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# “Zombie Attack_ An Introduction to Quantitative Modeling”

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```									                                     NATIONAL CENTER FOR CASE STUDY TEACHING IN SCIENCE

CASE TEACHING NOTES
for
“Zombie Attack! An Introduction to Quantitative Modeling”
by
Kyla M. Flanagan
Department of Biological Sciences,

INTRODUCTION / BACKGROUND                                               • Use a flow diagram to explain the meaning of the
In this case study, students are led through the process of               coupled ordinary differential equations describing
developing a mathematical model of a Zombie outbreak,                     the rates of change of state variables.
which they then use to evaluate different “management”                  • Develop and hypothesize links between the zombie
strategies to ensure human survival. Students are                         model and real biological problems, such as predator-
introduced to a flow diagram, learn how to explain how                    prey interactions, rates of spread of disease, rates of
equations are derived from a flow diagram, and evaluate                   spread of invasive species, etc.
predictions of the model. Links between a zombie                        • Explain the value of using a mathematical model
apocalypse and the spread of invasive species, infectious                 to develop predictions for biological problems and
disease outbreaks, and predator-prey relationships help                   evaluating potential management solutions.
the students to see how important biological processes                  • Examine the trade-off between complexity and
can be modeled quantitatively. Students see how different                 reality in modeling biological problems.
biological rates can control the predictions and outcomes
of a model and discuss the benefits and drawbacks                     CLASSROOM MANAGEMENT
of using a quantitative model to examine biological                   An ideal pre-lecture activity includes watching your
processes. Since the focal species (zombies) are fictitious           favorite zombie movie or TV series to brush up on your
(as far as we know), this case is an ideal way to introduce           basic Zombie Natural History. Starting the class by
students to the process of mathematical modeling                      watching the YouTube video of the zombie ant infection
without worrying about the system-specific details of any             from Planet earth (see References section below) may
particular organism.                                                  also be a great way to grab the students’ attention.
This case was designed for a second year introductory                 The case is to be presented in the interrupted case
ecology course. Ideally students will have finished a popu-           method. The introductory portion of the case (“Out-
lation ecology section and would be able to extend ideas              break!”), including the background information, ant
from population ecology to this case. Additionally, this case         story, and the zombie natural history, could be given as
would be appropriate for an epidemiology course, advanced             a handout to the students before class or given to read in
population ecology courses, applied ecology courses, and              the first few minutes of class. Once the students have read
quantitative biology/mathematical biology courses.                    the background information and the basic goals of the
It is my hope that by using an example which                     case are explained, Part I of the case should be distributed.
students find humorous they will be less intimidated by               Working in small groups, the students will discuss the
the modeling process, more interested in understanding                model proposed and do the task highlighted at the bottom
the development of predictions (will humans be wiped                  of Part I (summarize how individuals move from one state
out?), and thoroughly engaged!                                        to another). As a class, the answers to Part I can be briefly
mentioned and then students can be given Part II. In Part
Objectives                                                            II, the students examine the equations of the model and
Upon completing this case study, students should be able to:          then, working in small groups, answer the question at the
• Define a flow diagram, mass action, state variable,                bottom of Part II. In Part III, again in small groups, the
and parameter.                                                     students interpret a graph to determine the condition that

Case Teaching Notes for “Zombie Attack!” by Kyla M. Flanagan                                                                Page 1
NATIONAL CENTER FOR CASE STUDY TEACHING IN SCIENCE

allows for human survival. They then have to conclude              I also hope that through this modeling exercise
which management approach will be the most effective.          students will observe that in order to develop a model
Answers to Part III should be discussed as a class after the   we have to make simplifications or abstractions
group work to ensure the students are able to correctly        about the real world. These simplifications can allow
interpret the rates and the implications. In Part IV the       important generalities to emerge that are insightful;
students evaluate the model. This last part may be most        but some simplifications may obscure reality. I suspect
effectively done as a class discussion. Further topics to      that as students work through this exercise, they will
enhance this discussion are included below.                    have many “but what if…” ideas. In modeling, it is
important to consider these “but what ifs” and examine
the consequences, but also to recognize that it is
BLOCKS OF ANALYSIS                                             impossible to include all biological details in a model.
Model Development and Analysis                                 An interesting discussion could arise from this trade-off
A primary goal of this case is for students to experience      between simplicity and reality:
the process of developing and analyzing a model of a             • How do we determine the level of complexity
biological problem. Students should begin to see how               required in a model?
biological problems can be mathematically modeled to             • When is a model too simple? When is it not simple
make predictions about the future and guide decisions              enough?
regarding best courses of action.                                • How do you know if your model is an accurate
There are four stages in the modeling process:                portrayal of reality?
1. Identifying state variables (what are the different
“states” of the system we are examining?).                 Real Biological Processes
2. Producing a flow diagram of the state variables and       A second goal of this case is for students to make links
defining the transition rates between those state         between the developed zombie model and real biological
variables.                                                processes. This zombie model ends up being a kind of
3. Examining coupled ordinary differential equations         interesting hybrid between a predator-prey model and Sus-
(ODEs) that summarize the flow diagram and tran-          ceptible-Infected-Recovered (SIR) disease model. I think
4. Analyzing an outcome of the model for steady state or     in addition to the skills learned in developing the model.
equilibrium consequences (what are the relative abun-     Here are some potential discussion questions regarding
dances of each of the states once equilibrium has been    how this model relates to some basic biological principles:
reached?) and examining the consequences of chang-          • How does our zombie model relate to a predator-
ing the values of transition rates (aka the parameters         prey model? How would the models be similar?
of the model) on the equilibrium expectations.                 How would they be different?
Most biologists will use these four steps in some way or         • If we consider being a zombie to be a disease, how
form when developing a model.                                       might this model relate to models of other diseases
The four steps outlined above can either be made               such as influenza? How would the models be
explicit to the students or can just be implicit in the             similar? How would they be different? How might
exercise. For the beginner, the most important concept              immunity influence our understanding of rates of
learned from this case study should be that the relative            spread of zombies?
values of the different transition rates can have important      • In biology we are often concerned with the outcome
consequences for the outcome of the model predictions.              of a new species being introduced into a different
This is of course a very important concept when students            (novel) environment. How might this zombie model
begin to look at modeling problems with real world im-              relate to an invasive species?
plications, like predator-prey models used for conserva-         • In this model we have not explicitly considered space
tion efforts, or disease models used for predicting rates           (i.e., how close or isolated individuals are from one
of spread of real diseases (as opposed to a zombie plague),         another, how the density of people varies in space,
such as influenza. The ability to think about different             or what if humans can find a refuge from zombies).
rates controlling the numbers of individuals in different           All of these concepts are very important when exam-
states is critical to biology.                                      ining predator-prey relationships, rates of spread of

