一种COVID-19类传染病爆发的数学建模预测方法

A Mathematical Modeling and Prediction Method for COVID-19-Like Infectious Disease Outbreaks

摘要: 本文提出了一种基于离散时间递推方法的改进SIR (Susceptible-Infective-Removed)模型，用于预测COVID-19 (Coronavirus disease 2019)类传染病的传播趋势。模型将人群划分为易感者(S)、感染者(I)和移除者(R)三类，通过引入动态感染率(β)和移除率(γ)，建立了简化的、可实时计算的离散时间递推方程，以替代计算复杂的传统常微分方程。本文的建模以美国的实际疫情数据为案例，验证了模型在短期预测和长期趋势分析中的有效性。结果表明，模型能够较好地拟合疫情发展曲线，并准确捕捉关键传播指标(如基本繁殖数R₀和有效繁殖数R_e)的变化规律。特别地，当R_e > 1时疫情呈爆发趋势，而R_e < 1时疫情逐渐消退。此外，模型还定量分析了不同防控措施(如佩戴口罩、减少社交活动)对疫情传播的抑制效果，并探讨了群体免疫阈值的数学依据。本文的数学模型为公共卫生决策提供了科学支持，帮助评估防控措施效果并预测疫情发展趋势，同时展示了数学建模在应对突发传染病中的实用价值。

Abstract: This paper proposes an improved SIR (Susceptible-Infective-Removed) model based on a discrete-time recursive method for predicting the spread of COVID-19-like infectious diseases. The model divides the population into three categories: susceptible (S), infected (I), and removed (R). By introducing dynamic infection rates (β) and removal rates (γ), a simplified, real-time, discrete-time recursive equation is established, replacing the traditional, computationally complex ordinary differential equations. Actual epidemic data from the United States are used as a case study to validate the model’s effectiveness in short-term forecasting and long-term trend analysis. Results demonstrate that the model fits the epidemic curve well and accurately captures the dynamics of key transmission indicators, such as the basic reproduction number (R₀) and the effective reproduction number (R_e). Specifically, when R_e > 1, the epidemic exhibits an explosive trend, while when R_e < 1, the epidemic gradually subsides. Furthermore, the model quantitatively analyzes the effectiveness of different prevention and control measures (such as wearing masks and reducing social activities) on epidemic spread and explores the mathematical basis for the herd immunity threshold. This mathematical model provides scientific support for public health decision-making, helps evaluate the effectiveness of prevention and control measures, and predicts epidemic trends. It also demonstrates the practical value of this mathematical model in responding to infectious disease emergencies.