Optimal Control Problems in Public Health

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Feng Lin

Optimal Control Problems in Public Health

Doctor of Philosophy

Mark Lawley Ozan Akkus

Nan Kong Ann Rundell

Dulcy Abraham

Mark Lawley

George R. Wodicka 04/20/2010

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Optimal Control Problems in Public Health


Feng Lin

04/20/2010

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OPTIMAL CONTROL PROBLEMS IN PUBLIC HEALTH

A Dissertation

Submitted to the Faculty

of

Purdue University

by

Feng Lin

In Partial Fulfillment of the

Requirements for the Degree

of


May 2010

Purdue University

West Lafayette, Indiana

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UMI Number: 3413904

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To my beloved husband and loving parents

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ACKNOWLEDGMENTS

First of all, I am grateful to my advisor, Dr. Mark Lawley, for his continuous

support in the Ph.D. program. Dr. Lawley has explored tremendous opportunities

for me to learn and understand the essential of scientific research. He showed me

different means to approach a problem and the importance to be persistent. He made

me a better researcher. He always had confidence in me even when I had doubt

of myself. He taught me how to ask questions, how to express my ideas, and how

to write academic papers. He was always there to meet and talk about my ideas,

to improve my papers and reports, and to ask me good questions to help me think

through my problems. Without his encouragement and constant guidance, and the

tender loving care for me, I could not have finished this dissertation.

I thank Dr. Dulcy Abraham, Dr. Ozan Akkus, Dr. Nan Kong, and Dr. Ann Run-

dell for serving on my dissertation committee and providing valuable suggestions and

advise on my research. I thank Dr. Kumar Muthuraman for his technical guidance

on optimal control theory and sharing his research approach. I also thank Dr. Laura

Sands for sharing her perspective of the American’s long-term care system and her

insight of the public insurance programs. Special thanks to Prof. Pam Aaltonen for

sharing her expertise in public health and her continuous encouragement. She made

me believe that engineers could make significant contribution to public health.

I was delighted to have the opportunity of working at Purdue’s Healthcare Tech-

nical Assistance Program (HealthcareTAP) for several years. Many thanks to Dr.

David McKinnis and Mary Anne Sloan who allowed me to learn and understand

the hospital operations in the real world. I also thank Dr. Patricia Coyle-Rogers

for teaching me how nurses see and solve a problem in real world. I was also very

fortunate to have the opportunity of working at the Purdue Homeland Security Insti-

tute on Pandemic Exercise Preparedness Program. I thank Dave Hankins, Timothy

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Collins, and Martha Burns who generously shared their expertise in emergency re-

sponse and planning. Such experience not only helped me identify my dissertation

direction, but it also helped me tremendously in balancing academia pursuits and

practical implications.

Let me also say thank you to my fellow research colleagues and friends: Dr. Ping

Huang, Dr. Ayten Turkcan, Dr. Po-Ching DeLaurentis, Santanu Chakraborty, Ji

Lin, Dr. Renata Konrad, Rebeca Sandino, Brian Leonard, and Dr. Arun Chockaling.

We had many enjoyable and memorable moments shared over the past few years.

Last, but not least, I thank my family: my parents, Xiarong Lin and Liying Zhou,

for giving me life in the first place, for educating me with aspects from both arts

and sciences, for unconditional love and encouragement to pursue my interests; my

dearest husband, Chenzhou, for his love, encouragement, and never-ceasing faith in

me.

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TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Public health . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Public health achievements . . . . . . . . . . . . . . . . . . . 2

1.2.2 Public health challenges . . . . . . . . . . . . . . . . . . . . 4

1.3 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Optimal control and public health . . . . . . . . . . . . . . . . . . . 7

1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Outline of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER 2. AN OPTIMAL CONTROL THEORY APPROACH TO NON-PHARMACEUTICAL INTERVENTIONS . . . . . . . . . . . . . . . . . 11

2.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Optimal control model . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Computational examples . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 NPI policies assuming linear NPI implementation cost . . . 21

2.4.2 NPI policies assuming quadratic NPI implementation cost . 27

2.4.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.4 Sensitivity to exponential terminal time assumption . . . . . 38

2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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Page

