In a given population, the probabilities of dying in successive 10-year intervals, in percent, are [6, 4, 4, 4, 4, 8, 15, 20, 40, 100]. Assume that total deaths in each interval occur evenly throughout the interval.

i) Find the proportion of individuals in this population who survive to the end of each 10-year interval. Also find the approximate life expectancies in this population at birth, and at age 60.

Now suppose that there are two diseases that may affect the quality of life of individuals in this population. On their sixtieth birthday, surviving individuals (that is, those surviving to the end of interval 6) face a 10% risk of getting disease A, from which they will never recover, and which reduces the quality of their remaining life by 40% (but does not change their risk of dying). On their seventieth birthday, survivors who don't already suffer from A also face a 10 % risk of getting it, with the same consequences. Moreover, there is another disease, B, which, at any given time, afflicts 25% of all those in their 70s, 80s, and 90s, and which reduces the sufferer's quality of life by 30% but does not change their risk of dying.

ii) Calculate the fractions of this population who will suffer from disease A in the last four age intervals. Assuming that a person can have both A and B, and that the probabilities of the two kinds of illness are independent, also calculate what proportions in these age brackets will have both A and B

The h factors in the formula for the quality-adjusted life expectancy (see the powerpoint slides) in the last four age intervals in this population will be the weighted average of the h factors for four population sub-groups: Those who are well; those who suffer from disease A only; those who suffer from disease B only, and those who suffer from both A and B. From the information above, the h factors for the first three groups are 1, 0.6 and 0.7 respectively. For the last one, assume that the h factor is 0.6*0..42.

iii) From this information and your answer in ii), calculate the weighted-average h factors in this population, and the quality-adjusted life expectancy for the population as a whole

[Hint: In part ii) you have calculated the fractions with disease A in these age brackets, and you know that 20% of those in the last three age brackets have B. You have also calculated what fractions in the last three age brackets have both A and B. The fractions who have either "A only" or "B only" are calculated by subtracting "both A and B" from the total fractions with A and B. The fraction who are well is the residual.]
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iv) Without actually doing the calculations, explain how each of the following measures would influence this population's quality-adjusted life expectancy. Discuss what information, in addition to that provided below, you would need in order to decide which of these measures would be worthwhile.

reducing the death rates for those in their 30s and 40s from 4 to 3%;

reducing the death rates for those in their 80s from 40 to 30%

reducing the risk of getting A from 10 to 5%, on peoples' 60thand 70thbirthdays

a new drug that reduces the loss of life quality of those with disease B from 30% to 15%

Now consider 1000 people who have been diagnosed with a serious disease C. With current treatment methods, which cost $ 20,000 per patient, patients survive an average of 3 years, with a life quality that is only half that of a person in normal health. A new treatment method has been invented, which costs $70,000 per patient. It has a success rate of 60% (in unsuccessful cases, the patient dies). Among the survivors, 30% die within 5 years; half of those alive after 5 years die within the next five years; all those alive after 10 years die within the next 5 years. The life quality of survivors when the new treatment method is used is 70% of that of a person in normal health

v) Provide an estimate of the cost per additional QALY when the new treatment method is used, rather than the current one. In your calculations, again assume that total deaths during each five-year interval are evenly spread over the 5 years.