Tuberculosis models with fast and slow dynamics: the role of close and casual contacts

Song, Bingbing; Castillo‐Chavez, Carlos; Aparicio, Juan Pablo

doi:10.1016/s0025-5564(02)00112-8

Cited by 116 publications

(77 citation statements)

References 34 publications

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“…(Λ controls the total population sizes because the asymptotic carrying capacity of the population is µ Λ . This is set at 6 10 ≈ Λ µ [10] . For purposes of this study, we shall set the recruitment rate at 1,500 individuals per year.…”

Section: Numerical Simulation Of the Model: Below Are Some Of The Parmentioning

confidence: 99%

“…The Model: Now, using the mechanics of transmission as in the model of [6] , we formulate mathematical models that incorporate a parameter for the size of the area occupied by the population being studied in the transmission term so as to help analysis the densitydependence of the dynamics of tuberculosis. In our study, we assume that there is a homogeneous mixing of the population where all people are equally likely to be infected by the infectious individuals in case of contact, the population is distributed over the whole settlement (in the specified area) with a negligibly small area, compared to t he camp size, spared for other human activities and we further assume that all immigrants and newborns are uninfected and therefore join the susceptible category.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical Models for the Dynamics of Tuberculosis in Density-dependent Populations: The Case of Internally Displaced Peoplesâ€™ Camps (IDPCs) in Uganda

Ssematimba¹,

Mugisha²,

Luboobi³

2005

J. of Mathematics and Statistics

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Mathematical models are formulated to establish the conditions (restrictions) on the size of the area occupied required minimizing and thereafter eradicating tuberculosis. Both numerical and qualitative analyses of the model are done and the effect of variation in the area size and recruitment rate on the different epidemiological groups is investigated. The results of the analysis show that there exists a stable disease-free equilibrium point provided that the characteristic area is greater than the product of the probability of survival from the latent stage to the infectious stage and the number of latent infections produced by a typical infectious individual during his/her mean infectious period. The study recommends that the characteristic area per individual should be at least 0.25 square kilometres in order to minimize the tuberculosis incidence.

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Section: Numerical Simulation Of the Model: Below Are Some Of The Parmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical Models for the Dynamics of Tuberculosis in Density-dependent Populations: The Case of Internally Displaced Peoplesâ€™ Camps (IDPCs) in Uganda

Ssematimba¹,

Mugisha²,

Luboobi³

2005

J. of Mathematics and Statistics

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“…We resort to the center manifold theory described in (Feng et al, 2001) and used in (Song et al, 2002) to determine the stability of the endemic equilibrium point. We note that our system is of the form:…”

Section: Local Stability Of the Endemic Equilibriummentioning

confidence: 99%

Modeling the Dynamics of Tuberculosis Transmission in Children and Adults

Nyabadza¹,

Kgosimore²

2012

Journal of Mathematics and Statistics

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Problem statement:The fight Against Tuberculosis (TB) has mainly focused on the adult population because children were perceived to pose a very low risk in TB transmission. This assumption ignores the potential risk children had as reservoirs of latent infections from which future cases evolve when they become adults. It was therefore important to investigate the dynamics of TB taking into consideration, children. Approach: We formulated a compartmental model for TB with two age classes, children and adults. Qualitative analysis of the model was done to investigate the stability of the model equilibria in terms of the model reproduction number R 0 Numerical simulations were also done to investigate the role played by some key epidemiological parameters in the dynamics of the disease. Results: The model had two equilibria: The disease free equilibrium which was globally stable for R 0 <1 and the endemic equilibrium which was locally asymptotically stable for R 0 >1, for R 0 near 1. The study showed increased latent infections in the adult population as a result of increased latently infected children who mature to adulthood with latent infections. Conclusion/Recommendations: Progression to active TB among adults is epidemiologically significant and interventions should focus on the adult population. Anti-tuberculosis, treatment of adults is crucial in controlling the epidemic and should interventions be proposed, they should target progression to active TB for those latently infected. The fight against TB should also take into consideration tuberculosis among children.

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“…В 1997-2004 годах Карлосом Кастильо-Чавезом (Carlos Castillo-Chavez) и при его участии было опубликовано большое количество работ [49,50,51,52,53,37,54,55,56,57,58,59,60,61,62], посвященных математическим моделям распространения и контроля туберкулеза, а так же анализу их свойств. В подавляющем большинстве ста-тей этого цикла наибольшее внимание уделяется анализу положений равновесия си-стем уравнений, задающих динамику моделей, а также условиям их устойчивости.…”

Section: аналитические исследования моделей эпеидемиологии туберкулезаunclassified

“…В статье Жуана П. Апарицио (Juan P. Aparicio) с соавторами [53] произведена оценка вкладов этих двух механизмов. В более поздней работе [56] произведен анализ математической модели, построенной в [53].…”

Section: концепция "обобщенных семей" для описания структуры контактоunclassified

Mathematical Models of Tuberculosis Extension and Control of it

Avilov¹

2007

Math.Biol.Bioinf.

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Институт вычислительной математики РАН, Россия, Москва, 119991, ул.Губкина, д.8 Аннотация. В обзоре рассмотрены математические модели эпидемиологиче-ских процессов, определяющих динамику заболеваемости туберкулезом. Пер-вые модели эпидемиологии туберкулеза были разработаны и опубликованы в начале 60-х годов прошлого века. В этих и последующих работах 70-х годов сформулированы и описаны особенности эпидемиологии туберкулеза: (нали-чие длительной латентной фазы инфекции, очень низкая вероятность полно-го освобождения от инфекции, возможность быстрого развития заболевания после инфицирования и зависимость вероятности активации инфекции от со-стояния носителя и длительности латентного периода). Важнейшей областью приложений математических моделей эпидемиологии туберкулеза является оценка эффективности стратегий борьбы с этим заболеванием. Новая волна интереса к математическим моделям распространения туберкулеза связана с ростом заболеваемости в развивающихся странах из-за эпидемией ВИЧ и по-явлению штаммов микобактерий устойчивых к одному или нескольким препа-ратам. Модели 80-х и 90-х годов посвящены взаимодействию инфекции ВИЧ и микобактерий, формированию и распространению лекарственно устойчи-вых штаммов; большое внимание уделяется исследованию свойств моделей, оценке параметров, сравнению с реальными данными. В этот период моде-ли становятся важным средством для выработки и обоснования деятельности как национальных, так и международных организаций, отвечающих за борьбу с этой инфекцией. Для удобства в обзоре используется единая система обо-значений переменных и параметров, приводятся блок-схемы моделируемых процессов, оценки значений параметров, обсуждаются предположения и до-пущения, использованные при построении моделей. Работа является первым полным обзором этого класса моделей до 2006 года.

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Tuberculosis models with fast and slow dynamics: the role of close and casual contacts

Cited by 116 publications

References 34 publications

Mathematical Models for the Dynamics of Tuberculosis in Density-dependent Populations: The Case of Internally Displaced Peoplesâ€™ Camps (IDPCs) in Uganda

Mathematical Models for the Dynamics of Tuberculosis in Density-dependent Populations: The Case of Internally Displaced Peoplesâ€™ Camps (IDPCs) in Uganda

Modeling the Dynamics of Tuberculosis Transmission in Children and Adults

Mathematical Models of Tuberculosis Extension and Control of it

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