I'm having a difficult time understanding the intuition behind the rate that susceptible individuals become infected in SIR-type models.
The infection rate (number of susceptibles infected per time period) is often written as $a c S I$, where $c$ is the probability that susceptibles ($S$) interact with infected individuals ($I$) per infected individual per day, and $a$ is the probability that an interaction results in infection.
However, consider $a = 0.9$ and $c = 0.1$, and a population of $900$ susceptible people and $100$ infected people. Then, the number of people infected per day is $a c S I = 0.9 \cdot 0.1 \cdot 900 \cdot 100 = 8100$, which doesn't make sense (to me) because the number of infected is much larger than the number of susceptible people. I realise that in this case, the probability of contact per infected individual per day ($a$) is unrealistically high, but even if $a$ was smaller, and the populations were bigger, then we could run into problems.
I originally thought that this problem was answered by continuous time models, where the interactions take place in a tiny interval of time, but even discrete-time SIR models include this type of interaction for the infection rate.
Can anyone help me understand why the $a c S I$ formulation makes sense enough for it to be used?
This is the difference between models using population density and such using population counts. What you have in mind is the density equation $$ \dot s = -acs\imath $$ where $s=\frac SN$ and $\imath=\frac IN$ are densities. If you insert that you get for the equation of the counts $$ \dot S = -ac\frac{SI}N. $$