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Beretta E, Breda D.

Backward bifurcation and hysteresis in models of recurrent tuberculosis

Classical epidemic models therefore often have two intrinsic equilibria: Illustration of backward bifurcation when there is no recurrent TB i. A simplified model for the estimation of life-cycle greenhouse gas emissions of enhanced geothermal systems.

This form of incidence is considered to be more appropriate for infections in human populations [ 33 ]. While this excludes the realistic possibility of immigration of infectives either latent individuals or individuals with active TBit helps untangle the conditions that result in backward bifurcation.

However, if R 0 is increased slightly above unity then model Eq 1 has one positive endemic equilibrium point. This is supported by the fact that majority of new TB cases about ninety percent are as a result of reinfection rather than endogenous reactivation [ 1319 — 21 ]. Applying LCA to design alternatives. Regionalisation of Life Cycle Thinking: For a clear view, the bifurcation structure is plotted with semi-log axes.

To aid visualisation, bifurcation structure is plotted with semi-log axes. Now the roots of the cubic equation. The mathematics of infectious diseases. Yet, these pathways are critical to TB epidemiology, an aspect that was even pointed out by Lipsitch and Murray [ 18 ] as a weakness of the Feng et al.


In the literature it has been observed that individuals who previously have had active TB and who were xnalyse treated are more likely to gain active TB another time [ 13 ]. Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis.

Through the use of advanced molecular fingerprinting techniques, TB recurrence has been classified into two fundamental forms of infection: As such, Lipsitch and Murray [ 18 ] argued that backward bifurcations should not be expected in the real world. In the absence of recurrent TB due to reinfection i. Hence, this already suggests that R 0 alone is unable to completely quantify some key dynamical features of the TB epidemic, and is in fact the first sign that a backward bifurcation might be involved.

Transmission of tuberculosis with exogenous re-infection and endogenous reactivation.

Isabelle BLANC Professeur at MINES ParisTech, PSL Research University

Global sensitivity analysis in LCA of emerging technologies: A new technique for thermal modelling of building: In fact even just for the simplified reinfection pathways of the original Feng et al.

This relates to studies which have found that recovered individuals are more likely to be susceptible to future TB infection than TB-naive individuals [ 13 ].

In future work, it would be interesting ss1 additionally consider the qnalyse that some infected individuals who are treated do not become completely cured. Table 1 provides a detailed list and description of model parameters as well as typical parameter values used here, as obtained from the literature.


Dynamical models of tuberculosis and their applications. Journal of Mathematical Analysis and Applications.

CV HAL : Isabelle BLANC Professeur at MINES ParisTech, PSL Research University

Isabelle Blanc, Eric Labouze. Van den Driessche P, Watmough J. Shaker Verlag, Part 2: Similar to other infectious diseases, the ability for TB to persist in a population or go extinct is to a large degree governed by the basic reproduction number, R 0which is defined as the number of secondary cases an infected individual will generate when introduced into a completely susceptible population [ 14 — 17 ].

Estimating particulate matter health impact related to the combustion of different fossil s,ia. Acknowledgments All authors contributed equally and have no any competing interests.

A plot of the critical value of R c as a function of the level of exogenous reinfection p. It is interesting to compare the aalyse point p c for the reinfection threshold given in Eq 13 with that found by Feng et al.

Have you forgotten your login? The models of Feng et al. Fig 2 provides a typical bifurcation diagram that shows the key features of a backward bifurcation.

Drebszok, Artur Wyrwa, Isabelle Blanc. Environmental impacts of large-scale grid-connected ground-mounted PV installations. Global Tuberculosis Report When there are multiple stable equilibria coexisting simultaneously, the final equilibrium a population will reach is dependent on the initial conditions of its sub-populations.

Frontiers of Research in Industrial Ecology, Workshop 6: