COVID-19 epidemic: Power law spread and flattening of the curve

Mahendra K. Verma; Ali Asad; Soumyadeep Chatterjee

The other seven nations exhibit transition to their power law regimes after the initial exponential growth, with the power law exponents ranging from 2 to 4. Note however that the power law regimes are quite small, except for Italy. We believe that the curves will flatten further down from here, that is, they will transition to subsequent power laws with lower exponents. However, we need more data of subsequent days for a definitive conclusion. Figure 1 also contains plots forİ, derivatives of I(t), that represents daily new count of infections. Similar to I(t),İ increases exponentially in the beginning. After this, we observe a transition to power law regimes. For the power law I(t) ∼ Bt n , we derive thatİ ∼ I 1−1/n . Clearly, this slope is suppressed compared to the exponential regime by a factor of I −1/n . From time t 0 , I(t) doubles at t = 2 1/n t 0 . For South Korea, n = 3, hence for t 0 = 10, the count doubles at t = 10 × 2 1/3 ≈ 12.6 day, or in the interval of 2.6 days. This is a slower doubling rate than that in the exponential phase, which was one day. Note however that the epidemic growth in the power law regime is still very significant because I(t) is large. For large n (e.g., 4 or 5),İ ∝ I, which is same as the formula for the exponential growth (refer to the t 4 regime of Italy and Japan). Also note that in the linear regime,İ is constant, implying a constant number of new cases every day.