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An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis

机译:一种无条件稳定的非标准有限差分方法,解决描述内脏Leishmaniaisis的数学模型

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In this paper, a mathematical model of Visceral Leishmaniasis is considered. The model incorporates three populations, the human, the reservoir and the vector host populations. A detailed analysis of the model is presented. This analysis reveals that the model undergoes a backward bifurcation when the associated reproduction threshold is less than unity. For the case where the death rate due to VL is negligible, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. Noticing that the governing model is a system of highly nonlinear differential equations, its analytical solution is hard to obtain. To this end, a special class of numerical methods, known as the nonstandard finite difference (NSFD) method is introduced. Then a rigorous theoretical analysis of the proposed numerical method is carried out. We showed that this method is unconditionally stable. The results obtained by NSFD are compared with other well-known standard numerical methods such as forward Euler method and the fourth-order Runge-Kutta method. Furthermore, the NSFD preserves the positivity of the solutions and is more efficient than the standard numerical methods.
机译:在本文中,考虑了内脏利什曼病的数学模型。该模型包含三个人群,人,储层和载体宿主人群。提出了对模型的详细分析。这种分析表明,该模型经历向后分叉当相关联的再现阈值是小于1。对于由于VL引起的死亡率可忽略不计的情况,如果繁殖数量小于单位,则该模型的无疾病平衡显示为全局 - 渐近稳定。注意到控制模型是一种高度非线性微分方程的系统,其分析解决方案很难获得。为此,介绍了一种特殊的数值方法,称为非标准有限差(NSFD)方法。然后进行了对所提出的数值方法的严格理论分析。我们表明这种方法无条件稳定。将NSFD获得的结果与其他众所周知的标准数值方法进行比较,例如前向欧拉方法和第四阶跑为库方法。此外,NSFD保留了解决方案的积极性,并且比标准数值方法更有效。

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