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Found in: Page 282

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Prove that the infected population I(t)in the SIR model does not increase if S(0) is less than or equal to $$\frac{{\bf{k}}}{{\bf{a}}}$$.

Thus, it is proved that the infected population I(t)in the SIR model does not increase if S(0) is less than or equal to $$\frac{{\bf{k}}}{{\bf{a}}}$$.

See the step by step solution

## Step 1: Apply the SIR model.

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;

1. The number of infected individuals increases, and the number of S(t) decreases. And eventually, increases and eventually, the number of R(t) increases.

1. The number of S(t) n does not change, the number of I(t) decreases and they all eventually recover.

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$${\bf{i(t) = C}}$$.

## Step 2: Get the result.

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$${\bf{a}}\left( {{\bf{s - }}\frac{{\bf{k}}}{{\bf{a}}}} \right) \le 0$$.