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Optimal Control Theory for ComplexBiological Systems
Emily Dougherty, Mike Dunleaand Clinton Watton
Department of Mathematics and Computer ScienceUrsinus College
Mentor: Mohammed Yahdi
NSF REU Site in the Mathematical Sciences atUrsinus College
andUrsinus College Summer Fellows Program
July 22, 2011
2
Acknowledgements
The authors would like to thank the National Science Foundation for funding
part of this project, as well as Ursinus College for supporting part of this work
through the Summer Fellows Program. We are grateful to Dr. Mohammed
Yahdi for his guidance and patience throughout this project, and we extend
thanks to Sara Abdelmageed, Jon Lowden, and Lloyd Tannenbaum for their
work in the beginning stages of this project.
This material is based upon work supported in part by the National Science
Foundation under Grant No. DMS-1003972. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the National Science
Foundation.
Contents
0 Abstract 7
1 Introduction 9
1.1 Where The Problem Begins . . . . . . . . . . . . . . . . . . . 9
1.2 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . 11
1.3 Previous Model . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Modified VRE Model 19
3
CONTENTS 4
3 Optimal Control Techniques 23
3.1 Apply Optimal Control to VRE Model . . . . . . . . . . . . . 23
3.1.1 Objective Function . . . . . . . . . . . . . . . . . . . . 24
3.1.2 State Equations . . . . . . . . . . . . . . . . . . . . . . 26
3.1.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.4 Costate Equations . . . . . . . . . . . . . . . . . . . . 28
3.2 Minimization Principle . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Runge-Kutta nth Order . . . . . . . . . . . . . . . . . . . . . 30
3.4 Forward-backward sweep . . . . . . . . . . . . . . . . . . . . . 31
3.5 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . 32
4 Controlling k 33
4.1 Necessary Equations and Constraints . . . . . . . . . . . . . . 34
4.2 Mean Value Parameters . . . . . . . . . . . . . . . . . . . . . 36
CONTENTS 5
4.3 Parameters Representative of a Strong Infection . . . . . . . . 39
4.4 High Compliance . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Low Preventive Care Budget . . . . . . . . . . . . . . . . . . . 45
4.6 Analysis of k . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Controlling α: S to X 47
5.1 Necessary Equations and Constraints . . . . . . . . . . . . . . 48
5.2 Mean Value Parameters . . . . . . . . . . . . . . . . . . . . . 50
5.3 Parameters Representative of a Strong Infection . . . . . . . . 51
5.4 Influence of Compliance Rate p . . . . . . . . . . . . . . . . . 53
5.5 Analysis of α . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Controlling α and k 59
6.1 Necessary Equations and Constraints . . . . . . . . . . . . . . 60
CONTENTS 6
6.2 Average Conditions . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3 Extreme Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Controlling α, k, θ, and q 74
7.1 Necessary Equations and Constraints. . . . . . . . . . . . . . . 75
7.2 Average Conditions . . . . . . . . . . . . . . . . . . . . . . . . 78
7.3 Extreme Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8 Conclusion 83
Chapter 0
Abstract
The project aim is to determine the most efficient and economically favor-
able strategies to prevent outbreaks and to control the emergence of the life-
threatening antibiotic resistant VRE (Vancomycin Resistant Enterococci) in
hospital intensive care units. Optimal control theory provides optimization
methods for a dynamic system, with control functions and under certain
constraints, in order to achieve and optimize a certain output. Relevant op-
timal control concepts used include: single and multiple controls, isometric
and transversality conditions, Hamiltonian, Pontryagin’s maximum principle,
Hamilton-Jacobi-Bellman equation, Cayley-Hamilton Theorem, Bang-Bang
7
CHAPTER 0. ABSTRACT 8
Control, computer simulations using MatLab and Mathematica, and sys-
tems of eleven differential equations, twelve unknown functions and thirty
parameters. From recent models and sensitivity analysis results by Yahdi
et al. (2011), objective functions appropriate to given scenarios and goals
were formulated. The research work then focused on merging key methods
to provide necessary conditions for the existence, characterization and con-
struction of optimal controls. Computer simulations and analysis techniques
were used to visualize the optimal solutions and the role of critical param-
eters on reducing VRE infections and preventing outbreaks. Key controls
included the levels of special preventive care for colonized patients, the ICU
and health care workers compliance rates, and the health and economical
costs. The main conclusions include a collection of time depending functions
representing variable levels of special preventive care, rather than unchanged
high levels, recommended for achieving the most efficient and economically
favorable strategies to control VRE and prevent outbreaks.
Chapter 1
Introduction
1.1 Where The Problem Begins
Vancomycin resistant-enterococci infections are listed in the Center for Dis-
ease Control’s top ten health concerns. There are severe mortality and mon-
etary costs associated with the infections. Enterococci are spherical shaped
bacteria which can be found living in the bloodstream, wounds, and genital
and digestive tracts. They are notably resilient bacteria as they can with-
stand a wide range of temperatures and pHs, as well as salt concentrations.
9
CHAPTER 1. INTRODUCTION 10
They can tolerate temperatures from ten to forty-five degrees Celsius, as
well as a sixty degree Celsius environment for a period of up to ten minutes.
They can survive in up to a 6.5% NaCl solution and in pHs ranging from
4.5 to 10.0. Because of their resilience, enterococci infections are hard to
treat without antibiotics, which leads to the problem of antibiotic resistance.
Antibiotics are intended to treat infections by targeting bacterial organelles
and prevent cell replication. All bacteria, with the exception of spontaneous
mutants, are initially susceptible to antibiotics. Spontaneous mutants are the
exception to this rule; they are the few bacteria that are naturally immune
to the antibiotic. The bacteria can thus be separated into two strains: the
ancestral strain and the resistant strain. The resistant strain is composed of
the spontaneous mutants and their descendants, which are also spontaneous
mutants, and the ancestral strain is composed of the susceptible bacteria.
Initially, the ancestral strain is much larger than the resistant strain, but
once the antibiotic is used, the resistant strain increases and can replace the
ancestral strain. Vancomycin is an antibiotic used to treat bacteria which are
resistant to penicillin and its derivatives. VRE infections are extremely dan-
gerous because of their broad spectrum resistance. Linezoid and Tygacil are
two powerful drugs that can be used to treat VRE infections. This project
CHAPTER 1. INTRODUCTION 11
aimed at modeling the outbreak of VRE infections and the treatment sched-
ule using those drugs in order to optimize different goals. VRE infections
have been divided into three stages. The first stage is susceptible, which is
what everyone initially starts out as. From susceptible, people can be colo-
nized, meaning that they can produce a positive culture of the bacteria, but
do not portray symptoms of the VRE infection. People can then become
infected, meaning they show symptoms. Once a person recovers from their
VRE infection, they become susceptible again, as there is no immunity to
VRE.
1.2 Optimal Control Theory
Consider an arbitrary system whose dynamics are captured by ODEs, PDEs,
or discrete difference equations. If, in this system, there exist certain vari-
ables that can be controlled, then Optimal Control Theory is a tool for
determining how to control these variables to achieve a predetermined goal
or goals. Optimal Control Theory has many applications in finance, busi-
ness and marketing, physics, engineering, biology, and many more subjects.
CHAPTER 1. INTRODUCTION 12
One such application where Optimal Control Theory was successfully imple-
mented was at St. Jude’s Children’s Hospital to determine the best treatment
schedule and dosage of Topotecan (a chemotherapy drug) to fight Neuroblas-
toma, a type of cancer which affects nerve tissue. This study determined an
optimal treatment schedule and dosage for TPT [10]. The goal of Optimal
Control Theory is to optimize a dynamical system. With Optimal Control
Theory it is possible to create custom goals specific to the problem, such
as a best case scenario. It is also possible to use Optimal Control Theory
to discern the method for controlling a predetermined variable or parameter
in order to achieve said best case scenario. Optimal Control Theory uses a
control function u, a state function x and certain constraints to maximize
or minimize an objective function. The state function x(t) consists of the
dependent variables in a system of differential equations. The state function
represents the system’s characteristics and future. The control function is
introduced to affect the differential equations and to optimize the Objective
function.
