  In this work we study a kinetic model of active particles with delayed
dynamics, and its limit when the number of particles goes to infinity. This
limit turns out to be related to delayed differential equations with random
initial conditions. We analyze two different dynamics, one based on the full
knowledge of the individual trajectories of each particle, and another one
based only on the trace of the particle cloud, loosing track of the individual
trajectories. Notice that in the first dynamic the state of a particles is its
path, whereas it is simply a point in $\R^d$ in the second case. We analyse in
both cases the corresponding mean-field dynamic obtaining an equation for the
time evolution of the distribution of the particles states. Well-posedness of
the equation is proved by a fixed-point argument. We conclude the paper with
some possible future research directions and modelling applications.
