts.
ẋ(t) = Aσ(t)x(t)
whereI xi : i = 1...m concentration of viral strain i
I σ(t) ∈ {1, 2, . . . ,N} is drug therapy at time tI Aσ(t) = blockdiag{Ai ,σ(t)}+ µM
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Underlying Mathematical Problem
Equivalent positive switched discrete time system:
x(k + 1) = Φσ(k)x(k)
where
I x(k) is the state vector of all variables of interest
I Fixed treatment during intervalσ(t) = σk : ∀t ∈ (kT , (k + 1)T )
I Φσ(k) = eAσ(k)T : state transition matrix for treatment σ(k)
I σ(k) is our decision variable (drug regimen)
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Discrete Switched Systems problem
Design σ(k) as a causal function of x(k) to achieve
I Asymptotic Stability?
I Optimality?
I Guaranteed Performance?
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Sub Optimal (Guaranteed Cost) Control
Theorem (Guaranteed Cost - Finite Horizon)
Given q � 0, c � 0, suppose we can findαi (k) � 0, i = 1..N, k = 0, ..K and γ ≥ 0 such that αi (K ) = c and
Φ′iαi (k) + γ(αi (k)− αj(k)) + q � αi (k − 1)
then the treatment selection σ(k) = argmini∈{1,..N} {α′i (k)x(k)}ensures
K−1∑k=0
q′x(k) + c ′x(K ) ≤ mini{α′i (0)x(0)}
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Proof Outline - Guaranteed Cost Control
(Proof outline).
Define Lyapunov function:
V (k) = mini∈1,..N
{α′i (k)x(k)}
satisfiesV (k + 1) < V (k)− q′x(k) ∀x(k) � 0
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Comments
I Search for class of polytopic Lyapunov functions: Line searchover convex problems
I Guaranteed cost (upper bound on achievable performance)can be examined (also line search over convex)
I Extensions possible to generate lower bound on achievableperformance
I Finite horizon to ensure existence of an answer: highlyresistant mutant ⇒ uncontrollable growth
I Not clear how conservative the answer is...
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Comments
I Search for class of polytopic Lyapunov functions: Line searchover convex problems
I Guaranteed cost (upper bound on achievable performance)can be examined (also line search over convex)
I Extensions possible to generate lower bound on achievableperformance
I Finite horizon to ensure existence of an answer: highlyresistant mutant ⇒ uncontrollable growth
I Not clear how conservative the answer is...
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Comments
I Search for class of polytopic Lyapunov functions: Line searchover convex problems
I Guaranteed cost (upper bound on achievable performance)can be examined (also line search over convex)
I Extensions possible to generate lower bound on achievableperformance
I Finite horizon to ensure existence of an answer: highlyresistant mutant ⇒ uncontrollable growth
I Not clear how conservative the answer is...
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Comments
I Search for class of polytopic Lyapunov functions: Line searchover convex problems
I Guaranteed cost (upper bound on achievable performance)can be examined (also line search over convex)
I Extensions possible to generate lower bound on achievableperformance
I Finite horizon to ensure existence of an answer: highlyresistant mutant ⇒ uncontrollable growth
I Not clear how conservative the answer is...
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Comments
I Search for class of polytopic Lyapunov functions: Line searchover convex problems
I Guaranteed cost (upper bound on achievable performance)can be examined (also line search over convex)
I Extensions possible to generate lower bound on achievableperformance
I Finite horizon to ensure existence of an answer: highlyresistant mutant ⇒ uncontrollable growth
I Not clear how conservative the answer is...
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Problem
Terminal Cost only Problem
Given x0, c � 0,K , & positive linear switched system dynamics
x(k + 1) = Φσ(k)x(k) : k = 0, ..K − 1; x(0) = x0
Find σ(k), k = 0, ...K − 1 to minimise
J := c ′x(K )
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Theorem
Theoremσ(k) is an optimal switching sequence if and only if there existp(k) � 0 such that:
1. x(k + 1) = Φσ(k)x(k); x(0) = x0
2. p(k) = Φ′σ(k)p(k + 1); p(K ) = cand
3. σ(k) = argmini{p(k + 1)′Φix(k)}.
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Solution
I No simple way to solve optimality equations
I Forward ‘brute force’ search withΩk := set of all possible xk):
1. Initialise: Ω0 = {x0}2. Iterate: Ωk+1 = {Φ1Ωk , ...ΦNΩk}3. Select: argmini c
′ΩK ,i
I Complexity is NK .
I Speedup: remove redundant elements of Ωk at each step:Check for each i , and for all p(k) � 0:
p(k)′Ωk,i ≥ minj 6=i
p(k)′Ωk,j
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Solution
I No simple way to solve optimality equations
I Forward ‘brute force’ search withΩk := set of all possible xk):
1. Initialise: Ω0 = {x0}2. Iterate: Ωk+1 = {Φ1Ωk , ...ΦNΩk}3. Select: argmini c
′ΩK ,i
I Complexity is NK .
I Speedup: remove redundant elements of Ωk at each step:Check for each i , and for all p(k) � 0:
p(k)′Ωk,i ≥ minj 6=i
p(k)′Ωk,j
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Solution
I No simple way to solve optimality equations
I Forward ‘brute force’ search withΩk := set of all possible xk):
1. Initialise: Ω0 = {x0}2. Iterate: Ωk+1 = {Φ1Ωk , ...ΦNΩk}3. Select: argmini c
′ΩK ,i
I Complexity is NK .