CHAPTER 1. INTRODUCTION 13
1.3 Previous Model
Nineteen parameters are involved in the transitions between stages of VRE
infections. These parameters are shown below in Table 1.1, with their de-
scription, mean value, and range value [48]. Especially noteworthy are the
parameters k, α, θ, and q, for they are the parameters that are used to op-
timize later on in this study. k is the proportion of patients moving from
susceptible (S) to colonized without preventive care (X), α is the proportion
of patients moving from colonized without preventive care (X) to colonized
with preventive care (Y), θ is the proportion of patients moving from in-
fected without treatment (V) to infected with treatment (W), and q is the
proportion of patients moving from colonized with preventive care (Y) to
infected without treatment (V). These parameters were used to create the
mathematical model of differential equations representative of the growth of
each group. The five equations composing the model are also shown further
below in Equation 1.1.
CHAPTER 1. INTRODUCTION 14
Table 1.1: Summary Table for the Previous Parameters
Parameter Description MeanValue
Range Value
µ General Admission rate 0.0956 0.03 ≤ µ ≤ 0.14δ Contamination Rate 0.29845 0.2657 ≤ δ ≤ 0.3312m1 Admission Rate of Susceptible 0.7 0 ≤ m1 ≤ 1m2 Admission Rate of Colonized without Pre-
ventive Care0.1 0 ≤ m1 ≤ 1
m3 Admission Rate of Colonized with PreventiveCare
0.1 0 ≤ m1 ≤ 1
m4 Admission Rate of Infected without Treat-ment
0.05 0 ≤ m1 ≤ 1
β Rate of Spontaneous Curing 0.095 0.03 ≤ β ≤ 0.16τ Antibiotic Use Alone 0.302 0.07 ≤ τ ≤ 0.65γ Rate of Curing due to Treatment 0.46 0 < γ ≤ 0.46f Fitness Cost 0.25 0 ≤ f ≤ 0.5α Movement from Colonized without Preven-
tive Care to Colonized with Preventive Care0.2 0 < α < 0.5
αp Movement from Colonized with PreventiveCare to Colonized without Preventive Care
0.1 0 < αp < 0.5
θ Movement from Infected without Treatmentto Infected with Treatment
0.2 0 < θ < 0.5
θp Movement from Infected with Treatment toInfected without Treatment
0.1 0 < θp < 0.5
ε Factors Leading to Infection 0.2083 0 < ε < 1r People moving between Colonized without
Treatment and Infected without Treatment0.2 0 < r < 1
k People moving between Susceptible and Col-onized without Preventive Care
0.2 0 < k < 1
q People moving between Colonized with Pre-ventive Care and Infected without Treatment
0.5 0 < q < 1
p Compliance Rate 0.5 0 ≤ p ≤ 1
CHAPTER 1. INTRODUCTION 15
The pictorial model of VRE outbreak shown below in Figure 1.1 was made by
Yahdi, et al 2011, based off of these parameters. The various stages of VRE
infections described earlier were assigned variables as follows: S is susceptible,
X is colonized without preventive care, Y is colonized with preventive care,
V is infected without treatment, and W is infected with treatment.
The box surrounding the model is representative of an Intensive Care Unit.
The blue arrows represent transitions into and out of the ICU, the red arrows
show transitions that are good for VRE but bad for patients, the green
arrows show transitions that are bad for VRE but good for patients, and the
yellow arrows represent transitions within the same group (either starting or
stopping preventive care or treatment).
The VRE model of differential equations, shown below in Equation 1.1, is
based upon the parameters from Table 1.1. The following are initial condi-
tions for the VRE model.
S(0) = S0, X(0) = X0, Y (0) = Y0, V (0) = V0, W (0) = W0
and S0 +X0 + Y0 + V0 +W0 = 1.
CHAPTER 1. INTRODUCTION 16
Figure 1.1: Previous VRE Model.
CHAPTER 1. INTRODUCTION 17
dSdt
= µm1 + βY + βX + βV + γW − µS − k(δS(V +W +X + pY )
+τS)(1 − f) − (1 − k)(δS(V +W +X + pY ) + τS)(1 − f)
dXdt
= m2µ− µX + k(1 − f)(δS(X + pY + V +W ) + τS) + α1Y
−X(β + α + (1 − f)ε)
dYdt
= m3µ− µY + (1 − k)(1 − f)(δS(X + pY + V +W ) + τS) + αX
−Y (β + α1 + (1 − f)εp)
dVdt
= m4µ− µV + (1 − f)ε(rX + qpY ) + θ1W − (θ + β)V
dWdt
= (1 −m1 −m2 −m3 −m4)µ− µW + (1 − f)ε((1 − r)X+
(1 − q)pY ) + θV − (θ1 + γ)W
(1.1)
1.4 Motivation
VRE is a pressing issue as it is in the CDC’s top ten health concerns, and
no one is currently certain what the most efficient and effective manner of
dealing with an outbreak. The model consisting of differential equations
achieved in Yahdi, et al (2011), served as a motivation to apply optimal
control theory to VRE in hopes of minimizing an outbreak. Optimal control
CHAPTER 1. INTRODUCTION 18
theory allows us to pick an objective function in such a manner that we can
minimize certain variables and maximize others, making it a useful tool to
apply to VRE.
Chapter 2
Modified VRE Model
Since then, some of the parameters have slightly changed. The parameters k,
f , r, and q were switched to (1−k), (1−f), (1−r), and (1−q), respectively,
to make them more intuitive. Previously, k represented the number of people
sent to colonized without preventive care (X). However, one of the goals is
to optimize the number of people sent to colonized with preventive care (Y).
In this case, (1 − k) would need to be used, which makes the equations it
shows up in more complicated and confounding. Instead they were switched.
So, k now represents the number of people sent to colonized with preventive
care (Y). This not only simplifies mathematical formulae but it is also more
19
CHAPTER 2. MODIFIED VRE MODEL 20
intuitive. A similar justification exists for the switching of f , r, and q to
(1−f), (1−r), and (1−q), respectively. The following table summarizes the
changes to Table 1.1. The new parameter values are listed below in Table
2.1.
Table 2.1 lists the parameters whose value was changed from the previous
results. As shown, the values of f , r, k, and q were changed. These were
changed so that their values were more intuitive and more easily understood.
The parameter f represents fitness cost, r represents people transitioning be-
tween colonized without preventive care to infected with treatment, k stands
for people moving from susceptible to colonized with preventive care, and
q represents people moving from colonized with preventive care to infected
with treatment.
After the parameters f , r, k, and q were changed, the system of differential
equations also needed to be changed. The final list of parameters are provided
in Table 2.2 and from this table the final system of differential equations are
also shown, see Equation 2.1.
CHAPTER 2. MODIFIED VRE MODEL 21
Table 2.1: Parameters Whose Value Is Changed
Parameter Description MeanValue
Range Value
f Fitness Cost 0.75 0.5 ≤ f ≤ 1r People moving between Colonized without
Preventive Care and Infected with Treat-ment
0.8 0 < r < 1
k People moving between Susceptible and Col-onized with Preventive Care
0.8 0 < k < 1
q People moving between Colonized with Pre-ventive Care and Infected with Treatment
0.5 0 < q < 1
dSdt
= µm1 + βY + βX + βV + γW − µS − f(Sδ(V +W +X + pY )
+τS)f − k(δS(V +W +X + pY ) + τS)f
dXdt
= m2µ− µX + (1 − k)(f)(δS(X + pY + V +W ) + τS) + αpY
−X(β + α + fε)
dYdt
= m3µ− µY + kf(δS(X + pY + V +W ) + τS) + αX
−Y (β + αp + fεp)
dVdt
= m4µ− µV + fε((1 − r)X + (1 − q)pY ) + θpW − (θ + β)V
dWdt
= (1 −m1 −m2 −m3 −m4)µ− µW + fε(rX+
qpY ) + θV − (θp + γ)W
(2.1)
CHAPTER 2. MODIFIED VRE MODEL 22
Table 2.2: Summary Table for the New Parameters
Parameter Description MeanValue
Range Value
µ General Admission rate 0.0956 0.03 ≤ µ ≤ 0.14δ Contamination Rate 0.29845 0.2657 ≤ δ ≤ 0.3312m1 Admission Rate of Susceptible 0.7 0 ≤ m1 ≤ 1m2 Admission Rate of Colonized without Pre-
ventive Care0.1 0 ≤ m1 ≤ 1
m3 Admission Rate of Colonized with PreventiveCare
0.1 0 ≤ m1 ≤ 1
m4 Admission Rate of Infected without Treat-ment
0.05 0 ≤ m1 ≤ 1
β Rate of Spontaneous Curing 0.095 0.03 ≤ β ≤ 0.16τ Antibiotic Use Alone 0.302 0.07 ≤ τ ≤ 0.65γ Rate of Curing due to Treatment 0.46 0 < γ ≤ 0.46f Fitness Cost 0.75 0.5 ≤ f ≤ 1α Movement from Colonized without Preven-
tive Care to Colonized with Preventive Care0.2 0 < α < 0.5
αp Movement from Colonized with PreventiveCare to Colonized without Preventive Care
0.1 0 < αp < 0.5
θ Movement from Infected without Treatmentto Infected with Treatment
0.2 0 < θ < 0.5
θp Movement from Infected with Treatment toInfected without Treatment
0.1 0 < θp < 0.5
ε Factors Leading to Infection 0.2083 0 < ε < 1r People moving between Colonized without
Preventive Care and Infected with Treat-ment
0.8 0 < r < 1
k People moving between Susceptible and Col-onized with Preventive Care
0.8 0 < k < 1
q People moving between Colonized with Pre-ventive Care and Infected with Treatment
0.5 0 < q < 1
p Compliance Rate 0.5 0 ≤ p ≤ 1
Chapter 3
Optimal Control Techniques
3.1 Apply Optimal Control to VRE Model
The VRE system has ten differential equations, eleven unknown functions,
and thirty parameters. Because of its complexity, there is no analytical
solution. Only numerical solutions exist that must be solved with the aide
of computer programs such as Matlab and Mathematica. The code in these
programs can use different types of techniques to solve various optimal control
problems.
