Can Michaelis-Menten constant be zero in the Michaelis-Menten Equation, or is this a non-zero constant?
It does not make sense for the constant to be zero. The constant is usually expressed as follows:

$$ K_M = \frac{k_{\mathrm{reverse}} + k_{\mathrm{catalytic}}}{k_{\mathrm{forward}}} $$

Since the $k$ values are strictly non-negative, the constant would only be zero if $k_{\mathrm{reverse}} = k_{\mathrm{catalytic}} = 0$. In theory, $k_{\mathrm{reverse}}$ could just be $0$ (but that would not be called a Michaelis-Menten model if I want to nitpick), but if $k_{\mathrm{catalytic}}=0$ then you have no overall reaction at all.

Another way to see the same thing is to recall that the Michaelis-Menten constant has the same numeric value as the substrate concentration at the half of the maximum reaction rate. Can this be zero? It would not make any sense to me.

In any case, if you would extend your question to contain information about the epidemic model you would like to create, I am sure a lot of people (including me) would love to help finding a good reaction kinetics analogue for it.