Math 442 Assignment 2 - Spring 2015
Due Friday, February 6th
Directions: Use LATEX to generate the following output. You will print your solutions and
hand these in on Friday, February 6th in class.
1. George Box said “Essentially, all models are wrong, but some are useful”.
2. L’Hôpital’s Rule states that if f and g are differentiable functions such that lim |f (x)| =
x→a
lim |g(x)| where both limits equal 0 or ∞, then
x→a
f (x)
f 0 (x)
= lim 0
.
x→a g(x)
x→a g (x)
lim
3. MATLAB can be used to verify that
Z ∞
2
e−x dx =
√
(1)
π.
(2)
(x − a)n ,
(3)
−∞
4. The Taylor series for f (x) centered at x = a is
f (x) =
∞
X
f (n) (a)
n=0
n!
where f (n) (a) is the nth derivative of f at x = a.
5. The model consists of the following system of difference equations
dS
dt
dI
dt
dR
dt
dP
dt
= Λ − dS − β1 SI − β2 SP + νR
(4a)
= β1 SI + β2 SP − (d + µ + γ)I
(4b)
= γI − (d + ν)R
P
= αI + rP 1 −
− (ξ + δ)P
K
(4c)
(4d)
with non-negative initial conditions S(0), I(0), R(0), P (0) ≥ 0.
6. The feasible region is
Γ=
(S, I, R, P ) ∈
R4+ |S
Λ
+I +R≤ , P ≤C
d
(5)
where C is the unique positive root of the quadratic αΛ/d − (ξ + δ − r)P − (r/K)P 2 .
1
7. Linearization about the disease-free equilibrium (DFE) gives
dI~
~
= J I~ = (F − V )I,
dt
where J is the Jacobian matrix evaluated at the DFE. The matrices F and V are
β1 S̄ β2 S̄
d+µ+γ
0
F =
V =
.
α
r
0
ξ+δ
(Hint: The spacing between the matrices F and V is made with the command \qquad.)
8. Each of the measurements x1 < x2 < · · · < xr occurs p1 , p2 , . . . , pr times. The mean
value and standard deviation are then
v
u
r
r
X
u1 X
1
t
x=
p i xi ,
s=
pi (xi − x)2
n i=1
n i=1
where n = p1 + p2 + · · · + pr .
9. The following figure was generated by MATLAB.
1
0.8
0.6
y
0.4
0.2
0
-0.2
-0.4
-10
-5
0
5
10
x
Figure 1: Graph of f (x) =
sin x
on [−10, 10].
x
(Hint: Use MATLAB to generate this figure and save it as an Encapsulated Post Script
file. These types of figures look best in LATEX prepared documents.)
2
10. The following table lists the parameter values used for numerical simulations.
Table 1: Parameter values for our model.
Description
Recruitment rate
Natural death rate
Direct transmission
Indirect transmission
Disease-related death rate
Recovery rate
Immune period
Shedding rate
Pathogen replication rate
Pathogen death rate
Decontamination rate
Carrying capacity
3
Parameter
Value
Λ
0.1
d
0.001
β1
0.0006
β2
1.3 × 10−10
µ
0.001
γ
0.1
1/ν
100
α
5 × 107
r
0.2
ξ
0.051
δ
1.2
K
1.95 × 1014