Switched Positive Systems and Control of Mutation Rick Middleton - - PowerPoint PPT Presentation

Hamilton Institute 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

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute 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

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

HIV/AIDS: General Background

◮ High profile disease

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

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) ◮ .... Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

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)

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Integration, Transcription and Assembly

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Main drug classes and targets

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Basic Mathematical Model: Biochemical Reactions

Reaction Rate Description ∅ → T sT Production of T cells T → ∅ dTT Death of T cells T + V → T ∗ r := βTV Infection of T Cells T ∗ → ∅ dT ∗T ∗ Death of Infected Cells T ∗ → T ∗ + V pT ∗ Viral production V → ∅ dV V Viral death ˙ T = sT − dTT − r ˙ T ∗ = r − dT ∗T ∗ ˙ V = pT ∗ − dV V

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Notes on simplified model

◮ With appropriate parameters, explains reasonably well

• bservations of primary and asymptomatic phases of infection.

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Notes on simplified model

◮ With appropriate parameters, explains reasonably well

• bservations of primary and asymptomatic phases of infection.

◮ Many different model extensions possible to include a variety

• f 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 Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Key extension 1: Macrophages

Reaction Rate Description ∅ → M sM Production of Macrophages M → ∅ dMM Death of Macrophages M + V → M∗ r := βMMV Infection of Macrophages M∗ → ∅ 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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

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 V + T → V + 2T

ρT TV CT +V

Antigen stimulated proliferation V + M → V + 2M

ρMMV CM+V

Antigen stimulated proliferation Nonlinearity (Michelis-Menton) is important for appropriate model robustness.

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Problems with Anti Retroviral Therapy

◮ Cost, Side effects, Adherence ◮ Mutation and drug resistance:

◮ High mutation rate: probability of mutation = few % per

reverse transcription

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

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

• r more drugs)

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

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

• r 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).

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical 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 Cells M + Vi → M∗

i

rMi := βMiMVi Infection of macrophages T ∗

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 mutation M + Vi → M∗

j

rMji := µmjiβMiMVi Viral mutation

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions HIV/AIDS: General Background Mathematical Model

Simplified Mutation Model

During therapy, pre-virological failure, assume: Constant T-cell, macrophage, CTL etc. counts. ˙ x(t) = Aσ(t)x(t) where

◮ xi : i = 1...m concentration of viral strain i ◮ σ(t) ∈ {1, 2, . . . , N} is drug therapy at time t ◮ Aσ(t) = blockdiag{Ai,σ(t)} + µM

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Underlying Mathematical Problem

Equivalent positive switched discrete time system: x(k + 1) = Φσ(k)x(k) where

◮ x(k) is the state vector of all variables of interest ◮ Fixed treatment during interval

σ(t) = σk : ∀t ∈ (kT, (k + 1)T)

◮ Φσ(k) = eAσ(k)T : state transition matrix for treatment σ(k) ◮ σ(k) is our decision variable (drug regimen)

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Discrete Switched Systems problem

Design σ(k) as a causal function of x(k) to achieve

◮ Asymptotic Stability? ◮ Optimality? ◮ Guaranteed Performance?

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Sub Optimal (Guaranteed Cost) Control

Theorem (Guaranteed Cost - Finite Horizon)

Given q 0, c 0, suppose we can find αi(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) ≤ min

i {α′ i(0)x(0)}

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Proof Outline - Guaranteed Cost Control

(Proof outline).

Define Lyapunov function: V (k) = min

i∈1,..N{α′ i(k)x(k)}

satisfies V (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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Comments

◮ Search for class of polytopic Lyapunov functions: Line search

• ver convex problems

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Comments

◮ Search for class of polytopic Lyapunov functions: Line search

• ver convex problems

◮ Guaranteed cost (upper bound on achievable performance)

can be examined (also line search over convex)

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Comments

◮ Search for class of polytopic Lyapunov functions: Line search

• ver convex problems

◮ Guaranteed cost (upper bound on achievable performance)

can be examined (also line search over convex)

◮ Extensions possible to generate lower bound on achievable

performance

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Comments

◮ Search for class of polytopic Lyapunov functions: Line search

• ver convex problems

◮ Guaranteed cost (upper bound on achievable performance)

can be examined (also line search over convex)

◮ Extensions possible to generate lower bound on achievable

performance

◮ Finite horizon to ensure existence of an answer: highly

resistant mutant ⇒ uncontrollable growth

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Comments

◮ Search for class of polytopic Lyapunov functions: Line search

• ver convex problems

◮ Guaranteed cost (upper bound on achievable performance)

can be examined (also line search over convex)

◮ Extensions possible to generate lower bound on achievable

performance

◮ Finite horizon to ensure existence of an answer: highly

resistant mutant ⇒ uncontrollable growth

◮ Not clear how conservative the answer is...

