Science:Math Exam Resources/Courses/MATH102/December 2014/Question A 05

MATH102 December 2014

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[hide] Question A 05

The model given below on the left has been suggested for the spread of HIV within the immune system of an infected person. $C(t)$ is the density of healthy immune cells, $I(t)$ is the density of HIV-infected immune cells and $V(t)$ is the density of virus in the blood of a patient. Which of the options on the right gives a correct interpretation of some part of the model? ${\begin{aligned}{\frac {dC}{dt}}&=P-\alpha CV-\gamma _{1}C\\{\frac {dI}{dt}}&=\alpha CV-\gamma _{2}I\\{\frac {dV}{dt}}&=\beta I-\gamma _{3}V\end{aligned}}$ Healthy cells can become infected when they encounter infected cells. Healthy cells can become infected when they encounter virus. Virus is produced at a rate proportional to the current viral density. Infected cells die at a rate proportional to the viral density. Virus is killed/removed at a rate proportional to the density of healthy immune cells.

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!

[show] Hint
What does it mean to have a positive/negative term in the right hand side of a differential equation?

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[show] Solution 1

By observing the three given differential equations, we have the following interpretations Healthy cells have a constant increasing rate $P$. Healthy cells can become infected when they encounter infected virus(from the $\pm aCV$ term in equations of $C$ and $I$). Virus is produced at a rate proportional to the current infected cell density(from the $\beta I$ term in the equation for $V$). Healthy cells, infected cells and virus die at a rate proportional to its own density(from the $-\gamma _{1}C$, $-\gamma _{2}I$ and $-\gamma _{3}V$ terms). Thus the correct statement is (b).

[show] Solution 2

Another way through which this question can be solved involves analyzing the relationship between the variables of the equations. In the first equation, change in the number of healthy cells (${\frac {dC}{dt}}$) can be described as dependent on two variables ($C$ and $V$) and three constants($P$, $\alpha$, $\gamma _{1}$). This observation rules out option a) as it indicates the change in the density of healthy immune cells ($C(t)$) is only dependent on the density of healthy immune cells and the density of virus in the blood of the patient, and not the density of the infected cells ($I(t)$). (The correct answer of this question may also be deduced from this equation) Similarly, for the second question, change in the number of infected immune cells can be described as dependent on three variables ($C$,$V$, and $I$). Based on the equation, it can be deduced that an increase in $V$ will lead to an increase in $I(t)$, and thus, option c) can be ruled out. The third equation illustrates that change in viral density is dependent on two variables ($I$ and $V$). Based on the equation, option c) can be eliminated as an increase in $V$ will lead to smaller ${\frac {dV}{dt}}$. Option e) can also be ruled out as ${\frac {dV}{dt}}$ does not dependent on the variable $C$. The correct statement is therefore b).