23
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 24
3.1.1 Objective Function
In optimal control theory the goal is optimize some objective function, more
specifically, minimizing or maximizing an objective function. This objective
function is not given by optimal control theory, it is something that must
be constructed based upon the problem and the desired goals. For this
project the objective function was the quantity that must be minimized.
The objective function in this project had the basic form:
f(t) = S(t) +∫ T0
(aX(t) + bY (t) + cV (t) + dW (t) + e1u(t) + e2u2(t))dt
(3.1)
The quantities contained inside the integrand are the ones to be minimized,
such as colonized with and without preventive care, infected with and with-
out treatment, and the linear and quadratic versions of some control u(t).
The only variable outside of the integrand is the susceptible population, this
is because the susceptible population should not be minimized; preferably it
should only increase or remain the same. This is because the susceptible pop-
ulation is portion of patients in the intensive care unit do not test positively
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 25
for VRE, that is good. Equation 3.1 only has one control u(t), however more
than one control can be used at a time, if additional controls are used both
the linear and quadratic versions are added on inside the integrand. The co-
efficients of the variables inside the integrand are called weights. The values
of the weights themselves are seemingly meaningless, what is important is
one variable’s weight relative to another, in other words, if all the weights
have a value of 1 or all have a value of 100 the effect is the same, what is
important is the difference between one variable’s weight and another. The
larger the value of a variable’s weight the more important it is that that
variable is minimized over the specified time interval T . For this project a
set of average weights were derived: a = 5, b = 5, c = 10, d = 5, e1 = 1, and
e2 = 5. The highest weight is c = 10 the coefficient of the infected without
treatment (V), this should be the most important variable to minimize. This
is because a patient who is infected with VRE is at risk of additional health
complications and death, and if this patient is not receiving treatment for
VRE it is imperative that this variable be minimized. The controls always
receive the same weight in our project. The linear term receives a weight of
1 and the quadratic terms receives a weight of 5 this is because the quadratic
term grows more quickly than the linear and the the controls are more ex-
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 26
pensive to implement, so the more linear the cheaper it is for the hospitals.
The weights are not set in stone however, if a very different set of parameters
were introduced or extreme circumstances were encountered the weights can
easily be altered in order to produce results that cater more to the reality of
the problem.
3.1.2 State Equations
The state equations are simply the equations that form the VRE model, as
shown in Equation 2.1.
3.1.3 Hamiltonian
In order to optimize the objective function and solve for optimal control and
optimal states, it is required by optimal control theory to derive some further
equations. One such equation is the Hamiltonian H. The Hamiltonian is
constructed based off existing quantities. The Hamiltonian is defined as the
the integrand of the objective function plus the state equation multiplied
by the adjoint. The adjoint is not the same as, but similar in function to,
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 27
Lagrange multipliers in multivariable calculus. Equation 3.2 below shows the
definition of the Hamiltonian.
H = (Integrand of Objective Function) + (State Equation)(Adjoint)
(3.2)
Written in terms of Equation 3.1 (the objective function), where g(t) is the
state equation, the Hamiltonian can be further defined as shown in Equation
3.3 below.
H = aX(t) + bY (t) + cV (t) + dW (t) + e1u(t) + e2u2(t) + g(t)λ(t) (3.3)
Just as there can be multiple controls there can be multiple state equations.
In this project, the VRE model is described by five differential equations. In
this case, each of the five differential equations is paired with a corresponding
adjoint λi. At this point each λi is unknown, but from optimal control theory
there exist a differential equation for each unknown λi and final conditions
called the transversality condition. The differential equations are known and
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 28
can be derived from existing quantities, these equations are known as the
costate equations. Since the Hamiltonian is constructed from the objective
function, if the objective function changes so does the Hamiltonian.
3.1.4 Costate Equations
Each costate equation is defined as the negative first derivative of the Hamil-
tonian with respect to each of the five variables. Since the costate equations
are found from the Hamiltonian, if the Hamiltonian changes so do the costate
equations. The costate equations are defined below in Equation 3.4.
dλ1dt
= −dHdS
dλ2dt
= −dHdX
dλ3dt
= −dHdY
dλ4dt
= −dHdV
dλ5dt
= − dHdW
(3.4)
The transversality condition is defined as shown below in Equation 3.5:
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 29
λi(T ) = 0 (3.5)
,
where T is the final time and i is the index running through all five adjoints.
This transversality condition allows each adjoint to be solved for an exact
solution.
3.2 Minimization Principle
Pontryagin’s Minimization Principle 3.2.1. If u∗(t) and x∗(t) are op-
timal, then there exists a piecewise differentiable adjoint variable λ(t) such
that
H(t, x∗(t), u∗(t), λ(t)) ≤ H(t, x∗(t), u(t), λ(t)) (3.6)
for all controls u at each time t, where the Hamiltonian H is
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 30
H = f(t, x(t), u(t)) + λ(t)g(t, x(t), u(t)) (3.7)
and
λ′(t) = −∂H(t,x∗(t),u∗(t),λ(t))∂x
λ(tf ) = 0
(3.8)
Pontryagin’s Minimization Principle states the necessary conditions and ex-
istence of an optimal control [23].
3.3 Runge-Kutta nth Order
One such tool for solving optimal control problems is the Runge-Kutta nth
order method. This method lets us solve a differential equation numerically.
It is very accurate and well-behaved for a wide range of problems. This
method is very similar to the Euler method, in fact, the 1st order Runge-
Kutta is exactly the same as the Euler method. The number of terms n
represent the order of the approach, and the 4th order Runge-Kutta method
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 31
has been determined to be the most efficient in terms of results achieved per
work put in. 4th order Runge-Kutta operates by sampling the slopes at the
midpoint and endpoints of the interval. From these, the weighted average is
taken, placing more weight (a weight of 4) on the slope at the midpoint.
3.4 Forward-backward sweep
Another method which can be implemented in the Runge-Kutta method
is the forward-backward sweep method. This method solves for the state
equations forward in time and the co-state equations backward in time, each
according to their differential equation in their optimality system. As the
state and co-states are found, the control u is updated using their values,
which produces a new approximation of the state, co-state, and control (x,
λ, u). The loop terminates when there is sufficient agreement between the
states, co-states, and controls of the two passes through the approximation
loop. If the values from the new iteration and the previous iteration are
sufficiently close, then the current values are solutions.
CHAPTER 3. OPTIMAL CONTROL TECHNIQUES 32
3.5 Mathematical Modeling
Mathematical modeling can be used to simulate a biological system using
mathematical equations and expressions. For dynamic systems, systems that
change over time, often differential or discrete difference equations are the
best method to accurately simulate the system. Mathematical modeling
portrays biological events as the interactions of various parameters within the
confines of some constructed model. Such a model is comprised of a finite
number of differential or discrete difference equations which simulate the
system. Parameters function by explaining interactions between variables.
