Phenomenological model for the cumulative numbers of reported cases with $\lambda \in \mathbb{C}$

Assume that the eigenvalues are two conjugated complex numbers $\lambda =e^{\alpha \pm i\omega }\in \mathbb{C}$, for some $\alpha \in \mathbb{R}$ and $\omega \ge 0$. The phenomenological model used to fit the cumulative numbers of reported cases has the following form $$\mathrm{CR}\left(t\right)=e^{\alpha (t-t_0)}\left[Acos\left(\omega \left(t-t_0\right)\right)+Bsin\left(\omega \left(t-t_0\right)\right)\right]+C,\mathrm{for}t\in [t_0,+\infty ),$$ (18) where $\alpha \in \mathbb{R}$, $A\in \mathbb{R}$, $B\in \mathbb{R}$, $C\in \mathbb{R}$, and $\omega \ge 0$ are four real numbers. For discrete times, it is equivalent to say that $$\mathrm{CR}\left(n\right)={\displaystyle \frac{A-iB}{2}}{\lambda }^n+{\displaystyle \frac{A+iB}{2}}{\overline{\lambda }}^n+C,\mathrm{for}n=0,1,2,\dots .$$ (19) 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 $$\begin{array}{c}N\left(t\right)=e^{\alpha (t-t_0)}\left[\widehat{A}cos\left(\omega \left(t-t_0\right)\right)+\widehat{B}sin\left(\omega \left(t-t_0\right)\right)\right],\mathrm{for}t\in [t_0,+\infty ),\hfill \end{array}$$ (20) where $$\left\{\begin{array}{c}\widehat{A}=\alpha A+\omega B\hfill \\ \widehat{B}=-\omega A+\alpha B\hfill \end{array}\right.\iff \left\{\begin{array}{c}A={\displaystyle \frac{\alpha \widehat{A}-\omega \widehat{B}}{{\omega }^2+{\alpha }^2}}\hfill \\ B={\displaystyle \frac{\omega \widehat{A}+\alpha \widehat{B}}{{\omega }^2+{\alpha }^2}}\hfill \end{array}\right..$$ (21)