MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 10: An Epidemic
We assume an epidemic begins when a small number of infected individuals carrying a disease are introduced
into a community, that no one in the community is immune to the disease, and that the disease is
spread by direct contact of an infected individual with a non-infected, or susceptible, one. We assume, in
fact that during the epidemic there are three basic types of individual: those who are susceptible to the
disease (of which there are S = S(t)), those who are currently infected (of which there are I = I(t)), and
those who are post-infected -- those who are neither susceptible nor infected (of which there are P = P(t)).
As the epidemic runs its course, susceptible individuals catch the disease making them infected, and infected
individuals eventually become post-infected. Thus individuals move from one group to the other:
S  I  P,
and as long as the epidemic is of a relatively short duration,
S + I + P = S(t) + I(t) + P(t) = N
where N is a constant.
Three differential equations govern the transition of individuals from one group to the other:
dS
dt
= -k1SI,
dI
dt
= k1SI - k2I,
dP
dt
= k2I
The first equation states that the rate at which the number of susceptible individuals decreases is proportional
to the number SI of contacts between susceptible individuals and infected ones. The second equation says
that the rate at which the number of infected individuals changes increases because of susceptible individuals
becoming infected, and decreases because of infected individuals becoming post-infected. If the rate at which
infected individuals becomes post-infected is proportional to the number of infected individuals, then the
number of infectives decreases by the same amount the number of post-infectives increases; this is the third
equation.
Tasks
1. Using the fact that
dS
dP
=
dS/dt
dP/dt
=
-k1SI
k2I
= -
k1
k2
S
express S as a function of P.
2. Using the fact that
dI
dP
=
dI/dt
dS/dt
=
k1SI - k2I
k2I
=
k1
k2
S - 1
and your solution to part (1), express I as a function of P.
3. How would an epidemiologist determine k1 and k2 in practice?
4. Since the epidemic will "be over" when there is no further growth in the number of post-infectives, the
epidemic will be over when dP/dt = k2I = 0 or I = 0. Using your solution to part (2), find P when the
epidemic is over -- the extent of the epidemic -- assuming that k1 = 10-6
, k2 = 0.98, and N = 106
.
Comments
1. Observe that although we could not solve the three differential equations that describe the epidemic in
closed form, we could use the chain rule
dy
dx
=
dy/dt
dx/dt
to solve for S and I in terms of P, and that this led to our computation of a useful quantity: the extent
of the epidemic.
2. Also observe that the same type of model we used in this project to study the spread of an epidemic could
be adapted to the study of the spread of information, a topic of concern in a field such as advertising.