  Experimental results on the immune response to cancer indicate that
activation of cytotoxic T lymphocytes (CTLs) through interactions with
dendritic cells (DCs) can trigger a change in CTL migration patterns. In
particular, while CTLs in the pre-activation state move in a non-local search
pattern, the search pattern of activated CTLs is more localised. In this paper,
we develop a kinetic model for such a switch in CTL migration modes. The model
is formulated as a coupled system of balance equations for the one-particle
distribution functions of CTLs in the pre-activation state, activated CTLs and
DCs. CTL activation is modelled via binary interactions between CTLs in the
pre-activation state and DCs. Moreover, cell motion is represented as a
velocity-jump process, with the running time of CTLs in the pre-activation
state following a long-tailed distribution, which is consistent with a L\'evy
walk, and the running time of activated CTLs following a Poisson distribution,
which corresponds to Brownian motion. We formally show that the macroscopic
limit of the model comprises a coupled system of balance equations for the cell
densities whereby activated CTL movement is described via a classical diffusion
term, whilst a fractional diffusion term describes the movement of CTLs in the
pre-activation state. The modelling approach presented here and its possible
generalisations are expected to find applications in the study of the immune
response to cancer and in other biological contexts in which switch from
non-local to localised migration patterns occurs.
