  The incubation period of a disease is the time between an initiating
pathologic event and the onset of symptoms. For typhoid fever, polio, measles,
leukemia and many other diseases, the incubation period is highly variable.
Some affected people take much longer than average to show symptoms, leading to
a distribution of incubation periods that is right skewed and often
approximately lognormal. Although this statistical pattern was discovered more
than sixty years ago, it remains an open question to explain its ubiquity. Here
we propose an explanation based on evolutionary dynamics on graphs. For simple
models of a mutant or pathogen invading a network-structured population of
healthy cells, we show that skewed distributions of incubation periods emerge
for a wide range of assumptions about invader fitness, competition dynamics,
and network structure. The skewness stems from stochastic mechanisms associated
with two classic problems in probability theory: the coupon collector and the
random walk. Unlike previous explanations that rely crucially on heterogeneity,
our results hold even for homogeneous populations. Thus, we predict that two
equally healthy individuals subjected to equal doses of equally pathogenic
agents may, by chance alone, show remarkably different time courses of disease.
