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 $$ of change of the number of infected individuals $$ is given by an ellipse in the $$ - $$ plane which is supported in $$ . With $$ , it is shown that (1) when $$ , oscillation of the infection curve is self-organized and the period of the oscillation is in proportion to the ratio of the difference $$ and the geometric mean $$ of $$ and $$ , (2) when $$ , the infection curve shows a critical behavior where it decays obeying a power law function with exponent $$ in the long time limit after a peak, and (3) when $$ , 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 $$ .