Science:Math Exam Resources/Courses/MATH102/December 2015/Question 18 (c)

MATH102 December 2015

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9  •  Q10 (a)  •  Q10 (b)  •  Q11  •  Q12  •  Q13  •  Q14 (a)  •  Q14 (b)  •  Q15  •  Q16  •  Q17  •  Q18 (a)  •  Q18 (b)  •  Q18 (c)  •  Q18 (d)  •  Q19 (a)  •  Q19 (b)  •  Q19 (c)  •  Q19 (d)  •  Q19 (e)  •  

   Other MATH102 Exams

  •  December 2014  •  December 2011  •  December 2012  •  December 2013  •  December 2016  •  December 2015  •  

Question 18 (c)

A dangerous infectious disease spreads through Vancouver as described by the differential equation $I'=\beta I(N-I)-\mu I=\beta I\left(N-{\frac {\mu }{\beta }}-I\right)$ where $\beta >0$ is the transmission rate constant, $N$ is the total population size (constant), $\mu >0$ is the recovery rate and $I(t)$ is the number of infected individuals. Assume that $N>{\frac {\mu }{\beta }}$. (c) Sketch the phase line for the differential equation.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
For given differential equation $y'=f(y)$, if $f(y)>0$ on the interval $(a,b)$, then $y$ is increasing on the interval, while if $f(y)<0$ on the interval $(c,d)$, then $y$ is decreasing on it.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. First, draw the graph of the function $f(I)=\beta I\left(N-{\frac {\mu }{\beta }}-I\right)$. (In the figure below, the red line is the graph of $f$) Note that in the part (a), we find two steady state solutions $I_{1}=0$ and $I_{2}=N-{\frac {\mu }{\beta }}$. (i.e., $f(I_{1})=f(I_{2})=0$) Then, from the figure, we can easily see that on $(-\infty ,I_{1})$ and $(I_{2},\infty )$, $f(I)<0$, while on $(I_{1},I_{2})$, $f(I)>0$. By the hint, this means that on $(-\infty ,I_{1})$ and $(I_{2},\infty )$, $I$ is decreasing, so that we have the left arrows in these intervals on the phase line. On the other hand, on $(I_{1},I_{2})$, $I$ is increasing and hence we get the right arrow. Once we draw the arrows, it is obvious that $I_{1}$ is unstable (make empty hole), while $I_{2}$ is stable (fill up the hole). To summarize, we obtain the following phase line: