If $N(t)$ = quantity of nicotine at time $t$ in the lung,
$C(t)=$ number of smoked cigarettes at time $t$,
How to interprete each term of the model correctly? $$N'(t)=\beta N-\alpha NC$$ $$C'(t)=\alpha NC-\delta C$$
This is what I did: I am not sure in this term $\alpha NC$.
$N'(t)$=The rate at which the amount of nicotine in the lung changes at time t.
$\beta N$=Rate at which amount of nicotine is ingested at time t.
$-\alpha NC$=Rate at which nicotine decreases due to lack of cigarettes.
$C'(t)=$Rate at wich the number of smoked cigars changes.
$\alpha NC$=Rate at which nicotine increase due to cigarettes.??
$-\delta C$=Rate at which the use of cigarettes decreases.
You are forgetting the most central effect: Cigarette consumption brings nicotine into the lung. As this obviously has to depend on $C$, the only candidate for this is $-\alpha NC$. However, the sign on this is unusual, so we can expect more weird sign down the road. I also have no idea why this would depend on $N$. Is there some self-enhancing effect, i.e., the amount of nicotine in the lung increase the absorption somehow?
At the same time, the amount of nicotine consumed decreases the desire to consume more and thus the cigarette consumption, which leads to $\alpha NC$ in the second equation.
The remaining terms would then be the amount with which the body disposes nicotine ($\beta N$) and how much the addiction increase the desire to smoke ($-δN$).
The above is the only way I can make some sense of this. However, without further explanation and given the weird signs, I do not find the model very convincing. I would rather guess that somebody has taken the Lotka–Volterra model and blindly translated it.