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Random transmission events between individuals occurring at short scales control patterns emerging at much larger scales in natural and artificial systems. Examples range from the spatial propagation of an infectious pathogen in an animal population to the spread of misinformation in online social networks or the sharing of target locations between robot units in a swarm. Despite the ubiquity of information transfer events, a general methodology to quantify spatio-temporal transmission processes has remained elusive. The challenge in predicting when and where information is passed from one individual to another stems from the limited number of analytic approaches and from the large fluctuations and inherent computational cost of stochastic simulation outputs, the main theoretical tool available to study such processes so far. Here we overcome these limitations by developing an analytic theory of transmission dynamics between randomly moving agents in arbitrary spatial domains and with arbitrary information transfer efficiency. We move beyond well-known approximations employed to study reaction diffusion phenomena, such as the motion and reaction limited regimes, by quantifying exactly the mean reaction time in presence of multiple heterogeneous reactive locations. To demonstrate the wide applicability of our theory we employ it in different scenarios. We show how the type of spatial confinement may change by many orders of magnitude the time scale at which transmission occurs. When acquiring information represents the ability to capture, we use our formalism to uncover counterintuitive evasive strategies in a predatorprey contest between territorial animals. When information transmission represents the transfer of an infectious pathogen, we consider a population with susceptible, infected and recovered individuals that move and pass infection upon meeting and predict analytically the basic reproduction number. Finally we show how to apply the transmission theory semi-analytically when the topology of where individuals move is that of a network.

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Interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system. Examples range from molecules advancing along dendritic spines, to animal anti-predator displacements due to sparse vegetation, through to water vapour sifting across the pores of breathable materials. Estimating the effects of disorder on the transport characteristics in these and other systems has a long history. When the localised interactions are reactive, that is when particles may vanish or get irreversibly transformed, the dynamics is modelled as a boundary value problem with absorbing properties. The analytic advantages offered by such a modelling approach has been instrumental to construct a general theory of reactive interaction events. The same cannot be said when interactions are inert, i.e. when the environment affects only the particle movement dynamics. While various models and techniques to study inert processes across biology, ecology and engineering have appeared, many studies have been computational and explicit results have been limited to one-dimensional domains or symmetric geometries in higher dimensions. The shortcoming of these models have been highlighted by the recent advances in experimental technologies that are capable of detecting minuscule environmental features. In this new empirical paradigm the need for a general theory to quantify explicitly the effects of spatial heterogeneities on transport processes, has become apparent. Here we tackle this challenge by developing an analytic framework to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We do so by using a discrete space formulation whereby the interactions between an agent and the environment are modelled as perturbed dynamics between lattice sites. We calculate exactly how disorder affects movement due to reflecting or partially reflecting obstacles, regions of increased or decreased diffusivity, one-way gates, open partitions, reversible traps as well as long range connections to far away areas. We provide closed form expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return and exit probabilities and their respective means. The strength of an analytic formulation becomes evident as we uncover a surprising property, which we term the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi 1D systems. To demonstrate the widespread applicability of our formalism, we consider three examples that span different spatial and temporal scales. The first is an exploration of an enhancement strategy of transdermal drug delivery through the stratum corneum of the epidermis. In the second example we associate spatial disorder with a decision making process of a wandering animal to study thigmotaxis, i.e. the tendency to remain close to the edges of a confining domain. The third example illustrates th...