Dynamics of a reaction–diffusion waterborne pathogen model with direct and indirect transmission

In this paper a reaction–diffusion waterborne pathogen model, which incorporates both direct and indirect transmission pathways, is proposed. Then the basic reproduction number R0R0 defined as the spectral radius of the next generation operator is established for the model system, and an equivalent characterization of R0R0 is obtained in terms of an elliptic eigenvalue problem. Furthermore, we show that R0R0 serves as a threshold parameter which predicts whether the disease will persist. By using numerical simulations, we study the influence of diffusion coefficients, spatial heterogeneity and multiple transmission pathways.