Reed-Hughes mechanism

In this post we present a theoretical explanation for the power-law distributions observed in the spatial growth of COVID-19. From theoretical point-of-view it is interesting why power-law distribution is observed in the data, as typical epidemic spread is characterized by an exponential distribution (at least the SIR model predicts this). Yet, the explanation is pretty simple and is based on the Reed-Hughes mechanism [1].

So, SIR model predicts that in the initial stage of epidemic the growth of confirmed cases will be exponential. Let us consider only this initial stage and thus describe the growth of cumulative number of detected cases as follows:

\begin{equation} I \left( t \right) = \exp\left( \mu [t - t_0] \right) = \exp\left( \mu \tau \right) . \end{equation}

In the above \( t_0 \) is the start time of epidemic, while \( \tau \) is the elapsed time. This growth law applies for \( t \geq t_0 \) or \( \tau \geq 0 \).

Another obvious thing is that in a country as large as United States, epidemic would start at very different times in different locations. Let us assume that \( t_0 \) are exponentially distributed with the origin point being the time of the first confirmed case. In this case \( \tau \) will be also exponentially distributed:

\begin{equation} p_{\tau} \left( \tau \right) = \nu \exp\left( -\nu \tau \right) . \end{equation}

Let us now find \( p_I(I) \) by requiring that the probabilities to find observations in the infinitesimally small ranges would be equal:

\begin{equation} p_{\tau} \left( \tau \right) \mathrm{d} \tau = p_I \left( I \right) \mathrm{d} I . \end{equation}

After few trivial algebraic manipulations we get:

\begin{equation} p_I = p_{\tau} \frac{\mathrm{d} \tau}{\mathrm{d} I} = \frac{\nu}{\mu} I^{\frac{\nu}{\mu}-1} . \end{equation}

So we can reasonably expect that \( I \) should be power-law distributed. Note that as \( \tau > 0 \) is always true, \( I \) will take values large than \( 1 \). These observations as well as the restrictions are perfectly reasonable from the empirical point-of-view.

You can explore Reed-Hughes mechanism using the app below. Blue curve shows the probability density function of \( \tau \), while the red curve shows the probability density function of \( I \). The scale is double logarithmic so the red curve should be almost straight line (representing power-law dependence).

References

• W. J. Reed, B. D. Hughes. From gene families and genera to incomes and internet file sizes: Why power laws are so common in nature. Physical Review E 66: 0671033 (2002). doi: 10.1103/PhysRevE.66.06710.