Math 1180 - Mathematics for Life Scientists
Lab 1: Equilibria and Stability for a Disease Model
Due the week of January 29, 2007
Getting Started
Log in and open a terminal window. In the terminal window, make a new directory called lab1 by
typing mkdir lab1 and hitting enter. Then change into that directory by typing cd lab1 and hitting
enter. Now, open Maple by typing xmaple &. Once Maple is open, go to File, Save As, and save the
untitled worksheet as lab1.mws in the lab1 directory. Remember to save your worksheet periodically
as you go along, just in case Maple crashes.
A Model of a Disease
Reference: FR Adler, Modeling the Dynamics of Life, 2nd ed., pp. 433-435
Rate of recovery =
I
µ
1-I
Infected
Susceptible
Rate of becoming infected =
α I
Suppose a disease is circulating in a population. Individuals will recover but are then immediately
susceptible to another infection. Let I be the fraction of infected individuals in a population. Each
susceptible individual has a chance of getting infected upon contact with an infected individual. Assume
then that the rate of contact, i.e. the rate of becoming infected for each uninfected individual is
proportional to the number of those infected,
Per capita rate at which suscpetible individual becomes infected = I
Each individual is assumed to be either infected or susceptible, so
Fraction of susceptible individuals = 1 I
Thus, the total rate at which new individuals are infected is the product of the per capita rates and
the number of susceptibles,
Total rate at which infection occurs = I (1 I )
Now, suppose individuals recover from the disease at a rate proportional to the fraction of infected
individuals,
Rate at which infected individuals recover = I
Putting all of these together, we have
dI
= rate at which infection occurs rate of recovery
dt
dI
= I (1 I ) I
dt
Starting with a few infected individulas, we would like to nd out if the disease is going to persist. In
particular, how do changes in parameter values of and aect the outcome?
Exercises: Equilibria and Stability
First, type restart; to purge any saved values from Maple. (Do this before starting any assignment.)
Then, enter your dierential equation. Type
> de:= diff(i(t),t) = alpha*(1-i(t))*i(t) - mu*i(t);
In addition, you will want to set a function that corresponds to the right-hand side of the dierential
equation.
> f:= i -> alpha*(1-i)*i - mu*i;
Now, solve for when f equals zero to nd the equilibria.
> eq:= {solve(f(i)=0, i)};
One thing to note, capital I is reserved for the complex/imaginary number i in Maple. Try to avoid
using I in this lab.
1. Determine the equilibria using the Maple commands above. Try to explain what each equilibrium
represents. Think about what the state variable I physically represents: what are the range of I
values that would make sense? How many plausible equilibria are there when > ? How about
when < ?
2. Plot dIdt against I . You will do this for two cases: < (with = 1 and = 0:3) and > (with = 1 and = 1:8). The commands for the rst case are given below.
> mu:=1; alpha:=0.3;
> plot(f,-3..1);
For = 1:8, use plot(f,-1..1).
By hand, draw the phase line diagram for each case. Draw directional arrows, indicating the
direction of the ow of the system. Determine the stability of equilibria for the two cases. Do the
results that you obtain make sense biologically? Explain. When does the disease remain in the
population (become endemic)?
3. Using the command dsolve, draw the solution I (t) using the initial condition I (0) = 0:4 for the
two cases above. There is a trick that must be done to get Maple to give you a function back
after solving the equation. See below for the < case.
>
>
>
>
mu:= 1; alpha:= 0.3;
de1:= {de,i(0)=0.4};
I1:= unapply(rhs(dsolve(de1,i(t))),t);
plot(I1(t),t=0..5);
4. If you were to design a control measure to eradicate the disease, how would you do so? Would
you need to completely reduce the transmission rate to zero?