Stability and Sensitivity Analysis of the iSIR Model for Indirectly Transmitted Infectious Diseases with Immunological Threshold

Most pathogenic diseases remain epidemic and endemic in the world, causing thousands of deaths annually in less developed countries. Yet, their dynamics are still not fully understood. In this paper, we carry out a thorough stability and sensitivity analysis of an iSIR which incorporates an infection term that explicitly includes a minimum infection dose (MID), and determine an invariant domain. We discover that if the MID (denoted c) is less than the bacterial carrying capacity K, we may have two steady states: the endemic or epidemic steady state, and the disease-free and bacteria-free steady state. The latter is unstable and the former is globally stable under a certain condition. On the other hand, if c ≥ K, then up to four steady states may exist: an unstable endemic steady state, a locally stable endemic steady state, a conditionally globally stable disease-free steady state, and an unstable disease-free and bacteria-free steady state. We find that to control the period and intensity of the outbreaks, it might be better to focus on the bacterial carrying capacity rather than on the shedding rates.