一类病毒群体免疫最小免疫人口推断及检验
A Class of Virus Population Immunity Minimum Immune Population Inference and Testing

本文通过微分动力系统研究新冠病毒群体免疫屏障所需的最小免疫人口比率，并探究免疫人口比率对疫情爆发的影响。采用参数时变的SIR模型，通过一阶差分得到感染率和移出率的表达式，并对其进行曲线拟合。利用下一代矩阵法推导基本再生数的表达式，借助建立群体免疫屏障的临界基本再生数，确定新冠病毒群体免疫屏障所需的最小免疫人口比率。经过数值计算表明，即使随机感染率服从以原始最大感染率为均值，以0.1倍的原始最大感染率为标准差的正态分布，只要免疫人口比率不低于72.03%，人群依然能够建立起群体免疫屏障，有效控制新冠病毒的传播。这项研究的方法和结果也可以为控制其它动植物病毒提供有益的指导。
This study investigates the minimum immune population ratio required for achieving herd im-munity against the novel coronavirus and explores the impact of the immune population ratio on the outbreak of the epidemic using a differential dynamical system. The time-varying SIR model is employed, and the expressions for the infection rate and removal rate are obtained through first-order differencing and curve fitting. The expression for the basic reproduction number is de-rived using the next-generation matrix method. By establishing the critical basic reproduction number for achieving herd immunity, the minimum immune population ratio required for the nov-el coronavirus is determined. Numerical calculations demonstrate that even under the condition where the stochastic infection rate follows a normal distribution with the original maximum infec-tion rate as the mean and 0.1 times the original maximum infection rate as the variance, as long as the immune population ratio is not lower than 72.03%, the population can still establish herd im-munity and effectively control the transmission of the novel coronavirus. The methods and results of this study can provide valuable guidance for controlling other animal and plant viruses.