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