1. “Beanbag” biology: feelings count!
Holly Gaff
Old Dominion University
Biological Sciences

2. Who am I? My background: B.S. in Math and Environmental Science
PhD in Mathematics with an emphasis in mathematical ecology
Taught Mathematics for Life Sciences
Participated in math biology undergraduate curriculum development programs for 15 years

3. Where am I now? Old Dominion University Norfolk, Virginia
Tenured faculty in Biological Sciences
PI for many tick and tick-borne disease projects including long-term field study

4. Why do we need math to study biology?
Allows study beyond what is ethically or biologically feasible
Development process helps to formulate quantitative understanding of disease dynamics
Helps identify data gaps
Provides insights into key elements influencing disease spread leading to control options

5. My chosen areas of integration
Ecology
Environmental Science
Epidemiology
Public Health

6. Learning Styles
How do you learn?

7. Teaching Styles How is math taught? How is biology taught?
What are major differences?

8. Tactile learning
After learning to count, when is the last time you used a physical tool to learn a mathematical concept?

9. Bringing math into biology
To make math more
accessible
acceptable
comfortable
Use hands-on lab techniques to teach quantitative concepts!

10. This workshop Mark-recapture Disease spread
Predator-prey relationships
Founder Effect
Evolutionary strategies

11. Mark-Recapture Goal is to get an estimate of total population
Trap animals, mark and release
Repeat trapping and count captures of marked individuals
Important to know if closed or open population

12. Closed populations
No individuals enter or leave the population between surveys
Survey 1
Survey 2

13. Open populations
Individuals enter or leave the population between surveys
Survey 1
Survey 2

14. What makes a population closed?
Dispersal barriers
Philopatry
Large surveyed area
Slow reproductive/death rate
Short time between surveys

15. Lincoln-Petersen method: Closed population
Survey 1:
Survey 2:
Catch several animals
Catch C animals
Count recaptures (R)
Mark all M animals
Return animals to population
Return animals to population

16. Back to games! Remember rules Play nicely with others
Don’t spill the beads!

17. Mark-Recapture Game Guess how many total clear beads you have
Take one sample (one handful) of your population
Count clear beads sampled
Replace all of those clear beads with blue beads
Mix thoroughly!!!
Take another sample (one handful) of your population
Count number of clear and blue beads in re-sample
Resample at least 5 times (do NOT mark, just resample)

18. Mark-Recapture Game
Make sure each member of your group takes one handful in the resampling
How variable are your resamples?
What is the average number marked per sample?

19. Mark-Recapture Game
So how could we now estimate the total number of beads in your cup?
As you repeat the game
If you continue to mark, what would happen?
Does it matter if it is closed or open population?
Many approaches to calculating the total population

20. Survey 2:
C = 15
R = 4
Survey 1:
M = 12

21. Total Population
C = 15
R = 4
N = ???
M = 12

22. Lincoln-Peterson Method
Note that the proportion marked in the population equals the proportion marked in the 2nd sample

23. When would Lincoln-Petersen give you a bad estimate?
Population not closed
Marked animals likely to be re-trapped
Marked animals likely to die
Marks fall off
Tends to overestimate
Doesn’t work if no recaptured marks!

24. Bailey’s Modification
NB= Estimated population
S1= Initial marked population
S2= Recapture sample size
M = Recaptured marked sample size

25. Modified NC= Estimated population S1= Initial marked population
S2= Recaptured population size
M = Recaptured marked population

26. Multiple Samples
Schnabel
Schumacher-Eschmeyer
Bayesian Method

27. Schnabel N = Population Ct= number sampled at time t
Rt= number of recaptured individuals at time t
Ut= number of unmarked individuals at time t (these are marked and then returned)
t = time

28. Schnabel If marked >10% of population

29. Schumacher-Eschmeyer
Back to Lincoln-Peterson

30. Schumacher-Eschmeyer
Find slope of regression line
Take inverse of slope to find N

31. Jolly-Seber Method Works for open population
More steps but still simple calculations
Applets available on-line for calculations

32. To Excel

33. Epidemiology and math Epidemiology is the study of diseases
Uses all kinds of mathematics
Everyone can relate to being sick!

34. A little bit of history Daniel Bernoulli (1760) Ross (1908,1911)
Smallpox
Ross (1908,1911)
Malaria
Kermack and McKendrick (1930’s)
Classic SEIR formulation

35. Basic Model Concepts Identify all stages of a given disease
Susceptible
Exposed
Infectious
Recovered / Removed
Vaccinated
Etc.

36. Basic Model Concepts Identify disease progression
Link stages according to epidemiology of disease

37. Classic Models
SIR - Chicken Pox

38. Classic Models

39. Results of SIR model

40. Other types of models Stochastic models Individual-based models
Difference equation models
Partial differential equation models
Integro-difference equation models
Network models
Etc.

