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Question: recall the population model discussed in class with the goal...

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Recall the population model discussed in class, with the goal of predicting the number of cells in a tumor on the 9th, 14h, and 20h days, respectively, given the observed data: 1. Time, t (days) Number of cells, N (X 1000) 0 30 38 42 48 4 While the results of study are a guide to a physician for a total period of 3 to 5 years, it is desirable to have the predicted cell count to within 1.5% of actual value on the days named above. This will help to guide the physician at the initial stage. Recall that we agreed on the following assumptions A1: Let time t be on the continuous domain Ost 31825-5x 365. Since 1 day is very small compared to 5 years (1825+ days), the continuity assumption is reasonable. Moreover, while cell count was recorded daily, we know that the cell count changed continuously. Assume further that the rate of change of N (w.r.t. t) is a continuous function of t for t20. A2: Assume that, at any time t, the rate of change of N is proportional to N. It is well-known [technically, I need a reference here] that the rate of change of population N is a function of N, f(N), known as the growth function. That dN is, (N). This assumption spells out f(N) such is the constant of proportionality dN ,where k We solved the differential equation to arrive at a prototype: N)N, where No and k are parameters.
Your mini-project entails two distinct parts: .Adjusting the model, then responding to the questions below; and Proposing actions that would compound the model. Part1. Adjusting the Model: First, calculate the two parameters in the solution (model) using any two (pairs) of data points from the given set (as we crudely did in class). Using the formula (solution/model), compute the mean relative error, MREI. Identify and note any concerns i) i) Next, utilize the method of least squares (or a similar named procedure) to adjust the model obtained in i) above. Here, you want your procedure to be transparently clear. ii) Re-calculate your mean relative error MRE2 and compare it to the previous iv) If your adjusted model has less than 1.5% error (as expected), it is acceptable v) Beyond this, respond to the following: To what extent do you trust your one (MREI) in i) and ascertain the validity of your adjusted model. and you can now use it to predict the number of cells on the desired days. (edjusted) predictor formulal In partcalar if you observe that the predicted value of N(O) is not 30, provide a brief rationale (reason) for accepting this adjusted value (i.e. why is it reasonable to have N(0) different from 30?), and (1) (2) Given that the recorded cell count may have been off by approximately 250 cells, for how many days would you trust the use of your predictor formula to estimate N to within 2000 cells of the actual number? Part2. Compounding the Model. Propose a concrete manner in which the above model could be compounded goal is to show that you know the distinction between adjusting and compounding. Even as familiarity with adjusting is the target of this project, the distinction between adjusting and compounding is an expected byproduct. Defend your proposal by answering two questions and concluding with the directed action: . The
. Why are these proposed actions reasonable (i.e. provide rationale)?, and : Why would your proposal actually compound (refine) the original model? Formulate the differential equation in your compounded model (and stop here; i.e. do not solve or test). If you lift information from a source (not a crime), cite the source explicitly.
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