We consider state and parameter estimation for compartmental models having both time-varying and time-invariant parameters. Though the described Bayesian computational framework is general, we look at a specific application to the susceptible-infectious-removed (SIR) model which describes a basic mechanism for the spread of infectious diseases through a system of coupled nonlinear differential equations. The SIR model consists of three states, namely, the three compartments, and two parameters which control the coupling among the states. The deterministic SIR model with time-invariant parameters has shown to be overly simplistic for modelling the complex long-term dynamics of diseases transmission. Recognizing that certain model parameters will naturally vary in time due to seasonal trends, non-pharmaceutical interventions, and other random effects, the estimation procedure must systematically permit these time-varying effects to be captured, without unduly introducing artificial dynamics into the system. To this end, we leverage the robustness of the Markov Chain Monte Carlo (MCMC) algorithm for the estimation of time-invariant parameters alongside nonlinear filters for the joint estimation of the system state and time-varying parameters. We demonstrate performance of the framework by first considering a series of examples using synthetic data, followed by an exposition on public health data collected in the province of Ontario.

翻译：本文针对同时包含时变参数与时不变参数的仓室模型，研究了状态与参数估计问题。尽管所描述的贝叶斯计算框架具有通用性，但我们重点关注其在一类特定应用——易感-感染-移除（SIR）模型中的实现。该模型通过耦合非线性微分方程组描述传染病传播的基本机制，包含三个状态（即三个仓室）以及两个控制状态间耦合关系的参数。理论表明，采用时不变参数的确定性SIR模型在模拟疾病传播的复杂长期动态时过于简化。鉴于季节性趋势、非药物干预及其他随机效应会导致部分模型参数自然随时间变化，估计方法必须系统性地捕获这些时变效应，同时避免人为向系统引入虚假动态。为此，我们利用马尔可夫链蒙特卡洛（MCMC）算法的稳健性估计时不变参数，并联合非线性滤波器进行系统状态与时变参数的联合估计。我们首先通过一系列合成数据示例验证该框架的性能，随后将其应用于安大略省收集的公共卫生数据进行分析。