An Epidemiology Model (Differential Equations) Part 2

This module was developed through the support of a grant from the National Science Foundation (grant number DUE-9752555)

Let's consider a more complicated, more realistic model for the spread of a disease. Consider three populations,

S(t) „ 0, is susceptible people at time t (t is in days)

I(t) „ 0, is infected people at time t

R(t) „ 0, is recovered people at time t, who are at least temporarily protected from getting the disease again.

 

Now, these three populations vary according to the equations:

where the parameters are:

b > 0 is the chance that infection is spread when an infected person meets a susceptible person,

n > 0 is the rate at which people recover from the disease, and

g > 0 is the rate at which the protection from having the disease wanes.

Exploration 1

  1. What are the units on the parameters? (Use the equations to make sure they balance.)
  2. The total number of people, N, is the sum of the infected, susceptible, and recovered populations. Show that the model above assumes that N is a constant number by showing that its derivative is zero.
  3. What is an equilibrium solution of a system? Find the two equilibrium solutions for the system above. (Hint: Solve for variable values that make all derivatives equal to zero. Keep in mind that the total population must add up to N, so we can replace R with N-S-I.)
  4. What is the biological meaning of each equilibrium solution?
  5. In the non-zero equilibrium solution, there is an expression which could potentially give a negative population. Give a condition in terms of N, b , and n, which prevents this. State the condition in words. (If this condition is not met, the non-zero equilibrium solution does not exist.)

 

Exploration 2

Let's look at the Sunnyside Day Schools again.

  1. The Sunnyside Country Day School is a small school, and so it does not meet the condition you found in Exploration 1, #5. Make up values for the parameters (with units) that seem reasonable, but fail the condition for existence of the positive steady state. You will need a number for N (call it N1), as well as the others.
  2. Consider the school population, N1, to be constant, so we can use R = N - S - I to replace R in the model. Then we are left with only 2 DE’s, for S and I. Use a computer to draw the phase plane for the system, showing the direction field, the equilibrium solutions, and some other sample solutions. Can you guess what happens as t increases?
  3. Sunnyside City Day School is a larger school. Make up a new value for N (call it N2), and leave the other values the same, so that it now meets the condition from Exploration 1, #5.
  4. Again, using a computer, draw the phase plane for the system, as in #2 above. Can you guess what happens as t increases in this case? Does it depend on the initial conditions?
  5. How is the future different for the two schools? Considering your results, give some ideas for eliminating the disease from the schools.

 

© 2000 Meghan A. Burke, all rights reserved.