Robinson Brandon, Bisaillon Philippe, Edwards Jodi D, Kendzerska Tetyana, Khalil Mohammad, Poirel Dominique, Sarkar Abhijit
Department of Civil and Environmental Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada.
School of Epidemiology and Public Health, University of Ottawa and University of Ottawa Heart Institute, Ottawa, Ontario, Canada.
Infect Dis Model. 2024 May 3;9(4):1224-1249. doi: 10.1016/j.idm.2024.04.002. eCollection 2024 Dec.
We consider state and parameter estimation for compartmental models having both time-varying and time-invariant parameters. In this manuscript, we first detail a general Bayesian computational framework as a continuation of our previous work. Subsequently, this framework is specifically tailored 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 susceptible, infectious, and removed compartments. The coupling among these states is controlled by two parameters, the infection rate and the recovery rate. The simplicity of the SIR model and similar compartmental models make them applicable to many classes of infectious diseases. However, the combined assumption of a deterministic model and time-invariance among the model parameters are two significant impediments which critically limit their use for long-term predictions. The tendency of certain model parameters to vary in time due to seasonal trends, non-pharmaceutical interventions, and other random effects necessitates a model that structurally permits the incorporation of such time-varying effects. Complementary to this, is the need for a robust mechanism for the estimation of the parameters of the resulting model from data. To this end, we consider an augmented state vector, which appends the time-varying parameters to the original system states whereby the time evolution of the time-varying parameters are driven by an artificial noise process in a standard manner. Distinguishing between time-varying and time-invariant parameters in this fashion limits the introduction of artificial dynamics into the system, and provides a robust, fully Bayesian approach for estimating the time-invariant system parameters as well as the elements of the process noise covariance matrix. This computational framework is implemented by leveraging the robustness of the Markov chain Monte Carlo algorithm permits the estimation of time-invariant parameters while nested nonlinear filters concurrently perform the joint estimation of the system states 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.
我们考虑具有时变和时不变参数的 compartmental 模型的状态和参数估计。在本手稿中，我们首先详细介绍一个通用的贝叶斯计算框架，作为我们先前工作的延续。随后，该框架专门针对易感-感染-康复（SIR）模型进行了调整，该模型通过一个耦合非线性微分方程组描述了传染病传播的基本机制。SIR 模型由三个状态组成，即易感、感染和康复 compartments。这些状态之间的耦合由两个参数控制，即感染率和康复率。SIR 模型和类似的 compartmental 模型的简单性使其适用于许多类别的传染病。然而，确定性模型和模型参数之间的时间不变性这两个综合假设是两个重大障碍，严重限制了它们在长期预测中的使用。由于季节趋势、非药物干预和其他随机效应，某些模型参数随时间变化的趋势使得需要一个在结构上允许纳入此类时变效应的模型。与此互补的是，需要一种强大的机制来从数据中估计所得模型的参数。为此，我们考虑一个扩充的状态向量，它将时变参数附加到原始系统状态上，从而时变参数的时间演化以标准方式由一个人工噪声过程驱动。以这种方式区分时变参数和时不变参数限制了向系统中引入人工动力学，并为估计时不变系统参数以及过程噪声协方差矩阵的元素提供了一种强大的、完全贝叶斯方法。这个计算框架通过利用马尔可夫链蒙特卡罗算法的鲁棒性来实现，允许估计时不变参数，同时嵌套非线性滤波器并行执行系统状态和时变参数的联合估计。我们首先通过使用合成数据的一系列示例来展示该框架的性能，随后阐述在安大略省收集的公共卫生数据。
