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Infectious disease outbreak size If 99$\%$ of a population is initially uninfected and each initial infected person generates, on average, two new infections, then, according to the model we considered in Example 3.5 .13 ,$$0.99 e^{-2 A}=1-A$$where $A$ is the fraction of the population infected at the endof an outbreak. Use Newton's method to obtain an approximation (accurate to two decimal places) for the percentage of the population that is eventually infected.

$A=0.80020396$

Calculus 1 / AB

Chapter 3

Derivatives

Section 8

Linear Approximations and Taylor Polynomials

Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:26

Infectious disease outbrea…

01:06

08:07

Epidemic Model A single in…

04:12

Spread of a Disease The ra…

02:27

03:28

08:41

When studying the spread o…

in this problem were given I more. Where? Moralists Foreign 99 You, Judy, need to do a performance site. And where? Excuse me, if I wanted to find the percentage of population that is it lunch intact? So let's find a function of what her pride is. 0.99 You turning to a minus one bus? A unless plotted. So this one looked like this. Where are we have here? In the or here? So you have two section here. So this your coast XX is about 0.8. We're gonna need you. Just let it. So since we have dysfunction, we're gonna be fine. We're gonna find the car collected everything as well as the primary fight from this would be one minus one point named be jittery, native to a remember the New Testament. It says that the guests 40 food The next guess would be the previous guests minus the function have a way that the previous guest by five there, which will with you this guest And if you think our initial guest to be 0.8 and if you like 0.18 uh, for a N if you lived, there were even function At this point, we find the next does to be 0.8002040 and we find it then, if you like this in refined for guests to be point a deal's your 20 20397 A. People like this and we actually find that poor approximation for guesses about Simon Berlin. So this is your room that we're looking for. So from this week, insanity eventually about 80% off the population is infected.

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