Guy Romain, Larédo Catherine, Vergu Elisabeta
UR 341 Mathématiques et Informatique Appliquées, INRA, Jouy-en-Josas, France,
J Math Biol. 2015 Feb;70(3):621-46. doi: 10.1007/s00285-014-0777-8. Epub 2014 Mar 27.
Multidimensional continuous-time Markov jump processes [Formula: see text] on [Formula: see text] form a usual set-up for modeling [Formula: see text]-like epidemics. However, when facing incomplete epidemic data, inference based on [Formula: see text] is not easy to be achieved. Here, we start building a new framework for the estimation of key parameters of epidemic models based on statistics of diffusion processes approximating [Formula: see text]. First, previous results on the approximation of density-dependent [Formula: see text]-like models by diffusion processes with small diffusion coefficient [Formula: see text], where [Formula: see text] is the population size, are generalized to non-autonomous systems. Second, our previous inference results on discretely observed diffusion processes with small diffusion coefficient are extended to time-dependent diffusions. Consistent and asymptotically Gaussian estimates are obtained for a fixed number [Formula: see text] of observations, which corresponds to the epidemic context, and for [Formula: see text]. A correction term, which yields better estimates non asymptotically, is also included. Finally, performances and robustness of our estimators with respect to various parameters such as [Formula: see text] (the basic reproduction number), [Formula: see text], [Formula: see text] are investigated on simulations. Two models, [Formula: see text] and [Formula: see text], corresponding to single and recurrent outbreaks, respectively, are used to simulate data. The findings indicate that our estimators have good asymptotic properties and behave noticeably well for realistic numbers of observations and population sizes. This study lays the foundations of a generic inference method currently under extension to incompletely observed epidemic data. Indeed, contrary to the majority of current inference techniques for partially observed processes, which necessitates computer intensive simulations, our method being mostly an analytical approach requires only the classical optimization steps.
定义在(\mathbb{R}^n)上的多维连续时间马尔可夫跳跃过程({X(t)})构成了对类似(\mathcal{S})型流行病建模的常用框架。然而，面对不完整的疫情数据时，基于({X(t)})进行推断并非易事。在此，我们开始构建一个新框架，用于基于逼近({X(t)})的扩散过程的统计量来估计流行病模型的关键参数。首先，先前关于具有小扩散系数(\epsilon)（其中(N)为种群规模）的扩散过程对密度依赖型类似(\mathcal{S})模型的逼近结果被推广到非自治系统。其次，我们先前关于离散观测的小扩散系数扩散过程的推断结果被扩展到与时间相关的扩散过程。对于对应于疫情背景的固定观测次数(n)以及(N)，得到了一致且渐近高斯估计。还包含一个非渐近地产生更好估计的校正项。最后，在模拟中研究了我们的估计器相对于诸如(R_0)（基本再生数）、(N)、(\epsilon)等各种参数的性能和稳健性。分别对应单次和反复爆发的两个模型(\mathcal{SIR})和(\mathcal{SEIR})被用于模拟数据。研究结果表明，我们的估计器具有良好的渐近性质，并且对于实际的观测次数和种群规模表现显著良好。本研究为目前正在扩展到不完全观测疫情数据的通用推断方法奠定了基础。实际上，与当前大多数针对部分观测过程的推断技术不同，后者需要大量计算机模拟，而我们的方法主要是一种解析方法，仅需要经典的优化步骤。