2.5.1 Effect of NPI policies on the epidemic . . . . . . . . . . . . . 39


2.5.3 Linear v.s. quadratic cost functions . . . . . . . . . . . . . . 42

2.5.4 Exponential vaccine arrival time . . . . . . . . . . . . . . . . 43

2.5.5 Model limitations . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

CHAPTER 3. STOCHASTIC CONTROL WITH DEGRADING COMPLI-ANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


3.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Public compliance models . . . . . . . . . . . . . . . . . . . 52

3.2.2 Stochastic optimal control model with public compliance . . 55

3.3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Computational examples . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1 State-dependent compliance . . . . . . . . . . . . . . . . . . 59

3.4.2 Time-dependent compliance . . . . . . . . . . . . . . . . . . 61

3.4.3 State- and time-dependent compliance . . . . . . . . . . . . 62

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.1 Effect of compliance . . . . . . . . . . . . . . . . . . . . . . 65

3.5.2 Practical implication . . . . . . . . . . . . . . . . . . . . . . 67

3.5.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 68

3.5.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

CHAPTER 4. CAPACITY PLANNING OF PUBLICLY FUNDED COMMU-NITY BASED LONG-TERM CARE . . . . . . . . . . . . . . . . . . . . 72


4.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Admission rate modeling . . . . . . . . . . . . . . . . . . . . 80

4.3 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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Page

4.4 Computational example . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Application to population with dementia . . . . . . . . . . . 92


4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.5.1 Advantages of HCBS . . . . . . . . . . . . . . . . . . . . . . 100

4.5.2 Limited HCBS expansion . . . . . . . . . . . . . . . . . . . . 100


4.5.4 Methodology deficiency . . . . . . . . . . . . . . . . . . . . . 102

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

CHAPTER 5. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.1 Optimal NPI implementation during influenza pandemic . . . . . . 107

5.2 Capacity planning of publicly funded community based long-term care 109

5.3 Extensions to other public health problems . . . . . . . . . . . . . . 111

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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LIST OF TABLES

Table Page

2.1 Model notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Comparison of the means of the expected person-days lost per person dueto death and control intensity between the linear and quadratic models 29

2.3 Design of experiment for parameter effect analysis . . . . . . . . . . . . 31

2.4 Parameter ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Descriptive statistics from the uncertainty analysis . . . . . . . . . . . 34

2.6 Partial rank correlation coefficients . . . . . . . . . . . . . . . . . . . . 35

2.7 Descriptive statistics of percentage difference in cumulative deaths at ex-ponential and gamma terminal times . . . . . . . . . . . . . . . . . . . 38

3.1 Model notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1 Model notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3 Design of experiment for sensitivity analysis . . . . . . . . . . . . . . . 96

4.4 Partial rank correlation coefficients (PRCCs) for the optimal HCBS ca-pacity and the corresponding total expenditure over 1500 experiments . 97

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LIST OF FIGURES

Figure Page

2.1 Scheme of Susceptible-Infectious-Recovered Model. Boxes represent com-partments and arcs represent flux between compartments. . . . . . . . 13

2.2 Scheme of Susceptible-Infectious-Recovered/Death (SIRD) Model. Boxesrepresent compartments and arcs represent flux between compartments. 17

2.3 Optimal NPI policy and optimal isolation policy derived in [103] for aninfluenza characterized as β = 0.4, γ = 0.25, τ = 0.05, c = 0.05, andb = 0.2β. (a) Optimal NPI policy for the SIRD model. (b) Optimalisolation policy derived in [103]. . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Optimal NPI policy and optimal isolation policy derived in [103] for aninfluenza characterized as β = 0.6, γ = 0.25, τ = 0.05, c = 0.05, andb = 0.2β. (a) Optimal NPI policy for the SIRD model. (b) Optimalisolation policy derived in [103]. . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Epidemic curves of infectious and dead population with and without NPIimplementation in a pandemic characterized as β = 0.4, γ = 0.25, τ =0.05, c = 0.05, and b = 0.2β. (a) Epidemic curves with and withoutNPIs starting from x0 = (99%, 1%, 0, 0). (b) Epidemic curves with andwithout NPIs starting from x0 = (67%, 33%, 0, 0) (to be compared withFig. 2.5(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Epidemic curves of infectious and dead population with and without NPIimplementation in a pandemic characterized as β = 0.6, γ = 0.25, τ =0.05, c = 0.05, and b = 0.2β. (a) Epidemic curves with and withoutNPIs starting from x0 = (99%, 1%, 0, 0). (b) Epidemic curves with andwithout NPIs starting from x0 = (50%, 50%, 0, 0) (to be compared withFig. 2.6(a)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Optimal NPI policy obtained under quadratic control cost. (a) OptimalNPI policy assuming quadratic control cost for an influenza pandemiccharacterized as β = 0.4, γ = 0.25, τ = 0.05, c = 0.05, and b = 0.2β.(b) Optimal NPI policy assuming quadratic control cost for an influenzapandemic characterized as β = 0.6, γ = 0.25, τ = 0.05, c = 0.05, andb = 0.2β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Interpretation of the effect of parameter values on the size of control space. 30

