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Defective interfering particles (DIPs) are virus-like particles that occur naturally during virus infections. These particles are defective, lacking essential genetic materials for replication, but they can interact with the wild-type virus and potentially be used as therapeutic agents. However, the effect of DIPs on infection spread is still unclear due to complicated stochastic effects and nonlinear spatial dynamics. In this work, we develop a model with a new hybrid method to study the spatial-temporal dynamics of viruses and DIPs co-infections within hosts. We present two different scenarios of virus production and compare the results from deterministic and stochastic models to demonstrate how the stochastic effect is involved in the spatial dynamics of virus transmission. We compare the spread features of the virus in simulations and experiments, including the formation and the speed of virus spread and the emergence of stochastic patchy patterns of virus distribution. Our simulations simultaneously capture observed spatial spread features in the experimental data, including the spread rate of the virus and its patchiness. The results demonstrate that DIPs can slow down the growth of virus particles and make the spread of the virus more patchy. 

The Togashi Kaneko model (TK model) is a simple stochastic reaction network that displays discreteness-induced transitions between meta-stable patterns. Here we study a constrained Langevin approximation (CLA) of this model. This CLA, derived under the classical scaling, is an obliquely reflected diffusion process on the positive orthant and hence respects the constraint that chemical concentrations are never negative. We show that the CLA is a Feller process, is positive Harris recurrent and converges exponentially fast to the unique stationary distribution. We also characterize the stationary distribution and show that it has finite moments. In addition, we simulate both the TK model and its CLA in various dimensions. For example, we describe how the TK model switches between meta-stable patterns in dimension six. Our simulations suggest that, when the volume of the vessel in which all of the reactions that take place is large, the CLA is a good approximation of the TK model in terms of both the stationary distribution and the transition times between patterns.