Answered: A disease is spreading through the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A disease is spreading through the country. Let x be the number of people infected. Let the constant S be the number of people susceptible to infection. The infection rate dx is

dt proportional to the product of already infected people, x, and the number of susceptible but uninfected

people, S − x.

a) Write down the differential equation.

b) Supposing x(0) > 0, that is, some people are infected at time t ? 0, what is lim x(t). t→∞

c) Does the solution to part b) agree with your intuition? Why or why not?

Expert Solution

Step 1

Given that,

$\frac{d x}{d t} α x (S - x)$

So, the given equation becomes,

$\frac{d x}{d t} = k x (S - x)$

For some constant $K$ .

$\frac{d x}{x (S - x)} = k d t . . . . (1)$

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