So the equations should be (definition of vaccine efficacy from wikipedia)
.6 * p(sick2) = p(sick2) - p(sick1) p(sick1) - .4 p(sick2) = 0
.
i.e. efficacy is the difference be the unvaccinated and vacinated rates of infection divided by the unvaccinated rate. You have to assume there is no selective pressure in terms of who gets the vaccine (they have the same risk pool as the normal population for flu which is surely untrue) to get your assumtion that
This confusion is due to the fact that the system of two equations I wrote in my comment above was originally crammed onto one line, rather than being separated onto two lines. Sorry! This formatting error made the comment hard to read, and has since been corrected. The correct system of equations to solve is:
So the equations should be (definition of vaccine efficacy from wikipedia)
.6 * p(sick2) = p(sick2) - p(sick1)
p(sick1) - .4 p(sick2) = 0 . i.e. efficacy is the difference be the unvaccinated and vacinated rates of infection divided by the unvaccinated rate. You have to assume there is no selective pressure in terms of who gets the vaccine (they have the same risk pool as the normal population for flu which is surely untrue) to get your assumtion that
.42 p(sick1) + .58p(sick2) = .1 p(sick1) + 1.38p(sick2) = .238
or 1.78 p(sick2) = .238
p(sick2)=.13 (weird I getting a different result) p(sick1) = .05
Did I solve wrong or did you. I do math so I can’t actually manipulate numbers very well but I not seeing the mistake.
This confusion is due to the fact that the system of two equations I wrote in my comment above was originally crammed onto one line, rather than being separated onto two lines. Sorry! This formatting error made the comment hard to read, and has since been corrected. The correct system of equations to solve is:
p(sick1) x 0.42 + p(sick2) x 0.58 = 1 x 0.10
p(sick1) = p(sick2) x 0.60
Instead of:
.42 p(sick1) + .58p(sick2) = .238
.1 p(sick1) + 1.38p(sick2) = .238.