Case Teaching Notes for “Zombie Attack!” by Kyla M. Flanagan                                                        Page 2
NATIONAL CENTER FOR CASE STUDY TEACHING IN SCIENCE

disease, and rates of spread of invasive species. How      Websites
model would affect the model predictions?                      Planet Earth video of Zombie Ants.
• In this model we assume the movements of both              2. http://www.livescience.com/13046-zombie-fungus-
humans and zombies are random. How might                       carpenter-ant-brain-altering.html
incorporating more complex human and zombie                    More Zombie Ant video.
behaviors influence the predictions of the model?          3. http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf
A published Zombie model.
4. http://www.cdc.gov/phpr/zombies.htm
MODEL EQUATIONS                                                    CDC’s Zombie Preparedness Guidelines.
According to the model in the case, the four different         5. h t t p : / / n e w s . n a t i o n a l g e o g r a p h i c . c o m /
states a human can assume are susceptible (S), undead              news/2011/03/pictures/110303-zombie-ants-
(U), zombie (Z), and dead (D). The equations used to               fungus-new-species-fungi-bugs-science-brazil/
describe the rates of change of the four different state           National Geographic news, detailing the zombie ant biology.
variables are given below:                                     6. http://www.plosone.org/article/
dS/dt = bS − mS − SZ                                         info%3Adoi%2F10.1371%2Fjournal.pone.0017024
dZ/dt = zU − kSZ                                              Hidden Diversity Behind the Zombie-Ant Fungus
dU/dt = SZ − zU                                              Ophiocordyceps unilateralis: Four New Species Described
dD/dt = kSZ + mS                                              from Carpenter Ants in Minas Gerais, Brazil, PloS ONE
Interpretations of the model parameters are as follows:        7. http://www.nature.com/scitable/knowledge/
library/disease-ecology-15947677
Parameter Interpretation                                         “Disease Ecology,” by A. Marm Kilpatrick and Sonia
b     The number of babies born to each indi-                 Altizer, 2010, Nature Education Knowledge 2(12): 13.
vidual per year
m     The probability of death for each indi-             Print
vidual per year                                     Otto, S.P. and T. Day. A biologist’s guide to mathematical
     The probability that a human is infected                modeling in ecology and evolution. Princeton
in an encounter with a zombie                           University Press. New Jersey, USA. (http://press.
princeton.edu/chapters/s8458.pdf )
z     The probability that an undead human
Levin et al. 1997. Mathematical and computational
goes through the zombification process
challenges in population biology and ecosystem
to become a zombie
sciences. Science 275: 334–343.
k     The probability that a zombie is killed in          Jackson, L., A.S.Trebetz, and K.L. Cottingham. 2000.
an encounter with a human                               An introduction to the practise of ecological
modelling. BioScience 50: 694–706.
Answers to the questions posed in the case study are
provided in a separate answer key to the case. Those           Acknowledgements: This case was published with
for this case, go to the key. You will be prompted for a       under CCLI Award #0341279. Any opinions, findings
username and password. If you have not yet registered          and conclusions or recommendations expressed in this
with us, you can see whether you are eligible for an           material are those of the author and do not necessarily
account by reviewing our password policy and then              reflect the views of the National Science Foundation.
apply online or write to answerkey@sciencecases.org.
Copyright held by the National Center for Case Study
REFERENCES                                                     Teaching in Science, University at Buffalo, State University
These materials may be useful to you in preparing for this     of New York. Originally published September 11, 2012.
case study and may be of interest to students in the class.    Please see our usage guidelines, which outline our policy
concerning permissible reproduction of this work.

Case Teaching Notes for “Zombie Attack!” by Kyla M. Flanagan                                                                   Page 3

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