The use of mathematical methods to study the spread of contagious diseases goes back at least...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent years many mathematical models have been proposed and studied for many different diseases. 16 Problems 22 through 24 deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products. Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptible. Let z be the proportion of susceptible individuals and y the proportion of infectious individuals; then x+y=1. Assume that the disease spreads by contact between sick and well members of the population and that the rate of spread dy/dt is proportional to the number of such contacts. Further, assume that members of both groups move about freely among each other, so the number of contacts is proportional to the product of 2 and y. Since z = 1-y, we obtain the initial value problem dy/dt = ay(1 - y), y(0) = yo, where a is a positive proportionality factor, and yo is the initial proportion of infectious individuals. (i) (a) Find the equilibrium points for the differential equation (i) and determine whether each is asymptotically stable, semistable, or unstable. (b) Solve the initial value problem (i) and verify that the conclusions you reached in part (a) are correct. Show that y(t) → 1 as t→∞, which means that ultimately the disease spreads through the entire population. The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent years many mathematical models have been proposed and studied for many different diseases. 16 Problems 22 through 24 deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products. Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptible. Let z be the proportion of susceptible individuals and y the proportion of infectious individuals; then x+y=1. Assume that the disease spreads by contact between sick and well members of the population and that the rate of spread dy/dt is proportional to the number of such contacts. Further, assume that members of both groups move about freely among each other, so the number of contacts is proportional to the product of 2 and y. Since z = 1-y, we obtain the initial value problem dy/dt = ay(1 - y), y(0) = yo, where a is a positive proportionality factor, and yo is the initial proportion of infectious individuals. (i) (a) Find the equilibrium points for the differential equation (i) and determine whether each is asymptotically stable, semistable, or unstable. (b) Solve the initial value problem (i) and verify that the conclusions you reached in part (a) are correct. Show that y(t) → 1 as t→∞, which means that ultimately the disease spreads through the entire population.
Expert Answer:
Related Book For
Customer Service Career Success Through Customer Loyalty
ISBN: 978-0133056259
6th edition
Authors: Paul R. Timm
Posted Date:
Students also viewed these mathematics questions
-
A planar graph is a graph that can be drawn in a plane without any two edges intersecting. a. Show that neither of the graphs in Figure 9.87 is planar. b. Show that in a planar graph, there must...
-
A vessel, divided into two parts by a partition, contains 4 mol of nitrogen gas at 75C and 30 bar on one side und 2.5 mol of argon gas at 130C and 20 bar on the other. If the partition is removed and...
-
A segment divided into two parts and has a length of 24. one of the parts is 3 times the other part. How long is the smaller piece?
-
The following two equations have a common solution of (1, 2, 3). Which equation would complete a system of three linear equations in three variables having solution set {(1, 2, 3)}? x+y+z=6 2x = y +...
-
What is a debtor-in-possession reorganization case?
-
Determine whether the solid is a polyhedron. If it is, name the polyhedron. Explain your reasoning. I 1+
-
The comparative financial statements for Advanced Auto Technology, Inc., are in the Working Papers. The financial statements have been completed up to the financial analysis section. The following...
-
The comparative balance sheets for Zagloba Materials, Inc., for December 31, 2011 and 2010 follow. Additional information about Zagloba Materials operations during 2011 is as follows: (a) net income,...
-
Green Technologies is a leading global end-to-end technology provider, with a portfolio of hardware, software and service solutions. In a recent annual report, the balance sheet included the...
-
A partial adjusted trial balance of Manumaleuna Company at January 31, 2012, shows the following. InstructionsAnswer the following questions, assuming the year begins January 1.(a) If the amount in...
-
Re: Plate is a not-for profit grocery store that busy unused edible food from restaurants and for-profit grocery stores and sells the food at below-market prices. It also offers cooking classes for a...
-
Determine the steady-state amplitudes for the equation \[\ddot{x}+0.05 \dot{x}+x+0.01 x^{3}=0.03 \sin 0.33 t\]
-
What is meant by the statistical significance of differences? How does this notion underpin marketing research?
-
Determine the steady-state amplitudes for the equation \[\ddot{x}+0.05 \dot{x}+x+0.01 x^{3}=0.03 \sin 1.01 t\]
-
What is \(P(0)\) for the normalized Gaussian distribution?
-
A narrowband process has a power spectral density defined over a narrow band of frequencies. Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the...
-
How do you ensure compliance with data protection regulations (e.g., GDPR, HIPAA) in a relational database environment? What features and practices help meet these compliance requirements ?
-
Suppose the index goes to 18 percent in year 5. What is the effective cost of the unrestricted ARM?
-
Marcia and Lamar had just moved into a larger home and wanted to furnish the family room tastefully but without spending a lot of money. Saturday was the day for their furniture shopping, so off they...
-
Let's consider the flip side of good listening for a moment. How can we make it easier for other people to listen to us? Some things are easier to listen to than others. If we want to be listened to,...
-
What does the future of the Internet look like? Many people have written about what has been described as one of humanity's greatest technologies. But, time marches on and technology changes rapidly....
-
Refer to the information in Exercise 12-3B. During the year, (Appendix Melville Company purchased plant assets for cash, sold investments for cash (the entire \(\$ 7,000\) gain 12A) developed during...
-
Recently you read an article regarding a new product introduced late last year by 5G Inc., a consumer products company. The article's author, although praising \(5 \mathrm{G}\) Inc. for providing the...
-
The Thurston Company's current year income statement contains the following data: Thurston's comparative balance sheets show the following data (accounts payable relate to merchandise purchases):...
Study smarter with the SolutionInn App