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Figure Page

2.9 Interpretation of the effect of parameter values on the size of control space. 32

2.10 Empirical CDFs for the proportion of control area, ω, and mean cumula-tive death, dT , obtained from the 1000 LHS scenarios. (a) Empirical CDFfor the proportion of control area ω. (b) Empirical CDF for the meancumulative death dT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.11 Partial rank scatterplots of the ranks for ω and each of the five sampledinput parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Scheme of Susceptible-Infectious-Recovered/Death (SIRD) Model. Boxesrepresent compartments and arrows represent flux between compartments. 50

3.2 State-dependent compliance model . . . . . . . . . . . . . . . . . . . . 53

3.3 Time-dependent compliance model . . . . . . . . . . . . . . . . . . . . 54

3.4 State- and time-dependent compliance model . . . . . . . . . . . . . . 55

3.5 Optimal NPI policy derived for an influenza pandemic characterized asβ = 0.6, γ = 0.15, τ = 0.05, and b = 0.2β under the assumption ofstate-dependent compliance. (a) Optimal NPI policy under linear costassumption. (b) Optimal NPI policy under quadratic cost assumption. 60

3.6 Optimal NPI policy derived under the assumptions of time-dependentcompliance and linear control cost. (a) Optimal NPI policy at Week 1.(b) Optimal NPI policy at Week 3. (c) Optimal NPI policy at Week 5.(d) Optimal NPI policy at Week 7. . . . . . . . . . . . . . . . . . . . . 62

3.7 Optimal NPI policy derived under the assumptions of time-dependentcompliance and quadratic control cost. (a) Optimal NPI policy at Week1. (b) Optimal NPI policy at Week 3. (c) Optimal NPI policy at Week 5.(d) Optimal NPI policy at Week 9. . . . . . . . . . . . . . . . . . . . . 63

3.8 Optimal NPI policy derived under the assumptions of state- and time-dependent compliance and linear control cost. (a) Optimal NPI policy atWeek 1. (b) Optimal NPI policy at Week 3. (c) Optimal NPI policy atWeek 5. (d) Optimal NPI policy at Week 7. . . . . . . . . . . . . . . . 64

3.9 Optimal NPI policy derived under the assumptions of state- and time-dependent compliance and quadratic control cost. (a) Optimal NPI policyat Week 1. (b) Optimal NPI policy at Week 3. (c) Optimal NPI policy atWeek 5. (d) Optimal NPI policy at Week 9. . . . . . . . . . . . . . . . 66

3.10 Time line and milestones of 2009 H1N1 outbreak and public health re-sponses in the State of Indiana. . . . . . . . . . . . . . . . . . . . . . . 68

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Figure Page

4.1 The compartmental model for HCBS capacity planning. Boxes representcompartments and arrows represent flow between compartments. “H” -HCBS; “N” - institutional care; “E” - without LTC; “D” - death. . . . 77

4.2 Admission rates into nursing home and HCBS program vs. capacity ofHCBS program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 Annual spending on the studied population vs. capacity of HCBS programusing dynamic admission rates αH(S, x(t)) and αN(S, x(t)). . . . . . . . 93

4.4 Annual spending on the studied population vs. capacity of HCBS programusing relaxed admission rates α̃H(S, x(0)) and α̃N(S, x(0)). . . . . . . . 94

4.5 Optimal HCBS capacity S∗ against CH and CN under two HCBS infras-tructure costs (CS = 5000 and 10000) when the proportion of people to beadmitted to receiving LTC is 20% (β = 20%). (a) Optimal HCBS capac-ity S∗ against CH and CN under CS = 5000 and β = 20%. (b) OptimalHCBS capacity against CH and CN under CS = 10000 and β = 20%. . 104



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ABBREVIATIONS

SIR Susceptible-infectious-recovered model

SIRD Susceptible-infectious-recovered/death model

NPI Non-pharmaceutical interventions

LHS Latin hypercube sampling

LTC Long-term care

HCBS Home and community based services

IP Inpatient hospitalization care

ER Ambulatory care or emergency visit

ADL Activities of daily living

IADL Instrumental activities of daily living

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ABSTRACT

Lin, Feng. Ph.D., Purdue University, May 2010. Optimal Control Problems in PublicHealth. Major Professor: Mark Lawley.

