Classical epidemiological models assume homogeneous populations. There have been important extensions to model heterogeneous populations, when the identity of the sub-populations is known, such as age group or geographical location. Here, we propose two new methods to model the number of people infected with COVID-19 over time, each as a linear combination of latent sub-populations -- i.e., when we do not know which person is in which sub-population, and the only available observations are the aggregates across all sub-populations. Method #1 is a dictionary-based approach, which begins with a large number of pre-defined sub-population models (each with its own starting time, shape, etc), then determines the (positive) weight of small (learned) number of sub-populations. Method #2 is a mixture-of-$M$ fittable curves, where $M$, the number of sub-populations to use, is given by the user. Both methods are compatible with any parametric model; here we demonstrate their use with first (a)~Gaussian curves and then (b)~SIR trajectories. We empirically show the performance of the proposed methods, first in (i) modeling the observed data and then in (ii) forecasting the number of infected people 1 to 4 weeks in advance. Across 187 countries, we show that the dictionary approach had the lowest mean absolute percentage error and also the lowest variance when compared with classical SIR models and moreover, it was a strong baseline that outperforms many of the models developed for COVID-19 forecasting.

翻译：经典流行病学模型假设人群同质化。当子群体身份已知（如年龄组或地理位置）时，已有针对异质人群模型的重要扩展研究。本文提出两种新方法对COVID-19随时间变化的感染人数进行建模，每种方法均将感染人数表示为潜在子群体（即我们既不知道个体属于哪个子群体，唯一可观测数据仅为所有子群体的汇总总量）的线性组合。方法一为基于字典的方法：首先预定义大量子群体模型（每个模型具有独立的起始时间、形态等参数），随后确定少量（通过学习获得）子群体的（正）权重。方法二为可拟合曲线的混合模型（混合M条可拟合曲线），其中子群体数量M由用户指定。这两种方法均与任何参数模型兼容；本文分别采用（a）高斯曲线和（b）SIR轨迹进行实证。我们通过（i）观测数据建模和（ii）提前1至4周感染人数预测两个维度，实证展示了所提方法的性能。基于187个国家的数据表明：相较于经典SIR模型，基于字典的方法不仅平均绝对百分比误差最低且方差最小，同时作为强基线模型，其表现优于众多专为COVID-19预测开发的模型。