Using biological data these parameters are tested and refined until the current
value of parameters accurately simulate the system.
Chapter 4
Controlling k
Through sensitivity analysis, a previous team found that the parameter k,
the proportion of patients moving from susceptible (S) to colonized with pre-
ventive care (Y), to be a critical parameter. A critical parameter is one such
that changes made to this parameter greatly affect the differential model.
For this reason k was chosen to function as our first optimal control model.
33
CHAPTER 4. CONTROLLING K 34
4.1 Necessary Equations and Constraints
Included below are all of the necessary equations and constraints that are
needed to perform optimal control theory on this specific simulation with
the control k. Equation 4.1 is the objective function which will be minimized
using optimal control theory.
∫ T0
(aX + bY + cV + dW + ek2 + gk)dt (4.1)
The Hamiltonian, which is a function of 11 functions and time, is given by
Equation 4.2.
CHAPTER 4. CONTROLLING K 35
H(X, Y, V,W, k) = aX + bY + cV + dW + gk + ek2 + λ1(β(V +X + Y ) +Wγ − Sµ−
fS((V +W +X + pY )δ + τ) + µm1) + λ2(m2µ− µX+
(1 − k)f(δS(X + pY + V +W ) + τS) + α1Y −X(β + α + fε))+
λ3(m3µ− µY + kf(δS(X + pY + V +W ) + τS) + αX−
Y (β + α1 + fεp)) + λ4(m4µ− µV + fε((1 − r)X + (1 − q)pY )
+θ1W − (θ + β)V ) + λ5((1 −m1 −m2 −m3 −m4)µ− µW
+fεrX + qpY ) + θV − (θ1 + γ)W )
(4.2)
The equation for k∗, as used in the optimality condition, is shown below in
Equation 4.3. The k∗ equation is used to find the critical functions for the
Hamiltonian. This is useful to help find the optimal control function k(t) as
a function of all variables and parameters. As in calculus, this is similar to
finding a critical point, but instead of a point we have a function.
k∗ = − (e+fS(V+W+X+pY )δ+τ)(−λ2+λ3))2g
(4.3)
The co state equations are depicted in Equation 4.4 below. These are used
to solve the adjoint.
CHAPTER 4. CONTROLLING K 36
dλ1dt
= −(−µ− f((V +W +X + pY )δ + τ))λ1 − f(1 − k)((V +W +X + pY )δ + τ)λ2
−fk((V +W +X + pY )δ + τ)λ3
dλ2dt
= −a− (β − fSδ)λ1 − (−α− β + f(1 − k)Sδ − fε− µ)λ2 − (α + fkSδ)λ3
−f(1 − r)ελ4 − frελ5
dλ3dt
= −b− (β − fpSδ)λ1 − (f(1 − k)pSδ + αp)λ2 − (−β + fkpSδ − fpε− µ− αp)λ3
−fp(1 − q)ελ4 − pqλ5
dλ4dt
= −c− (β − fSδ)λ1 − f(1 − k)Sδλ2 − fkSδλ3 − (−β − θ − µ)λ4 − θλ5
dλ5dt
= −d− (γ − fSδ)λ1 − f(1 − k)Sδλ2 − fkSδλ3 − θpλ4 − (−γ − µ− θp)λ5
(4.4)
4.2 Mean Value Parameters
Figure 4.1 displays four plots from a single simulation. This simulation used
the mean value of all parameters, average initial populations: S0 = 0.7,
X0 = 0.1, Y0 = 0.1, V0 = 0.05, W0 = 0.05, and the weights used for this
figure (and all figures unless otherwise stated) are a = 1, b = 5, c = 10,
d = 5, e = 1, g = 5. Figure 4.1(a) shows the proportion of patients being
sent from susceptible (S) to colonized with preventive care (Y), this of course
CHAPTER 4. CONTROLLING K 37
is the control. Figure 4.1(b) displays the susceptible population over time.
The susceptible population is important, however, it is by no means the most
important population. Overall, the susceptible population should increase or
level off; the more patients that are no longer infected or colonized, the more
susceptible increases. As long as susceptible is not rapidly decreasing on day
20 then the model is okay.
Figure 4.1(c) shows the total colonized (X+Y) population evolving over time.
The colonized population is more important than the susceptible population,
though still not the most important. The total colonized population can
increase, though it is preferred that it levels off, it is better to reach an
equilibrium point. The more patients that are colonized, fewer will wind
up in infected states. Figure 4.1(d) represents the total infected population
changing over time. The total infected population is the most important, in
the sense that it is the population we wish to minimize. In hospital intensive
care units patients are not in the best of shape and many of the patients are on
immune-suppressants which make patients more vulnerable to infection. For
this reason it is extremely important to minimize the infected populations;
the more patients that stay in infected populations, the more will die either
from VRE or complications due to VRE. The total infected population, in
CHAPTER 4. CONTROLLING K 38
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Co
ntr
ol: k
(a) Control: k
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Su
sce
ptib
le
(b) Susceptible
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
To
tal C
olo
niz
ed
(c) Total Colonized
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
To
tal In
fecte
d
(d) Total Infected
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Ou
tbre
ak R
isk
(e) Outbreak Risk
Figure 4.1: Mean Value Parameters. Initial populations: S0 = 0.7, X0 = 0.1,
Y0 = 0.1, V0 = 0.05, W0 = 0.05. Weights: a = 1, b = 5, c = 10, d = 5, e = 1,
and g = 5.
CHAPTER 4. CONTROLLING K 39
all cases, should decrease or level off. In this simulation the total infected
population is low from the start and only grows a little, and levels off. Lastly,
Figure 4.1(e) models the Outbreak Risk factor. If the Outbreak Risk is below
1 there is not a large chance of an outbreak, if it is about 1 then there is a
good chance an outbreak will occur, and if it is well above 1 then there is
a good chance an epidemic will occur. The Outbreak Risk starts off at 0.5
and quickly decreases. One goal of the model is optimize final conditions
as well as conditions during the simulation. Figure 4.1(e) shows us that the
Outbreak Risk factor is the lowest at the end of the simulation, this is good.
4.3 Parameters Representative of a Strong
Infection
Figure 4.2 represents an environment that is considerably different than the
average day-to-day intensive care unit. The value of some parameters were
changes, some raised some lowered, to help produce this different environ-
ment. General admission rate µ increased, the contamination rate δ de-
creased, the rate of spontaneous curing β increased slightly, the previous
CHAPTER 4. CONTROLLING K 40
antibiotic use factor τ decreased slightly, the rate of curing due to treatment
γ decreased moderately, the rate of patients moving from colonized with
preventive care (Y) to colonized without preventive care (X) αp decreased,
factors leading to infection ε increased dramatically to its maximum value,
fitness level f also increased to its maximum value, and compliance rate p in-
creased to its maximum value as well. This increases and decreases describe a
unique environment for the breeding grounds of VRE. This particular blend
of parameters yields a highly contagious and strong strain of bacteria. This
intensive care unit has more patients being admitted than normal, this in-
crease in bodies increases the chance of the bacteria spreading to another
susceptible patient. Whether or not a patient is cured is more unpredictable,
as the rate of spontaneous curing increased while the rate of curing due to
treatment decreased. The three most important chances in parameters are
ε, f , and p. Factors leading to infection ε is at its maximum value, mean-
ing there dramatically many more ways to get infected than normal. The
fitness cost of the bacteria is very high, at its maximum value, this means
the bacteria can reproduce more effectively and faster and the chance for a
antibiotic resistant strain to emerge is increased. The compliance rate p is
also at its maximum value which means the impact and compliance of special
CHAPTER 4. CONTROLLING K 41
preventive care is least effective and hygiene regulations are not being fol-
lowed which ultimately decreases the effectiveness of special preventive care
and increases the chance of contamination and the spread of the infection.
In Figure 4.2(a) 100% of patients are moved from susceptible (S) to colonized
with preventive care (Y). If it were possible to know everything about the bac-
teria, and the hospital could plan for such a strong strain and contaminant-
prone environment, it would perhaps make the mistake of applying full con-
trol to try to quell the spread of bacteria. The top plot, the dotted blue
line in Figure 4.2(a), is the basic reproductive number which describes the
chance of an outbreak or epidemic at any time throughout the 20 days. If
this number is beneath 1 then the chance of an outbreak is very slim, if it
is over 1 there is a high chance of an outbreak. In Figure 4.2(a) the basic
reproductive number remains constant over the 20 days at about 1.75, which
is well over 1 and there is a very high chance of an outbreak. This is because
the hospital is placing a large proportion of patients into preventive care, but
with p = 1, preventive care is virtually not effective at all. It ends up being
a waste of money and resources for the hospital.
