Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE)

Kamgang, Jean Claude; Sallet, Gauthier

doi:10.1016/j.mbs.2008.02.005

Cited by 139 publications

(163 citation statements)

References 28 publications

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“…Define φ(h) according to (32), (34), and (62) so that (61) holds. Let θ andθ be such that either θ = 0 or θ +θ ≥ 1.…”

Section: Theorem 14mentioning

confidence: 99%

“…The function φ is chosen according to (32), (34), and (62) so that (61) holds for these values of θ andθ. More precisely, we choose…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…However, to construct such Lyapunov functions is very challenging in most problems of interest. These considerations led Kamgang and Sallet [32] to seek another way to prove that the DFE is GAS. In their paper, they obtained a necessary and sufficient condition for the global asymptotic stability of the DFE under reasonable assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…In this work we follow Kamgang and Sallet [32], and reformulate (1), using the notation x = (y, z), in the following manner,…”

Section: Introductionmentioning

confidence: 99%

“…Kamgang and Sallet, [32], also presented applications of Theorem 1 to various epidemiological problems, such as models for Tuberculosis and HIV, and we note that it was used in the study of models for vector-borne diseases, [21,22].…”

Section: Introductionmentioning

confidence: 99%

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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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“…Define φ(h) according to (32), (34), and (62) so that (61) holds. Let θ andθ be such that either θ = 0 or θ +θ ≥ 1.…”

Section: Theorem 14mentioning

confidence: 99%

“…The function φ is chosen according to (32), (34), and (62) so that (61) holds for these values of θ andθ. More precisely, we choose…”

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In this work we follow Kamgang and Sallet [32], and reformulate (1), using the notation x = (y, z), in the following manner,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

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The impact of demography in a model of malaria with transmission‐blocking drugs

Ouifki,

Banasiak,

Tchoumi

2024

Math Methods in App Sciences

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In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with transmission‐blocking drugs (TBDs). The paper's main aim is to demonstrate the impact the chosen model for demographic growth has on the disease's transmission and the effect of its treatment with TBDs. We calculate the model's control reproduction number and equilibria and perform a global stability analysis of the disease‐free equilibrium point. The mathematical analysis reveals that, depending on the model's demography, the model can exhibit forward, backward, and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short‐time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe.

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Modeling and stability analysis for a cholera model with vaccination

Zhou

Cui

2011

Math. Meth. Appl. Sci.

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In this paper, we consider a cholera model with vaccination. The disease‐free equilibrium of the system is globally asymptotically stable when the basic reproduction number scriptR0MathClass-rel≤1. If scriptR0MathClass-rel>1, the disease persists and the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region under some conditions, which is obtained by compound matrices and geometric approaches. We perform sensitivity analysis of scriptR0 on the parameters in order to determine their relative importance to disease transmission and prevalence. Numerical simulations are presented to illustrate the results. Copyright © 2011 John Wiley & Sons, Ltd.

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Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE)

Cited by 139 publications

References 28 publications

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Dynamically consistent nonstandard finite difference schemes for epidemiological models

The impact of demography in a model of malaria with transmission‐blocking drugs

Modeling and stability analysis for a cholera model with vaccination

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