I am a student at the early stage of learning differential equations. In my textbook there is an introduction of the SIR model, and later I also found this Covid prediction model published in 2020.
After seeing these models, I got confused with the current hot topic of whether the Covid explosion in a given region has 'exponential' case increase rate. Because from both models mentioned above (which I assume serve as somewhat a base model of reality), the solution for the infected population size as a function of time seems to be inevitably an exponential function.
For example, the SIR model states that for the susceptible population fraction S and infected population fraction I with contagion rage $\alpha$ and recover rate $\beta$:
$\frac{dS}{dt} = -\alpha SI$
$\frac{dI}{dt} = \alpha SI - \beta I$
And the 2020 paper linked above simply states $\frac{dN}{dt} = \lambda N$ for infected population $N$ and growth rate $\lambda\enspace $, for which I believe the solution would be $N(t) = Ae^{\lambda t}, \quad A \in \Bbb R$.
When we talk about the fortunate situations where Covid case increases are not exponential, are we assuming a different model, or different parameters whose curve compared with the initial parameters (at the start of the explosion) seem quantitatively shifted in magnitude?
Any direction would be really appreciated. Many thanks :)