基于接种的新型冠状病毒模型及其分析：以上海为例

Novel Coronavirus Model Based on Vaccination and Its Analysis: A Case of Shanghai

摘要: 新冠病毒感染是一种由新型冠状病毒引起的急性传染性疾病，自2019年12月新冠病毒感染爆发以来，不仅危及人类的生命健康，还对全世界的经济发展造成了严重的影响，由于新冠病毒可在复制过程中不断适应宿主而产生突变，其具有更强的传播力和免疫逃逸能力，导致其对疫苗的敏感性降低，接种疫苗的人群仍旧有感染新冠病毒的风险，同时接种的疫苗在一定时间后会对人体的保护作用逐渐减弱，导致疫苗的有效性逐渐减弱，因此研究接种疫苗对新冠病毒的影响具有一定的意义，本文将接种人群单独考虑为一个仓室建立了数学模型。文中给出了模型的基本再生数以及正平衡点，通过构造Lyapunov函数证明了无病平衡点以及Rv > 1时唯一的正平衡点的稳定性。以上海2022年2月26日~5月31日的数据为例，结合当时上海所采取的政策，对模型进行数据拟合，并给出了上海在此期间的基本再生数为2.5709、5.1583、0.2982，最后通过数值模拟表明：提高每日的接种率、增加初始接种比例、能够有效降低确诊病例，从而降低新冠病毒的传染规模。但是即使初始接种比例或每日接种比例为100%，并不能完全阻断新冠病毒的传播。

Abstract: Novel coronavirus infection is an acute infectious disease caused by novel coronavirus. Since novel coronavirus infection broke out in December 2019, it not only endangers human life and health, but also has a serious impact on the economic development of the whole world. As novel coronavirus can constantly adapt to the host in the process of replication, it has a stronger transmission power and immune escape ability. Resulting in the reduction of its sensitivity to the vaccine, the vaccinat-ed population is still at risk of infection with novel coronavirus. At the same time, the protective ef-fect of the vaccinated vaccine on the human body will gradually weaken after a certain period of time, resulting in the gradual weakening of the effectiveness of the vaccine. Therefore, it is of certain significance to study the impact of vaccination on novel coronavirus. In this paper, the basic regen-eration number and the positive equilibrium point of the model are given, and the stability of the disease-free equilibrium point and the unique positive equilibrium point when Rv > 1 are proved by constructing the Lyapunov function. Taking the data of Shanghai from February 26 to May 31, 2022 as an example, and combining the policies adopted by Shanghai at that time, the model is fitted to the data. The basic regeneration numbers of Shanghai during this period are 2.5709, 5.1583 and 0.2982. Finally, numerical simulation shows that increasing the daily vaccination rate and increas-ing the initial vaccination rate can effectively reduce the confirmed cases, thereby reducing the in-fection scale of the new coronavirus. However, even if the initial vaccination rate or the daily vac-cination rate is 100%, it will not completely block the transmission of the novel coronavirus.