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A Mathematical Model of the Tuberculosis Epidemic. Acta Biotheor 2021; 69:225-255. [PMID: 33877474 DOI: 10.1007/s10441-020-09406-8] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2020] [Accepted: 12/24/2020] [Indexed: 10/21/2022]
Abstract
Tuberculosis has continued to retain its title as "the captain among these men of death". This is evident as it is the leading cause of death globally from a single infectious agent. TB as it is fondly called has become a major threat to the achievement of the sustainable development goals (SDG) and hence require inputs from different research disciplines. This work presents a mathematical model of tuberculosis. A compartmental model of seven classes was used in the model formulation comprising of the susceptible S, vaccinated V, exposed E, undiagnosed infectious I1, diagnosed infectious I2, treated T and recovered R. The stability analysis of the model was established as well as the condition for the model to undergo backward bifurcation. With the existence of backward bifurcation, keeping the basic reproduction number less than unity [Formula: see text] is no more sufficient to keep TB out of the community. Hence, it is shown by the analysis that vaccination program, diagnosis and treatment helps to control the TB dynamics. In furtherance to that, it is shown that preference should be given to diagnosis over treatment as diagnosis precedes treatment. It is as well shown that at lower vaccination rate (0-20%), TB would still be endemic in the population. As such, high vaccination rate is required to send TB out of the community.
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Nadim SS, Chattopadhyay J. Occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: A case study on COVID-19. CHAOS, SOLITONS, AND FRACTALS 2020; 140:110163. [PMID: 32834647 PMCID: PMC7430254 DOI: 10.1016/j.chaos.2020.110163] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2020] [Revised: 07/15/2020] [Accepted: 07/26/2020] [Indexed: 05/20/2023]
Abstract
The outbreak of COVID-19 caused by SARS-CoV-2 is spreading rapidly around the world, which is causing a major public health concerns. The outbreaks started in India on March 2, 2020. As of April 30, 2020, 34,864 confirmed cases and 1154 deaths are reported in India and more than 30,90,445 confirmed cases and 2,17,769 deaths are reported worldwide. Mathematical models may help to explore the transmission dynamics, prediction and control of COVID-19 in the absence of an appropriate medication or vaccine. In this study, we consider a mathematical model on COVID-19 transmission with the imperfect lockdown effect. The basic reproduction number, R 0, is calculated using the next generation matrix method. The system has a disease-free equilibrium (DFE) which is locally asymptotically stable whenever R 0 < 1. Moreover, the model exhibits the backward bifurcation phenomenon, where the stable DFE coexists with a stable endemic equilibrium when R 0 < 1. The epidemiological implications of this phenomenon is that the classical epidemiological requirement of making R 0 less than unity is only a necessary, but not sufficient for effectively controlling the spread of COVID-19 outbreak. It is observed that the system undergoes backward bifurcation which is a new observation for COVID-19 disease transmission model. We also noticed that under the perfect lockdown scenario, there is no possibility of having backward bifurcation. Using Lyapunov function theory and LaSalle Invariance Principle, the DFE is shown globally asymptotically stable for perfect lockdown model. We have calibrated our proposed model parameters to fit daily data from India, Mexico, South Africa and Argentina. We have provided a short-term prediction for India, Mexico, South Africa and Argentina of future cases of COVID-19. We calculate the basic reproduction number from the estimated parameters. We further assess the impact of lockdown during the outbreak. Furthermore, we find that effective lockdown is very necessary to reduce the burden of diseases.
