Mathematical modeling and control of Tuberculosis: Backward and Hopf bifurcations under case-finding and case-holding interventions

Abstract

Reinfection and relapse are known to induce backward bifurcation (a subcritical pitchfork bifurcation) in many infectious disease models, particularly tuberculosis. This phenomenon implies the coexistence of multiple equilibria when the basic reproduction number is below one: a stable disease-free equilibrium and two endemic equilibria, one stable and one unstable. As a result, hysteresis may arise, meaning that reducing the reproduction number below one may not eliminate the disease given that the system remains in the basin of attraction of the endemic state. Another relevant phenomenon, often associated with intervention delays, is the occurrence of supercritical Hopf bifurcations. In this case, the endemic equilibrium loses stability for reproduction numbers above one, leading to sustained oscillations in the form of stable limit cycles. Although less explored in the tuberculosis literature, this behavior has important implications for disease control. In this work, we analyze a tuberculosis model to investigate these dynamical effects and show that Hopf bifurcation can arise even without delays or saturation effects. The bifurcation analysis focuses on two key public health interventions -case finding and case holding- and their combined effect on disease dynamics. Numerical simulations, based on epidemiological data from Salta, Argentina -a province with high TB burden- illustrate the challenges these dynamics pose for the design and implementation of public health policies.