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We consider an SIR-type (Susceptible $\to$ Infected $\to$ Recovered)stochastic epidemic process with multiple modes of transmission on a contactnetwork. The network is given by a random graph following a multilayerconfiguration model where edges in different layers correspond to potentiallyinfectious contacts of different types. We assume that the graph structureevolves in response to the epidemic via activation or deactivation of edges. Wederive a large graph limit theorem that gives a system of ordinary differentialequations (ODEs) describing the evolution of quantities of interest, such asthe proportions of infected and susceptible vertices, as the number of nodestends to infinity. Analysis of the limiting system elucidates how the couplingof edge activation and deactivation to infection status affects diseasedynamics, as illustrated by a two-layer network example with edge typescorresponding to community and healthcare contacts. Our theorem extends someearlier results deriving the deterministic limit of stochastic SIR processes onstatic, single-layer configuration model graphs. We also describe precisely theconditions for equivalence between our limiting ODEs and the systems obtainedvia pair approximation, which are widely used in the epidemiological andecological literature to approximate disease dynamics on networks. Potentialapplications include modeling Ebola dynamics in West Africa, which was themotivation for this study.

The large graph limit of a stochastic epidemic model on a dynamic multilayer network