Backward bifurcation and hysteresis in models of recurrent tuberculosis

Isaac Mwangi Wangari, Lewi Stone

An epidemiological model is presented that provides a comprehensive description of the transmission pathways involved for recurrent tuberculosis (TB), whereby cured individuals can become reinfected. Our main goal is to determine conditions that lead to the appearance of a backward bifurcation. This occurs when an asymptotically stable infection free equilibrium concurrently exists with a stable non-trivial equilibria even though the basic reproduction number R0 is less than unity. Although, some 10-30% cases of TB are recurrent, the role of recurrent TB as far as the formation of backward bifurcation is concerned, is rarely if ever studied. The model used here incorporates progressive primary infection, exogenous reinfection, endogenous reactivation and recurrent TB as transmission mechanisms that contribute to TB progression. Unlike other studies of TB dynamics that make use of frequency dependent transmission rates, our analysis provides exact backward bifurcation threshold conditions without resorting to commonly applied approximations and simplifying assumptions. Exploration of the model through analytical and numerical analysis reveal that recurrent TB is sometimes capable of triggering hysteresis effects which allow TB to persist when R0 < 1 even though there is no backward bifurcation. Furthermore, recurrent TB can independently induce backward bifurcation phenomena if it exceeds a certain threshold.