In Figure 4.2(b) the colonized and infected populations do not differ signifi-
CHAPTER 4. CONTROLLING K 42
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Po
pu
latio
n
Colonized
Infected
k
Outbreak Risk
(a) Full Control
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Po
pu
latio
n
Colonized
Infected
k
Outbreak Risk
(b) Optimal Control
Figure 4.2: Bad Case Scenario Simulation. Five parameters related to the
spread of infection and the curing of infections were altered in order to sim-
ulate a bad case scenario. Parameters altered: µ = 0.14, δ = 0.2, m3 = 0,
β = 0.16, τ = 0.3, γ = 0.3, f = 1, αp = 0.05, ε = 1, p = 1. Initial popula-
tions: S0 = 0.7, X0 = 0.1, Y0 = 0.1, V0 = 0.05, W0 = 0.05. Weights: a = 1,
b = 5, c = 10, d = 5, e = 1, and g = 5.
cantly however, the basic reproductive number is significantly lower than in
Figure 4.2(a) with an initial value slightly above 0.4 and quickly dropping
to 0 on or about day 1 of the simulation, and remains at 0 throughout the
simulation. The optimal control was used here instead of placing everyone
into preventive care. This saved the hospital money, resources, space, and
most importantly lives.
CHAPTER 4. CONTROLLING K 43
4.4 High Compliance
Another parameter which is important is p, which is the compliance rate. The
parameter p describes the effectiveness of hygiene regulations and preventive
care effectiveness. When p is small, close to zero, compliance is high and
regulations are being followed and treatment via preventive care is effective.
In Figure 4.3 p was the only parameter altered, normally p has an average
value of p = 0.5, for this example p was lowered to p = 0.1. The effect on the
model is dramatically less control is needed throughout the 20 days. This
agrees with what our intuition tells us, if the hygiene regulations are being
followed and preventive care is effective less patients will need to be sent
there because it is not necessary if hygiene regulations are being correctly
followed.
CHAPTER 4. CONTROLLING K 44
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
k
Outbreak Risk
(a) No Control
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
k
Outbreak Risk
(b) Optimal Control
Figure 4.3: High Compliance. Mean value parameters were used with the
exception that p was lowered to p = 0.1. Initial populations: S0 = 0.7,
X0 = 0.1, Y0 = 0.1, V0 = 0.05, W0 = 0.05. Weights: a = 1, b = 5, c = 10,
d = 5, e = 1, and g = 5.
CHAPTER 4. CONTROLLING K 45
4.5 Low Preventive Care Budget
Preventive care is expensive, it costs the hospital more money to put patients
and keep patients in preventive care. Some hospitals do not have the funding
necessary to place a lot of patients into preventive care. With the use of
weights it is possible to weigh down the usage of preventive care, and thereby
reducing the number of patients that are going into preventive care from
susceptible. Figure 4.4 depicts what an optimal control would look like if
the hospital has low resources and/or budget for special preventive care. By
placing a heavier weight (e = 25, g = 25) on the cost function for preventive
care we can still achieve optimal results even with low resources.
4.6 Analysis of k
As stated, the parameter k, representative of the proportion of patients mov-
ing from susceptible to colonized with preventive care (Y), is a critical pa-
rameter, thus it was the first parameter chosen to control. In doing this,
the total infected population was aimed to be minimized by using the opti-
mal value of k. As shown above in Figure 4.2(a) and Figure 4.2(b), using
CHAPTER 4. CONTROLLING K 46
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
k
Outbreak Risk
(a) No Control
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
k
Outbreak Risk
(b) Optimal Control
Figure 4.4: Low Preventive Care Budget. Mean value parameters. Initial
populations: S0 = 0.7, X0 = 0.1, Y0 = 0.1, V0 = 0.05, W0 = 0.05. Weights:
a = 1, b = 5, c = 10, d = 5, e = 25, and g = 25.
a constant full control is not the most efficient, and even more importantly
the most effective, method. Using the optimal control of k in Figure 4.2(b)
saved the hospital lives, money, resources, and space. Figure 4.4 shows how
optimal results can be achieved while having limited resources (thus keeping
k low). K is a critical parameter, so it is important to monitor it and keep
it at its optimal value.
Chapter 5
Controlling α: S to X
The parameter α is the proportion of patients moving from colonized without
preventive care (X) to colonized with preventive care (Y). Previously, when
k was the control, the only patients that were abled to be controlled were
patients in susceptible (S); the proportion of patients moving from X to Y
was fixed with a value α = 0.2. For this analysis we no longer treat k as a
control and it will return to parameter status with it’s mean value k = 0.8.
Instead we investigate the control of α.
47
CHAPTER 5. CONTROLLING α: S TO X 48
5.1 Necessary Equations and Constraints
Included below in this section are the necessary equations and constraints
needed to apply optimal control to this specific simulation of controlling α.
The objective function is show in Equation 5.1 below, which is what will be
minimized.
∫ T0
(aX + bY + cV + dW + n0α + nα2)dt (5.1)
The Hamiltonian is portrayed below in Equation 5.2. As before, the Hamil-
tonian is a function of 11 functions and time.
CHAPTER 5. CONTROLLING α: S TO X 49
H(X, Y, V,W, α) = aX + bY + cV + dW + n0α + nα2 + λ1(β(V +X + Y ) +Wγ − Sµ−
fS((V +W +X + pY )δ + τ) + µm1) + λ2(m2µ− µX+
(1 − k)f(δS(X + pY + V +W ) + τS) + α1Y −X(β + α + fε))+
λ3(m3µ− µY + kf(δS(X + pY + V +W ) + τS) + αX−
Y (β + α1 + fεp)) + λ4(m4µ− µV + fε((1 − r)X + (1 − q)pY )
+θ1W − (θ + β)V ) + λ5((1 −m1 −m2 −m3 −m4)µ− µW
+fεrX + qpY ) + θV − (θ1 + γ)W )
(5.2)
The equation for α∗, as used in the optimality condition, is shown below in
Equation ??. The α∗ equation is used to find the critical functions for the
Hamiltonian. This is useful to help find the optimal control function α(t) as
a function of all variables and parameters. As in calculus, this is similar to
finding a critical point, but instead of a point we have a function.
α∗ = −n0−λ2X+λ3X)2n
(5.3)
Depicted in Equation 5.4 are the co state equations used for the control of
α. These are used to solve the adjoint.
CHAPTER 5. CONTROLLING α: S TO X 50
dλ1dt
= −f(1 − k)λ2(τ + δ(V +W +X + pY ))
−fkλ3(τ + δ(V +W +X + pY )) − λ1(−mu− f(τ + δ(V +W +X + pY )))
dλ2dt
= −a− εfλ4(1 − r) − εfλ5r − λ1(β − δfpS)
−λ2(−α− β − εf − µ+ δf(1 − k)S − λ4(α + δfkS)
dλ3dt
= −b− εfλ4p(1 − q) − εfλ5pq − λ1(β − δfpS) − λ2(α1 + δf(1 − k)pS)
−λ3(−α1 − β − µ− εfp+ δfkpS)
dλ4dt
= −c− δf(1 − k)λ2S − δfkλ3S − λ1(β − δfS) − λ4(−β − µ− θ) − λ5θ
dλ5dt
= −d− δf(1 − k)λ2S − δfkλ3S − λ1(γ − δfS) − λ5(−γ − µ− θ1) − λ4θ1
(5.4)
5.2 Mean Value Parameters
Figure 5.1 depicts the average intensive care unit, with parameters set to their
mean values, average initial populations, and normal weights. Compared to
Figure 4.1 the basic reproductive number remains high throughout. The
reason for this is because in these α-control simulations, k takes on its mean
value of k = 0.8, which is very high, since α is also sending patients into
preventive care, this simulation concentrates more patients in preventive care
CHAPTER 5. CONTROLLING α: S TO X 51
than did the k-control simulation in Figure 4.1. This higher concentration of
patients in colonized with preventive care (Y) is the cause for the elevated
basic reproduction number. Perhaps a second control could be introduced
to help lower the basic reproductive number, or use a different control all
together in this instance. From this simulation, controlling α may not be the
best option, however while the basic reproductive number is high, it is not
at or above 1, meaning this is not the worst option to control.
