一类非标准离散霍乱动力学模型

A Nonstandard Numerical Methods for a Mathematical Model for Cholera

本文旨在利用非标准有限差分方法离散并求解一个包含预防接种的霍乱传染病模型.该离散模型具有和对应的原连续模型一致的平衡点,正性和有界性等性质.其次本文证明当基本再生数小于1时,无病平衡点是局部渐近稳定和全局渐近稳定的;当基本再生数大于1时,通过构造适当的Lyapunov函数,地方病平衡点也是全局渐近稳定的.最后利用离散模型可以成功模拟2008年津巴布韦霍乱,并可数值证明离散模型的稳定性,且与步长和初始条件等无关,再与其他离散方法比较验证NSFD方法的优势所在.

By applying a nonstandard finite difference scheme, we construct and solve a discretized cholera epidemic model. The scheme can ensure that equilibrium points, the positivity and boundedness of solutions to the discrete model is the same as the original mathematical model. We have proved that when the basic reproduction number is less than 1, the disease-free equilibrium is locally and globally asymptotically stable. When the basic reproduction number is greater than 1, we prove that the endemic equilibrium is globally asymptotically stable by constructing a suitable Lyapunov function. Finally, the NSFD scheme can be well suited to numerically solve the cholera outbreak in Zimbabwe.