New Horizons
VMI Journal of Undergraduate Research
Volume 3 Issue 1 April 2009
TABLE OF CONTENTS
1 From the Executive Editor Sciences
5 Mathematical Model of Rabbit Haemorrhagic Disease Cadet Marshall H. Jarrett (Civil Engineering, ’11) Faculty Mentor: Dr. Lea R. Lanz, Assistant Professor of Mathematics and Computer Science
15 Adaptive Numerical Analysis of Laser Pulses Cadet Thomas M. Shaffner (Physics, ’08) Faculty Mentors: Dr. John R. Thompson, Professor and Head, Department of Physics and Astronomy and Dr. Troy J. Siemers, Associate Professor of Mathematics and Computer Science
31 A Kinematic Model for Hand Movements Cadet Christopher M.P. Leach (Mechanical Engineering, ’10) Faculty Mentor: Dr. Vonda K. Walsh, Professor of Mathematics and Computer Science Engineering
41 Two-Dimensional Transient Heat Transfer Experiment Cadet Hsin-sheng, Lee (Mechanical Engineering, ’09) Faculty Mentor: Dr. Robert L. McMasters, Professor of Mechanical Engineering
49 Thermal Distortion of a Subscale Membrane Mirror Cadet Scott T. MacDonald (Mechanical Engineering, ’10) Faculty Mentor: Dr. Joseph R. Blandino, Professor of Mechanical Engineering Interdisciplinary
59 The Rhetoric of Science: A Case Study of Susumu Tonegawa’s Landmark Discovery Cadet Joshua C. Kenny (Biology, ’09) Faculty Mentor: Dr. Christina R. McDonald, Institute Writing Director Humanities
67 Learning to See: The Black Mountain College Experiment Cadet Even T. Rogers (English and Fine Arts, ’10) Faculty Mentor: Dr. Robert L. McDonald, Professor of English
83 Kitchener to the Somme: British Strategy on the Western Front during the Great War Cadet Gregory E. Lippiatt (History and English, ’09) Faculty Mentor: Dr. Charles F. Brower IV, Acting Director, VMI Center for Leadership and Ethics
93 Marshall and the Politics of Command: 1906-June 6, 1944 Cadet John M. Curtis (History, ’10) Faculty Mentor: Dr. Malcolm Muir, Henry King Burgwyn, Jr. Boy Colonel of the Confederacy Chair in Military History
111 About the Contributing Editors
115 Undergraduate Research at VMI
117 In Memoriam New Horizons r Volume 3 r Number 1 r 2009
From the Executive Editor
No hay lı´mites salvo el cielo. [The sky’s the limit.] Miguel de Cervantes
hree years have passed since New through any concerted effort by the editorial T Horizons editorial board first presented board, however, but rather by the level of its vision for the new journal to the VMI academic excellence established by the cadet faculty and Corps of Cadets. With only a authors themselves in volumes 1 (2007) and blueprint in our collective conscience, we had 2 (2008) of New Horizons. little to show our first audience except a Like their predecessors, the nine cadets conceptual diagram and a bulleted list of goals whose research appears in this year’s print and aspirations. We had set demanding edition—as well as the three cadets whose standards, too demanding many commented, work will appear in the electronic version of when they learned of the multi-layered process New Horizons, volume 3—have successfully we had set forth for prospective authors. met the demands of an eight-month review/ “Daunting,” said some, “too ambitious,” revise process, including recommendation for retorted others, while still others just shook publication by an anonymous third-party their heads with gentle kindness and reader. More likely than not, the reviewer’s sympathetically expounded on the futility of response is the first ungraded qualitative tilting at windmills. evaluation of their work these cadets have Undoubtedly, we have learned many lessons ever received, and at first glance they may about publishing a journal of undergraduate have found the comments overly critical or research since 2006 and even tilted at the even dispiriting. Academic review, after all, occasional windmill. But we have never had does not tend towards gentle kindness, but to compromise the standards we put forth at rather constructive criticism, which can the inception of the journal. To the contrary, sometimes overwhelm even the most the bar for publication has been raised, not seasoned writer.
