Suppose the following data represents the temporal sequence of circles in antigenic 2D space Year Center Radius Year 1 (1.21, 1.03) 0.625 Year 2 (2.80, 1.52) 1.42 Year 3 (1.48, 3.25) 0.75 Year 4 (3.25, 3.47) Year 5 (4.58, 4.89 0145 1.18 Using an application such as Excel, plot this data. Then determine if a single vaccine might work for a given year and its subsequent year for each pair. For example, a single vaccine may work for year 1 and 2 but not 2 and 3 but again for 3 and 4, ete. Include this plot and your findings in your report. We can also use vectors to map the evolutionary progress of a pathogen by using the coordinate system underlying the antigenic space. 5. Using the 3D data given in 1., calculate the 4 vectors representing the evolutionary progress of the influenza strains (i. e. 1 - 2. 2 3. etc.). Additionally, calculate the corresponding magnitude for each of the evolutionary vectors 6. Vectors have a property called The Triangle Inequality which states for vectors a and 5. If a represents the vector you found for the antigenic change from Year 1 to Year 2, and b represents the vector you found for the antigenic change from Year 2 to Year 3. explain what the Triangle Inequality means in terms of this antigenic change. Vaccine escape refers to the loss of effectiveness of a vaccine due to the antigenic evolution of the pathogen. To calculate the rate of vaccine escape you can use a vector projection calculation. Given a vector u and a vector y, if you wish to calculate the scalar projection of U onto T (ie. the "length of the shadow" of u on 7), we do so by computing, the scalar projection compū = | T|| cos Q = * ved (i. e. the "length of the shadow") 7. Referring back to the data presented in 4., using your calculated vector ū= 1 + 2, calculate the vector projection of ū onto the vector v = 15. Now, returning to the rate of vaccine escape, suppose that a vaccine protects against all pathogens within a circle of radius 4 centered at the origin in a 2D antigenic space. in Also, suppose that the vector v which gives the instantaneous velocity of evo- lution in antigenic space is always of fixed magnitude k and angle from a line connecting the pathogen to the origin, with 0 < 0 < 5, as shown at right.