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Kuddus MA, McBryde ES, Adekunle AI, White LJ, Meehan MT. Mathematical analysis of a two-strain tuberculosis model in Bangladesh. Sci Rep 2022; 12:3634. [PMID: 35256670 PMCID: PMC8901732 DOI: 10.1038/s41598-022-07536-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2021] [Accepted: 02/18/2022] [Indexed: 12/02/2022] Open
Abstract
Tuberculosis (TB) is an airborne infectious disease that causes millions of deaths worldwide each year (1.2 million people died in 2019). Alarmingly, several strains of the causative agent, Mycobacterium tuberculosis (MTB)—including drug-susceptible (DS) and drug-resistant (DR) variants—already circulate throughout most developing and developed countries, particularly in Bangladesh, with totally drug-resistant strains starting to emerge. In this study we develop a two-strain DS and DR TB transmission model and perform an analysis of the system properties and solutions. Both analytical and numerical results show that the prevalence of drug-resistant infection increases with an increasing drug use through amplification. Both analytic results and numerical simulations suggest that if the basic reproduction numbers of both DS (\documentclass[12pt]{minimal}
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\begin{document}$${\text{R}}_{{0{\text{s}}}}$$\end{document}R0s) and DR (\documentclass[12pt]{minimal}
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\begin{document}$${\text{R}}_{{0{\text{r}}}}$$\end{document}R0r) TB are less than one, i.e. \documentclass[12pt]{minimal}
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\begin{document}$$\max \left[ {{\text{R}}_{{0{\text{s}}}} ,{\text{ R}}_{{0{\text{r}}}} } \right] < 1,$$\end{document}maxR0s,R0r<1, the disease-free equilibrium is asymptotically stable, meaning that the disease naturally dies out. Furthermore, if \documentclass[12pt]{minimal}
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\begin{document}$${\text{R}}_{{0{\text{r}}}} > {\text{max}}\left[ {{\text{R}}_{{0{\text{s}}}} ,1} \right]$$\end{document}R0r>maxR0s,1, then DS TB dies out but DR TB persists in the population, and if \documentclass[12pt]{minimal}
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\begin{document}$${\text{R}}_{{0{\text{s}}}} > {\text{max}}\left[ {{\text{R}}_{{0{\text{r}}}} ,1} \right]$$\end{document}R0s>maxR0r,1 both DS TB and DR TB persist in the population. Further, sensitivity analysis of the model parameters found that the transmission rate of both strains had the greatest influence on DS and DR TB prevalence. We also investigated the effect of treatment rates and amplification on both DS and DR TB prevalence; results indicate that inadequate or inappropriate treatment makes co-existence more likely and increases the relative abundance of DR TB infections.
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Affiliation(s)
- Md Abdul Kuddus
- Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, QLD, Australia. .,College of Medicine and Dentistry, James Cook University, Townsville, QLD, Australia. .,Department of Mathematics, University of Rajshahi, Rajshahi, 6205, Bangladesh.
| | - Emma S McBryde
- Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, QLD, Australia.,College of Medicine and Dentistry, James Cook University, Townsville, QLD, Australia
| | - Adeshina I Adekunle
- Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, QLD, Australia.,Decision Sciences Program, Victoria University, Melbourne, Australia
| | - Lisa J White
- Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, Nuffield Department of Medicine, University of Oxford, Oxford, UK
| | - Michael T Meehan
- Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, QLD, Australia
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Jabbari A, Castillo-Chavez C, Nazari F, Song B, Kheiri H. A two-strain TB model with multiple latent stages. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2016; 13:741-785. [PMID: 27775384 DOI: 10.3934/mbe.2016017] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
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.
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Affiliation(s)
- Azizeh Jabbari
- Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.
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IBARGÜEN-MONDRAGÓN EDUARDO, ROMERO-LEITON JHOANAP, ESTEVA LOURDES, BURBANO-ROSERO EDITHMARIELA. MATHEMATICAL MODELING OF BACTERIAL RESISTANCE TO ANTIBIOTICS BY MUTATIONS AND PLASMIDS. J BIOL SYST 2016. [DOI: 10.1142/s0218339016500078] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Diversity of drugs against bacterial infections, and development of resistance to such drugs are increasing. We formulate and analyze a deterministic model for the population dynamics of sensitive and resistant bacteria to multiple bactericidal and bacteriostatic antibiotics, assuming that drug resistance is acquired through mutations and plasmid transmission. Model equilibria are determined from qualitative analysis, and numerical simulations are used to assess temporal dynamics of sensitive and drug-resistant bacteria. The model presents three possibilities: elimination of bacteria, persistence of only resistant bacteria, or coexistence of sensitive and resistant bacteria. Evolution to one of these scenarios depends on thresholds numbers involving sensitive and resistant bacteria.
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Affiliation(s)
| | - JHOANA P. ROMERO-LEITON
- Est. Doc., Instituto de Matemáticas, Universidad de Antioquia, Cll 67 Cra 52, Medellín, Colombia
| | - LOURDES ESTEVA
- Departamento de Matemáticas, Universidad Nacional Autónoma de México, 04510 México DF, México
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