The Biology and Math Interface
Group Presents…
Our Teachable Tidbit Topic: Exponential
growth and decay with applications to
biology
Learning Outcomes From Tidbit:
The student should be able to:
•
•
•
•
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Fit an exponential model to data
Make predictions using an exponential model
Interpret components of a exponential model
Do sensitivity analysis
Read, understand, and know how to graph an
exponential function
What the instructors and students
need to know
Overall Goals for Unit: Students will appreciate the
importance of mathematics in modeling biological
processes
Skill Level of Students: This tidbit can be used in a
college calculus course or an introductory biology class
What we assumed: Students have little to no prior
experience with college level biology, but students have
been exposed to logarithms and exponentials
Content Presented Prior to Tidbit
Video
http://contagionmovie.warnerbros.com/index.h
tml#/home
Activity Introduction...
If we let P define the number of people
infected lets look at a way we can
model the spread of infection starting
with Gwyneth Paltrow…
Each group should have two cups, one
with pennies and one without. We will
be simulating a model of exponential
growth, that is the spread of disease, by
flipping the pennies and adding a penny
for every head.*
Begin with one penny, the initial
infected individual. Your group will need
a scribe, someone to flip pennies, and
someone to add pennies.
*Exponential decay can also be modeled using pennies just
begin with all pennies instead of one and remove a penny
for every heads or tails.
How Long Until Global Infection?
Introduction:
Penny Penny-Worth has become infected with a virus. Every day she
has a 50% chance of infecting another person. Similarly each
subsequently infected individual has a 50% chance of infecting another
person. In this activity, you will explore exponential growth models and
answer the question of how long it will take Penny to infect the world?
Materials:
 Two cups: Label one cup infected and the second cup healthy
 One-hundred pennies
Activity directions:
Let heads be an infected individual and tails a healthy individual. Put
one penny (representing Penny Penny-Worth) in the “infected” cup and
all other pennies in the healthy cup. Dump out the infected cup onto
the table and record the number of heads. Add one penny from the
healthy cup to the infected cup for each head that you get. Create a
table and record the new total number of infected people. Repeat this
process eight times, each time recording the amount of infected
individual. For example,
time (t)
0
1
2
3
4
5
6
7
8
Number of
infected
people
1
1
2
4
7
11
12
15
23
time (t)
0
1
2
3
4
5
6
7
8
Analysis:
1) Using the average growth ratio (i.e. growth factor), find an
exponential model for the number of infected people as a
function of time (i.e. 𝑃(𝑡) = 𝐾𝑒 𝑟𝑡 ).
2) What does k mean? What does r mean? What are the units
of each?
3) If infection would continue at the current rate when would
the infected population double?
Number of
infected
people
In the year 1995 the population equals
75 if the growth rate is 0.65, calculate
the population in the year 2011.
Suppose we have two diseases spreading
through two different populations. Disease A’s
propagation is identical to disease B, except
disease B has a k value twice as big as disease A.
How is the population of people affected by
disease A compared to disease B?
A. Population of A is twice the population of B
B. Population of B is twice the population of A
C. Both populations are the same
D. Not enough information given to determine populations
E. None of the above
Find the equation for
the exponential
population curve
above.
Homework and Extension Ideas
Example of an isomorphic homework assignment
Acroporid Coral Infection
Introduction:
A particular colony of Acroporid corals is infected with white band disease. Every hour it infects five
other coral colonies. Similarly each subsequently infected colony infects five other coral colonies.
1) Complete the following population table for this disease spread.
time (t)
0
1
2
3
4
5
6
7
8
Number of
infected
coral
colonies
1
2) Using the average growth ratio (i.e. growth factor), find an exponential model for the number of
infected people as a function of time (i.e. 𝑃(𝑡) = 𝐾𝑒 𝑟𝑡 ).
3) What does k mean? What does r mean? What are the units of each?
4) If infection would continue at the current rate when would the infected population double?
5) How sensitive is the equation that you have found?
Extension ideas?????