91Ó°ÊÓ

If S(t) is the number of susceptible individuals, I(t) the number of the currently infected individuals, and R(t) the number of individuals who have recovered from the infection.

Then N = I+S+R.

The SIR model assumes that N does not change with time then the possibilities are;

In both cases\({\bf{S(0}}) \ge {\bf{S(t)}}\).

The change in the numbers is:

\(\frac{{{\bf{di}}}}{{{\bf{dt}}}}{\bf{ = a}}\left( {{\bf{s - }}\frac{{\bf{k}}}{{\bf{a}}}} \right){\bf{i}}\)

And

\({\bf{i = }}\frac{{\bf{I}}}{{\bf{N}}}{\bf{,s = }}\frac{{\bf{S}}}{{\bf{N}}}\)

The solution of the equation is\({\bf{i(t) = }}{{\bf{e}}^{{\bf{(as - k)t}}}}\)or\({\bf{i(t) = C}}\).

The solution is\({\bf{a}}\left( {{\bf{s - }}\frac{{\bf{k}}}{{\bf{a}}}} \right){\bf{ = 0}}\)and the first form.

Now, the infected population will not decrease if and only if\({\bf{a}}\left( {{\bf{s - }}\frac{{\bf{k}}}{{\bf{a}}}} \right) \le 0\).