ME/ECE859Spring 2008 Homework 2, Due date: 1/30/08 Wed
The spread of infective diseases can be modeled as a nonlinear system.
We introduce here
the SIRS model. The population consists of three disjoint groups. The population of susceptible
individuals is denoted by
S
, the infected population by
I
, and the recovered population by
R
. We
assume that the total population, which is denoted by
τ
is constant, so that
d
dt
(
τ
=
S
+
I
+
R
) = 0.
We assume that the rate of transmission of the disease (denoted by
β
) is proportional to the
number of encounters between susceptible and infected individuals. Also assume that the return of
recovered individuals to the class
S
occurs at a rate (denoted by
μ
) proportional to the population
of recovered individuals (like malaria and tuberculosis).
Finally assume that the rate at which
infected individuals recovers (denoted by
ν
) is proportional to the number of infected. Hence the
SIRS model is given by
dS
dt
=

βSI
+
μR
dI
dt
=
βSI

νI
dR
dt
=
νI

μR,
where
β, μ, ν
and
τ
are positive real numbers. Since
τ
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 Spring '08
 CHOI
 Bifurcation diagram, µR dt dI

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