Online information is increasingly linked to real-world instability, especially as automated accounts and LLM-based agents help spread and amplify news. In this work, we study how information spreads on networks of Large Language Models (LLMs) using mathematical models. We investigate how different types of offline events, along with the "personalities" assigned to the LLMs, affect the network dynamics of online information spread of the events among the LLMs. We introduce two models: 1) a stochastic agent-based network model and 2) a system of differential equations arising from a mean-field approximation to the agent-based model. We fit these models to simulations of the spread of armed-conflict news on social media, using LLM agents each with one of 32 personality trait profiles on k-regular random networks. Our results indicate that, despite the complexity of the news events, personalities, and LLM behaviors, the overall dynamics of the system are well described by a Susceptible-Infected (SI) type model with two transmission rates.