Large Language Models (LLMs) offer new opportunities to accelerate complex interdisciplinary research domains. Epidemic modeling, characterized by its complexity and reliance on network science, dynamical systems, epidemiology, and stochastic simulations, represents a prime candidate for leveraging LLM-driven automation. We introduce EpidemIQs, a novel multi-agent LLM framework that integrates user inputs and autonomously conducts literature review, analytical derivation, network modeling, mechanistic modeling, stochastic simulations, data visualization and analysis, and finally documentation of findings in a structured manuscript, through five predefined research phases. We introduce two types of agents: a scientist agent for planning, coordination, reflection, and generation of final results, and a task-expert agent to focus exclusively on one specific duty serving as a tool to the scientist agent. The framework consistently generated complete reports in scientific article format. Specifically, using GPT 4.1 and GPT 4.1 Mini as backbone LLMs for scientist and task-expert agents, respectively, the autonomous process completes with average total token usage 870K at a cost of about $1.57 per study, successfully executing all phases and final report. We evaluate EpidemIQs across several different epidemic scenarios, measuring computational cost, workflow reliability, task success rate, and LLM-as-Judge and human expert reviews to estimate the overall quality and technical correctness of the generated results. Through our experiments, the framework consistently addresses evaluation scenarios with an average task success rate of 79%. We compare EpidemIQs to an iterative single-agent LLM, benefiting from the same system prompts and tools, iteratively planning, invoking tools, and revising outputs until task completion. The comparisons suggest a consistently higher performance of EpidemIQs.