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Two problem for math 1.

Two problem for math 

1.The Kermack-McKendrick model of an epidemic states that the number S = S(t)of individuals susceptible to a disease and I = I(t), the number of infected individuals in a population, aregoverned by

S' = −rIS

 I' = rIS − aI     r > 0, a > 0

In words, this model assumes the per capita rate at which susceptibles become infected is proportional tothe number of infected individuals present, no new susceptibles enter the population, and the per capita rateat which infected individuals recover is constant a. We are interested in the phase portrait of the system inthe first quadrant, S ≥ 0, I ≥ 0.

(a) Find the equilibria of the system. Are they hyperbolic?

(b) Sketch the nullclines and the vector field on the nullclines.

(c) Form a differential equation for dI/dS and solve it in order to find the shape of the solution trajectories.

(d) Sketch the phase portrait. What happens as t → ∞?

(e) Let S0 and I0 be the initial conditions. An epidemic is said to occur if I increases initially. Under whatconditions on S0 and I0 does an epidemic occur?

2.The system
x'= x − by

y' = −cx + y
x(0) = x0>0,  y(0) = y0>0
models the numbers of two groups moving into and out of a city. Here b and c are positive
constants. ( See Exercises 784-786). Suppose initially there are no members of population x in the
city. Suppose individuals of group x are added to (immigrate into) the city at a constant rate r > 0
Then we have the initial value problem
x' = x − by + r
y' = −cx + y
x(0) = 0, y(0) = y0>0
Exercise 787. Use a computer to study solutions of this initial value problem. Organize your
exploration as follows. Choose and fix an initial condition y0 > 0 and graph x and y for an
increasing sequence of immigration rates r. Repeat this for several choices of y0 > 0. What do you
conclude about the long term group composition of the city?
Exercise 788. Find a formula for the solution of this initial value problem.
Exercise 789. Use your answer in Exercise 788 to verify your conclusions in Exercise 787.


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