1. A general model for the spread of an epidemic with S as the proportion of the population that is susceptible, I as the proportion of the population that is infected, and R as the proportion of the population that is recovered can be modeled as the following set of

                                     ds/dt = r(1 - S) - bSI

differential equations: -dI/dt = bSI - cl Whether the recovered proportion becomes

                                      dR/dt = cl

susceptible or not depends on the disease, but in either case they can be factored in without explicit description by adjusting r so we can concentrate on just the first two equations for our analysis.

a. Explain what the parameters r, b, and c are in this model.

b. Using just the first two equations, find the two relevant equilibrium points. What has to be true about the parameters for two relevant equilibrium points to exist?

c. Sketch the nullclines for this system along with directions.

d. Conduct a linearized analysis for this system and discuss the stability of the two equilibrium points. What has to be true about the parameters so that there is a locally stable equilibrium point.

2. Find a Lyapunov function of the form V(x, y) = ax^2m + by²ⁿ to show the global stability of the origin for the following system :

 dx/dt = -3x³ - y

 dy/dt = x⁵ - 2y³

                              dx/dt = 4x+ 4y- x(x² + y²)

3. For the system; dx/dt = - 4x+ 4y - y (x²+y²) transform the system to polar coordinates and

                              dy/dt = -4x + 4y - y(x2 + y2)

show there is a Closed solution between the circles r = 1 and r = 4. Does this verify the Poincare-Bendixon theorem?

4. For the initial value problem dy/dt = t²y, y(0) = 1 over 0 ≤ t ≤ 2 (a) determine the analytic solution, (b) use the three methods (Euler's, Improved Euler's and Runge-Kutta) to approximate y(2) for n =4 steps and compute en, (c) how many steps would you expect to use for each method to obtain an approximation for which en 5 .0001

5. For the difference equation x_{n +1} = F(xn) where F(x)= 6 - x² Find all fixed points and all periodic points of period 2. Determine whether the fixed points are attracting or repelling. Sketch web diagrams to support your conclusions.