Self-organization of oscillation in an epidemic model for COVID-19

On the basis of a compartment model, the epidemic curve is investigated when the net rate $\lambda$ of change of the number of infected individuals $I$ is given by an ellipse in the $\lambda$-$I$ plane which is supported in $[I_{\ell}, I_h]$. With $a \equiv (I_h - I_{\ell})/(I_h + I_{\ell})$, it is shown that (1) when $a < 1$ or $I_{\ell} >0$, oscillation of the infection curve is self-organized and the period of the oscillation is in proportion to the ratio of the difference $(I_h - I_{\ell})$ and the geometric mean $\sqrt{I_h I_{\ell}}$ of $I_h$ and $I_{\ell}$, (2) when $a = 1$, the infection curve shows a critical behavior where it decays obeying a power law function with exponent $-2$ in the long time limit after a peak, and (3) when $a > 1$, the infection curve decays exponentially in the long time limit after a peak. The present result indicates that the pandemic can be controlled by a measure which makes $I_{\ell} < 0$.