3.8. One method to reduce the spread of a disease is to vaccinate those who are susceptible. A model that attempts to account for the vaccination of the is pr rend susceptible group is 3.1 S+1 -21, I R, S - R R-S. Here S, I, and R are the same groups used in the SIR model described in Section 3.1.3. Assume in the problem that S(0) = 30,1(0) = 1,, R(0) = 0, where So and Iare positive. (a) For each reaction give a sentence or two that explains what assumptions were made to produce the given reaction. Make sure to identify the reac- tion(s) used to account for vaccinations. (b) Using the Law of Mass Action, write down the initial value problem that comes from the above reactions. After this use a conservation law to reduce this to a problem for just S and I. (c) Nondimensionalize the reduced problem in part (b), using No = So +1, to scale both S and Use s and i as the nondimensional dependent vari- ables. The final problem, including the initial conditions, should only con- tain four nondimensional parameters. Identify which one is the vaccination parameter (d) What are the steady-states for the scaled problem? (e) Explain why the solution must satisfy 058 51 and 0 Sis 1. What re- strictions, if any, do you need to impose on the nondimensional so the steady-states you found in part (d) satisfy these inequalities? parameters (f) One of the steady-states you found has is = 0. Under what conditions on the nondimensional parameters is this steady-state asymptotically stable? (g) One of the steady-states you found has i, 70, what is called an epidemic equilibrium. Under what conditions on the nondimensional parameters is this steady-state asymptotically stable? (h) With the long-term objective of keeping the number of infected individuals down to a minimum, what conditions, if any, should be imposed on the vaccination rate constant