Applications of Systems of Equations 471the number of susceptibles and the number of infectives. Thus, it is assumed
that(2)dS
dt = - (JS(t)I(t) for all twhere the positive constant of proportionality (3 is called the infection rate.
The product S(t)I(t) represents the rate of contact between susceptibles and
infectives while the product (JS(t)I(t) represents the proportion of contacts
which result in the infection of susceptibles.
Next, we assume that the rate of change of the removeds is proportional to
the number of infectives. Thus, we assume(3)dRdt = rI(t) for all t
where the positive constant r is called the removal rate.
Solving equation (1) for I, we get I = N - S - Rand differentiating, we
find upon substitution from (2) and (3)
dI dS dR- = --- - = (JS(t)I(t) - rI(t) = ((JS(t) - r)I(t).
dt dt dt
Hence, a system of three first-order differential equations for modelling an
epidemic is
(4)dS = - (JSI
dtdIdt = ((JS - r)I
dR = rI
dtwhere (3 and r are positive constants. Appropriate initial conditions for this
system are the initial number of susceptibles S(O) =So > 0, the initial number
of infectives I(O) = 10 > 0, and the initial number of removeds R(O) = Ro = 0.
By analyzing the equations of system ( 4), we can determine some interestinggeneral facts regarding this epidemic model. Since (3 > 0, S(t) ?". 0, and
I(t) ?". 0, we have dS/dt = -(JSI:::; 0 for all t. In fact, dS/dt < 0 unless S = 0
or I = 0. Because dS/ dt < 0 , the number of susceptibles, S(t), is a strictly
decreasing function of time until S becomes 0 or I becomes 0. Likewise, since
r > 0 and I(t) ?". 0, we have dR/dt = rI ?". 0 and, in fact, dR/dt > 0 unless
I = 0. So the number of removeds is a strictly increasing function of time
until the number of infectives becomes 0. Since I(t) ?". 0 the sign of the rate