Feng Xiao-Mei, Liu Li-Li, Zhang Feng-Qin
School of Mathematics and Informational Technology, Yuncheng University, Yuncheng, 044000 China.
School of Mathematics and Informational Sciences, Shaanxi Normal University, Xi'an, 710062 China.
Acta Math Appl Sin. 2022;38(2):282-303. doi: 10.1007/s10255-022-1075-7. Epub 2022 Apr 9.
For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model. For the age-structured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.
对于一些传染病，如腮腺炎、乙肝病毒，有证据表明接种疫苗的个体总是会根据接种时间以不同的速率失去免疫力。在本文中，我们提出了一个年龄结构的流行病模型，使用阶跃函数来描述接种疫苗个体失去免疫力的速率，并将年龄结构的流行病模型简化为延迟微分模型。对于年龄结构模型，我们考虑半流的正性、有界性和紧致性，并通过构造适当的李雅普诺夫泛函来研究平衡点的全局稳定性。此外，对于简化后的延迟微分方程模型，我们研究地方病平衡点的存在性并证明平衡点的全局稳定性。最后，提供了一些数值模拟来支持我们的理论结果并给出了简要讨论。
