The SIR model is a rst-order system

dS/dt = IS N dI/dt = IS N I, dR/dt = I

where S is the number susceptible, I the number infected, and R the number recovered in a population of N individuals.

Here = 1 Tc where Tc is the typical time between contacts and = 1 Tr where Tr is the typical time for an infected individual to recover.

Adding the equations gives

d/dt(S + I + R) = 0

so S + I + R = N, the total population.

Using S = N I R

dI/dt = I(N I R) N I.

Let x = I/N be the fraction of the total population infected and let y = R/N the fraction of the total population recovered.

Then dx/dt = ((1xy))x, dy/dt = x

We call this the IR model for the fractions of infected and recovered.

We assume one unit in t represents the typical time to recovery.

Then = 1/Tr = 1.

The fraction R0 = / is the basic reproduction ratio, the expected number of new infections from a single infection in a population where all subjects are susceptible.

We assume R0 = 2.5 so = 2.5

Consider the IR model:

dx/dt = (2.5(1xy)1)x dy/dt = x.

Construct a single gure including 1. Title IR model Direction Field and Eulers Method solutions by FirstName LastName with your actual rst and last names, along the top

2. Labels for the axes

3. a direction eld for 0.2 x 1.0, 0.2 y 1.0

4. Eulers method solution using stepsize h = 0.1 for initial condition x(0) = 0.0001M, y(0) = 0 where M is the integer corresponding to the rst letter of your last name. If your last name begins with A, M=1, x(0) = 0.0001 and y(0) = 0. If your last name begins with Z, M=26, x(0) = 0.0026 and y(0) = 0.

5. A label (x(0),y(0)) = (x0,y0) but with the actual initial values instead of x0 and y0.

6. A label x(tmax) = xmax but with the actual value of the time tmax at which the maximum value of x occurs and the value of x at that time

7. A label x(tsmall) = 0.0001 but with the actual value of the time tsmall > tmax at which x has the small value 0.0001.