We consider the outbreak of a transmittable disease in an infinite population. The disease is characterised by its ability η ∈ [0,1) to infect. Each individual has a specific level of resistance R distributed on the interval [0,1]. The resistance of an individual is independent of the other individuals and constant in time, except that an infected person becomes immune, i.e. R for that individual becomes equal to 1. Once a year an infection of strength η drawn as i.i.d. from [0,1) hits all individuals in the population and those with R

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(a) We treat time as a discrete variable measured in years. Assume a person succumbs to the disease at year t. What is the probability that the next incident occurs at t? (b) Consider the number of outbreaks Q that have occurred between two years t and 12. Assume Q 1 and compute to a good approximation the probability that Q=q. Now we consider some details of the spreading of the disease. The probability that an infected person A directly transmits the disease to k other people is given by Pk = (1-e)eak, k = 0,1,2,... and a > 0. (1) (c) Determine the average number of direct transmissions caused by the infected person A. This is the first generation of people infected by A.