  The conventional voter model is modified so that an agent's switching rate
depends on the `age' of the agent, that is, the time since the agent last
switched opinion. In contrast to previous work, age is continuous in the
present model. We show how the resulting individual-based system with
non-Markovian dynamics and concentration-dependent rates can be handled both
computationally and analytically. Lewis' thinning algorithm can be modified in
order to provide an efficient simulation method. Analytically, we demonstrate
how the asymptotic approach to an absorbing state (consensus) can be deduced.
We discuss three special cases of the age dependent switching rate: one in
which the concentration of voters can be approximated by a fractional
differential equation, another for which the approach to consensus is
exponential in time, and a third case in which the system reaches a frozen
state instead of consensus. Finally, we include the effects of spontaneous
change of opinion, i.e., we study a noisy voter model with continuous ageing.
We demonstrate that this can give rise to a continuous transition between
coexistence and consensus phases. We also show how the stationary probability
distribution can be approximated, despite the fact that the system cannot be
described by a conventional master equation.
