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

Abstract

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 $a<1$ , 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 $I_{\ell }<0$ .

Keywords: COVID-19; Control of the epidemic; Infection curve; Oscillation; SIQR model; Self-organization.