39. Antigenic evolution and vaccination Antigenic data like...

39. Antigenic evolution and vaccination Antigenic data like those in Figure 14 can be summarized by taking the points from each year and drawing the smallest possible circle that encompasses these data. This results in a temporal sequence of circles in antigenic space, one for each year. Similarly, if the antigenic data are in three dimensions, spheres can be drawn around the data for each year,as shown in the figure. If the circles (or spheres) from years$x$ and $x+1$ overlap, then the amount of antigenic change between these years is relatively small. In such cases we might expect a single vaccine to work for both years. If the circles or spheres do not overlap, then we might need different vaccines for each year. (a)Suppose the antigenic data are two-dimensional,and the circles for two successive years are givenby the equations $(x-2)^{2}+(y-3)^{2}=1$ and $(x-3)^{2}+(y-2)^{2}=\frac{1}{4} .$ Would a single vaccine work for both vears? (b) Suppose the antigenic data are three-dimensional, and the spheres for two successive years are given by the equations and $$\begin{aligned}(x-2)^{2}+(y-3)^{2}+(z-1)^{2} &=1 \\(x-3)^{2}+(y-2)^{2}+z^{2} &=\frac{1}{4} \end{aligned}$$ Would a single vaccine work for both years? (c) Notice that the x- and y-coordinates of the centers of the circles in part (a) are the same as the x- and y-coordinates of the centers of the spheres in part (b), and the radii are the same as well. What is the relationship between the plot of the circles in part (a) and the plot of the spheres in part (b)?

39. Antigenic evolution and vaccination Antigenic data like those in Figure 14 can be summarized by taking the points from each year and drawing the smallest possible circle that encompasses these data. This results in a temporal sequence of circles in antigenic space, one for each year. Similarly, if the antigenic data are in three dimensions, spheres can be drawn around the data for each year,as shown in the figure. If the circles (or spheres) from years$x$ and $x+1$ overlap, then the amount of antigenic change between these years is relatively small. In such cases we might expect a single vaccine to work for both years. If the circles or spheres do not overlap, then we might need different vaccines for each year.

(a)Suppose the antigenic data are two-dimensional,and the circles for two successive years are givenby the equations $(x-2)^{2}+(y-3)^{2}=1$ and $(x-3)^{2}+(y-2)^{2}=\frac{1}{4} .$ Would a single vaccine work for both vears?
(b) Suppose the antigenic data are three-dimensional, and the spheres for two successive years are given by the equations

and $$\begin{aligned}(x-2)^{2}+(y-3)^{2}+(z-1)^{2} &=1 \\(x-3)^{2}+(y-2)^{2}+z^{2} &=\frac{1}{4}
\end{aligned}$$

Would a single vaccine work for both years?
(c) Notice that the x- and y-coordinates of the centers of the circles in part (a) are the same as the x- and y-coordinates of the centers of the spheres in part (b), and the radii are the same as well. What is the relationship between the plot of the circles in part (a) and the plot of the spheres in part (b)?

Key Concepts

Antigenic Evolution

Antigenic evolution refers to the process by which the antigenic properties of pathogens, such as viruses, change over time due to mutations. This concept is central to understanding why vaccines may need updates or changes, as even slight changes can affect the immune system’s ability to recognize and neutralize the pathogen effectively.

Antigenic Space

Antigenic space is a conceptual framework used to represent the relationships between different antigenic variants of a pathogen. In this space, each point corresponds to a distinct antigenic profile, and the distance between points quantifies the level of antigenic change. This abstraction helps in visualizing and analyzing how far apart two strains are and estimating the impact on vaccine effectiveness.

Geometric Representation of Data

The method of summarizing antigenic data by enclosing the data points within the smallest possible circle (in two dimensions) or sphere (in three dimensions) provides a visual and quantitative measure of the spread of antigenic variation within a specific time period. This geometric approach simplifies complex multi-dimensional data into an interpretable form that can be compared across time periods.

Overlap Analysis in Antigenic Space

Overlap analysis examines whether the geometric shapes (circles or spheres) representing antigenic data from successive time periods intersect. If there is an overlap, it indicates that the antigenic change between those periods is small, suggesting that a single vaccine could potentially remain effective. In contrast, non-overlapping shapes suggest significant antigenic drift, which may necessitate updated or separate vaccines.

Dimensionality in Analysis

Dimensionality refers to the number of variables or axes used to represent antigenic data. While two-dimensional plots (using circles) provide a simplified view of antigenic relationships, three-dimensional representations (using spheres) allow for a more comprehensive depiction by incorporating an additional variable. Understanding how these different dimensions relate helps in interpreting the consistency and robustness of the analysis across varying levels of data complexity.

Transcript

00:01 Hello and welcome.

00:02 We are looking at chapter 8, section 1, problem 39.

00:07 So the gist of this problem is we want to find out if circles overlap.

00:11 So we have these two circles.

00:14 Notice that this first circle has a center of two, three, and a radius of one.

00:23 The second circle here has a center of three, two, and a radius of one half.

00:38 So one half.

00:39 Not one -fourth.

00:40 Remember whatever it, this is the square of the radius.

00:45 So one -half squared is one -fourth.

00:46 So we have these two circles.

00:49 We want to find out how they overlap.

00:50 So we can graph them and i'll show you with the graphs in a second, but i want to show you how we can do this algebraically.

00:57 So if we have two circles, there's some distance between their centers.

01:03 So these are the centers of the circles.

01:05 We call this center one and center two.

01:08 So there's some distance between the circles.

01:11 Those circles also have their own respective radii called r1 and r2.

01:20 So if the distance between the centers is greater than the sum of those two radii, it means the two radii can't reach, can't reach each other.

01:30 There's too much distance to cover.

01:32 So they would not overlap.

01:33 However, if the distance is less than the sum of the radii, then they will overlap.

01:38 And so that this is, this right here is the logic we're going to use so for part a we just need to find the sum of the radii so r1 let's do this over here r1 plus r2 equals 1 .5 and then we need to find the distance between the two points the two centers so this will be 2 minus 3 squared plus 3 minus 2 squared that is a square root of 2 and so square root of 2 is about 1 .4.

02:19 So the distance is less than the sum of the radii.

02:24 So it must overlap.

02:35 So part a, they have to overlap, and the reasoning is from this logic up here, essentially.

02:45 So we did the calculation for the distance here and the sum of the radii there.

02:50 So that's part a...

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