Global stability of a multiple infected compartments model for waterborne diseases

• Feedback mechanism from contaminated water is incorporated.
• The global behavior is determined by the basic reproduction number.
• The disease always dies out if R0R0 is less than or equal to one.
• The decay rate of pathogens has a significant impact on the epidemic growth rate.

In this paper, mathematical analysis is carried out for a multiple infected compartments model for waterborne diseases, such as cholera, giardia, and rotavirus. The model accounts for both person-to-person and water-to-person transmission routes. Global stability of the equilibria is studied. In terms of the basic reproduction number R0R0, we prove that, if R0⩽1R0⩽1, then the disease-free equilibrium is globally asymptotically stable and the infection always disappears; whereas if R0>1R0>1, there exists a unique endemic equilibrium which is globally asymptotically stable for the corresponding fast–slow system. Numerical simulations verify our theoretical results and present that the decay rate of waterborne pathogens has a significant impact on the epidemic growth rate. Also, we observe numerically that the unique endemic equilibrium is globally asymptotically stable for the whole system. This statement indicates that the present method need to be improved by other techniques.