SIRI+Q model with a limited capacity of isolation

We construct and analyze an SIRI+Q model with a piecewise smooth system of ordinary differential equations for the epidemic dynamics of a reinfectious disease, in which a limited capacity of isolation is incorporated. To consider the relation of the limited isolation capacity to the epidemic consequence, we derive the condition that the isolation reaches the capacity at finite time along the path of the epidemic process, and that the disease becomes endemic. We investigate in particular how the endemicity, the endemic size, or the final epidemic size could depend on the isolation capacity. From the obtained mathematical results, we find theoretical implications on the relevance of the isolation capacity and the difficulty of its measure to control the spread of the disease in the community.