The advantage of being discrete: analytical predictions of reactive and inert interaction events, from infection transmission to movement in disordered media

Reaction diffusion processes with multiple targets and movement in spatially heterogeneous space are notoriously difficult problems to solve. With continuous variables in dimensions higher than one they lead to complex boundary value problems that can only be tackled via numerical solutions of PDE or via stochastic simulations. With spatially discrete variables it is possible to bypass this challenge thanks to recent advances in lattice random walk theory both in homogeneous and heterogeneous space as well as in different topologies, e.g. Cartesian, hexagonal and triangular. In this talk I will present two general lattice random walk formalisms. (i) The first allows to describe analytically the movement of a diffusing particle in the presence of inert spatial heterogeneities, i.e. heterogeneities that affect the movement but not the lifetime of a particle, e.g. areas with different diffusivities, permeable or impenetrable barriers and long-range connections between distant sites. (ii) With the second formalism, by accounting for the co-occurrence of multiple random processes, it is possible to quantify exactly the spatio-temporal dynamics of reaction diffusion events in a multi-target environment, e.g. for encounter and infection transmission. I will present various applications of the formalisms when the movement is Markovian (diffusive), and, if time allows, some results for when the walker steps are correlated, the latter representing the simplest case of a non-Markov walk.