发表机构
Departamento de Matemática, Centro de Ciências Tecnológicas; Universidade do Estado de Santa Catarina（数学系，科技中心; 圣卡塔琳娜州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文以SIR模型为核心，教学性地阐述分室流行病学建模，区分密度与频率依赖传播，推导离散与连续动态方程，并提供Python代码及参数估计示例，助力传染病建模教学。
AI 中文摘要
我们围绕SIR框架，对分室疾病传播建模进行了教学性的阐述，旨在服务于数学教育。我们关注的重点是建模选择如何编码传播机制和数据收集实践。我们首先陈述分室模型的结构性假设，包括种群划分、均匀混合、流闭合和马尔可夫转移。然后，我们在连续时间（微分方程）和离散时间（差分方程）两种设定下推导出基本的SIR动态方程。一个关键目标是阐明密度依赖传播与频率依赖传播之间的区别。我们通过从接触率、患病率和每次接触的传播性来构建感染力，同时跟踪单位和解释来实现这一点。我们超越了通常的质量平衡SIR模型，通过累积感染力引入离散的Kermack-McKendrick视角。这自然导致了带有指数逃逸概率项的离散SIR更新。我们还为密度依赖和频率依赖传播推导了基于伯努利的指数形式，在观测区间内对这些形式进行概率解释，并获得相关的离散传染性核。这突出了几何传染病存活如何从每步移除中产生。本文包含可复现的、入门级的Python代码，用于实现基本的离散SIR模型、连续SIR模型以及指数离散变体。我们为教师提供了一个使用最小二乘法从患病率数据估计参数的示例，并附有关于解释和局限性的指导。该框架旨在帮助教师将模型假设、单位、模拟和推断整合到一个连贯的传染病建模导论中。
英文摘要
We present a didactic treatment of compartmental disease spread modeling centered on the SIR framework, aimed at mathematics education. Our focus is on how modeling choices encode transmission mechanisms and data-collection practices. We begin by stating the structural hypotheses of compartmental models, including population partitioning, homogeneous mixing, flow closure, and Markovian transitions. We then derive the basic SIR dynamic equations in both continuous-time (differential equations) and discrete-time (difference equations) settings. A key goal is to clarify the distinction between density-dependent and frequency-dependent transmission. We do this by constructing the force of infection from the contact rate, prevalence, and per-contact transmissibility while tracking units and interpretations. We go beyond the usual mass-balance SIR model by introducing the discrete Kermack-McKendrick perspective through the cumulative force of infection. This naturally leads to discrete SIR updates with an exponential escape probability term. We also derive Bernoulli-based exponential formulations for both density- and frequency-dependent transmission, interpret these probabilistically over observation intervals, and obtain the associated discrete infectivity kernel. This highlights how geometric infectious survival emerges from per-step removal. The paper includes reproducible, introductory-level Python code to implement the basic discrete SIR model, the continuous SIR model, and the exponential discrete variants. We provide instructors with an example of parameter estimation from prevalence data using least squares, along with guidance on interpretation and limitations. This framework is designed to help teachers integrate model assumptions, units, simulation, and inference in a coherent introduction to infectious disease modeling.
Journal refLetters in Biomathematics 13(3), 224-242 (2026)