5.3 Parameters Representative of a Strong
Infection
Again, parameters were altered in order to create a simulation where infection
can easily spread, see Figure 5.2. In the two subfigures two different methods
of control are shown. Figure 5.2(a) shows full control, which is sending 100%
of patients from colonized without preventive care (X) into colonized with
preventive care (Y). The top plot, the blue dotted line, is the familiar basic
reproductive number which is just under the value of 1.8, which indicates
there is a very high chance an outbreak will occur. Figure 5.2(b) uses the
CHAPTER 5. CONTROLLING α: S TO X 52
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
Colonized
Infected
Control: alpha
Outbreak Risk
Figure 5.1: Optimal Control of α. Initial populations: S0 = 0.7, X0 = 0.1,
Y0 = 0.1, V0 = 0.05, W0 = 0.05. Weights: a = 1, b = 5, c = 10, d = 5, e = 1,
and g = 5.
CHAPTER 5. CONTROLLING α: S TO X 53
optimal control instead of full control. Again, the basic reproductive number
is high, well above one, it is lower than in Figure 5.2(a), starting out just
under 1.6 and decreasing until about 1.3. This control cannot lower the
basic reproductive number significantly lower than 1 given the parameters
and initial conditions. This is not a flaw in the model, it is simply not
possible with this control alone to sufficiently lower the basic reproductive
number. Perhaps introducing a second control or perhaps using a different
control altogether may solve the inevitable outbreak.
5.4 Influence of Compliance Rate p
The four graphs in Figure 5.3 portray the strong influence of the compliance
rate p. The top two graphs Figure 5.3(a) and Figure 5.3(b) were simulated
with a compliance rate p = 0, representative of full compliance. The bottom
two graphs, Figure 5.3(c) and Figure 5.3(d) were simulated with a compliance
rate p = 1, representative of no compliance. The graphs on the left, Figure
5.3(a) and Figure 5.3(c), are the optimal controls, and the graphs on the
right, Figure 5.3(b) and Figure 5.3(d), are constant controls, with α = 0.2.
CHAPTER 5. CONTROLLING α: S TO X 54
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Po
pu
latio
n
Colonized
Infected
Control: alpha
Outbreak Risk
(a) High Constant
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Po
pu
latio
n
Colonized
Infected
Control: alpha
Outbreak Risk
(b) Optimal
Figure 5.2: Parameters Representative of a Strong Infection. Parameters
altered: µ = 0.14, δ = 0.2, m3 = 0, β = 0.16, τ = 0.3, γ = 0.3, f = 1,
αp = 0.05, ε = 1, p = 1. Initial populations: S0 = 0.7, X0 = 0.1, Y0 = 0.1,
V0 = 0.05, W0 = 0.05. Weights: a = 1, b = 5, c = 10, d = 5, e = 1, and
g = 5.
CHAPTER 5. CONTROLLING α: S TO X 55
It is clear that in both the optimal control and the constant control, the
cases where the compliance rate p = 0 or full compliance, had a higher colo-
nized population, lower infected population, and a drastically lower outbreak
risk. This shows that a full compliance rate is very beneficial to fighting the
outbreak of VRE. These are the expected results, as full compliance makes
preventive care more effective, whereas poor compliance makes preventive
care incredibly less effective. Unfortunately compliance is hard to control
as this parameter is not only dependent upon the apparent effectiveness of
treatments but also of the hygiene of the hospital staff.
5.5 Analysis of α
The above examples Figures 5.1 - 5.3 depict the various environments, influ-
ential parameters, and possible controls a hospital may experience. In each
figure the method in α is controlled changes and so does its effect on different
intensive care unit populations and the basic reproductive number. From a
few of these simulations one notices that while controlling α, there still is a
high basic reproductive number throughout, such as Figure 5.1. In Figure 5.1
CHAPTER 5. CONTROLLING α: S TO X 56
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
Control: alpha
Outbreak Risk
(a) p = 0, optimal control.
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
Control: alpha
Outbreak Risk
(b) p = 0, α = 0.2.
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
Control: alpha
Outbreak Risk
(c) p = 1, optimal control.
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Po
pu
latio
n
Colonized
Infected
Control: alpha
Outbreak Risk
(d) p = 1, α = 0.2.
Figure 5.3: Optimal Control of α vs. mean value α = 0.2 with extreme
compliance rates.
CHAPTER 5. CONTROLLING α: S TO X 57
the basic reproductive number stays just above 0.7. In this simulation, the
parameters are set at their mean value, the initial populations are average,
and the weights are normal. So, why is the basic reproductive number so
high? Both plots in Figure 5.2 have high basic reproductive numbers, though
this is to be expected initially because of the environment in this simulation.
This environment is created by altering the parameters in such a manner that
the infection flourishes more easily. In these simulations though, while it is
expected to have a high basic reproductive number, even the optimal control
doesn’t lower it considerably. From Figure 5.1 and Figure 5.2 it seems that
α is not the best parameter to control alone. Introducing optimal control
of α into these two simulations helped the situation when compared to full
control or no control however the basic reproductive number is still too high
to be considered sufficiently controlled. From these simulations α does not
prove to be an extremely useful control, at least not in extreme or mean
value conditions. α may prove useful when paired with an additional con-
trol. In an average intensive care unit, it would be more beneficial to control
k rather than α if a goal is to reduce the value of the basic reproductive
number. In Figure 4.1 and Figure 5.1 the only graph that drastically differs
is the basic reproductive number; the control schedules are different because
CHAPTER 5. CONTROLLING α: S TO X 58
the two controls are different. Figure 4.1 uses a higher level of its control
throughout the simulation, achieves nearly the same results in the infected
and colonized populations as Figure 5.1 while keeping the basic reproductive
number well below 1. In Figure 5.1 the control is not used as much, the
infected and colonized populations are similar to Figure 4.1 yet in this case
the basic reproductive number is high. So what is more important, a higher
cost for placing patients into preventative care or a lower basic reproduc-
tive number? To reduce the chance of an outbreak throughout simulation it
would be best to control k in an average intensive care unit environment.
Chapter 6
Controlling α and k
After controlling α and k separately, the two were controlled together. The
goal was to further minimize our infected populations by optimizing the flow
of patients from both susceptible and colonized without preventive care into
colonized with preventive care. Controlling more than one parameter allows
for a more efficient and optimized result. In this case, all movement into
colonized with preventative care is controlled.
59
CHAPTER 6. CONTROLLING α AND K 60
6.1 Necessary Equations and Constraints
For the simulation of controlling α and k, the necessary equations and con-
straints are depicted below. Equation 6.1 shows the objective function which
will be minimized.
∫ T0
(aX + bY + cV + dW + ek2 + gk + n0α + nα2)dt (6.1)
Equation 6.2 shows the Hamiltonian, which is now a function of 12 functions
and time.
H(X, Y, V,W, k, α) = aX + bY + cV + dW + ek2 + gk + n0α + nα2 + λ1(β(V +X + Y )
+Wγ − Sµ− fS((V +W +X + pY )δ + τ) + µm1) + λ2(m2µ− µX+
(1 − k)f(δS(X + pY + V +W ) + τS) + α1Y −X(β + α + fε))+
λ3(m3µ− µY + kf(δS(X + pY + V +W ) + τS) + αX−
Y (β + α1 + fεp)) + λ4(m4µ− µV + fε((1 − r)X + (1 − q)pY )
+θ1W − (θ + β)V ) + λ5((1 −m1 −m2 −m3 −m4)µ− µW
+fεrX + qpY ) + θV − (θ1 + γ)W )
(6.2)
CHAPTER 6. CONTROLLING α AND K 61
The equations for k∗ and α∗, as used in the optimality condition, are shown
below in Equation 6.3 and Equation 6.4, respectively. They are used to find
the critical functions for the Hamiltonian.
k∗ = −e+f(λ2−λ3)S(τ+δ(V+W+X+pY ))2n0
(6.3)
α∗ = X(λ3−λ2)+n2n
(6.4)
The costate equations for the k and α control are shown below in Equation
6.5.
