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  1. The length of shifts worked by medical residents and other hospital staff is always of great public concern. CBC's Marketplace recently investigated the issue and published their findings. In this OSH, you will consider a mathematical model of the problem to learn what influences the optimal choice of shift length for public safety.

    In deciding how long a resident's shift in the emergency room should be, the Chief of Staff at Vancouver General Hospital would like to minimize the average rate at which errors are made. Let $E(t)$ be the number of errors made by a resident from the start of a shift until $t$ hours into the shift. The instantaneous rate of change of errors made is $E'(t)=4-t+\frac{1}{16}t^2$.
    1. During what interval of time is $E'(t)$ increasing? decreasing? For each region of increase/decrease, suggest an explanation for why the error rate is going up or down.
    2. To build an expectation of the formal results to come, state whether you expect the optimal choice of shift length to come before, at or after the minimum of $E'(t)$ (this expectation will not be evaluated for correctness - the point is to encourage you to establish an expectation).
    3. What is the total number of errors, $E(t)$, made $t$ hours into a shift?
    4. What is the average rate of change of $E(t)$ from the start of a shift ($t=0$) up until time $t$? Call it $A(t)$.
    5. How long should a resident's shift be in order to minimize the average rate of change of errors made (i.e. minimize $A(t)$)?
    6. Sketch $E(t)$. Label any minima, maxima and/or inflection points. On the same axes, draw a line that shows when the average error rate is minimized. Label any important points on this line.
    7. Does your expectation from part b match your result in part e? There is something arguably counter-intuitive about this optimum. Explain why the actual minimum occurs where it does (with regard to the regions of increase and decrease in $E'(t)$).

  2. A drug used to treat cancer is effective at low doses with an efficacy that increases with the quantity of the drug. However, at sufficiently high doses, the drug becomes lethal. For positive values of the constants $k_1$ and $k_2$, the fraction of patients surviving cancer with this drug treatment is given by $$S(q) = \dfrac{q}{k_1+q}-\dfrac{q}{k_2+q}$$ where $q$ is the drug quantity in milligrams/day given to the patient. Assume that $k_1<k_2$ (think about why this assumption is necessary).
    1. Find $S(0)$ and $\lim_{q\to\infty}S(q)$ and in each case explain what your findings mean in medical terms.
    2. What is the optimal daily drug quantity to administer in terms of $k_1$ and $k_2$ so as to maximize the fraction of patients surviving?
    3. Suppose Health Canada has only approved the use of the drug of up to $45$ mg/day and suppose $k_1=25$ mg/day is the same for all patients but $k_2$ varies from patient to patient. To determine a personalized treatment strategy it would be useful for physicians to have a plot of the optimal daily drug quantity as a function of $k_2$, call it $q_{*}(k_2)$. Sketch a plot of $q_{*}(k_2)$ and explain why you drew it that way. Hint: don't forget that $k_1<k_2$!