The health care delivery system in the United States is poorly planned to meet

the growing needs of its population. This research establishes the foundations of

developing decision-support tools in the emerging field of health care engineering,

with special emphasis on public health. It demonstrates the potential of applying

engineering methods, especially optimal control theory, to facilitate decision making

in the complex health care delivery systems. Two compelling problems of public

health are studied: 1) how to optimally implement non-pharmaceutical interventions

to mitigate an influenza pandemic; and 2) how to allocate limited long-term care

budget effectively.

1. Pandemic planning: Optimal implementation of non-pharmaceutical

interventions during influenza pandemic

Non-pharmaceutical interventions (NPIs) are the first line of defense against pan-

demic influenza. These interventions dampen virus spread by reducing contact be-

tween infected and susceptible persons. Because they curtail essential societal ac-

tivities, NPIs must be applied judiciously. Their effectiveness also depends on the

degree of public compliance, as NPIs require people to change their daily behaviors.

The public “buy-in” depends on their awareness and perception of the severity of the

outbreak. It is also likely to degrade as time evolves due to compliance fatigue.

In this work, we use an epidemiologic compartmental model to develop optimal

triggers for NPI implementation. The objective is to minimize the expected person-

days lost from influenza related deaths and NPI implementation. In the first part of

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this work, optimal policies for a deterministic control model are derived. A multi-

variate sensitivity analysis is performed to study the effects of input parameters on

the optimal control policy. Additional studies investigate the effects of departures

from the modeling assumptions, including exponential terminal time and linear NPI

implementation cost.

Next, a stochastic control model is developed from the deterministic model to

investigate the effect of public compliance and uncertainties of system dynamics on the

NPI policies. The public compliance is modeled as functions of time and incidence of

infection. Diffusion terms are introduced to capture the uncertainties in the dynamic

of the system. Optimal NPI policies are derived for different compliance functions

and diffusion terms.

Numerical results for interpreting policy characteristics are presented along with

guidelines for practical implementation. Our findings highlight the importance of

timely surveillance and effective risk communications during pandemic outbreak. The

application of optimal control theory can provide valuable insight to develop effective

control strategies for pandemic.

2. Long-term care planning: Capacity planning of publicly funded

community based care

Long-term care (LTC) provides medical and non-medical services to people with

chronic disease or disability, many of whom are older adults eligible for receiving

care through public funding sources. At present, the annual spending on LTC in

the U.S. is over $200 billion and this number is increasing rapidly. The federal and

state governments paying for LTC are under increasing financial pressure. Although

nursing home care has been a viable option, it often provides expensive and more

than necessary care. Home and community based services (HCBS) offers a flexible

alternative by providing care at home and in the community. However, little is known

on how much infrastructure is needed for providing community-based care.

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This research formulates an optimal control problem to determine the optimal

infrastructure capacity of HCBS program from a societal expenditure viewpoint. A

compartmental model is established to describe the population dynamics in the pub-

licly funded LTC system. Two models are considered in determining whether to

provide LTC in the community or in a nursing home. The objective of the optimal

control problem is to minimize the overall expenditure, including spending on long-

term and acute care services, over a given time period. We consider two alternative

models when determining whether to provide LTC in the community or in a nurs-

ing home. Analytical properties are presented along with computational examples

for dementia patients based on published data. A full-factorial sensitivity analysis is

performed to study the sensitivity of various parameters.

The compartmental model is validated against the published data, which indicates

that it is a reasonable abstraction of the LTC system for the elderly. Reduction in

total expenditure suggested by the model indicates that future development of the

LTC system should increase HCBS capacity, but unrestricted HCBS expansion is not

desirable. Also, HCBS cost should not exceed a certain proportion of nursing home

cost for the HCBS program to remain economic.