CHAPTER 6. CONTROLLING α AND K 62
dλ1dt
= −f(1 − k)λ2(τ + δ(V +W +X + pY ))
−fkλ3(τ + δ(V +W +X + pY )) − λ1(−µ− f(τ + δ(V +W +X + pY )))
dλ2dt
= −a− εfλ4(1 − r) − εfλ5r − λ1(β − δfS) − λ2(−α− β − εf − µ+ δf(1 − k)S)
−λ3(α + δfkS)
dλ3dt
= −b− εfλ4p(1 − q) − εfλ5pq − λ1(β − δfpS) − λ2(α1 + δf(1 − k)pS)
−λ3(−alpha1 − β − µ− εfp+ δfkpS)
dλ4dt
= −c− δf(1 − k)λ2S − δfkλ3S − λ1(β − δfS) − λ4(−β − µ− θ) − λ5θ
dλ5dt
= −d− δf(1 − k)λ2S − δfkλ3S − λ1(γ − δfS) − λ5(−γ − µ− θ1) − λ4θ1
(6.5)
6.2 Average Conditions
The first set of graphs used the following initial populations: S0 = 0.45,
X0 = 0.2, Y0 = 0.15, V0 = 0.1, W0 = 0.1, a normal set of initial conditions,
and a = 5, b = 5, c = 10, d = 5, e = 1, g = 5, n0 = 1, n = 5 for the weights.
Figure 6.1 shows a scenario of no control. This means that Y=0 for the initial
population and everyone in colonized with preventative care (Y) is put into
CHAPTER 6. CONTROLLING α AND K 63
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.1: No Control. Initial populations: S0 = 0.45, X0 = 0.35, Y0 = 0.0,
V0 = 0.1, W0 = 0.1, and weights: a = 5, b = 5, c = 10, d = 5, e = 1, g = 5,
n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 64
colonized without preventative care (X=0.35). The results showed that no
one was sent to preventative care. Outbreak prevention stayed at a constant.
Figure 6.2 shows the scenario of initial conditions with optimal control. In
this scenario the controls show more movement. When the optimal control is
achieved for this case, the controls should not be constant and they should be
vary over time. These optimal controls also affect the states. When compared
to one control the optimized total infection dropped almost 1%. It also
lowered the total number of colonized. Also, patients sent from susceptible
to preventative care decrease when the second control is introduced.
In Figure 6.3, full control was tested. This was to see if sending everyone to
preventative care would be beneficial. Sending 100% of the colonized popula-
tion to preventative care is very expensive and resource heavy. It also raises
the outbreak risk compared to the optimal control from peaking at 0.4 up
to 0.7. The total infected remains the same, meaning the hospital is wasting
resources that do not actually help to reduce the number of infected in the
intensive care unit and it causes an increase in the chance of an outbreak.
Figure 6.4 was created to compare to previous research. Previously, it was
discovered that 60% of patients should be sent to preventative care. The
CHAPTER 6. CONTROLLING α AND K 65
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.2: Optimal Control. Initial populations: S0 = 0.45, X0 = 0.2,
Y0 = 0.15, V0 = 0.1, W0 = 0.1, and weights: a = 5, b = 5, c = 10, d = 5,
e = 1, g = 5, n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 66
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.3: Full Control. Initial populations: S0 = 0.45, X0 = 0.0, Y0 = 0.35,
V0 = 0.1, W0 = 0.1, and weights: a = 5, b = 5, c = 10, d = 5, e = 1, g = 5,
n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 67
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.4: Constant Control of 60%. Initial populations: S0 = 0.45, X0 =
0.2, Y0 = 0.15, V0 = 0.1, W0 = 0.1, and weights: a = 5, b = 5, c = 10, d = 5,
e = 1, g = 5, n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 68
figure shows how the population evolves with a constant control of 60%. Our
optimal control differs from the previous because we have created a dynamic
optimization that changes instead of staying constant. Figure 6.4 shows that
with a constant control, outbreak risk increases, total infected increases, and
once again expensive resources are wasted.
6.3 Extreme Case
Figure 6.5, was created to show the effects of an extreme scenario. This
scenario was modified to have the initial conditions S0 = 0.2, X0 = 0.1, Y0 =
0.1, V0 = 0.3, W0 = 0.3. This has a significantly higher rate of total infected
then the previous scenario, which is a lot more dangerous. The graph shows
how quickly the total infected decreases when under the optimal control. It
is also interesting to note that more people are sent from colonized without
preventative care to with preventative care than straight from susceptible.
Figure 6.6 shows the effects of full control on the extreme scenario. The
results are similar to before. Applying full control simply does not use the
resources efficiently and also increases the outbreak risks significantly. In this
CHAPTER 6. CONTROLLING α AND K 69
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.5: Optimal Control. Initial populations: S0 = 0.2, X0 = 0.1,
Y0 = 0.1, V0 = 0.3, W0 = 0.3, and weights: a = 5, b = 5, c = 10, d = 5,
e = 1, g = 5, n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 70
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.6: Full Control. Initial populations: S0 = 0.2, X0 = 0.0, Y0 = 0.2,
V0 = 0.3, W0 = 0.3, and weights: a = 5, b = 5, c = 10, d = 5, e = 1, g = 5,
n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 71
case, there is no good reason to use full control if you can achieve the same
results without using as many resources.
In previous research, the formula for the outbreak risk R was formulated. R
represents the reproductive number which translates into the outbreak risk
of the VRE infection. This is the level of outbreak risk in the intensive care
unit. If it is under 1 there is low risk of an outbreak, if it is over 1 there is
a high chance of outbreak, an epidemic. Therefore, the value of R must be
kept low. Figure 6.7 shows a set of parameters calculated to raise the value
of R over 1 to represent an epidemic. Figure 6.8 shows that extreme case
except with optimal control on admission to preventative care. The graph
shows the spike in R due to the extreme parameters, but the optimal control
is able to lower the level of total infected and more importantly it decreases
the level of outbreak risk to below the dangerous level.
CHAPTER 6. CONTROLLING α AND K 72
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.7: Mean Value Control, k = 0.8 and α = 0.2. Initial populations:
S0 = 0.2, X0 = 0.2, Y0 = 0.0, V0 = 0.3, W0 = 0.3, and weights: a = 5, b = 5,
c = 10, d = 5, e = 1, g = 5, n0 = 1, n = 5.
CHAPTER 6. CONTROLLING α AND K 73
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Popula
tion
k
alpha
Total Infected
Outbreak Risk
Figure 6.8: Optimal Control. Initial populations: S0 = 0.2, X0 = 0.2,
Y0 = 0.0, V0 = 0.3, W0 = 0.3, and weights: a = 5, b = 5, c = 10, d = 5,
e = 1, g = 5, n0 = 1, n = 5.
Chapter 7
Controlling α, k, θ, and q
In the previous chapters, controlling two parameters has a greater effect on
the infected population. It is easier to achieve our goals if we control more
than one parameter at a time. In this chapter, a four-control optimal control
is investigated. The two new controls introduced are analogous to α and k but
with respect to the infected populations. The parameter θ is the movement
from infected without treatment (V) to infected with treatment (W), and
the parameter q is the proportion of patients moving from colonized with
preventive care (Y) to infected with treatment (W).
74
CHAPTER 7. CONTROLLING α, K, θ, AND Q 75
7.1 Necessary Equations and Constraints.
The necessary equations and constraints for the control of α, k, θ, and q
are portrayed in the equations below. Equation 7.1 depicts the objective
function, which is clearly much more complex than the previous ones as this
simulation has four controls. This objective function will be minimized.
∫ T0
(aX + bY + cV + dW + e1k + e2k2 + e3α + e4α
2 + e5θ + e6θ2 + e7q + e8q
2)dt
(7.1)
Equation 7.2 shows the Hamiltonian, which is now a function of 14 functions
and time.
