  In contrast to the common assumption in epidemic models that the rate of
infection between individuals is constant, in reality, an individual's viral
load determines their infectiousness. We compare the average and individual
reproductive numbers and epidemic dynamics for a model incorporating
time-dependent infectiousness and a standard SIR model for both fully-mixed and
category-mixed populations. We find that the reproductive number only depends
on the total infectious exposure and the largest eigenvalue of the mixing
matrix and that these two effects are independent of each other. When we
compare our time-dependent mean-field model to the SIR model with equivalent
rates, the epidemic peak is advanced and modifying the infection rate function
has a strong effect on the time dynamics of the epidemic. We also observe
behavior akin to a traveling wave as individuals transition through infectious
states.
