Problem 1:

For this model,

For and initial conditions X(0)=999 and Y(0)=1, run the following simulations:

• , i.e.
• , i.e.
• , i.e.

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:

• is the background death rate constant.
• k is the excess death rate constant due to the disease.
• is the rate constant for recovery.
• is the effective contact rate per person that would lead to disease if the contact of a susceptible were with an infectious person.

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

• , i.e.
• , i.e.
• , i.e. .

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?

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