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Smallpox Controversy To Inoculate or Not to Inoculate Problem Questions
Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program agains ...
Smallpox Controversy To Inoculate or Not to Inoculate Problem Questions
Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health.Question 1. Who was Daniel Bernoulli?Why was he working on this project?Bernoulli's model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity.Consider the cohort of individuals born in a given yearπ‘=0t=0, and letπ(π‘)n(t) be the number of these individuals surviving π‘t years later.Let π₯(π‘)x(t)be the number of members of this cohort who have not had smallpox by year π‘t and who are therefore still susceptible. Let π½Ξ² be the rate at which susceptibles contract smallpox, and let πΞ½ be the rate at which people who contract smallpox die from the disease. Finally, let π(π‘)ΞΌ(t) be the death rate from all causes other than smallpox. Then ππ₯ππ‘dxdt, the rate at which the number of susceptibles declines, is given byππ₯ππ‘=β[π½+π(π‘)]π₯dxdt=β[Ξ²+ΞΌ(t)]x (1)The first term on the right of Eq.1 is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also,ππππ‘=βππ½π₯βπ(π‘)πdndt=βΞ½Ξ²xβΞΌ(t)n, (2)where ππππ‘dndt is the death rate of the entire cohort, and the two terms on the right side are the death rates due to smallpox and to all other causes, respectively.Introduce a new variable π§(π‘)=π₯(π‘)π(π‘).z(t)=x(t)n(t).Question 2.Formulatetheinitial value problem for π§(π‘)z(t).Question 3. Find the complete general solution to differential equation in Q.2 equation.Question 4. Find the solution to the initial value problem.Question 5. Draw the slope field( by hand, or with DField oruse MATLAB, for example, ode45) for this equation and draw some representative solutions. What does this picture tell youabout the solution to this initial value problem.Question 6.Does the solutionin Q.5 agree with your analysis in Q. 4?Question 5. Bernoulli estimated that π=π½=1/8Ξ½=Ξ²=1/8.With that in mind, determine the proportion of 20-year-olds who have not had smallpox.On the basis of the model just described and the best mortality data available at the time, Bernoulli calculated that if deaths due to smallpox could be eliminated, π=0Ξ½=0, then the life expectancy among inoculated will increase by almost 10%.Question 6. Check the internet for the information about what was the life expectancy in 1760 in the country where Bernoulli lived. How would the inoculation program change it?As a result of this study,Bernoullisupported the inoculation program.
Imperial College London Discrete Math Questions
support with abstract math introductory exem -sets and functions, prime numbers ,, functions and limits of functions, cont ...
Imperial College London Discrete Math Questions
support with abstract math introductory exem -sets and functions, prime numbers ,, functions and limits of functions, continuity, groups
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Please see attachment of excel spreadsheet, on the first tab of the spreadsheet has the directions.
MATH 144 GCU Math Budget Expense Loans and Statistics Worksheet
Please see attachment of excel spreadsheet, on the first tab of the spreadsheet has the directions.
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Most Popular Content
5 pages
20191111055210stat 200 Week 4 Homework Problems 2
The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during ...
20191111055210stat 200 Week 4 Homework Problems 2
The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak rush hour periods of ...
5 pages
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A company has estimated the standard deviation of individual sales transactions to be $2000. Approximately 100 sales trans ...
Homework Help
A company has estimated the standard deviation of individual sales transactions to be $2000. Approximately 100 sales transactions take place each ...
Smallpox Controversy To Inoculate or Not to Inoculate Problem Questions
Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program agains ...
Smallpox Controversy To Inoculate or Not to Inoculate Problem Questions
Daniel Bernoulli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox, which at that time was a major threat to public health.Question 1. Who was Daniel Bernoulli?Why was he working on this project?Bernoulli's model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity.Consider the cohort of individuals born in a given yearπ‘=0t=0, and letπ(π‘)n(t) be the number of these individuals surviving π‘t years later.Let π₯(π‘)x(t)be the number of members of this cohort who have not had smallpox by year π‘t and who are therefore still susceptible. Let π½Ξ² be the rate at which susceptibles contract smallpox, and let πΞ½ be the rate at which people who contract smallpox die from the disease. Finally, let π(π‘)ΞΌ(t) be the death rate from all causes other than smallpox. Then ππ₯ππ‘dxdt, the rate at which the number of susceptibles declines, is given byππ₯ππ‘=β[π½+π(π‘)]π₯dxdt=β[Ξ²+ΞΌ(t)]x (1)The first term on the right of Eq.1 is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also,ππππ‘=βππ½π₯βπ(π‘)πdndt=βΞ½Ξ²xβΞΌ(t)n, (2)where ππππ‘dndt is the death rate of the entire cohort, and the two terms on the right side are the death rates due to smallpox and to all other causes, respectively.Introduce a new variable π§(π‘)=π₯(π‘)π(π‘).z(t)=x(t)n(t).Question 2.Formulatetheinitial value problem for π§(π‘)z(t).Question 3. Find the complete general solution to differential equation in Q.2 equation.Question 4. Find the solution to the initial value problem.Question 5. Draw the slope field( by hand, or with DField oruse MATLAB, for example, ode45) for this equation and draw some representative solutions. What does this picture tell youabout the solution to this initial value problem.Question 6.Does the solutionin Q.5 agree with your analysis in Q. 4?Question 5. Bernoulli estimated that π=π½=1/8Ξ½=Ξ²=1/8.With that in mind, determine the proportion of 20-year-olds who have not had smallpox.On the basis of the model just described and the best mortality data available at the time, Bernoulli calculated that if deaths due to smallpox could be eliminated, π=0Ξ½=0, then the life expectancy among inoculated will increase by almost 10%.Question 6. Check the internet for the information about what was the life expectancy in 1760 in the country where Bernoulli lived. How would the inoculation program change it?As a result of this study,Bernoullisupported the inoculation program.
Imperial College London Discrete Math Questions
support with abstract math introductory exem -sets and functions, prime numbers ,, functions and limits of functions, cont ...
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support with abstract math introductory exem -sets and functions, prime numbers ,, functions and limits of functions, continuity, groups
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Please see attachment of excel spreadsheet, on the first tab of the spreadsheet has the directions.
MATH 144 GCU Math Budget Expense Loans and Statistics Worksheet
Please see attachment of excel spreadsheet, on the first tab of the spreadsheet has the directions.
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