In a college hostel accommodating 1000 students, one of the hostellers came in carrying a flu virus, and the hostel - Maths - Differential Equations

Question

In a college hostel accommodating 1000 students, one of the
hostellers came in carrying a flu virus, and the hostel was
isolated If the rate at which the virus spreads is assumed to be
proportional to the product of the number N of infected students
and the number of non-infected students, and if infected students
are 50 after 4 days then show that more than 95% of the hostellers
will be infected after 10 days If Shyam was the first student to be
infected what precautions he should have initiated to avoid this
situation

Lovina Kansal · Accepted Answer

Dear student
We have,dNdt∝N(1000-N)⇒dNdt=λN(1000-N) where λ is a constant⇒1N(1000-N)dN=λdt⇒∫dNN(1000-N)=λ∫dt⇒11000∫11000-N+1NdN=λ∫dt⇒11000logN-log(1000-N)=λt+C⇒11000logN1000-N=λt+C     
(1)At t=0,we have N=111000log1999=C⇒C=11000log1999⇒C=-log9991000Putting the value of C in (1), we get11000logN1000-N=λt-log9991000⇒11000logN1000-N+11000log999=λt⇒11000log999N1000-N=λt      
(2)It t=4 then N=50Putting these values in (2), we get11000log49950950=4λ⇒λ=14000log499595⇒λ=14000log99919Putting the value of λ in (2), we get11000log999N1000-N=14000log99919t⇒4log999N1000-N=log99919tWhen t=10,the value of N is given by4log999N1000-N=10×log99919⇒log999N1000-N=52log99919⇒log1000-N999N=-52log99919=log99919-52⇒log1000-N999N=log99919-52⇒1000-N999N=99919-52⇒1000999N-1999=99919-52⇒1000N=1999+99919-52⇒1000N=1+(999)-32×195/2⇒N=10001+(999)-32×195/2⇒N=952 approx∴N1000×100=N10=95
2Hence, more than 95% students will be infected after 10 days
Regards