Math 5110/6830
Homework 1
1. (ex. 1.4.2 from dV) Study the continuous and discrete models for the recovery
of the infected individuals (from class):
a) For the discrete model
I(t + ∆t) = I(t) − p∆t I(t)
choose a time unit, say, ∆t = 21 day. Then p∆t = p 12 = α ·
In = I(n · ∆t), then the equation becomes
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2
day. If we define
In+1 = (1 − p 21 )In .
Assume that you know the initial number of infected individuals I0 . Find the
solution In (the number of infected individuals at subsequent time points). First,
you should find a general formula for the solution. Second, compute in Matlab
and plot specific values for n up to 15 assuming α = 0.3, I0 = 100 (provided
Matlab code should help you). What happens to the number of infected people
as time goes on?
b) For the continuous model
I ′ (t) = −αI(t)
find the formula for the solution. Plot the solution for the same parameter
values as in a).
c) Compare the discrete and the continuous solutions (numerically). Vary the
time increment ∆t (e.g. try ∆t= 0.25 day, 1/8 day, 1 day, 10 day). What do
you observe? Which choice of the parameter gives the best and which one gives
the worst agreement? Can you explain why?
d) What happens to the solutions as you vary α? Can you explain why?
You can do the above analytically, plotting the graphs yourself. However, it
is a good idea to use this problem to start learning about Matlab. Learning some
Matlab is a part of this class and it will be needed in subsequent homeworks.
Use www.math.utah.edu/ borisyuk/5110/compute.html to get started. You can
run the code that I wrote for this problem (the file is next to this homework).
See what it does, try to understand how and then modify it to do what you
need.
2. (ex. 1.4.1 from dV) Assume you have a culture of bacteria growing in
a petri dish, and each cell divides into two identical copies of itself every 10
minutes.
(a) Choose a unit of time and find the corresponding probability of cell division.
(b) Write down a discrete-time model which relates the amount of cells at time
t and t + ∆t
(c) Define the growth rate, and derive the corresponding continuous-time model.
(d) Solve both models and compare the solutions.
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