  What role do asymptomatically infected individuals play in the transmission
dynamics? There are many diseases, such as norovirus and influenza, where some
infected hosts show symptoms of the disease while others are asymptomatically
infected, i.e. do not show any symptoms. The current paper considers a class of
epidemic models following an SEIR (Susceptible $\to$ Exposed $\to$ Infectious
$\to$ Recovered) structure that allows for both symptomatic and asymptomatic
cases. The following question is addressed: what fraction $\rho$ of those
individuals getting infected are infected by symptomatic (asymptomatic) cases?
This is a more complicated question than the related question for the beginning
of the epidemic: what fraction of the expected number of secondary cases of a
typical newly infected individual, i.e. what fraction of the basic reproduction
number $R_0$, is caused by symptomatic individuals? The latter fraction only
depends on the type-specific reproduction numbers, while the former fraction
$\rho$ also depends on timing and hence on the probabilistic distributions of
latent and infectious periods of the two types (not only their means). Bounds
on $\rho$ are derived for the situation where these distributions (and even
their means) are unknown. Special attention is given to the class of Markov
models and the class of continuous-time Reed-Frost models as two classes of
distribution functions. We show how these two classes of models can exhibit
very different behaviour.