I Speedup: remove redundant elements of Ωk at each step:Check for each i , and for all p(k) � 0:
p(k)′Ωk,i ≥ minj 6=i
p(k)′Ωk,j
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Solution
I No simple way to solve optimality equations
I Forward ‘brute force’ search withΩk := set of all possible xk):
1. Initialise: Ω0 = {x0}2. Iterate: Ωk+1 = {Φ1Ωk , ...ΦNΩk}3. Select: argmini c
′ΩK ,i
I Complexity is NK .
I Speedup: remove redundant elements of Ωk at each step:Check for each i , and for all p(k) � 0:
p(k)′Ωk,i ≥ minj 6=i
p(k)′Ωk,j
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Backward Search
I Reverse time ‘brute force’ search withΠk := set of all possible pk :
1. Initialise: ΠK = {c}2. Iterate: Πk−1 = {Φ′1Πk , ...Φ′NΠk}3. Select: argmini x
′0Π0,i
Complexity is NK , but can also search for redundant columnsvia LP
I Also can combine forward and backward searches.
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Backward Search
I Reverse time ‘brute force’ search withΠk := set of all possible pk :
1. Initialise: ΠK = {c}2. Iterate: Πk−1 = {Φ′1Πk , ...Φ′NΠk}3. Select: argmini x
′0Π0,i
Complexity is NK , but can also search for redundant columnsvia LP
I Also can combine forward and backward searches.
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Tightening the LPs
I Define a simple superset on the co-state variables:
pk ∈[Φ′
K−kc ,Φ′
K−kc]
(1)
Check for each i , and subject to (1):
p(k)′Ωk,i ≥ minj 6=i
p(k)′Ωk,j
I Forward, backward searches also permit further tightening.E.g., if I know Π`, tighten (1) to:
pk ∈[Φ′`−k
Π`,Φ′`−kΠ`
]
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Guaranteed Cost ControlOptimal Control
Optimal Control - Tightening the LPs
I Define a simple superset on the co-state variables:
pk ∈[Φ′
K−kc ,Φ′
K−kc]
(1)
Check for each i , and subject to (1):
p(k)′Ωk,i ≥ minj 6=i
p(k)′Ωk,j
I Forward, backward searches also permit further tightening.E.g., if I know Π`, tighten (1) to:
pk ∈[Φ′`−k
Π`,Φ′`−kΠ`
]
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Computer Simulations: Idealised Problem
4 Viral Genotypes, 2 Treatment Options (Symmetric)Genotype (i) Description λi ,1 λi ,2
1 Wild Type -0.19 -0.19
2 Resistant to Drug 1 0.16 -0.19
3 Resistant to Drug 2 -0.19 0.16
4 Highly Resistant Mutant 0.06 0.06
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Mutations
Circular, Symmetric Mutations(1) ⇔ (2)m m
(3) ⇔ (4)
M =
0 1 1 01 0 0 11 0 0 10 1 1 0
µ = 3× 10−5
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Simulation Results
Simulation for between 200 and 400 days, with 30 days betweentests/decisions.Costs based on total viral load.
Control Total Viral Load at t = 200 Time to Escape
Optimal 11.7 312
Guaranteed Cost 11.7 312
Switch on Rebound 112, 000 184
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Simulation Results: (sub) Optimal Control
0 50 100 150 200 250 300 350 4000
1
2
3
σ
Control Law for (sub) Optimal Control
0 50 100 150 200 250 300 350 400
100
105
10−5
xTα i
Time (days)
Decision Variables
i=1i=2
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Simulation Results: Optimal Control
0 50 100 150 200 250 300 350 40010
−4
10−2
100
102
104
Time (days)
Guaranteed Cost Performance
Vira
l Loa
d
312
WTRes.#1Res.#2HRMTotal
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Control based on Viral rebound
0 50 100 150 20010
−4
10−2
100
102
104
Time (days)
Switch on Virological Failure Strategy
Vira
l Loa
d
184
WTRes.#1Res.#2HRMTotal
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
A Less Idealised problem
I 14 Total State variables
I Significant asymmetry in viral fitness landscape
I Non-uniform mutation rates
I Non-linear model, control based on approximate linearisation
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Control based on Virological Failure
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
Guaranteed Cost Control
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Idealised Problem (4 state)A Less Idealised Problem
MPC - 2 year horizon
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Conclusions
I Particular class of switching control design problemsmotivated by limiting viral mutation.
I For this class of systems, stabilising and guaranteed costcontrols can be computed efficiently
I Optimal control potentially very complex to compute, thoughmay be tractable in some examples
I In a specific case, (Simple, symmetric,...) Guaranteed costturns out to be optimal. Not true in general.
I Exact optimal controls may be prohibitive in terms of detailedknowledge of state and rates and mutation tree....
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Some interesting dynamics and control questions:
I Modelling: More rigorous approach to model building.
I Robust switching control.
I Output feedback control problem for uncertain switchedsystems.
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
-
Hamilton Institute
Introduction & Motivating ProblemSwitched Systems Theory
Computer SimulationsDiscussion & Conclusions
Discussion - possible implications for treating mutation?
All else being equal...
I Optimal, or suboptimal controls, for a variety of simplifiedmodels, seem to switch frequently.
I However, standard practice in treating HIV is to wait tillvirological failure is observed, then switch.
I Perhaps it would be better to switch more regularly, possiblyin a periodic pattern? Possibly with some consideration ofpossible future viral rebound?
Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation
Introduction & Motivating ProblemHIV/AIDS: General BackgroundMathematical Model
Switched Systems TheoryGuaranteed Cost ControlOptimal Control
Computer SimulationsIdealised Problem (4 state)A Less Idealised Problem
Discussion & Conclusions
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