Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada.
Department of Mathematics, Imperial College London, London, UK.
J Math Biol. 2023 Aug 1;87(2):35. doi: 10.1007/s00285-023-01958-w.
Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman-Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox.
更新方程是一种在暴发中模拟新感染（即发病率）数量的常用方法。我们基于时变的 Crump-Mode-Jagers 分支过程变体开发了一种暴发的随机模型。该模型可容纳时变的繁殖数和时变的世代间隔分布。然后，我们在该模型下推导出发病率、累积发病率和患病率的更新似的积分方程。我们证明了发病率和患病率的方程与所谓的回溯关系一致。我们分析了这些积分方程的两个特殊情况，一个来自 Bellman-Harris 过程，另一个来自传染病传播的非齐次泊松过程模型。我们还表明，这两个特定模型产生的发病率积分方程与传染病建模中普遍使用的更新方程一致。我们提出了一种数值离散化方案来求解这些方程，并使用该方案来估计从 SARS-CoV-2 的血清流行率和流感、麻疹、SARS 和天花的历史发病率数据中得出的传播率。