1 2 New Horizons / April 2009
The cadet authors and cover designer science, and philosophy in representation of whose work comprise this third volume of six different institutions. The New Horizons New Horizons—along with their faculty Editorial Board and the Institute are indebted mentors—represent eight departments across to them for the generosity of their time and VMI’s three academic divisions. And while expertise on behalf of our cadets. interdisciplinarity is not a new feature in our We are equally grateful to all our colleagues journal, the extent and breadth of the at the Institute who were no less generous with collaborative efforts across the curriculum and their time and intellect in their service to cadet throughout the inquiry/writing/review development by serving as research mentors process in this year’s issue merit special and contributing editors for this third volume mention. Chapeau bas as well to the VMI of New Horizons. The unconditional support Department of Physics and Astronomy, who and enthusiasm we receive from faculty make their drum-roll debut in New Horizons colleagues, the dean’s office, our “Friends of with nothing less than one article, a New Horizons” and especially Dr. Jim contributing editorship, the cover design, and Turner, Director of the VMI Undergraduate the newest member of the New Horizons Research Initiative, continue to inspire us, as Editorial Board, Dr. George M. Brooke, IV, cadets set and re-set the standard of Assistant Professor of Physics. undergraduate research at the Institute In addition to the cross-disciplinary evermore skyward. investigative endeavors that define the 2009 Finally, my heartfelt thanks to my fellow edition of our journal, the editorial board editors Alexis Hart, Bob McMasters, and takes exceptional pride in the number of Merce Brooke, without whom this intellectual colleagues who graciously served as quest would be only an endless row of extramural reviewers. This year’s directory windmills on the horizon. of contributing editors includes non-VMI teacher/scholars from the fields of rhetoric, Mary Ann Dellinger engineering, American literature, political Executive Editor, New Horizons
New Horizons is published annually through the VMI Undergraduate Research Initiative. For information, contact: [email protected] or Ms. Leslie Joyce, Undergraduate Research, 309 Science Building, VMI, Lexington, VA 24450. SCIENCES New Horizons r Volume 3 r Number 1 r 2009
Mathematical Model of Rabbit Haemorrhagic Disease
Cadet Marshall H. Jarrett
Faculty Mentor: Dr. Lea R. Lanz, Assistant Professor of Mathematics
ABSTRACT Scientists and mathematicians have developed mathematical models to describe the spread of diseases in populations. An epidemic recently modeled is Rabbit Haemorrhagic Disease (RHD), a disease that first surface in China in the early 1980’s. There are many different mathematical models employed to describe epidemics, and specifically RHD. One type of classical epidemic model, the MSEIR, divides an infected population into different subclasses and uses differential equations to represent population changes in each subclass. From history of the disease, disease characteristics, and mathematical analysis of the model, variations of the MSEIR model, the SIR and SIRS models, are considered as appropriate models for RHD. However, because a replenishment of the susceptible class is not a characteristic of RHD, the SIRS model is more applicable to RHD.
INTRODUCTION numbers of domestic rabbits on farms (Cooke 2002). The loss of millions of In 1984, an epidemic severely attacked consumer rabbits could have crippled millions of Angora rabbits in the People’s businesses relying on the rabbits for meat Republic of China. Spreading rapidly, the and fur (Cooke 2002). However, the disease killed millions of wild rabbits (Cooke uncontrolled spread of the disease led to 2002; Mitro & Krauss 1993). During the research geared toward using the disease to next twelve years, reports of the disease control inflated rabbit populations causing killing thousands of both wild and domestic harm to an area’s natural ecosystem. During rabbits traveled from Asia to Europe, Africa, the mid 1990’s, a research center was and Central America (Cooke 2002). Caused established on Wardang Island in Australia by a calicivirus, the now classified Rabbit to see how the climate could affect the Haemorrhagic Disease (RHD) has become spread of the disease. Before research could endemic among domestic and wild rabbit be completed, the virus escaped to mainland populations. Because of its worldwide Australia and began to spread through wild prominence, research continues to uncover rabbit populations all over the country. At new information about how fast the disease the time of the unwanted release, the spreads and what factors contribute to its epidemic raised concern for Australia’s wild proliferation. rabbit population, but actually turned out to Initially, researchers studied RHD to find be a successful means of biologically ways to prevent the virus from killing large controlling the millions of rabbits that
5 6 New Horizons / April 2009
