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Mixed Time Series Quasi-Likelihood Models for Uncovering Covid-19 Viral Load and Mortality Dynamics

This paper introduces a new mixed-valued time series quasi-likelihood (MixTSQL) model that jointly analyzes continuous viral load and count mortality data without strict distributional assumptions, demonstrating its application in São Paulo, Brazil, to establish that viral load Granger-causes Covid-19 deaths while providing statistical guarantees for estimator consistency and asymptotic normality.

Original authors: Kejin Wu, Raanju R. Sundararajan, Michel F. C. Haddad, Luiza S. C. Piancastelli, Wagner Barreto-Souza

Published 2026-03-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kejin Wu, Raanju R. Sundararajan, Michel F. C. Haddad, Luiza S. C. Piancastelli, Wagner Barreto-Souza

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict a storm. Traditionally, meteorologists look at how many people are already getting wet (reported cases) or how many umbrellas are broken (hospital admissions). But there's a problem: people often forget to report they are wet, or they report it late. By the time you know everyone is soaked, the storm has already passed.

This paper introduces a smarter way to predict the storm by looking at the clouds before the rain starts. In the world of viruses, these "clouds" are called Viral Load—how much virus is actually inside a person's body.

Here is the story of the paper, broken down into simple parts:

1. The Problem: The "Mixed" Mess

The researchers wanted to study two things happening at the same time in Brazil:

Viral Load: A number that tells you how much virus is in a sample. This is a continuous number (like 0.45 or 0.82) and it has a limit (it can't go above 1 or below 0).
Deaths: A count number (1, 2, 3... you can't have 2.5 deaths).

Most statistical tools are like specialized hammers: they are great at hitting nails (continuous data) or great at driving screws (count data), but they break if you try to use them on both at once. The existing tools also assumed the data followed a perfect "bell curve" shape, which real-world virus data rarely does.

2. The Solution: The "MixTSQL" Swiss Army Knife

The authors invented a new statistical tool called MixTSQL (Mixed Time Series Quasi-Likelihood).

Think of this model as a Swiss Army Knife for data. Instead of demanding the data fit a strict, rigid shape (like a perfect bell curve), it only asks two simple questions:

"What is the average?" (The Mean)
"How much does it bounce around?" (The Variance)

Because it doesn't care about the strict shape of the data, it can handle the "messy" mix of viral loads and death counts without breaking. It's flexible, like water taking the shape of whatever container you pour it into.

3. The Detective Work: Who Causes Whom?

The big question was: Does the amount of virus in people's bodies cause the number of deaths later? Or is it the other way around?

In statistics, this is called Granger Causality. Imagine you are watching a row of dominoes. If you see the first domino fall, you know the second one will fall soon.

The researchers used their new "Swiss Army Knife" model to see if high viral loads (the first domino) reliably predicted future deaths (the second domino).
They found a clear link: High viral loads today predict more deaths about 6 weeks from now.

This makes sense biologically. It takes time for a virus to make someone sick enough to go to the hospital and, tragically, pass away. The model successfully caught this "time delay" that other methods might have missed.

4. The Proof: Simulations and Real Data

Before trusting their new tool, the authors played a game of "make-believe."

They created fake virus data using computers (simulations) to see if their model could find the patterns they planted.
They found that their model was accurate, even with small amounts of data, and worked just as well as more complicated, rigid models—but without the headaches.

Then, they applied it to real data from São Paulo, Brazil, covering over two years of the pandemic.

The Result: The model successfully tracked the three major "waves" of the pandemic in Brazil.
The Prediction: When they tried to predict future deaths, their new model was more accurate than the standard models used by statisticians. It didn't just guess; it understood the unique "personality" of the virus data.

5. Why This Matters

Think of this paper as upgrading the dashboard of a car.

Old Dashboard: Only showed you how fast you were going after you hit a bump (reported cases).
New Dashboard (MixTSQL): Shows you the road conditions ahead (viral load), allowing you to slow down before the crash.

By using Cycle Threshold (Ct) values from lab tests as a proxy for viral load, this method gives public health officials a crystal ball. It tells them, "Hey, viral loads are rising in the community; expect more severe cases and deaths in about six weeks."

This allows governments to act faster—implementing social distancing or boosting hospital readiness before the hospitals get overwhelmed, rather than reacting after the damage is done.

In a Nutshell

The authors built a flexible, new statistical engine that can drive two different types of data (virus levels and death counts) simultaneously. They proved that by watching the "engine noise" (viral load), we can predict the "crash" (deaths) weeks in advance, giving us a crucial head start in fighting future pandemics.

1. Problem Statement

The paper addresses the challenge of jointly modeling two distinct types of time series data to understand the dynamics of the Covid-19 pandemic:

Viral Load: Derived from Cycle Threshold (Ct) values of RT-qPCR tests. This is a continuous, bounded variable (typically normalized to the unit interval).
Mortality: The daily/weekly count of deaths. This is a discrete, count variable.

Key Challenges:

Data Limitations: Traditional surveillance metrics (reported cases, hospital admissions) suffer from reporting delays, selection bias, and under-ascertainment. Viral load (via Ct values) offers a more direct, real-time proxy for infectivity and disease progression.
Modeling Gap: Existing time series models for Granger causality often rely on linear Vector Autoregressive (VAR) models assuming multivariate normality, which is unsuitable for mixed data types (bounded continuous + count). Previous mixed-valued models (e.g., Generalized Linear Models) often require strict exponential family distributional assumptions, which cannot accommodate bounded variables like Ct-derived viral load without modification.
Goal: To develop a flexible framework that can jointly model these mixed-type series, test for Granger causality in both mean and variance, and provide robust statistical inference without requiring full distributional assumptions.

