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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, University of Calgary, Alberta, Canada 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 sition rates. explicit links between some basic biology should be made, 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 do you think explicitly thinking about space in this 1. http://www.youtube.com/watch?v=XuKjBIBBAL8 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. ANSWER KEY • 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 answers are password-protected. To access the answers additional support from the National Science Foundation 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