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Affiliation(s)
- Sk Shahid Nadim
- Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata 700 108, India
| | - Joydev Chattopadhyay
- Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata 700 108, India
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Martins J, Pinto A, Stollenwerk N. The maximum curvature reinfection threshold. ECOLOGICAL COMPLEXITY 2019. [DOI: 10.1016/j.ecocom.2019.100791] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Quantifying TB transmission: a systematic review of reproduction number and serial interval estimates for tuberculosis. Epidemiol Infect 2018; 146:1478-1494. [PMID: 29970199 PMCID: PMC6092233 DOI: 10.1017/s0950268818001760] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022] Open
Abstract
Tuberculosis (TB) is the leading global infectious cause of death. Understanding TB transmission is critical to creating policies and monitoring the disease with the end goal of TB elimination. To our knowledge, there has been no systematic review of key transmission parameters for TB. We carried out a systematic review of the published literature to identify studies estimating either of the two key TB transmission parameters: the serial interval (SI) and the reproductive number. We identified five publications that estimated the SI and 56 publications that estimated the reproductive number. The SI estimates from four studies were: 0.57, 1.42, 1.44 and 1.65 years; the fifth paper presented age-specific estimates ranging from 20 to 30 years (for infants <1 year old) to <5 years (for adults). The reproductive number estimates ranged from 0.24 in the Netherlands (during 1933-2007) to 4.3 in China in 2012. We found a limited number of publications and many high TB burden settings were not represented. Certain features of TB dynamics, such as slow transmission, complicated parameter estimation, require novel methods. Additional efforts to estimate these parameters for TB are needed so that we can monitor and evaluate interventions designed to achieve TB elimination.
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Gerberry DJ. Practical aspects of backward bifurcation in a mathematical model for tuberculosis. J Theor Biol 2016; 388:15-36. [DOI: 10.1016/j.jtbi.2015.10.003] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2015] [Revised: 08/20/2015] [Accepted: 10/06/2015] [Indexed: 10/22/2022]
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Mushayabasa S, Bhunu CP. Modeling the impact of early therapy for latent tuberculosis patients and its optimal control analysis. J Biol Phys 2013; 39:723-47. [PMID: 23975671 DOI: 10.1007/s10867-013-9328-6] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2013] [Accepted: 07/01/2013] [Indexed: 10/26/2022] Open
Abstract
Effective tuberculosis (TB) control depends on case findings to discover infectious cases, investigation of contacts of those with TB, as well as appropriate treatment. Adherence and successful completion of the treatment are equally important. Unfortunately, due to a number of personal, psychosocial, economic, medical, and health service factors, a significant number of TB patients become irregular and default from treatment. In this paper, a mathematical model is developed to assess the impact of early therapy for latent TB and non-adherence on controlling TB transmission dynamics. Equilibrium states of the model are determined and their local stability is examined. With the aid of the center manifold theory, it is established that the model undergoes a backward bifurcation. Qualitative mathematical analysis of the model suggests that a high level of latent tuberculosis case findings, coupled with a decrease of defaulting rate, may be effective in controlling TB transmission dynamics in the community. Population-level effects of organized campaigns to improve early therapy and to guarantee successful completion of each treatment are evaluated through numerical simulations and presented in support of the analytical results.
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Affiliation(s)
- S Mushayabasa
- Department of Mathematics, University of Zimbabwe, PO Box MP 167, Harare, Zimbabwe.
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7
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Global Dynamics of a Tuberculosis Epidemic Model and the Influence of Backward Bifurcation. ACTA ACUST UNITED AC 2012. [DOI: 10.1007/s10852-012-9210-8] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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MUSHAYABASA S, BHUNU CP. MODELING THE IMPACT OF VOLUNTARY TESTING AND TREATMENT ON TUBERCULOSIS TRANSMISSION DYNAMICS. INT J BIOMATH 2012. [DOI: 10.1142/s1793524511001726] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A deterministic model for evaluating the impact of voluntary testing and treatment on the transmission dynamics of tuberculosis is formulated and analyzed. The epidemiological threshold, known as the reproduction number is derived and qualitatively used to investigate the existence and stability of the associated equilibrium of the model system. The disease-free equilibrium is shown to be locally-asymptotically stable when the reproductive number is less than unity, and unstable if this threshold parameter exceeds unity. It is shown, using the Centre Manifold theory, that the model undergoes the phenomenon of backward bifurcation where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The analysis of the reproduction number suggests that voluntary tuberculosis testing and treatment may lead to effective control of tuberculosis. Furthermore, numerical simulations support the fact that an increase voluntary tuberculosis testing and treatment have a positive impact in controlling the spread of tuberculosis in the community.
