COMPLEX DYNAMICS EMERGING FROM INTERACTING DYNAMICAL SYSTEMS - An Invitation to Biomathematics - page 66

The assumption that the population remains constant is:

S ð t Þþ I ð t Þþ R ð t Þ¼ N ;

and thus dS

dI
dt þ

dR
dt ¼ dt þ

0 for all t.

We need equations describing how each group changes. As with
the SIS model, we assume the rate of new infections is given by

SI.In
this case, however, there is no flow into S, because the recovered are
immune. We then have:

a

dS
dt ¼a

SI :

Individuals join the recovered group after they have been infected. We
assume, as before, that the infectives recover at a constant per capita rate
b

. Then we would have:

dR
dt ¼ b

I :

The schematic representation of the SIR model is presented in
Figure 2-5:

b I

a SI

S I R

FIGURE 2-5.
Schematic representation of the SIR model. S, I, and R represent susceptible, infected, and
recovered, a is the infection rate, and b is the per capita recovery rate.

E XERCISE 2-4

(a) Will S(t) be increasing or decreasing? Give two reasons: one
based on physical considerations, the other on knowing the
derivative.

(b) Repeat part (a) for R(t).
(c) Show that dI

dt ¼ a

SI b I :

In summary, the SIR model is described by the following system of
differential equations:

dS
dt ¼a

SI

dI
dt ¼ a

SI b I

(2-4)

dR
dt ¼ b

I :