Exercise A. The overall goal of this exercise is to see two different ways of modeling vaccination. Consider the SIR model for measles we built in the last homework. The model is given by S' = -BSI I' = BSI -TI R = TI. a) How would the model change if the disease started to be more contagious (eg. because of a new variant)? b) Suppose that we introduce vaccination into our model. Vaccinated people are completely immune to the disease. So, we can think of vaccination as going directly from being susceptible from being immune (ie. in this case recovered). Only susceptible people get vaccinated. Suppose that the likelihood of an infection when a susceptible meets an infectious person is 0.1. Suppose that on average, people are infectious for 4 weeks. Finally, the per capita rate of vaccination is 0.02. Write down the new box-and-arrow diagram and the equations for this situation. Describe what each new term in your model represents. Make sure to indicate the time units. c) Same situation as before but now we vaccinate 5,000-people per week. What part of the model would you change? Make sure to indicate the time units. d) Finally, suppose that the new vaccine is imperfect in the sense that vaccinated people can still get the disease though at a smaller rate than susceptible people. How would you model this situation? Suppose that recovered people are completely immune to the disease and that the per capita vaccination rate is v. Write down the new box-and-arrow diagram and the equations for this situation. Describe what each new term in your model represents. Hint: you will need to introduce a new compartmentew variable in the model