-Assume that N = S + I + R is the total population and show that the above set o

Question

-Assume that N = S + I + R is the total population and show that the above set of ODEs ensures that N remains constant for all time.

- An important question is whether the infection will spread or not, for given r, a, and initial values S and I at t = 0. Let ? = a/r and show that (i) the number of infected I decreases for all times and goes to zero as time goes to infinity, if S < ?. (This means no epidemic occurs.) (ii) the number of infected I increases for all times, if S > ?. (Thus, an epidemic occurs.)

Explanation / Answer

1)
N = S+I+R
dN/dt = dS/dt + dI/dt +dR/dt
= -rSI+ (rSI-aI) + (aI)
= 0

so, N is constant or population doesn't change.

2)
i)
P= a/r
dI/dt = rsI - aI
If I has to decrease:
dI/dt should be negative
or,
rSI - aI<0
S< a/r
S<P
so, no epidemic occurs

ii)
If I has to increase:
dI/dt should be positive
or,
rSI - aI > 0
S> a/r
S>P
so, epidemic occurs

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