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：本文研究兼具时变与时不变参数的房室模型的状态估计与参数估计问题。尽管所描述的贝叶斯计算框架具有通用性，我们重点关注其在易感-感染-移除（SIR）模型中的具体应用——该模型通过耦合非线性微分方程组描述了传染病传播的基本机制。SIR模型包含三个状态（即三个房室）以及控制状态间耦合关系的两个参数。研究表明，固定参数的确定性SIR模型在建模疾病传播的复杂长期动力学特征时过于简化。考虑到部分模型参数会因季节性趋势、非药物干预及其他随机效应自然产生时变性，估计方法必须系统性地捕获这类时变效应，同时避免向系统引入人为动力学干扰。为此，我们利用马尔可夫链蒙特卡洛（MCMC）算法对时不变参数进行估计的稳健性，并结合非线性滤波器对系统状态与时变参数进行联合估计。本文通过合成数据示例验证框架性能，随后展示了其在安大略省公共卫生数据上的应用效果。