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Affiliation(s)
- S. MUSHAYABASA
- Department of Applied Mathematics, Modeling Biomedical Systems Research Group, National University of Science and Technology, P. O. Box 939 Ascot, Bulawayo, Zimbabwe
| | - C. P. BHUNU
- Department of Mathematics, University of Zimbabwe, P. O. Box MP 167, Harare, Zimbabwe
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Wang J, Liao S. A generalized cholera model and epidemic-endemic analysis. JOURNAL OF BIOLOGICAL DYNAMICS 2012; 6:568-589. [PMID: 22873606 DOI: 10.1080/17513758.2012.658089] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R (0)=1 for the combined human-environment epidemiological model under biologically reasonable conditions.
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Affiliation(s)
- Jin Wang
- Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA.
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Yang HM, Raimundo SM. Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis. Theor Biol Med Model 2010; 7:41. [PMID: 21059256 PMCID: PMC2995482 DOI: 10.1186/1742-4682-7-41] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2010] [Accepted: 11/08/2010] [Indexed: 11/10/2022] Open
Abstract
In order to achieve a better understanding of multiple infections and long latency in the dynamics of Mycobacterium tuberculosis infection, we analyze a simple model. Since backward bifurcation is well documented in the literature with respect to the model we are considering, our aim is to illustrate this behavior in terms of the range of variations of the model's parameters. We show that backward bifurcation disappears (and forward bifurcation occurs) if: (a) the latent period is shortened below a critical value; and (b) the rates of super-infection and re-infection are decreased. This result shows that among immunosuppressed individuals, super-infection and/or changes in the latent period could act to facilitate the onset of tuberculosis. When we decrease the incubation period below the critical value, we obtain the curve of the incidence of tuberculosis following forward bifurcation; however, this curve envelops that obtained from the backward bifurcation diagram.
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Affiliation(s)
- Hyun M Yang
- UNICAMP-IMECC, Departamento de Matemática Aplicada, Praça Sérgio Buarque de Holanda, 651, CEP: 13083-859, Campinas, SP, Brazil.
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Hilker FM. Population collapse to extinction: the catastrophic combination of parasitism and Allee effect. JOURNAL OF BIOLOGICAL DYNAMICS 2010; 4:86-101. [PMID: 22881072 DOI: 10.1080/17513750903026429] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Infectious diseases are responsible for the extinction of a number of species. In conventional epidemic models, the transition from endemic population persistence to extirpation takes place gradually. However, if host demographics exhibits a strong Allee effect (AE) (population decline at low densities), extinction can occur abruptly in a catastrophic population crash. This might explain why species suddenly disappear even when they used to persist at high endemic population levels. Mathematically, the tipping point towards population collapse is associated with a saddle-node bifurcation. The underlying mechanism is the simultaneous population size depression and the increase of the extinction threshold due to parasite pathogenicity and Allee effect. Since highly pathogenic parasites cause their own extinction but not that of their host, there can be another saddle-node bifurcation with the re-emergence of two endemic equilibria. The implications for control interventions are discussed, suggesting that effective management may be possible for ℛ(0)≫1.
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Affiliation(s)
- Frank M Hilker
- Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Lisboa, Portugal.
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Bacaër N, Ouifki R, Pretorius C, Wood R, Williams B. Modeling the joint epidemics of TB and HIV in a South African township. J Math Biol 2008; 57:557-93. [PMID: 18414866 DOI: 10.1007/s00285-008-0177-z] [Citation(s) in RCA: 64] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2007] [Revised: 03/22/2008] [Indexed: 10/22/2022]
Abstract
We present a simple mathematical model with six compartments for the interaction between HIV and TB epidemics. Using data from a township near Cape Town, South Africa, where the prevalence of HIV is above 20% and where the TB notification rate is close to 2,000 per 100,000 per year, we estimate some of the model parameters and study how various control measures might change the course of these epidemics. Condom promotion, increased TB detection and TB preventive therapy have a clear positive effect. The impact of antiretroviral therapy on the incidence of HIV is unclear and depends on the extent to which it reduces sexual transmission. However, our analysis suggests that it will greatly reduce the TB notification rate.
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Affiliation(s)
- Nicolas Bacaër
- Institut de Recherche pour le Développement, 32 avenue Henri Varagnat, 93143 Bondy, France.
