Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, USA.
School of Public Health, Georgia State University, Atlanta, GA, USA.
Bull Math Biol. 2019 Nov;81(11):4343-4365. doi: 10.1007/s11538-017-0284-3. Epub 2017 May 2.
Deterministic and stochastic methods relying on early case incidence data for forecasting epidemic outbreaks have received increasing attention during the last few years. In mathematical terms, epidemic forecasting is an ill-posed problem due to instability of parameter identification and limited available data. While previous studies have largely estimated the time-dependent transmission rate by assuming specific functional forms (e.g., exponential decay) that depend on a few parameters, here we introduce a novel approach for the reconstruction of nonparametric time-dependent transmission rates by projecting onto a finite subspace spanned by Legendre polynomials. This approach enables us to effectively forecast future incidence cases, the clear advantage over recovering the transmission rate at finitely many grid points within the interval where the data are currently available. In our approach, we compare three regularization algorithms: variational (Tikhonov's) regularization, truncated singular value decomposition (TSVD), and modified TSVD in order to determine the stabilizing strategy that is most effective in terms of reliability of forecasting from limited data. We illustrate our methodology using simulated data as well as case incidence data for various epidemics including the 1918 influenza pandemic in San Francisco and the 2014-2015 Ebola epidemic in West Africa.
近年来，基于早期病例数据的确定性和随机性方法在预测疫情爆发方面受到了越来越多的关注。从数学角度来看，由于参数识别的不稳定性和有限的数据可用性，疫情预测是一个不适定的问题。虽然之前的研究主要通过假设特定的函数形式（例如指数衰减）来估计时变传播率，这些函数形式取决于少数几个参数，但在这里，我们引入了一种通过投影到由勒让德多项式构成的有限子空间来重建非时变传播率的新方法。这种方法使我们能够有效地预测未来的病例数，与在当前可用数据的时间段内有限的网格点上恢复传播率相比，这是一个明显的优势。在我们的方法中，我们比较了三种正则化算法：变分（Tikhonov）正则化、截断奇异值分解（TSVD）和改进的 TSVD，以确定在利用有限数据进行可靠预测方面最有效的稳定策略。我们使用模拟数据以及包括旧金山 1918 年流感大流行和 2014-2015 年西非埃博拉疫情在内的各种疫情的病例数据来说明我们的方法。
