Chapter 5.1, Problem 19E

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8th Edition

James Stewart

ISBN: 9781305270343

Textbook Problem

In someone infected with measles, the virus level *N* (measured in number of infected cells per mL of blood plasma) reaches a peak density at about *t* = 12 days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of *N*(*t*) from *t* = 0 to *t* = 12 (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells × time). The function *N* has been modeled by the function

Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles.

*Source:* J. M. Heffernan et al., “An In-Host Model of Acute Infection: Measles as a Case Study,” *Theoretical Population Biology* 73 (2006): 134–47.

To determine

The total amount of infection needed to develop symptoms of measles.

Explanation

**Given information:**

The virus level reaches the peak density at

The function *N* is modeled by the function

The subintervals is

**Calculation:**

Draw the graph of

Find the width

Here, the upper limit is *b*, the lower limit is *a*, and the number of rectangles is *n*.

Substitute 12 for *b*, 0 for *a*, and 6 for *n* in Equation (2).

The intervals are 0, 2 days, 4 days, 6 days, 8 days, 10 days, and 12 days.

The midpoints of the intervals are 1 day, 3 days, 5 days, 7 days, 9 days, and 11 days.

The expression to find the total amount of infection needed to develop symptoms of measles

Rearrange Equation (2) for the

Refer Figure 1

Sect-5.1 P-1ESect-5.1 P-2ESect-5.1 P-3ESect-5.1 P-4ESect-5.1 P-5ESect-5.1 P-6ESect-5.1 P-7ESect-5.1 P-8ESect-5.1 P-13ESect-5.1 P-14E

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P-1RECh-5 P-2RECh-5 P-3RECh-5 P-4RECh-5 P-5RECh-5 P-6RECh-5 P-7RECh-5 P-8RECh-5 P-9RECh-5 P-10RECh-5 P-11RECh-5 P-12RECh-5 P-13RECh-5 P-14RECh-5 P-15RECh-5 P-16RECh-5 P-17RECh-5 P-18RECh-5 P-19RECh-5 P-20RECh-5 P-21RECh-5 P-22RECh-5 P-23RECh-5 P-24RECh-5 P-25RECh-5 P-26RECh-5 P-27RECh-5 P-28RECh-5 P-29RECh-5 P-30RECh-5 P-31RECh-5 P-32RECh-5 P-33RECh-5 P-34RECh-5 P-35RECh-5 P-36RECh-5 P-37RECh-5 P-38RECh-5 P-39RECh-5 P-40RECh-5 P-41RECh-5 P-42RECh-5 P-43RECh-5 P-44RECh-5 P-45RECh-5 P-46RECh-5 P-47RECh-5 P-48RECh-5 P-49RECh-5 P-50RECh-5 P-51RECh-5 P-52RECh-5 P-53RECh-5 P-54RECh-5 P-55RECh-5 P-56RECh-5 P-57RECh-5 P-58RECh-5 P-59RECh-5 P-60RECh-5 P-61RECh-5 P-62RECh-5 P-63RECh-5 P-66RECh-5 P-69RECh-5 P-70RECh-5 P-71RECh-5 P-72RECh-5 P-1PCh-5 P-2PCh-5 P-3PCh-5 P-5PCh-5 P-6PCh-5 P-7PCh-5 P-8PCh-5 P-9PCh-5 P-10PCh-5 P-11PCh-5 P-14PCh-5 P-15PCh-5 P-18PCh-5 P-19P

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