Optimal Control Theory for Complex Biological Systems

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


5.2 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


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


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.


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.


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.


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


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.


Figure 1.1: Previous VRE Model.


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


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.


Table 2.1: Parameters Whose Value Is Changed


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)


Table 2.2: Summary Table for the New Parameters


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


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-


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,


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


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:


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


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


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.


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.


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.


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


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


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.


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


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


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-


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.


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.


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.


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


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


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.


H(X, Y, V,W, α) = aX + bY + cV + dW + n0α + nα2 + λ1(β(V +X + Y ) +Wγ − Sµ−





+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.


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


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


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,


and g = 5.


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.


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.


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


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.


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


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


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+




+θ1W − (θ + β)V ) + λ5((1 −m1 −m2 −m3 −m4)µ− µW

+fεrX + qpY ) + θV − (θ1 + γ)W )

(6.2)


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.


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


dλ5dt


(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


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.


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


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.


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,


n0 = 1, n = 5.


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.


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


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



e = 1, g = 5, n0 = 1, n = 5.


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,


n0 = 1, n = 5.


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.


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.


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



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.


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−





+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)


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


dλ5dt


(7.7)


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?


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



and g = 5.


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


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).


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,


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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