Neco 2019 Biology Question 20

During an outbreak of a contagious disease in a community of 10\,000 people, 20 new cases are reported on day\,1. Each infected individual infects, on average, k others on the day they are infectious. Due to control measures, the effective transmission factor decreases by 20\% each subsequent day (i.e., on day\,2 the average number infected per case is 0.8k, on day\,3 it is 0.8^{2}k, etc.). Assuming the outbreak lasts exactly 5 days and the cumulative number of cases after day\,5 is 500, determine (i) the initial daily transmission factor k (to two decimal places) and (ii) the basic reproduction number R_{0}, defined as the total number of secondary cases generated by a typical case over the 5\-day infectious period (to one decimal place).