5 Data suggest that the number of cases of infection from a particular disease tends to oscillate between two values over a period of approximately 6 months.
- Suppose that the number of cases, \(P\) thousand, after time \(t\) months is modelled by the equation \(P = \frac { 2 } { 2 - \sin t }\). Thus, when \(t = 0 , P = 1\).
- By considering the greatest and least values of \(\sin t\), write down the greatest and least values of \(P\) predicted by this model.
- Verify that \(P\) satisfies the differential equation \(\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ^ { 2 } \cos t\).
- An alternative model is proposed, with differential equation
$$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } \left( 2 P ^ { 2 } - P \right) \cos t$$
As before, \(P = 1\) when \(t = 0\).
- Express \(\frac { 1 } { P ( 2 P - 1 ) }\) in partial fractions.
- Solve the differential equation (*) to show that
$$\ln \left( \frac { 2 P } { P } \right) = \frac { 1 } { 2 } \sin t$$
This equation can be rearranged to give \(P = \frac { 1 } { 2 \mathrm { e } ^ { \frac { 1 } { 2 } \sin t } }\).
- Find the greatest and least values of \(P\) predicted by this model.