Chapter 15.4, Problem 33E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the distance between them. Consider a circular city of radius 10 miles in which the population is uniformly distributed. For an uninfected individual at a fixed point *A* (x_{0}, y_{0}), assume that the probability function is given by

where *d (P,* A) denotes the distance between points *P* and *A.*

(a) Suppose the exposure of a person to the disease is the sum of the probabilities of catching the disease from all members of the population. Assume that the infected people are uniformly distributed throughout the city, with *k* infected individuals per square mile. Find a double integral that represents the exposure of a person residing at *A.*

(b) Evaluate the integral for the case in which *A* is the center of the city and for the case in which *A* is located on the edge of the city. Where would you prefer to live?

(a)

To determine

**To find:** The double integral that represents the exposure of a person residing at the point *A*.

Explanation

**Given:**

The given city is in the shape of circle of radius 10 miles.

The probability function is given by

The infected individuals per square mile is denoted by *k.*

The exposure of a person to the disease is the sum of the probabilities of catching the disease from all members of the population.

**Calculation:**

Let the probability of an individual at the point *A* infected by another individual at the point *P* be denoted by
*A* be denoted by
*P* be denoted by
*P* and *A* is given by,

(b)

To determine

**To evaluate:** The integral for the case in which *A* is the center of the city and the edge of the city and to tell the preference of living.

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