2. Methodology: The MixTSQL Model

The authors introduce the Mixed-valued Time Series Quasi-Likelihood (MixTSQL) model.

Core Framework:

Quasi-Likelihood (QL) Approach: Instead of specifying a full probability distribution (e.g., Normal, Poisson, Beta), the model only specifies the first two moments (mean and variance) of the conditional distribution.
Model Specification:
- Let $Y_{1t}$ be the viral load (bounded) and $Y_{2t}$ be the mortality count.
- The conditional means $\mu_{1t}$ and $\mu_{2t}$ are linked to linear predictors via link functions $g_1(\cdot)$ and $g_2(\cdot)$ .
- The linear predictors include autoregressive terms (past values of the same series) and cross-lagged terms (past values of the other series).
- Variance Functions:
  - For Viral Load ( $Y_{1t}$ ): $V(\mu_1) = \mu_1(1-\mu_1)$ (mimicking a Beta distribution).
  - For Mortality ( $Y_{2t}$ ): $V(\mu_2) = \mu_2$ (mimicking a Poisson/Double Poisson distribution).
Estimation: Parameters are estimated using Quasi-Maximum Likelihood Estimation (QMLE). The quasi-log-likelihood function is constructed based on the specified mean-variance relationship.
Granger Causality Test:
- The authors define causality in both mean and variance.
- They propose a Quasi-Likelihood Ratio (QLR) test statistic to test the null hypothesis that cross-lagged coefficients ( $\gamma$ ) are zero.
- The test statistic asymptotically follows a Chi-squared distribution, allowing for the detection of causal effects even when the underlying data distribution is unknown or misspecified.

Theoretical Guarantees:

The paper provides rigorous proofs for the consistency and asymptotic normality of the QMLE estimators under specific regularity conditions (stationarity, ergodicity, and correct mean specification).
It establishes that the QLR test is valid for detecting causality in mixed-valued time series.

3. Key Contributions

Novel Model (MixTSQL): A new class of models capable of handling multivariate time series with mixed types (continuous bounded, count, positive, binary) without full distributional assumptions.
Simultaneous Causality Testing: The ability to test Granger causality in both the mean and variance simultaneously, which is crucial for understanding how viral load fluctuations affect not just the average number of deaths but also the volatility of mortality.
Robust Inference: Demonstration that QMLE provides consistent estimates even if the true data-generating process differs from the assumed variance function, provided the mean structure is correct.
Empirical Validation: Application to a large-scale, real-world dataset (Sao Paulo, Brazil) involving 342,699 PCR records, bridging the gap between theoretical time series econometrics and epidemiological surveillance.

4. Results

Simulation Studies:

Extensive Monte Carlo simulations were conducted with sample sizes ( $n=100$ ) mimicking the real data.
Performance: The QMLE estimators were found to be unbiased and normally distributed even in small samples.
Standard Errors: Both theoretical asymptotic standard errors and bootstrap-based standard errors performed well. The theoretical approach was deemed preferable for its computational efficiency, with negligible loss in accuracy compared to bootstrapping.
Robustness: The model remained robust even when the data-generating process (e.g., Bessel-Poisson) differed from the model specification (Beta-Double Poisson).

Real-World Application (Sao Paulo, Brazil):

Data: Weekly aggregated data from March 2020 to May 2022 ( $n=112$ ).
Model Fit: The MixTSQL model successfully captured the three major epidemiological waves in Brazil (Ancestral, Gamma, and Omicron variants).
Granger Causality:
- The test rejected the null hypothesis ( $H_0: \gamma_6 = 0$ ) with a p-value $< 10^{-5}$ .
- Finding: Viral load Granger-causes mortality with a lag of 6 weeks. This aligns with biological expectations (time from infection to hospitalization to death).
Forecasting Performance:
- An out-of-sample forecasting exercise (recursive, 1-step-ahead) compared MixTSQL against a standard Gaussian linear model (with square-root transformation).
- Result: The MixTSQL model achieved a lower Root Mean Forecasting Error (RMFE) than the Gaussian model.
- Prediction Intervals: The Gaussian model produced narrower intervals that frequently failed to cover the true counts, whereas the MixTSQL model provided better-calibrated prediction intervals, particularly for the count data.

5. Significance and Implications

Public Health Utility: The study validates the use of Ct values (viral load) as a leading indicator for mortality. The 6-week lead time offers a critical window for policymakers to implement interventions (e.g., social distancing, resource allocation) before mortality peaks.
Methodological Advancement: The MixTSQL framework fills a significant gap in time series analysis by allowing the joint modeling of heterogeneous data types without restrictive distributional assumptions. This is highly relevant for future pandemics where data types may vary.
Policy Relevance: By demonstrating that viral load dynamics precede mortality, the paper supports the integration of molecular surveillance data into real-time epidemic control strategies, moving beyond reliance on lagging indicators like reported deaths or hospitalizations.

In conclusion, the paper presents a robust, theoretically grounded, and practically effective methodology for analyzing complex epidemiological data, successfully linking viral load dynamics to mortality outcomes in Brazil and providing a template for future mixed-valued time series analysis.