A two-strain TB model with multiple latent stages

Abstract

A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving ``arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below 1.

Mathematical analysis of the dynamics of a fractional‐order tuberculosis epidemic in a patchy environment under the influence of re‐infection

“…The more real the data, the closer the parameters extracted from them will be to reality. Considering that the way most diseases develop, especially infectious diseases, are almost the same all over the world, for this reason, we use the values of the parameters that were reported in the valid articles [7,15,34]. Table 1 lists the associated variables and parameters.…”

“…The majority of the mathematical modeling of TB with the occurrence of exogenous re‐infection is famous for exhibiting an event called backward bifurcation. This phenomenon is such a bifurcation that provides conditions that disease‐free fixed point and endemic equilibria exist simultaneously, and both of them are stable [6, 7, 9–14]. Nonetheless, Sharomi et al [15] proved it is not possible for exogenous re‐infection all the time to lead to the phenomenon of backward bifurcation, so this phenomenon authorizes the disease to continue in the case $$$ {R}_0\le 1 $$$.…”

“…However, according to the prevalence level of TB, it is best to consider it as a slow-moving disease. In other words, according to the distribution of long and changeable latent periods of TB, and comparing it with the short, not-so-changeable infectious periods, it can be considered as a slow-moving disease [6,7]. Return of the TB to the host people is defined by the happening of the next episode of the disease after treatment results in individuals under therapy not sufficiently having potent immunity against the TB.…”

“…Then, we show that the TB model, together with the exogenous re-infection, with some conditions, is possible to show backward bifurcation. Although some of the authors work on the influence of exogenous re-infection on backward bifurcation in TB epidemic models in formulation with integer order and single patch [6,7,9,15,34], we investigate this effect by considering modeling in the fractional order and two patches. We state the conditions that give rise to show that the model of TB has backward bifurcation.…”

Dynamics of Tuberculosis (TB) with Drug Resistance to First-Line Treatment and Leaky Vaccination: A Deterministic Modelling Perspective

“…In 1993, the WHO decreed TB a global epidemic [ 3 ]. It is estimated that the risk of contracting active TB after coming into contact with an infected person is between 5% and 10%, with a greater proportion of the disease playing a crucial role which happens in the very first few years after the initial infection with the arrival of HIV [ 4 ].…”