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Gabriela M Gomes M, Rodrigues P, Hilker FM, Mantilla-Beniers NB, Muehlen M, Cristina Paulo A, Medley GF. Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions. J Theor Biol 2007; 248:608-17. [PMID: 17669435 DOI: 10.1016/j.jtbi.2007.06.005] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2006] [Revised: 05/15/2007] [Accepted: 06/06/2007] [Indexed: 11/26/2022]
Abstract
One-third of the world population (approximately 2 billion individuals) is currently infected with Mycobacterium tuberculosis, the vast majority harboring a latent infection. As the risk of reactivation is around 10% in a lifetime, it follows that 200 million of these will eventually develop active pulmonary disease. Only therapeutic or post-exposure interventions can tame this vast reservoir of infection. Treatment of latent infections can reduce the risk of reactivation, and there is accumulating evidence that combination with post-exposure vaccines can reduce the risk of reinfection. Here we develop mathematical models to explore the potential of these post-exposure interventions to control tuberculosis on a global scale. Intensive programs targeting recent infections appear generally effective, but the benefit is potentially greater in intermediate prevalence scenarios. Extending these strategies to longer-term persistent infections appears more beneficial where prevalence is low. Finally, we consider that susceptibility to reinfection is altered by therapy, and explore its epidemiological consequences. When we assume that therapy reduces susceptibility to subsequent reinfection, catastrophic dynamics are observed. Thus, a bipolar outcome is obtained, where either small or large reductions in prevalence levels result, depending on the rate of detection and treatment of latent infections. By contrast, increased susceptibility after therapy may induce an increase in disease prevalence and does not lead to catastrophic dynamics. These potential outcomes are silent unless a widespread intervention is implemented.
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Affiliation(s)
- M Gabriela M Gomes
- Instituto Gulbenkian de Ciência, Apartado 14, 2781-901 Oeiras Cedex, Portugal.
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Colijn C, Cohen T, Murray M. Emergent heterogeneity in declining tuberculosis epidemics. J Theor Biol 2007; 247:765-74. [PMID: 17540410 PMCID: PMC2652758 DOI: 10.1016/j.jtbi.2007.04.015] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2006] [Revised: 04/20/2007] [Accepted: 04/23/2007] [Indexed: 11/30/2022]
Abstract
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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Affiliation(s)
- Caroline Colijn
- Department of Epidemiology, Harvard School of Public Health, Boston, MA, USA.
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Segovia-Juarez JL, Ganguli S, Kirschner D. Identifying control mechanisms of granuloma formation during M. tuberculosis infection using an agent-based model. J Theor Biol 2004; 231:357-76. [PMID: 15501468 DOI: 10.1016/j.jtbi.2004.06.031] [Citation(s) in RCA: 178] [Impact Index Per Article: 8.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2004] [Revised: 06/28/2004] [Accepted: 06/30/2004] [Indexed: 11/17/2022]
Abstract
Infection with Mycobacterium tuberculosis is a major world health problem. An estimated 2 billion people are presently infected and the disease causes approximately 3 million deaths per year. After bacteria are inhaled into the lung, a complex immune response is triggered leading to the formation of multicellular structures termed granulomas. It is believed that the collection of host granulomas either contain bacteria resulting in a latent infection or are unable to do so, leading to active disease. Thus, understanding granuloma formation and function is essential for improving both diagnosis and treatment of tuberculosis. Granuloma formation is a complex spatio-temporal system involving interactions of bacteria, specific immune cells, including macrophages, CD4+ and CD8+ T cells, as well as immune effectors such as chemokine and cytokines. To study this complex dynamical system we have developed an agent-based model of granuloma formation in the lung. This model combines continuous representations of chemokines with discrete agent representations of macrophages and T cells in a cellular automata-like environment. Our results indicate that key host elements involved in granuloma formation are chemokine diffusion, prevention of macrophage overcrowding within the granuloma, arrival time, location and number of T cells within the granuloma, and an overall host ability to activate macrophages. Interestingly, a key bacterial factor is its intracellular growth rate, whereby slow growth actually facilitates survival.
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Affiliation(s)
- Jose L Segovia-Juarez
- Department of Microbiology and Immunology, University of Michigan, Ann Arbor, MI 48109, USA
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