Australian farmers and naturalists considered populations, B.D. Cooke outlines the as pests (Cooke 2002). The introduction of importance of understanding death rates of the disease dramatically decreased the infected rabbits to better “assist the number of rabbits to a more desirable level. management of wild rabbit populations either Although the disease is relatively new, for conservation or pest control purposes” continuing study of outbreaks confirm certain (Cooke 2002). Because the disease is facts about Rabbit Haemorrhagic Disease. endemic in certain parts of the world, other The Center for Food Security and Public studies continue to produce valuable Health at Iowa State University publishes a information about RHD by using collected web page dedicated to all aspects of the data and mathematical modeling. Using these disease such as species affected, geographic models, researchers can simulate how certain distribution transmission, etc. RHD only factors may or may not lead to higher affects the European rabbit (Oryctolagus mortality rates within an infected population cuniculus) which is prominent in all parts of of European rabbits. In this study, a simple the world from Europe to New Zealand and model of an infected rabbit population will be Australia. When a population becomes constructed to better understand how exposed to the virus, some rabbits contract mathematics can be applied to a natural the disease either directly or indirectly. Direct phenomenon like the spread of Rabbit transmission occurs during oral, nasal, or Haemorrhagic Disease. conjuctival contact with an infected host. Indirect transmission occurs from free virus particles remaining in an infected carcass or 1 MODERN EPIDEMIC MODELING “most or all excretions including urine, feces, Epidemic modeling first appeared in 1766 and nasal secretions” deposited by an infected when Daniel Bernoulli formulated a rabbit (Iowa State 2007). Upon contraction, mathematical model to study the spread of the disease remains latent for a period smallpox (Hethcote 2000). Since then, ranging anywhere between one and three epidemic models evolved into many different days (Cooke 2002). During this incubation shapes and forms. The first step in constructing stage the host can spread the virus, but is not an epidemic model is to divide the members of suffering from the disease. At the end of the the dynamic affected population into different latent period, a rabbit either survives without classes. Then a system of differential equations experiencing any effects of the disease or dies is created. After the model is designed, within 12–36 hours from hemorrhages on its researchers analyze how each subclass changes internal organs ranging from the trachea and during the course of an epidemic. Modern lungs to the liver and kidneys. A portion of models divide a population into many different surviving rabbits develop immunity to the classes. The existence and application of each disease while others remain susceptible. class is purely situational and varies with each However, newborn populations experience a disease and the affected population. Some period of resistance to the disease for around models may include all five subclasses, and as 8 weeks; some experimental results show few as two subclasses adequately represent 40% of young rabbits do not contract RHD some epidemic scenarios. while the adult population experiences 90% mortality (Cooke 2002). The data collected from hundreds of studies 1.1 Population Classes involving the spread of RHD can help people While mathematical modeling cannot understand how to control the spread or exactly replicate the reality of an epidemic, elimination of the disease depending on results provide qualitative analysis of the certain circumstance. In the study Rabbit situation (Murray 2002). As mentioned haemorrhagic disease: field epidemiology above, some epidemic models divide a and the management of wild rabbit population into five different classes: passive Jarrett / Mathematical Model of Rabbit Haemorrhagic Disease 7 immune (M), susceptible (S), exposed (E), A disease’s characteristics determine what infected (I), and removed (R) (Hethcote types of individuals are considered removed. At 2000). Figure 1 is a diagram of the MSEIR the end of the infection period, the individual model and dynamics between classes. may die, gain immunity from experiencing the Looking at the diagram of MSEIR model in disease, or simply recover and return to the figure 1, classification of a population susceptible class. member begins immediately after birth. The M class represents newborn infants possessing some form of antibodies, 2 EPIDEMIC MODELS AND RHD temporarily preventing this class from 2.1 The Kermack-McKendrick (SIR) acquiring the disease. The existence of antibodies, their source, and the duration of Model immunity depends solely on the disease and While more complicated models with the effected species. For example, one extensive parameters may better simulate species exposed to specific disease may disease growth and decay, this section covers experience infant immunity for a lengthy the classic Kermack-Mckendrick model first period of time while another species