41. Learn by doing Time for hands on games Rules for games
Divide into groups of 4-6 students
Share responsibility for tasks
Don’t spill the beads!!
Ask any questions you have

42. Game #1 – Disease Modeling
Tasks: Cup holder, scribe, clear bead manager, blue bead manager, bead selector
Rules:
Start with 20 clear beads and 1 blue bead
Bead selector pulls out 2 beads (no peeking!!)
If 2 clear or 2 blue – put both back, if 1 clear and 1 blue – put 2 blues back
Repeat until time is up
Scribe counts final numbers

43. What did you get? How many of each bead did you get?
Did everyone get the same results?
Why or why not?

44. Game #2 – Disease Modeling Revisited
Tasks: Cup holder, scribe, clear bead manager, blue bead manager, bead selector
Rules:
Start with 20 clear beads and 1 blue bead
Bead selector pulls out 2 beads (no peeking!!)
If 2 clear or 2 blue – put both back
If 1 clear and 1 blue, flip a coin. If heads, put 2 blues back, if tails, put 1 clear and 1 blue back
Repeat until time is up
Scribe counts final numbers

45. Modifications
Not always sick forever so could replace sick people with recovered people at some time
Could vaccinate people so they can’t get sick
Other ideas?

46. SIR Model

47. What about ecology? Ecology has benefited from math for a longer time
Many ecology concepts are natural models such as predator-prey and competition
Again, looking at populations and flow rates between them

48. More games!!
Again often start from a simple hands on experiment links the math and biology more closely
Often send students out to
measure length and width of leaves
measure length of middle finger to height
Anything that teaches relationships

49. Game #3 Predator-Prey Tasks: Rabbit breeder, Lynx, scribe Rules:
Start with 3 rabbits spread across the meadow
Toss the lynx square once to catch rabbits
3 rabbits = lynx survives and reproduces
All rabbits breed so double the number of rabbits and disperse across the meadow
If lynx doesn’t get 3 rabbits, it dies
If no lynx, one immigrates. If no rabbits, 3 immigrate
Repeat

50. Predator-Prey
Plot the numbers you got for lynx and rabbits in each generation
Can you predict how many there would be in the next generations?

51. Game #4 Bottleneck Effect
Tasks: Cup holder, scribe, clear bead manager, blue bead manager, bead selector
Rules:
Start with one blue bead and one clear bead
Bead selector pulls out 1 bead (no peeking!!)
If pull a blue bead, put two blues back into cup. If pull a clear bead, put two clear beads back into cup.
Repeat until time is up
Scribe counts final numbers

52. Bottleneck What were the results?
What does that imply for genetics in isolated populations?

53. Population Genetics

54. Game Theory thru ABM Competition between players
Focus on pairwise competition
John Maynard Smith – pioneer of biological application of game theory

55. Game Theory Strategies Contests and payoffs
Particular behavior that a player uses
Can be “pure” (always play the same) or “mixed” (alternates between strategies)
Contests and payoffs
Two players interact
Payoff matrix tells who wins/loses based on strategies of both

56. Payoffs Payoff = Benefit from win – cost of loss
Payoff = chance of win * value of win
- chance of loss * cost of loss
Chance of winning or losing depends on frequency of strategies in population

57. Payoff Matrix Opponent’s Strategy A B Strategy E(A,A) E(A,B) E(B,A)
E(B,B)

58. Evolutionary Stable Strategy
Does system evolve to a stable equilibrium
Pure ESS – one strategy outcompetes all others
Mixed ESS – two strategies permanently exist at a set frequency
All individuals have the same fitness at ESS

59. Hawk v Dove Hawk – always fight, never flee
Dove – always flee, never fight
Who wins, i.e., what is the ESS?

60. Hawk v Dove Game Without choice
Everyone gets a card, must play that strategy
Everyone starts with 120 points

61. Hawk v Dove Game Each day Costs 30 points
One encounter with randomly assigned partner
“Die” when you have 0 points

62. Hawk v Dove Rules Hawk encounters Dove Hawk encounters Hawk
Hawk gets 180 points
Dove gets 0 points
Hawk encounters Hawk
Flip a coin to see who wins
Both lose 60 points to encounter
Winner gets 180 points (so net 120)
Dove encounter Dove
Split 180 to get 90 each

63. Hawk v Dove Game Who survived? Hawks or Doves?
Would that change if it cost more fight? To live each day?

64. Hawk v Dove So what are agents? What are rules? What is world?
What is missing?

65. Hawk v Dove adaptive
Play same people in same order, but now you can choose at each encounter if you want to play hawk or dove

66. Hawk v Dove Rules Hawk encounters Dove Hawk encounters Hawk
Hawk gets 180 points
Dove gets 0 points
Hawk encounters Hawk
Flip a coin to see who wins
Both lose 60 points to encounter
Winner gets 180 points (so net 120)
Dove encounter Dove
Split 180 to get 90 each

67. Hawk v Dove Adaptive Who wins now?
What would change that? Cost of fighting? Benefits of winning?

68. Payoff Matrix Opponent’s Strategy H D Strategy E(H,H) E(H,D) E(D,H)
E(D,D)

69. Hawk-Dove Payoff
Opponent’s Strategy
Hawk
Dove
Strategy
-25
+50
+15

70. Hawk v Dove E(H,H) = 0.5*50 + 0.5*-100 = -25 E(H,D) = 1.0*50 -0 = 50
E(D,H) = 0*50+1.0*0 = 0
E(D,D) = 0.5*(50-10) + 0.5*(-10) = 15

71. Make your students into Math Biologists!
Encourage them to learn a lot of math and biology!
Help them learn to play well with others
In the process they will help solve problems and can improve people’s lives

72. QUBES FMN
Beanbag Biology: Teaching Quantitative Biology through Hands-on Activities
Starting Fall 2017
Opportunity for mentoring while you implement these in YOUR class!

73. Questions?