Computer Project

Computer project 1 Solution

The goal of this project is to investigate the contagious behavior of a disease, which is

modeled by the following iterative non-linear model

2

In+1 = f (In ) = In − rIn + sIn 1 −

In

N

,

where In is the number of infected people at n-th week, N is the total population, r is

the recovery rate, and s is a positive constant. Note that this model is different from the

non-linear infection model we’ve discussed in class.

1. Suppose that the recovery rate r is 0.7 and the total population is 10000. Find the

condition for s that f (x) defines a dynamical system (note that the image of f

must be in the domain of f ).

For this model, the domain is [0, 10000]. So In+1 = f (In ) is an unbreakable dynamical system if the image of f is in [0, 10000].

If we plot a general graph of f (x) (with an appropriate s), the graph looks like

the following:

plot(x-0.7*x+0.0005*x^2*(1-x/10000),(x,0,10000),aspect_ratio=1)

+plot(10000,(x,0,10000))

1

MATH 1700 Homework

Han-Bom Moon

So the image of f is in [0, 10000] if the maximum occurring when x is around 7000

is less than 10000. By sketching graphs with various s, we can conclude that if s

is in (0, 0.000537], then f defines an unbreakable dynamical system.

2. In addition to conditions in 1, suppose that s = 0.00051. Find all fixed points.

solution_set = solve(x-0.7*x+0.00051*x^2*(1-x/10000) == x,x)

for s in solution_set:

print float(s.right_hand_side())

1642.24631578

8357.75368422

0.0

Thus we can find three fixed points 1642.24631578, 8357.75368422, and 0.

3. By making a table of In for 0 ≤ n ≤ 20 with various I0 , investigate the stability of

each fixed point.

I = [1500]

for j in range(20):

I.append(I[j]-0.7*I[j]+0.00051*I[j]^2*(1-I[j]/10000))

for j in range(21):

print "I_%(index)s = %(value)f" % {"index" : j, "value" : I[j]}

I0

I1

I2

I3

I4

I5

I6

I7

I8

I9

I10

I11

I12

I13

I14

I15

I16

I17

I18

I19

I20

1500

1425.37500000000

1316.08417457192...