Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

Question

I'm trying to visualize what the following equation is saying:

$$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$

where $S$ is a probability-density, but I think you can assume any other quantity of interest (by the way, my variable of interest is the susceptibility of a person to a given disease), $a$ is a variable such as age and $t$ is time.

I understand a simple solution for this equation is given by $S(a,t) = c (a - t)$. However, after plotting this plane, I can't intuitively grasp what the differential equation is actually doing.

As I understand it, the left-hand side is describing the variation of $S(a,t)$ when we change a bit the time $t$. This should be equal to the variation of $S(a,t)$ when we change a bit the age $a$. In this particular example, I expect a reduction in $S(a,t)$ as we move through time (because here the susceptibility should decrease when people get older.) Therefore, I negative sign is appropriate. However, I don't see that intuition in the solution $S(a,t) = c (a - t)$ when I plot it.

As a side note, the equation above doesn't describe the full dynamics of susceptibility. I dropped a few terms in order to simplify the analysis.

UPDATE:

I guess what is confusing me is that, in the left-hand side equation, we have the variation along $a$: $\displaystyle \frac{S(a + \Delta, t) - S(a,t)}{\Delta a}$, so we stay at the same time $t$ and move a bit in the $a$ direction . Similarly, the right side $\displaystyle \frac{S(a, t + \Delta) - S(a,t)}{\Delta t}$ means that we stay at the same (age) $a$ and move a bit in the $t$ direction. However, when I plot the solution $S(a,t)$, I can't see this solution expressing that a change in $t$ should be equal to a change in $a$ with opposite sign.

Thanks in advance.

Answer 1

If you plot $S(a,t)$ with $a,t$ as your $x,y$ axes and $S$ as the $z$-axis, you should get a sloped plane that increases toward the $(+,-)$ quadrant. Think of the plane as the slope of a mountain; when you increase in the $a$ direction, you move up the mountain. When you increase in the $t$ direction, you move down the mountain. The differential equation just says that the speeds of increase are related to one another.

Does that make sense?

Answer 2

Divide both sides by $S_a$. From multivariable calculus, you obtain $\frac{da}{dt}=1$. What does this mean? It may not be obvious, but it at least suggests that curves of the form $a-t=c$ are important to the geometry of the problem. Let's play with this by eliminating $a$ and considering $S(c+t,t)$. Now we're onto something -- with $f(t) = S(c+t,t)$, we have $f'(t) = \frac{\partial S}{\partial a} \frac{da}{dt} + \frac{\partial S}{\partial t} = 0$.

What does this mean? That means for any $c$, $S(c+t,t)$ is constant (for all $t$). This demonstrates that the correct "frame of reference" is actually along the traveling wave $(c+t,t)$.

[edit] In an even more hand-wavy fashion, you would agree that $1 \cdot g'(x)=0$ specifies $g(x)$ is constant. In your particular example, you have $\langle1,1\rangle \cdot \nabla S(a,t) = 0$. By analogy, then $S(a,t)$ is constant along the direction $\langle1,1\rangle$, but since we leaving the starting coordinate unspecified, that introduces a degree of freedom that is then propagated along rays of direction $\langle1,1\rangle$.