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal 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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Theorem

Theorem

σ(k) is an optimal switching sequence if and only if there exist p(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) = c and

• 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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Solution

◮ No simple way to solve optimality equations

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Solution

◮ No simple way to solve optimality equations ◮ Forward ‘brute force’ search with

Ωk := set of all possible xk):

• 1. Initialise: Ω0 = {x0}
• 2. Iterate: Ωk+1 = {Φ1Ωk, ...ΦNΩk}
• 3. Select: argmini c′ΩK,i

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Solution

◮ No simple way to solve optimality equations ◮ Forward ‘brute force’ search with

Ωk := set of all possible xk):

• 1. Initialise: Ω0 = {x0}
• 2. Iterate: Ωk+1 = {Φ1Ωk, ...ΦNΩk}
• 3. Select: argmini c′ΩK,i

◮ Complexity is NK.

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Solution

◮ No simple way to solve optimality equations ◮ Forward ‘brute force’ search with

Ωk := set of all possible xk):

• 1. Initialise: Ω0 = {x0}
• 2. Iterate: Ωk+1 = {Φ1Ωk, ...ΦNΩk}
• 3. Select: argmini c′ΩK,i

◮ Complexity is NK. ◮ Speedup: remove redundant elements of Ωk at each step:

Check for each i, and for all p(k) 0: p(k)′Ωk,i ≥ min

j=i p(k)′Ωk,j

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Backward Search

◮ Reverse time ‘brute force’ search with

Πk := 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 columns via LP

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Backward Search

◮ Reverse time ‘brute force’ search with

Πk := 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 columns via LP

◮ Also can combine forward and backward searches.

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Tightening the LPs

◮ 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 ≥ min

j=i p(k)′Ωk,j

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Guaranteed Cost Control Optimal Control

Optimal Control - Tightening the LPs

◮ 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 ≥ min

j=i p(k)′Ωk,j ◮ Forward, backward searches also permit further tightening.

E.g., if I know Πℓ, tighten (1) to: pk ∈

• Φ′ℓ−kΠℓ, Φ′ℓ−kΠℓ
• Rick Middleton and Esteban Hernandez

Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & 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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Idealised Problem (4 state) A Less Idealised Problem

Mutations

Circular, Symmetric Mutations (1) ⇔ (2)

• (3)

⇔ (4) M =     1 1 1 1 1 1 1 1     µ = 3 × 10−5

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Idealised Problem (4 state) A Less Idealised Problem

Simulation Results

Simulation for between 200 and 400 days, with 30 days between tests/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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Idealised Problem (4 state) A Less Idealised Problem

Simulation Results: (sub) Optimal Control

50 100 150 200 250 300 350 400 1 2 3 σ Control Law for (sub) Optimal Control 50 100 150 200 250 300 350 400 10 10

5

10

−5

xTαi Time (days) Decision Variables i=1 i=2 Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Idealised Problem (4 state) A Less Idealised Problem

Simulation Results: Optimal Control

50 100 150 200 250 300 350 400 10

−4

10

−2

10 10

2

10

4

Time (days) Guaranteed Cost Performance Viral Load 312 WT Res.#1 Res.#2 HRM Total

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Idealised Problem (4 state) A Less Idealised Problem

Control based on Viral rebound

50 100 150 200 10

−4

10

−2

10 10

2

10

4

Time (days) Switch on Virological Failure Strategy Viral Load 184 WT Res.#1 Res.#2 HRM Total

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions Idealised Problem (4 state) A Less Idealised Problem

A Less Idealised problem

◮ 14 Total State variables ◮ Significant asymmetry in viral fitness landscape ◮ Non-uniform mutation rates ◮ Non-linear model, control based on approximate linearisation

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & 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 Problem Switched Systems Theory Computer Simulations Discussion & 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 Problem Switched Systems Theory Computer Simulations Discussion & 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 Problem Switched Systems Theory Computer Simulations Discussion & Conclusions

Conclusions

◮ Particular class of switching control design problems

motivated by limiting viral mutation.

◮ For this class of systems, stabilising and guaranteed cost

controls can be computed efficiently

◮ Optimal control potentially very complex to compute, though

may be tractable in some examples

◮ In a specific case, (Simple, symmetric,...) Guaranteed cost

turns out to be optimal. Not true in general.

◮ Exact optimal controls may be prohibitive in terms of detailed

knowledge of state and rates and mutation tree....

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions

Some interesting dynamics and control questions:

◮ Modelling: More rigorous approach to model building. ◮ Robust switching control. ◮ Output feedback control problem for uncertain switched

systems.

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation

Hamilton Institute Introduction & Motivating Problem Switched Systems Theory Computer Simulations Discussion & Conclusions

Discussion - possible implications for treating mutation?

All else being equal...

◮ Optimal, or suboptimal controls, for a variety of simplified

models, seem to switch frequently.

◮ However, standard practice in treating HIV is to wait till

virological failure is observed, then switch.

◮ Perhaps it would be better to switch more regularly, possibly

in a periodic pattern? Possibly with some consideration of possible future viral rebound?

Rick Middleton and Esteban Hernandez Switched Positive Systems and Control of Mutation