On the Computation of R 0 and its Role on Global Stability

Castillo‐Chavez, Carlos; Feng, Zhilan; Huang, Wenzhang

doi:10.1007/978-1-4757-3667-0_13

Cited by 444 publications

(400 citation statements)

References 12 publications

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“…The quantity R 0 (see, for instance, [2,8,9,12,23,47]) measures the expected number of secondary cases generated by an index case in an unvaccinated, equilibrium population (S = N o , V = 0). With this definition for R 0 and introducing…”

Section: 1mentioning

confidence: 99%

An sveir model for assessing potential impact of an imperfect anti-SARS vaccine

Gumel

McCluskey

Watmough

2006

Mathematical Biosciences and Engineering

108

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Abstract. The control of severe acute respiratory syndrome (SARS), a fatal contagious viral disease that spread to over 32 countries in 2003, was based on quarantine of latently infected individuals and isolation of individuals with clinical symptoms of SARS. Owing to the recent ongoing clinical trials of some candidate anti-SARS vaccines, this study aims to assess, via mathematical modelling, the potential impact of a SARS vaccine, assumed to be imperfect, in curtailing future outbreaks. A relatively simple deterministic model is designed for this purpose. It is shown, using Lyapunov function theory and the theory of compound matrices, that the dynamics of the model are determined by a certain threshold quantity known as the control reproduction number (Rv ). If Rv ≤ 1, the disease will be eliminated from the community; whereas an epidemic occurs if Rv > 1. This study further shows that an imperfect SARS vaccine with infection-blocking efficacy is always beneficial in reducing disease spread within the community, although its overall impact increases with increasing efficacy and coverage. In particular, it is shown that the fraction of individuals vaccinated at steady-state and vaccine efficacy play equal roles in reducing disease burden, and the vaccine must have efficacy of at least 75% to lead to effective control of SARS (assuming R 0 = 4). Numerical simulations are used to explore the severity of outbreaks when Rv > 1.

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Section: 1mentioning

confidence: 99%

An sveir model for assessing potential impact of an imperfect anti-SARS vaccine

Gumel

McCluskey

Watmough

2006

Mathematical Biosciences and Engineering

108

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“…It is instructive to start with the conservation law (16). On taking t = t k+1 and setting M k+1 = M (t k+1 ) in (18), simple manipulations show that the exact scheme (see [44]) for (16) is either…”

Section: Nonstandard Finite Difference Schemesmentioning

confidence: 99%

“…They proved the following theorem, which can also be found in [18] under some restrictive assumptions:…”

Section: Introductionmentioning

confidence: 98%

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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This work is the numerical analysis and computational companion of the paper by Kamgang and Sallet (Math. Biosc. 213 (2008), pp. 1-12) where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens' rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.

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“…We will use the theorem by Castillo-Chavez et al [38], to prove the global stability result. Theorem 4.2 If the model system (1.1) can be written in the form :…”

Section: Global Stability Of Disease-free Equilibriummentioning

confidence: 99%

Mathematical Analysis of the Transmission Dynamics of HIV/AIDS: Role of Female Sex Workers

Kaur¹,

Ghosh²,

Bhatia³

2014

Appl. Math. Inf. Sci.

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Abstract:The transmission of 'Human Immunodeficiency Virus (HIV)' that causes the 'Acquired Immunodeficiency Syndrome (AIDS)' is strongly associated with un-protected sex and at the present understanding this epidemic can reach higher prevalence threshold level when there are extensive sexual contacts between the sex workers and general population. In the present work, we investigate a nonlinear model for studing the transmission dynamics of HIV/AIDS epidemic with emphasis on the role of female sex workers. Here, we consider only the heterosexual transmissions of HIV/AIDS and formulate the mathematical model by dividing the total adult population under consideration into three different classes: male, female and female sex workers. We assume different rates of recruitment for different classes of the population. The equilibria of the model and their stability are discussed in detail. The basic reproduction number R 0 of the model is computed and it is shown that the disease-free equilibrium is stable only when R 0 < 1. When the associated reproduction number R 0 > 1, the endemic equilibrium is globally stable. Finally, the numerical simulations are reported to support the presented analytical results.

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On the Computation of R 0 and its Role on Global Stability

Cited by 444 publications

References 12 publications

An sveir model for assessing potential impact of an imperfect anti-SARS vaccine

An sveir model for assessing potential impact of an imperfect anti-SARS vaccine

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Mathematical Analysis of the Transmission Dynamics of HIV/AIDS: Role of Female Sex Workers

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