BIO 5910: Homework Assignment 5
Due on October 1, 2013
1. There are several ways to include births and deaths in the basic disease model
dS
dt
dI
dt
= −αIS
= αIS − µI.
Suppose that all births are into the susceptible group at rate bS, and that there are deaths at
rate kI from the infected group and at rate kS from the susceptible group.
a. What does the birth term mean? Do infected individuals reproduce?
b. Write the system of differential equations.
c. Find the nullclines and equilibria of this model if b > k (try the case with b = 2.0, k = 1.0,
α = 2.0 and µ = 1.0).
d. Find the nullclines and equilibria of this model if b < k. (try the case with b = 0.5, k = 1.0,
α = 2.0 and µ = 1.0).
e. Draw direction arrows, a sample solution, and explain.
2. Suppose that everything is the same as in the previous exercise, but that births are into the
susceptible class at rate b(S + I).
a. Explain in biological terms how the two models differ.
b. Write the system of differential equations.
c. Find the nullclines and equilibria of this model if b > k. (try the case with b = 2.0, k = 1.0,
α = 2.0 and µ = 1.0).
d. Find the nullclines and equilibria of this model if b < k. (try the case with b = 0.5, k = 1.0,
α = 2.0 and µ = 1.0).
e. Draw direction arrows, a sample solution, and explain.
3. Recall our model of healthy and sick proteins (prions) in a cell:
dS
dt
dI
dt
= r − βIS − uS
= βIS − µI.
Use the R code provided to simulate the system. Confirm the results derived in class by choosing
parameter values so that R0 < 1 and R0 > 1. Turn in plots of the phase plane and time series
for each case with an explanation of what the populations are doing.