may introduced in 1927 (Murray 2002; Hethcote have a shortened period of infant immunity 2000). or no infant immunity at all (Hethcote 2000). This Kermack-McKendrick model applied to Individuals capable of contracting the disease RHD studies the effects of the disease on a are classified into the susceptible class (S). During constant population caused by equivalent any epidemic, members of the susceptible class birth rates and death rates. While birth rate interact with members of other classes, including and death rate occur in natural populations, the infective class from which they acquire the but are excluded in this model, a relatively disease. Once acquired, a susceptible individual accurate representation of the disease is becomes “exposed” to the disease and is attainable (Murray 2002). Often referred to as classified into class (E). Members in this class a SIR model, the Kermack and McKendrick experience a period of latency where the disease model separates a population into three exists in an individual but cannot be spread. At different classes: susceptible class (S), infective the end of the latent period, the length of which class (I), and removed class (R) as shown in varies for each disease, an exposed individual figure 2. Many studies commonly omit the (M) becomes infected (Hethcote 2000). Members and (E) classes because they do not affect the classified in this class (I) are capable of infecting interaction between the susceptible and the susceptible class. Many models end with the infective classes. The susceptible class in a removed class (R) which contains individuals population affected by RHD included rabbits recovered from the disease with immunity. capable of becoming infected with the disease,
Figure 1. Representation of the MSEIR model developed by R. W. Hethcote (Hethcote 2000). 8 New Horizons / April 2009
Based on the above assumptions, the SIR model for RHD is as follows dS Figure 2. SIR Schematic with transfer rates ; ¼ SI; ð1Þ and ; dt dI ¼ SI I; ð Þ the infective class consists of rabbits in the dt 2 population that have the disease and are able dR to spread it, and the removed class includes ¼ I: ð Þ dt 3 rabbits who experienced the disease, recover, dS gain immunity, or isolate themselves from the In the SIR Model, dt is the rate of change in population. Essentially, the removed class is the the number of rabbits in the susceptible class, dI remainder of the population after the susceptible dt is the rate of change in the number of dR or infective rabbits are accounted for. rabbits in the infective class, and dt is the rate In this model as shown in figure 2.1, S(t), of change in the number of rabbits in the I(t), and R(t) represent the number of rabbits removed class. Furthermore, parameters and in the susceptible, infective, and removed parameter units are found in table 1. classes, respectively, at any given time t.To derive the differential equations describing 2.2 Analysis of the SIR Model the interactions between the classes, the following conditions are assumed: The following mathematical analysis come from papers by Murray and Hethcote. In r even spacial distribution among order to solve the system of differential susceptible and infected rabbits, equations (1)–(3), it is assumed that at t =0, r the infection rate of susceptible rabbits S(0) = S0 > 0, I(0) = I0 > 0, R(0) = R0 =0.To depends on the interaction between simplify the system, it is also assumed the total rabbits in the susceptible and infective population, N, is always constant. This means classes. Therefore, the rate of transfer of rabbits from the susceptible to infective SðtÞþIðtÞþRðtÞ¼N; SI class is determined by b , where b is the dS dI dR contact rate between the susceptible class þ þ ¼ : which implies dt dt dt 0 and infective class, r rabbits transfer from the susceptible to More specifically, at t =0,S0 + I0 = N. the infective class at the same rate they When analyzing an epidemic model, the key are removed from the susceptible class, point dI/dt determines the occurrence of an r rabbits transfer from the infective class to epidemic. For an epidemic to occur it is the the removed class at the same rate necessary that, dI/dt > 0. Remember the they are removed from the infective class assumption I0 > 0 must exist for any infection and this rate is proportional to the to occur. Looking at equation (2), evaluated at number of infected rabbits. This is t = 0, it follows denoted by gI, where g is the transfer rate from the infective to removed class, Table 1. Parameter values and table design based r because the incubation period of the on R. W. Hethcote (Hethcote 2000). RHD virus is short—between one and three days—the model operates under Variable Description (Units) the assumptions that a susceptible rabbit S Number in the susceptible class (rabbits) will contract the disease immediately after I Number in the infective class (rabbits) R Number in the removed class (rabbits) a direct contact with an infected rabbit. Contact rate (per rabbits per day) Essentially, the latent period is ignored Removal rate (rabbits per day) (Murray 2002). Jarrett / Mathematical Model of Rabbit Haemorrhagic Disease 9