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Chapter 1. INTRODUCTION

1.1 Background

The health care delivery system in the United States is poorly planned to meet

the growing needs of its population [1,2]. In 2007, the health care spending consumed

over 15% of the U.S. gross domestic product (GDP) [3] and this spending is expected

to grow at over 6.7% annually in the future [4, 5]. However, the performance of such

expensive service is not competent [6, 7], as the system is riddled with inefficiencies,

excessive expenses, and poor management [8]. Notably, the average growth in national

health spending is projected to be 6.2%, which is 2.1% faster than average annual

growth in GDP [9,10]. The public, government, insurers, and industries are straining

under a serious financial and societal crisis [11, 12].

Many of the problems that threaten health care in the U.S. are the result of

uninformed, irrational decision making throughout the system, where data driven

systems modeling and analysis activities are virtually unknown. Engineering tools,

especially mathematical modeling and analysis techniques, are considered as one of

the potential means to facilitate and improve decision making in the complex systems

[13, 14]. Over the past 30 years, the engineering methods have been successfully

applied in the manufacturing, logistics, distribution, and transportation [2, 14, 15].

These methods, properly modified and applied, can provide similar high-level impacts

in health care.

This dissertation employs engineering methodologies, especially optimal control,

to facilitate decision making in health care systems, with special emphasis in public

health. Two compelling problems in public health are studied: one focuses on devel-

opment of response plans for large-scale infectious disease outbreaks, and the other

focuses on allocation of limited resources for the long-term care system. This work

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helps showcase the potential of applying engineering approaches, including operations

research, systems engineering, and applied mathematics, to improve decision making

in health care systems.

1.2 Public health

Public health is “the science and art of preventing disease, prolong-

ing life, and promoting physical health and efficiency through organized

community efforts for the sanitation of the environment, the control of

community infections, the education of the individual in principles of per-

sonal hygiene, the organization of medical and nursing service for the early

diagnosis and preventive treatment of disease, and the development of the

social machinery which will ensure to every individual in the community

a standard of living adequate for the maintenance of health”.

–Charles-Edward A. Winslow 1920 [16]

The goals of public health focus on proactive, interventional, and collective activ-

ities to prevent diseases and maintain the health and well-being of the population.

Public health is a combination of science, technologies, and practice to maintain and

improve the health of all people. It deals with preventive care at population-level,

which differs from clinical medical care that focuses on curative care at individual-

level health issues. While medicine tends to treat a disease when it occurs to a person,

public health focus on identifying and implementing interventions to prevent a disease

through surveillance of cases and the promotion of healthy behaviors [17].

1.2.1 Public health achievements

Over decades, public health has many successes. In the 20th century, public health

in the U.S. made significant accomplishments in responding to infectious diseases and

increasing the lifespan of Americans. The Morbidity and Mortality Weekly Report

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(MMWR) highlights the ten great public health achievements in the 20th century [18].

These achievements include:

• Vaccination [19]

• Motor-vehicle safety [20]

• Safer workplaces [21]

• Control of infectious diseases [22]

• Decline in deaths from coronary heart disease and stroke [23]

• Safer and healthier foods [24]

• Healthier mothers and babies [25]

• Family planning [26]

• Fluoridation of drinking water [27]

• Recognition of tobacco use as a health hazard [28].

These achievements indicate that the major health threats in the 20th century

were infectious diseases associated with poor hygiene and poor sanitation, diseases as-

sociated with poor nutrition, poor maternal and infant health, and diseases or injuries

associated with unsafe workplaces or hazardous occupations [29]. They document the

established roles of public health in responding to the major causes of morbidity and

mortality during that period of time. Nonetheless, they also reflect the science and

technology advances in public health, which involve a variety of fields, including en-

vironmental health, epidemiology, biostatistics, nutrition, behavioral science, health

education, and health services administration and management.

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1.2.2 Public health challenges

Although public health continues to expand its established roles, it is taken for

granted and most people are unaware of its critical activities. Government expen-

ditures on health are alarmingly high, but little of the spending is directed towards

public health. Meanwhile, new health problems have emerged, such as AIDS/HIV epi-

demic, environmental pollution, chronic diseases, aging population, and social prob-

lems [17, 30]. All these issues have become the major concerns and challenges to

public health.

An important issue challenging the public health system is emerging infectious dis-

eases, such as severe acute respiratory syndrome (SARS), influenza, and AIDS/HIV.

