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I am getting confused on how to find the vertical and horizontal nullclines for a set of ODEs. I understand I have to set them to 0, and then solve but whether I rearrange/solve for $x$ or $y$ (in this case $s$ and $i$) is what's confusing me.

For example: I am trying to draw a s-i plane for (susceptible-infected) the following two ODEs: $$ds/dt = -0.7si$$ $$di/dt = 0.7si - 0.2i$$

For the vertical nullcine I set $ds/dt = 0$: $$0=-0.7si $$

For the horizontal nullcline I set $di/dt=0$: $$0=0.7si-0.2i$$

In each case, what am I rearranging for? Do I have to get $i$ in terms of $s$? Or $s$ in terms of $i$?

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2 Answers 2

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The equation $-0.7 si = 0$ is equivalent to $s=0$ or $i=0$. So the nullcline for $s$ will just be the union of the two straight lines whose equations are $s=0$ and $i=0$.

Can you now find the nullcline for $i$ similarly? Write the equation as $i(0.7s-0.2)=0$.

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  • $\begingroup$ There will be two horizontal nullclines? $i=0$ and $s=2/7$? $\endgroup$
    – Curious
    Commented Nov 13, 2017 at 8:47
  • $\begingroup$ Yes, that's right! $\endgroup$ Commented Nov 13, 2017 at 9:00
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In the case $0=−0.7si$ the nullclines is set of points such that either $s$ or $i$ is zero, hence it is axis cross. In the second case you get $s$-axis ($i=0$) and the constant $s=\frac{0.7}{0.2}$ for every $i$. The intersections of these sets are stationary points.

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