Problem 1:
For this model,
For and initial conditions X(0)=999 and Y(0)=1, run the
following simulations:
Problem 2:
In the SIR model, U is a constant rate of recruitment of new susceptibles, X is the number of susceptibles, Y is the number of infecteds (and infectious) and Z is the number of immunes. N = X+Y+Z. The parameters are:
The equations for this system are:
Notice that the equation for the infecteds can be rearranged to give
Here .
For your simulations, let U=10, X(0)=1000, Y(0)=1, and Z(0)=0. Using
the parameter values , k=.01, and
, again make runs for
What if, as a result of vaccination, a percentage of the population were immune right at the start, when the infection is introduced? What would Z(0) or Z(0)/N have to be to prevent an epidemic from taking off?