CHAPTER 7. CONTROLLING α, K, θ, AND Q 76
H(X, Y, V,W, k, α, θ, q) = aX + bY + cV + dW + e1k + e2k2 + e3α + e4α
2 + e5θ+
e6θ2 + e7q + e8q
2 + λ1(β(V +X + Y ) +Wγ − Sµ−
fS((V +W +X + pY )δ + τ) + µm1) + λ2(m2µ− µX+
(1 − k)f(δS(X + pY + V +W ) + τS) + α1Y −X(β + α + fε))+
λ3(m3µ− µY + kf(δS(X + pY + V +W ) + τS) + αX−
Y (β + α1 + fεp)) + λ4(m4µ− µV + fε((1 − r)X + (1 − q)pY )
+θ1W − (θ + β)V ) + λ5((1 −m1 −m2 −m3 −m4)µ− µW
+fεrX + qpY ) + θV − (θ1 + γ)W )
(7.2)
The equations for k∗, α∗, θ∗, and q∗, as used in the optimality condition, are
shown below in Equations 7.3-7.6, respectively. They are used to find the
critical functions for the Hamiltonian.
k∗ = −e1+f(λ2−λ3)S(τ+δ(V+W+X+pY ))2e2
(7.3)
α∗ = e3−λ2X+λ3X2e4
(7.4)
CHAPTER 7. CONTROLLING α, K, θ, AND Q 77
θ∗ = e5−λ4V+λ5V2e6
(7.5)
q∗ = −e7+εf(λ4−λ5)pY2e8
(7.6)
Shown in Equation 7.7 are the costate equations for the control of α, k, θ,
and q. These are used to solve for the adjoint.
dλ1dt
= −f(1 − k)λ2(τ + δ(V +W +X + pY )) − fkλ3(τ + δ(V +W +X + pY ))
−λ1(−µ− f(τ + δ(V +W +X + pY )))
dλ2dt
= −a− εfλ4(1 − r) − εfλ5r − λ1(β − δfS) − λ2(−α− β − εf − µ+ δf(1 − k)S)
−λ3(α + δfkS)
dλ3dt
= −b− εfλ4p(1 − q) − εfλ5pq − λ1(β − δfpS) − λ2(α + δf(1 − k)pS)
−λ3(−α1 − β − µ− εfp+ δfkpS)
dλ4dt
= −c− δf(1 − k)λ2S − δfkλ3S − λ1(β − δfS) − λ4(−β − µ− θ) − λ5θ
dλ5dt
= −d− δf(1 − k)λ2S − δfkλ3S − λ1(γ − δfS) − λ5(−γ − µ− θ1) − λ4θ1
(7.7)
CHAPTER 7. CONTROLLING α, K, θ, AND Q 78
7.2 Average Conditions
Figure 7.1 shows two graphs side by side, the graph on the left is the control
graph, where each control α, k, θ, and q is shown. The graph on the right is
the population and outbreak risk graph; shown are the infected and colonized
populations and the basic reproductive number which represents the risk of
outbreak. This simulation is identical to the simulation in Figure 4.1, that
is the initial populations, parameters, and weights used are all the same;
the only difference is the number of controls used. In Figure 4.1 only k is
being controlled, while in Figure 7.1 α, k, θ, and q are being controlled.
The two figures have similar population trends, especially the total infected
population. In comparison, notice that the four-control simulation lowers the
total colonized population, raises the total infected a little, and the outbreak
risk begins decreasing immediately rather than after day 10. For this average
case, it seems that controlling four parameters led to a small risk of an
outbreak but a slightly higher infected population, on the order of 1 − 2%.
If the ultimate goal is minimize the total infected populations, than perhaps
having more controls does not necessarily imply a better outcome. Which is
more desirable, total infected population or lower risk of outbreak?
CHAPTER 7. CONTROLLING α, K, θ, AND Q 79
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
Alpha
Theta
q
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
Colonized
Infected
Outbreak Risk
Figure 7.1: Optimal Control. Initial populations: S0 = 0.7, X0 = 0.1,
Y0 = 0.1, V0 = 0.05, W0 = 0.05. Weights: a = 1, b = 5, c = 10, d = 5, e = 1,
and g = 5.
CHAPTER 7. CONTROLLING α, K, θ, AND Q 80
7.3 Extreme Case
Figure 7.2 shows the same simulation ran as in Figure 6.8 except with the
four-control. We again investigate wether more controls are better for a
given simulation, or better in general. In Figure 7.2(a) full control was used.
This means that each control was set to 100%. Notice, at the top of the
graph on the left all four plots overlap one another because they all have the
same constant value of 1. The graph on the right of Figure 7.2(a) shows the
total infected and colonized population along with the outbreak risk. At full
control the outbreak risk is about 2.25, its precise value is not important,
what is important is that is far greater than 1, this is not good. Figure ??
shows the same simulation but with optimal control. The graph on the left
shows the control functions and the graph on the left shows the infected and
colonized populations along with the outbreak risk. The control graph of
Figure 7.2(b) shows that it is not best to keep the controls at full control,
rather it is better to vary them over time. The graph on the right shows
a large spike in the outbreak risk, peaking at just over 1.6. This spike is
considerably lower than the spike in Figure 7.2(a). This spike is unavoidable.
Recall that this simulation is an extreme case where the infection has a
CHAPTER 7. CONTROLLING α, K, θ, AND Q 81
great probability of spreading, the bacteria are able to reproduce quickly and
efficiently, and there is literally no compliance. Given the specific parameters
of this simulation the outbreak risk should be extremely high. The optimal
control lowers the outbreak risk which was the largest problem in Figure
7.2(a).
CHAPTER 7. CONTROLLING α, K, θ, AND Q 82
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
Alpha
Theta
q
0 5 10 15 20
0
0.5
1
1.5
2
2.5
3
Time
Popula
tion
Colonized
Infected
Outbreak Risk
(a) Full Control
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Time
Popula
tion
k
Alpha
Theta
q
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Popula
tion
Colonized
Infected
Outbreak Risk
(b) Optimal Control
Figure 7.2: Extreme Case. Initial Populations: S0 = 0.2, X0 = 0.2, Y0 = 0.0,
V0 = 0.3, W0 = 0.3, and weights: a = 5, b = 5, c = 10, d = 5, e = 1, g = 5,
n0 = 1, n = 5.
Chapter 8
Conclusion
The results achieved in this study support the idea that a full control is not
necessarily the best. Using an optimal control can result in the same number
of lives saved and same level of outbreak risk as a constant control, while
saving valuable resources such as money. It has also been found that the
optimal control is extremely sensitive to initial conditions and the choice of
parameters. Altering the parameters, even slightly, can have a large effect
on how the system evolves over time. Controlling only one parameter at a
time is useful, but not the most useful. Some parameters, when controlled
alone, are better than others. Comparing the simulations of controlling only
83
CHAPTER 8. CONCLUSION 84
k and only α, controlling k seems to be more effective than only controlling
α. Why is this? Perhaps it is the fact that if patients are sent directly from
susceptible to special preventive care, they will not make contact with other
colonized patients without preventive care, thus reducing their exposure to
patients who are not receiving preventive care. There may be other less
obvious reasons. Regardless, it is found that α by itself is not as effective
as k is by itself. However, when α and k are paired together and controlled,
the so called ’α-k’ simulations, produce good consistent results. In these
series of simulations both k and α are utilized less than when they are both
controlled alone. This leads to the notion, that when two or more controls
are paired together each has to ’work’ less, and there is more ’cooperation’
between the controls, all working together to achieve the same common goal.
It is better to have a combination of controls working together instead of
just one control, especially two controls that deal with two different types
of the population, such as controls with the colonized populations and the
infected populations. Another benefit of using multiple controls is the weight
of importance or influence on any one control is reduced and shared amongst
the other controls. Consider the graphs where k is the only control; k does
a lot of work, in many simulations the value of k may remain close to or
CHAPTER 8. CONCLUSION 85
at 100% for some duration of the simulation. If more than just k is being
controlled the value of each control does not reach or maintain such a high
value compared to if it were being controlled alone. For this reason, in the
four-control simulations of α, k, θ, and q the controls have values far less than
100% and tend to maintain lower values throughout the simulation. This is
cost effective and good for the hospitals. Future work with these results
include implementing the optimal methods of action into hospitals, as well
as applying our programs to different situations. This research focused more
simply applying optimal control theory to the VRE model. Some parameters,
weights, and initial conditions were changed to see what effect there might
be, but extensive studies into these alterations is still needed. In particular,
alterations to the weights of the objective function could lead to some very
interesting simulations where the idea of cost itself may be challenged. This
research focused mainly the presumed belief that a patient’s life is extremely
important, it is more important to reduce the cost of lives over the cost of
treatment. Future research may want to consider the optimal controls with
weights where this distinction between cost becomes blurred. There are many
interesting angles left to investigate with this research.
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