A Model for Tuberculosis with Exogenous Reinfection

Feng, Zhilan; Castillo‐Chavez, Carlos; Capurro, Ángel F.

doi:10.1006/tpbi.2000.1451

Cited by 330 publications

(312 citation statements)

References 37 publications

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“…1 The authors restate that no epidemiological consequences can be assigned to the reinfection threshold because R 0 ¼ 1=s is not a bifurcation point. They claim that our findings concerning reinfection reduce to a nonlinear increase in the prevalence of infection as R 0 increases, as previously reported by Blower et al (1998) and Feng et al (2000). This is not the case.…”

Section: Final Remarkssupporting

confidence: 83%

The reinfection threshold

Gomes

White

Medley

2005

Journal of Theoretical Biology

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Abstract' ' ' r 2005 Published by Elsevier Ltd.Keywords: '; '; ' Breban and Blower (2005) challenge our use of the word ''threshold'' to name the ''Reinfection Threshold '' (Gomes et al., 2004a) on the basis that this is not a bifurcation point. Assuming that a misnomer of the concept would invalidate its implications, they deduce that our results have no epidemiological consequences. Here we explain the terminology but, more importantly, we emphasize that the choice of name does not affect the implications of the concept. We assert that the epidemiological behaviour associated with R 0 ¼ 1=s remains unchanged.The recognition that partial immunity divides the transmissibility axis into two distinct modes of transmission emerged almost simultaneously from two studies: a systematic analysis of microparasite transmission under suboptimal immunity (Gomes et al., 2004a); and a representation of the transmission dynamics of tuberculosis leading to a hypothesis for the widely debated variable efficacy of the bacille Calmette-Gue´rin (BCG) vaccine (Gomes et al., 2004b). The two studies overlap through a modified SIR modelwhere S and I are the proportions of fully susceptible and infected individuals, respectively, and R ¼ 1 À S À I is the proportion recovered with partial immunity. In units of average duration of infection, births and deaths occur at a rate e; and a proportion v of the population is vaccinated at birth to enter compartment R: Individuals in S are infected at a rate, R 0 I; and individuals in R are infected at a reduced rate, sR 0 I; where 0psp1: The existence of these two distinct rates of infection is the key to our findings.The endemic equilibria of model (1) are represented in Fig. 1 as a function of the basic reproduction number, R 0 : The proportion of infectious individuals is represented in linear (a) and logarithmic (b) scales, and the four curves correspond to different vaccination coverage: v ¼ 0; 0.67, 0.89, 1, from left to right. For the purpose of illustration we chose s ¼ 0:25; but the conclusions that follow remain valid for any value between 0 and 1. We set e ¼ 0:0012; which is attained under a life expectancy of 70 years and an average duration of infection of 1 month. Shorter infectious periods would enhance the effects that follow. Breban and Blower (2005) carry out a bifurcation analysis of the v ¼ 0 model, and correctly conclude that

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Section: Final Remarkssupporting

confidence: 83%

The reinfection threshold

Gomes

White

Medley

2005

Journal of Theoretical Biology

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“…The reason that reinfection does not play a role in the location of the R 0 = 1 threshold is that near the bifurcation, by definition the system is near the point S = 1, L = I = R = 0, where S >> L + R so that the contribution of reinfection to disease is negligible. Consistent with the findings of several previous modelers (Feng et al, 2000;Singer and Kirschner, 2004), an endemic equilibrium exists in the model even for R 0 < 1 when the reinfection progression rate p r is sufficiently large (i.e. there is a backward bifurcation reflecting the fact that reinfection parameters do not occur in the expression for R 0 ).…”

Section: Dynamics Of the Delayed Modelsupporting

confidence: 87%

“…Mathematical modeling has proven a valuable tool for understanding TB dynamics (Blower et al, 1995;Vynnycky and Fine, 1997;Feng et al, 2000;Singer and Kirschner, 2004) and has served as the basis for establishing control targets and assessing policy strategies (Blower et al, 1996;Dye et al, 1998;Cohen et al, 2006). However, most such models, with occasional exceptions (Schinazi, 1999), have been differential equation susceptible-exposed-infected-recovered (SEIR) models that assume a homogenously mixed population.…”

Section: Introductionmentioning

confidence: 99%

Emergent heterogeneity in declining tuberculosis epidemics

Colijn

Cohen

Murray

2007

Journal of Theoretical Biology

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Tuberculosis is a disease of global importance: over 2 million deaths are attributed to this infectious disease each year. Even in areas where tuberculosis is in decline, there are sporadic outbreaks which are often attributed either to increased host susceptibility or increased strain transmissibility and virulence. Using two mathematical models, we explore the role of the contact structure of the population, and find that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneity even in the absence of host or strain variability. We discuss the implications of this finding for tuberculosis control in low incidence settings.

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“…However, often the constructions necessary to produce these dynamics require biologically tenuous assumptions such as in the model above where the rate of reinfection from the vaccinated state is greater than the rate of returning to the susceptible state. Another example is a model for Tuberculosis in which a backward bifurcation will occur when infection with re-exposure is more likely than infection with primary exposure (Feng, Castillo-Chavez, & Capurro, 2000;Martcheva, 2015). To our knowledge, however, there exists no real-world experimental evidence of the occurrence of backwards bifurcation in infectious disease systems.…”

Section: Backward Bifurcationsmentioning

confidence: 99%