Assignment: In this assignment, we will look at the real data from the H1N1 2009 flu pandemic showing the cumulative deaths in Europe, weeks 36-68 of the outbreak. The origin of the word epidemic is Greek, where epi means "u pon\" and demic means "the people\". When the disease affects several geographic regions at the same, it is called a pandemic (from Greek pan meaning "all\".) Epidemiology is the scientific study of contagious diseases, and it deals with the essential features of the disease, in particular, the math behind modeling the rate of infection. In this assignment, we will model and study the infection rates for the H1N1. 1. Open the spreadsheet H1N1.xls. (Table 1) This data tracks the cumulative number of deaths in Europe from the H1N1 virus during weeks 36-68. Plot any three different trendline options in excel to plot against this data (except moving average). For this data, briefly describe why the three choices you chose are not good fits to the data based on the behavior ofthe function. Include the three plots in your report Briefly describe the logistic regression and why it would be a good fit for this data. Notice (sadly) that excel does not have a logistic regression option. Let's use calculus to help find this regression curve. On Table 1 in Excel, let N(t) be the function that gives the cumulative number of deaths in week t. Add a new column to Table 1 where you approximate the \"death rate" of the disease (the div AN rate of change) using; =3 E' Round your values to one decimal place. Leave the first cell blank . . . AN as there Is no prior data to fInd E' 4. In Excel, create a new two column table (Table 2) to analyze the rate of change % as a function of time. a. Label the first column, \"NewWeek", and the second column "Death Rate\". b. Start "NewWeek" at 0 and populate the column by subtracting 36 from each Week value. c. Use the death rate values to populate this second column from the prior step. (Week 0 will not have a death rate.) d. Once the table is complete, create a scatterplot ofthe data. Label your axes and title the graph. 5. Explain the properties of the rate of change of Mt) based on this scatterplot. Relate it to the spread of a disease. Include the scatterplot in your report. 6. Create a new two column table in Excel (Table 3). a. The first column header will be "N(t)*(AN(t))\10. 11. 12. 13. 14. 15. Solve this differential equation to find the Logistic Regression with the initial condition N(0) = 7 from the data. Show all the steps of your solution. Since Excel is not great at plotting continuous functions, open desmos.com. Copy and paste the original Table 1 values from Excel into Desmos. This will create a table in Desmos and a scatterplot. In Desmos, graph the solution you found to the differential equation. See how well your logistic equation fits the data. Because the initial condition used was N(0) : 7shifted the date by 36. your graph will be shifted left by 36 units. Fix this by horizontally shifting your function right by 36 to match the data. (Hint: what substitution do you make for x to do this?) Checkpoint: If your logistic regression fits the data pretty well, congratulations! You have done what excel cannot. Include a picture of your curve fit to the data in your report. Unlike Excel, Desmos can do Logistic Regressions. In Desmos, below the table of data, enter the following formula using \"X1\" and \"y1\16. 17. 18. 19. 20. Explain what the sign of the first derivative N'(t) tells us about the model. Explain what the sign of the second derivative N\"(t) tells us about the model. Find the week where the death rate starts to slow down. Describe the domain and range of the two functions. Explain the differences between the coefficients A and B in the logistic function you found by solving the differential equation and the one found by Desmos. Which curve best models the data: the one found by solving the differential equation or the logistic regression found using desmos? Why is one better than the other? Give a reason other than \"it goes through more data points.\