2011 CJC C2 BT Q5

It has been suggested that the outbreak of H1N1 in Mexico in 2009 can be modelled by a special type of differential equation called Logistic Equation. In this model, \(x\) denotes the number of infected individuals in the population of a town at any time \(t\) days.

The differential equation is given by \(\frac{\mathrm{d}x}{\mathrm{d}t}=0.2x\left( 1-\frac{x}{N} \right)\), where \(0.2\) indicates the initial growth rate of the virus and \(N\) can be taken to be the population. It was known that a small town has a population of \(1000\) and initially \(20\) people were infected with H1N1.

1. Show that \(x=\frac{1000}{P+Q{{e}^{-0.2t}}}\), where \(P\) and \(Q\) are positive integers to be determined.[5]
2. Hence sketch the solution curve, indicating clearly asymptotes and axial intercepts if any.[2]
3. Deduce the number of days when \(75\%\) of the population will be infected with H1N1. State one assumption you have made about the population in your calculation.[3]