A disease is spreading through the country. Let x be the number of people infect

Question

A disease is spreading through the country. Let x be the number of people infected. Let the constant S be the total population. Here if one is not infected, then he or she is susceptible to infection. The infection rate dx/dt is proportional to the product of already infected people, x, and the number of susceptible but uninfected people, S - x. Write down the differential equation for x. Supposing x(0) > 0, that is, some people are infected at time t = 0, what is lim_t rightarrow infinity x(t). Does the solution to part (b) agree with your intuition? Why or why not?

Explanation / Answer

Ans.dx/dt =kx(S-x), where k is constant.

on solving,

dx/x(S-x)=kdt=>1/S[1/x +1/(S-x)]=kdt

integrating both sides w.r.t't',we get

S[logx+log(S-x)]=kt+c=>logx(S-x)=kt+c=>x(S-x)=e^kt+c

x(0)>0,let x(0)=1,then 1(S-1)=1+c=>c=S-2

so,x(S-X)=e^kt(S-2)

at t=0,x(S-x)=1=>Sx-x^2=1orx^2-Sx=1

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