This simple phenomenological model relies on two parameters to characterize the early trajectory of an epidemic and to generate short-term epidemic forecasts (Chowell et al.
2016a; Pell et al.
2016; Viboud et al.
2016). The model incorporates epidemic growth patterns that range from sub-exponential (e.g., polynomial) to exponential by estimating two parameters: (1) the intrinsic growth rate (
r) and (2) a dimensionless “deceleration of growth” parameter with quantified uncertainty (
p). The latter modulates growth patterns ranging from constant incidence rates to exponential epidemic growth (Tolle
2003). It is useful to characterize the scaling of the growth pattern of the epidemic. In particular, this parameter is helpful to distinguish between sub-exponential (
\( p < 1 \)) and exponential growth dynamics (
\( p = 1 \)). Previous analyses highlighted the presence of early sub-exponential growth patterns in infectious disease data across a diversity of disease outbreaks including the 2013–2015 Ebola epidemic in West Africa, influenza, smallpox, plague, measles, foot-and-mouth disease, and HIV/AIDS (Viboud et al.
2016). The GGM model is given by the following differential equation (Tolle
2003; Viboud et al.
2016):
$$ \frac{{{\text{d}}C\left( t \right)}}{{{\text{d}}t}} = C^{{\prime }} \left( t \right) = rC\left( t \right)^{p} $$
where
\( C^{{\prime }} \left( t \right) \) describes the incidence curve over time
\( t \). The cumulative number of cases at time
\( t \) is given by
\( C\left( t \right) \), while
\( r \) is a positive parameter denoting the growth rate (1/time) and
\( p \in [ 0 , 1 ] \) is a “deceleration of growth” parameter. As described in Viboud et al. (
2016), if
\( p = 0 \), this equation describes a constant incidence over time, while if
\( p = 1 \) the equation becomes the well-known exponential growth model (EXPM) (Chowell and Viboud
2016). Intermediate values of
\( p \) (
\( 0 < p < 1 \)) describe sub-exponential (e.g., polynomial) growth patterns. For sub-exponential growth, the closed-form solution of this equation is given by the following polynomial of degree
\( m \) (Tolle
2003):
$$ C\left( t \right) = \left( {\frac{r}{m}t + A} \right)^{m} $$
where
\( m \) is a positive integer,
\( A = C(0)^{1/m} \), and the deceleration parameter is given by
\( p = 1 - 1/m \) (Tolle
2003). An equivalent formulation of the GGM model is given by (Chowell et al.
2016b):
$$ C^{{\prime }} \left( t \right) = \mu^{{\prime }} \left( t \right)C\left( t \right) $$
where
\( \mu^{{\prime }} \left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{r}{{r\left( {1 - p} \right)t + e^{{{ \log }\left( {C\left( 0 \right)} \right)\left( {1 - p} \right)}} }}} \hfill & {0 \le p < 1} \hfill \\ r \hfill & {p = 1} \hfill \\ \end{array} } \right. \)