Metodos Matematicos da Fisica

We present here a method to identify the numbers of waves of contamination and death by Covid-19 in a given case data. Each wave is represented by a hyperbolic tangent function. A linear superposition of single waves represents a case data having accumulated cases over \(N\) weeks, \begin{equation} \label{eq:mf} Z_{N}(n)=\sum\limits_{i=1}^{l}a_{i}\tanh(b_{i}n-c_{i})+d. \end{equation} This model function has the growth rates \begin{equation} \label{eq:va} V(n)=\frac{dZ}{dn}= \sum\limits_{i=1}^{l}\frac{a_{i}b_{i}}{\cosh^{2}(b_{i}n-c_{i})},\quad A(n)=\frac{dV}{dn}= -2\sum\limits_{i=1}^{l}a_{i}^{2}b_{i} \frac{\sinh(b_{i}n-c_{i})}{\cosh^{3}(b_{i}n-c_{i})}, \end{equation} speed and acceleration, respectively.

At the inflection points $i_{p}$, by definition, the acceleration $A(i_{p})$ is null (and about to become negative), and the speed $V(i_{p})$ is at a local maximum. Each inflection point belongs to one wave. Since the acceleration becomes negative after the inflection point, the speed diminishes and the stabilization can be reached if a new wave is not on the way. When both velocity and acceleration are practically zero on both sides of the inflection point, a wave is called complete. In order to have multiple waves in play, the acceleration must be zero at the points connecting them and the velocity must be at a local minimum.

Parameters $\{a,b,c,d\}$ appearing in the model function were obtained by a (local) non-linear fitting minimizing the root mean square (rms) deviations. All case data points are equally weighted. Case data from every state are available at CDC.