These diseases have continually threatened the health of populations worldwide. The

2003 SARS outbreak caused severe widespread societal disruption and significant eco-

nomic losses, apart from the direct costs of medical care and control measures [31–34].

The 2009 H1N1 pandemic has infected over 246,571 humans worldwide, resulting in

at least 12,220 deaths in 2009 [35], and its severity is considered as mild-moderate.

AIDS/HIV is a major global health emergency, causing millions of deaths and suf-

fering to millions more worldwide [34, 36, 37]. Although the treatment of AIDS has

been improved, it remains the leading infectious cause of adult death and it also fuels

reemerging tuberculosis epidemics of global concern [34,37].

Besides infectious diseases, chronic and degenerative diseases have become leading

causes of death and disability [38, 39]. Different from acute conditions, the chronic

conditions, such as heart disease, stroke, cancer, diabetes, and arthritis, are long-

lasting and/or recurrent. They require persistent attention and daily assistance for

the patients and their families. Nearly 133 million Americans have at least one

chronic condition [39–41], and these illnesses cause about 70% of deaths in the U.S.,

accounting for about 75% of the costs each year [39, 40, 42]. As aging population is

highly correlated with such conditions, the size of population with chronic conditions

is expected to grow significantly because of longer life spans and aging baby boomers

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[43]. This number is projected to increase to 171 million by 2030 [40], resulting in

20% increase in the nation’s health care spending [44–46]. Furthermore, people with

chronic illnesses are often associated with racial and ethinic diversities. This without

doubt places significant challenges to health care delivery and access.

In addition to disease control and prevention, public health is also responsible

for emergency preparedness and response, maintenance of healthy environment, and

promotion of healthy lifestyle. The tasks of public health have been not only to iden-

tify health problems, but also to find effective interventions to solve these problems

within a certain social and political structure. All these issues have placed significant

financial burden on the federal and state governments, and posed serious threats to

national security.

Unfortunately, public health is typically underfunded, yet it is responsible for

critical functions that affect the entire population. In 2004, public health activities

accounted for only 3% of total health spending in the U.S., which was much lower than

the spending on personal services and even less than the administration cost [47–49].

Many vital societal functions that public health is responsible for, including disease

control and prevention, environment health, healthy lifestyle promotion, are always

carried out under tight budgets and limited resources. These resources are often un-

evenly distributed, resulting in wide disparities geographically and demographically.

In addition, there exists wide variation among communities, as different communities

are facing distinct health problems and they have different demographic, political and

social structures. Hence, even the approaches to investigate a similar health problem

might differ significantly in different communities. Finally, public health interven-

tions always involve many stakeholders with competing objectives. It is complicated

to balance these competing objectives to achieve the best benefit for the community

as a whole.

To summarize, public health agencies are required to make decisions at various

scenarios from time to time within various political and societal structures. The

decision making is a complicated process that involves the demographic, political,

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and economic factors, and different stakeholders in a given community. It has become

increasingly important to help the public health agencies quantify and understand

the economic and social impact caused by various threats, as well as to inform the

decisions on allocation and management of constrained health care resources.

1.3 Optimal control

Optimal control is a mathematical method for finding optimal means to control a

dynamic system. For many problems, a natural question is how to intervene in the

system to produce the “best” possible outcome, as measured by some predetermined

goals. Optimal control theory is one of the approaches to address such questions.

An optimal control problem is composed of two components, the system dynamics

and the objective. The system is described by a set of state variables and control

variables, which are governed by a set of differential equations. The cost function

measures the performance of the system. Optimal control theory governs strategies

for maximizing/minimizing a performance measure as the state of the dynamic system

evolves.

Optimal control has been successfully applied to many problems in manufactur-

ing, aerospace and defense, automotive systems, structural and mechanical design,

environmental control, and biological and biomedical systems. It has covered a wide

range of interdisciplinary and complex problems, in which balancing the benefit and

the cost of control strategies is important yet nonintuitive. This subject has been

enriched rapidly since the early works of Balakrishnan [50], Butkovskii [51], and Fat-

torini [52]. In the remainder of this section, we review the applications of optimal

control theory in economics and management, engineering, and biological systems.

Optimal control has also been widely used to find control mechanisms for au-

tomotive, and aerospace and defense systems. For example, Nearly [53] develops

throughput control strategies that minimize energy expenditure while satisfying a set

of power constraints for wireless networks.

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Optimal Control Problems in Public Health

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