Accurate real-time monitoring of disease transmission is crucial for epidemic control, which has conventionally relied on reported cases or hospital admissions. Such metrics are frequently susceptible to delays in reporting, various forms of bias, and under-ascertainment. Cycle threshold values obtained from reverse transcription quantitative polymerase chain reaction offer a promising alternative, serving as a proxy for viral load. In this paper, we aim to jointly model the viral load and the number of deaths (mortality), which involves a continuous bounded and a count time series, and therefore, a proper mixed-type model is needed. This is the motivation to introduce a new mixed-valued time series quasi-likelihood (MixTSQL) model capable of analyzing multivariate time series of different types, like continuous, discrete, bounded, and continuous positive. The MixTSQL model only requires a mean-variance specification with no distributional assumptions needed, and allows for testing Granger causality. Statistical guarantees are provided to ensure consistency and asymptotic normality of the proposed quasi-maximum likelihood estimators. We analyze weekly viral load and Covid-19 death counts in São Paulo, Brazil, using our MixTSQL model, which not only establishes the temporal order in which viral load Granger-causes mortality but also offers a comprehensive joint statistical analysis.

翻译：对疾病传播进行准确的实时监测对流行病控制至关重要，传统上依赖报告病例或住院人数。这些指标常受报告延迟、多种偏差及低估的影响。从逆转录定量聚合酶链反应获得的循环阈值作为病毒载量的替代指标，提供了一种有前景的选择。本文旨在联合建模病毒载量与死亡人数（死亡率），涉及连续有界时间序列和计数时间序列，因此需要合适的混合类型模型。为此，我们引入一种新的混合值时间序列拟似然（MixTSQL）模型，能够分析不同类型（如连续型、离散型、有界型和连续正型）的多元时间序列。MixTSQL模型仅需均值-方差设定，无需分布假设，并允许检验格兰杰因果关系。我们提供了统计保证，确保所提出的拟最大似然估计量的一致性和渐近正态性。利用MixTSQL模型，我们分析了巴西圣保罗的每周病毒载量和新冠死亡人数，不仅确定了病毒载量格兰杰导致死亡的时间顺序，还提供了全面的联合统计分析。