Revisiting the Linear Chain Trick in epidemiological models: Implications of underlying assumptions for numerical solutions

TL;DR

This paper critiques the common ODE approach in infectious disease modeling for assuming exponential dwell times, and promotes the Linear Chain Trick to realize Erlang-distributed stay times within an eight-state, age-resolved SECIR framework. It systematically analyzes how distributional assumptions affect change-point dynamics, peak timing, and transient behavior, while showing that final epidemic size remains governed by the reproduction number. The authors provide a detailed parameterization workflow, validate the model against SARS-CoV-2 dynamics in Germany, and demonstrate substantial runtime efficiency and scalability via the MEmilio software for large ensembles. They also highlight the benefits of incorporating age structure for realism and policy relevance, and discuss extensions to more flexible phase-type distributions to further relax distributional assumptions. Overall, the work emphasizes careful consideration of stay-time distributions in epidemic forecasts and contributes a modular, scalable tool for rapid, data-informed scenario assessment.

Abstract

In order to simulate the spread of infectious diseases, many epidemiological models use systems of ordinary differential equations (ODEs) to describe the underlying dynamics. These models incorporate the implicit assumption, that the stay time in each disease state follows an exponential distribution. However, a substantial number of epidemiological, data-based studies indicate that this assumption is not plausible. One method to alleviate this limitation is to employ the Linear Chain Trick (LCT) for ODE systems, which realizes the use of Erlang distributed stay times. As indicated by data, this approach allows for more realistic models while maintaining the advantages of using ODEs. In this work, we propose an advanced LCT SECIR-type model incorporating eight infection states with demographic stratification. We review key properties of the corresponding LCT model and demonstrate that predictions derived from a simple ODE-based model can be significantly distorted, potentially leading to wrong political decisions. Our findings demonstrate that the influence of distribution assumptions on the behavior at change points and on the prediction of epidemic peaks is substantial, while the assumption has no effect on the final size of the epidemic. With respect to prior findings in literature, we demonstrate that the influence of the number of subcompartments on the timing and size of the epidemic peak is nontrivial and that a general statement cannot be obtained. We, then, show how these age-resolved LCT SECIR-type models capture the spread of SARS-CoV-2 in Germany in 2020. Eventually, we study the implications on the time-to-solution for different LCT models using fixed and adaptive step-size Runge-Kutta methods and provide computational performance for these models in the MEmilio software framework, also using distributed memory parallelism to speed up ensemble runs.

Figures (19)

• Figure 1: Structure of the LCT-SECIR model, omitting age groups visualization. Schematic illustration of the possible transitions between compartments and subcompartments according to the LCT-SECIR model. For the sake of clarity, we have omitted the indices for age groups. The subcompartments, in which individuals are infectious and can infect people from the Susceptible compartment, are highlighted in red. A description of the model parameters can be found in \ref{['tab:parameters']}.
• Figure 2: Density and survival function in an LCT model. Representation of the density function $f_{n/T,\,n}(\tau)$ of the Erlang distribution (left) and the associated survival function $1-F_{n/T,\,n}(\tau)$ (right) for different choices of the parameter $n$. The average stay time $T=10$ is set for all functions. Here, we omit the indices for compartments and age groups.
• Figure 3: Daily new transmissions around change points. Comparison of the daily new transmissions, i.e., the number of people transiting from compartment $S$ to $E$ within one simulation day, of different LCT models against a simple ODE model at change points. The contact rate $\phi(t)$ is halved (left) or doubled (right) after the second simulation day. The naming of the LCT models refer to different assumptions regarding the number of subcompartments, e.g., LCT$3$ refers to an LCT model with $n_Z=3$ subcompartments for each compartment $Z\in\mathcal{A}$ and ODE corresponds to LCT$1$. LCTvar refers to an LCT model with $n_{Z}\approx T_{Z}$ for each compartment $Z\in\mathcal{A}$.
• Figure 4: Number of individuals in Carrier and Infected state around different change points. Comparison of the number of individuals in the Carrier (left) and Infected (right) compartment of different LCT models against an ODE model for the case of a halved (top) or doubled (bottom) contact rate $\phi(t)$ after two simulation days. Further notation as in \ref{['fig:changepoints']}.
• Figure 5: Distribution of individuals in the subcompartments for an increased contact rate. The figures depict the number of individuals in the $n=10$ (top) or $n=50$ (bottom) subcompartments of the Exposed compartment (left), the Carrier compartment (center), and the Infected compartment (right) for each simulation day in the event of a doubled contact rate $\phi(t)$ after two days. The data for day $0$ are omitted, as no change can be observed in comparison to day $1$.

Theorems & Definitions (9)

• Remark 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Corollary 3.5
• Remark 3.6
• Remark 4.1