Kengne Jacques Ndé, Tadmon Calvin
Committed Mathematics Team, Research Unit in Mathematics and Applications, Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67 Dschang, Cameroon.
Institute of Mathematics, University of Mainz, Staudingerweg 9, 55128, Mainz, Germany.
Infect Dis Model. 2024 Apr 9;9(3):775-804. doi: 10.1016/j.idm.2024.03.007. eCollection 2024 Sep.
This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018-2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.
本文研究了一个具有指数非线性发生率函数的埃博拉疫情模型，该函数考虑了防控效果和行为变化。当前模型还纳入了一种新的密度依赖型治疗方法，以捕捉疾病传播对治疗的影响。首先，我们对所得到的非线性微分方程模型进行了理论研究。更确切地说，我们推导了有效再生数，并在适当条件下证明了平衡点的稳定性。之后，我们表明，当分岔参数和再生数小于1时，该模型呈现后向分岔现象。我们发现，双稳定性和后向分岔在疫情模型中并非自动关联。事实上，当出现后向分岔时，无病平衡点可能是全局稳定的。在数值方面，我们使用著名的标准工具将模型拟合到2018 - 2020年基伍埃博拉疫情报告的数据，并进行敏感性分析。为了控制埃博拉疫情，我们的研究结果建议将快速的行为改变与实施具有高效能的适当治疗策略相结合。其次，我们提出并分析了一个分数阶埃博拉疫情模型，它是所研究的第一个模型的扩展。我们使用卡普托算子并构建了 Grünwald - Letnikov 非标准有限差分格式，并展示了其优势。
