Exact solution of infection dynamics with gamma distribution of generation intervals

Infectious disease outbreaks are expected to grow exponentially in time but their initial dynamics is less known. Here I derive analytical expressions for the infectious disease dynamics with a gamma distribution of generation intervals. Excluding the exponential distribution, the outbreak grows as a power law at short times. At long times, the dynamics is exponential with a growth rate determined by the basic reproductive number and the parameters of the generation interval distribution. These analytical expressions can be deployed to do better estimates of infectious disease parameters.

Figure 1

Gamma distribution with shape parameters $s$.

Figure 2

Shape of $f(x,s)$ for different values of $s$.

Figure 3

Population reproductive number $P$ as a function of $s$ for $R>1$ (solid line) and $R<1$ (dashed line).

Figure 4

Numerical simulations of disease transmission in a virtual city. (a) Average number of new infectious as a function of time for different number of intermediate states (symbols). The lines are fits to the exponential function in Eq. (27), providing an estimate of $P$. (b) Population reproductive number $P$ as a function of $s$, as obtained from the fitting to the numerical data (symbols) and from the expected theoretical line (28) (line).