Math 19. Lecture 7
Phase Plane Analysis
T. Judson
Fall 2006
1
SIR Models
Systems of differential equations are very useful in epidemiology. We can use
differential equations to model how a disease will spread through a popula
tion. Let us assume that we have a closed population of size
N
and that each
individual in the population falls into one of the following categories:
S
(
t
)
=
Susceptible individuals
I
(
t
)
=
Infected individuals
R
(
t
)
=
Removed individuals
Susceptible individuals who do not yet have the disease and can catch the
disease from infected individuals. Individuals enter the removed population
by either recovering from the disease or dying.
If an infected individual
recovers, then the individual is immune to the disease. Since the population
is closed, we know that
S
(
t
) +
I
(
t
) +
R
(
t
) =
N.
We can model how the disease acts with the following system of equations,
dS
dt
=

αSI
dI
dt
=
αSI

βI.
1
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2
An Epidemic Model
Consider the model of a viral epidemic that moves through an isolated pop
ulation. We make the following assumptions.
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 Spring '11
 KOP
 Epidemiology, null clines, null cline

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