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In a population at high risk of developing cancer, each individuali classified as: Undiagnosed (U), Positive (P), Treatment (T) or Deceased (D). There is a test for the cancer, but there are false positive and false negative test results. Each month: • Each Undiagnosed person will either test positive and move to the Positive category (10 % likelihood), return a false negative test result and Die (3 %) or return a true negative test result (87 %) and remain Undiagnosed. • Each Positive person will move to the Treatment category (70 % likelihood), or move to the Undiagnosed category (30 %) because their earlier test result was a false positive. • Each person in the Treatment category will either recover and move to the Undiagnosed category (40 % likelihood), require treatment for another month (40 %), or Die (20 %). Researchers wish to develop a model based on a system of differential equations. 1. Draw a life-cycle diagram showing all possible transitions between the four categories "Undiagnosed" U, "Positive" P, "Treatment" T, and "Dead" D, including the probability of each transition. 2. Let U(t), P(t), T(t) and D(t) be the number of people in each category at any time t. Equations for U' and D' are as follows. Write similar equations for P' and T'. U' -0.13U+0.3P+0.4T D'= 0.2T+0.03U

In a population at high risk of developing cancer, each individuali classified as: Undiagnosed (U), Positive (P), Treatment (T) or Deceased (D). There is a test for the cancer, but there are false positive and false negative test results. Each month: • Each Undiagnosed person will either test positive and move to the Positive category (10 % likelihood), return a false negative test result and Die (3 %) or return a true negative test result (87 %) and remain Undiagnosed. • Each Positive person will move to the Treatment category (70 % likelihood), or move to the Undiagnosed category (30 %) because their earlier test result was a false positive. • Each person in the Treatment category will either recover and move to the Undiagnosed category (40 % likelihood), require treatment for another month (40 %), or Die (20 %). Researchers wish to develop a model based on a system of differential equations. 1. Draw a life-cycle diagram showing all possible transitions between the four categories "Undiagnosed" U, "Positive" P, "Treatment" T, and "Dead" D, including the probability of each transition. 2. Let U(t), P(t), T(t) and D(t) be the number of people in each category at any time t. Equations for U' and D' are as follows. Write similar equations for P' and T'. U' -0.13U+0.3P+0.4T D'= 0.2T+0.03U

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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In a population at high risk of developing cancer, each individual is classified as: Undiagnosed (U), Positive (P), Treatment (T) or Deceased (D). There is a test for the cancer, but there are false positive and false negative test results. Each month: • Each Undiagnosed person will either test positive and move to the Positive category (10 % likelihood), return a false negative test result and Die (3 %) or return a true negative test result (87%) and remain Undiagnosed. • Each Positive person will move to the Treatment category (70 % likelihood), or move to the Undiagnosed category (30 %) because their earlier test result was a false positive. • Each person in the Treatment category will either recover and move to the Undiagnosed category (40 % likelihood), require treatment for another month (40 %), or Die (20 %). Researchers wish to develop a model based on a system of differential equations. 1. Draw a life-cycle diagram showing all possible transitions between the four categories "Undiagnosed" U, "Positive" P, "Treatment" T, and "Dead" D, including the probability of each transition. 2. Let U(t), P(t), T(t) and D(t) be the number of people in each category at any time t. Equations for U' and D' are as follows. Write similar equations for P' and T'. U' = -0.13U+0.3P+0.4T D' = 0.2T+0.03U 3. In a particular population, at time t = 0, the functions and derivatives have the following values. Apply two steps of Euler's method with a step size of one month to estimate the number of people in each category at time t = 2 months. ● Initial function values: U(0) = 800, P(0) = 100, T(0) = 50, D(0) = 0 • Initial derivative values: U'(0) = −54, P′(0) = −20, T'(0) = 40, D′(0) = 34
Transcribed Image Text:In a population at high risk of developing cancer, each individual is classified as: Undiagnosed (U), Positive (P), Treatment (T) or Deceased (D). There is a test for the cancer, but there are false positive and false negative test results. Each month: • Each Undiagnosed person will either test positive and move to the Positive category (10 % likelihood), return a false negative test result and Die (3 %) or return a true negative test result (87%) and remain Undiagnosed. • Each Positive person will move to the Treatment category (70 % likelihood), or move to the Undiagnosed category (30 %) because their earlier test result was a false positive. • Each person in the Treatment category will either recover and move to the Undiagnosed category (40 % likelihood), require treatment for another month (40 %), or Die (20 %). Researchers wish to develop a model based on a system of differential equations. 1. Draw a life-cycle diagram showing all possible transitions between the four categories "Undiagnosed" U, "Positive" P, "Treatment" T, and "Dead" D, including the probability of each transition. 2. Let U(t), P(t), T(t) and D(t) be the number of people in each category at any time t. Equations for U' and D' are as follows. Write similar equations for P' and T'. U' = -0.13U+0.3P+0.4T D' = 0.2T+0.03U 3. In a particular population, at time t = 0, the functions and derivatives have the following values. Apply two steps of Euler's method with a step size of one month to estimate the number of people in each category at time t = 2 months. ● Initial function values: U(0) = 800, P(0) = 100, T(0) = 50, D(0) = 0 • Initial derivative values: U'(0) = −54, P′(0) = −20, T'(0) = 40, D′(0) = 34
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