Phenomenological model for the cumulative numbers of reported cases with $\lambda >0$

We start with a first eigenvalue $\lambda =e^{\alpha }>0$, for some $\alpha \in \mathbb{R}$. The phenomenological model used to fit the cumulative numbers of reported cases has the following form $$\mathrm{CR}\left(t\right)=A{e}^{\alpha (t-t_0)}+C,\mathrm{for}t\in [t_0,+\infty ),$$ (15) where $A\in \mathbb{R}$, $\alpha \in \mathbb{R}$, and $C\in \mathbb{R}$ are real numbers. For discrete times, it is equivalent to say that $$\mathrm{CR}\left(n\right)=A{\lambda }^n+C,\mathrm{for}n=0,1,2,\dots .$$ (16) By computing the first derivative of $t\mapsto \mathrm{CR}\left(t\right)$, we obtain a model for the daily number of cases of the following form $$N\left(t\right)=A\alpha e^{\alpha (t-t_0)},\mathrm{for}t\in [t_0,+\infty ).$$ (17)