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A nonstandard numerical scheme for a novel SECIR integro-differential equation-based model allowing nonexponentially distributed stay times

Anna Wendler, Lena Plötzke, Hannah Tritzschak, Martin J. Kühn

TL;DR

The paper addresses the limitation of exponential stay times in ODE-based infectious disease models by introducing a SECIR-type IDE that allows arbitrary stay-time distributions across eight compartments. It develops a nonstandard discretization with two equivalent schemes (sum and update) and proves mass conservation, positivity, and convergence properties, complemented by convergence tests. Numerical experiments reveal a linear convergence rate and a realistic lag in IDE response at change points, contrasting with the immediate response of ODE models. In a COVID-19–inspired German scenario, the IDE provides improved predictions for new transmissions, mild infections, and ICU demand, illustrating the practical value of incorporating realistic stay-time distributions.

Abstract

Ordinary differential equations (ODE) are a popular tool to model the spread of infectious diseases, yet they implicitly assume an exponential distribution to describe the flow from one infection state to another. However, scientific experience yields more plausible distributions where the likelihood of disease progression changes accordingly with the duration spent in a particular state of the disease. Furthermore, transmission dynamics depend heavily on the infectiousness of individuals. The corresponding nonlinear variation with the time individuals have already spent in an infectious state requires more realistic models. The previously mentioned items are particularly crucial when modeling dynamics at change points such as the implementation of nonpharmaceutical interventions. In order to capture these aspects and to enhance the accuracy of simulations, integro-differential equations (IDE) can be used. In this paper, we propose a generalized model based on IDEs with eight infection states. The model allows for variable stay time distributions and generalizes the concept of ODE-based models as well as IDE-based age-of-infection models. In this, we include particular infection states for severe and critical cases to allow for surveillance of the clinical sector, avoiding bottlenecks and overloads in critical epidemic situations. On the other hand, a drawback of IDE-based models is that efficient numerical solvers are not as widely available. We extend a recently introduced nonstandard numerical scheme. This scheme is adapted to our more advanced model and we prove important mathematical and biological properties. Furthermore, we validate our approach numerically by demonstrating the convergence rate. Eventually, we also show that our novel model is intrinsically capable of better assessing disease dynamics upon the introduction of nonpharmaceutical interventions.

A nonstandard numerical scheme for a novel SECIR integro-differential equation-based model allowing nonexponentially distributed stay times

TL;DR

The paper addresses the limitation of exponential stay times in ODE-based infectious disease models by introducing a SECIR-type IDE that allows arbitrary stay-time distributions across eight compartments. It develops a nonstandard discretization with two equivalent schemes (sum and update) and proves mass conservation, positivity, and convergence properties, complemented by convergence tests. Numerical experiments reveal a linear convergence rate and a realistic lag in IDE response at change points, contrasting with the immediate response of ODE models. In a COVID-19–inspired German scenario, the IDE provides improved predictions for new transmissions, mild infections, and ICU demand, illustrating the practical value of incorporating realistic stay-time distributions.

Abstract

Ordinary differential equations (ODE) are a popular tool to model the spread of infectious diseases, yet they implicitly assume an exponential distribution to describe the flow from one infection state to another. However, scientific experience yields more plausible distributions where the likelihood of disease progression changes accordingly with the duration spent in a particular state of the disease. Furthermore, transmission dynamics depend heavily on the infectiousness of individuals. The corresponding nonlinear variation with the time individuals have already spent in an infectious state requires more realistic models. The previously mentioned items are particularly crucial when modeling dynamics at change points such as the implementation of nonpharmaceutical interventions. In order to capture these aspects and to enhance the accuracy of simulations, integro-differential equations (IDE) can be used. In this paper, we propose a generalized model based on IDEs with eight infection states. The model allows for variable stay time distributions and generalizes the concept of ODE-based models as well as IDE-based age-of-infection models. In this, we include particular infection states for severe and critical cases to allow for surveillance of the clinical sector, avoiding bottlenecks and overloads in critical epidemic situations. On the other hand, a drawback of IDE-based models is that efficient numerical solvers are not as widely available. We extend a recently introduced nonstandard numerical scheme. This scheme is adapted to our more advanced model and we prove important mathematical and biological properties. Furthermore, we validate our approach numerically by demonstrating the convergence rate. Eventually, we also show that our novel model is intrinsically capable of better assessing disease dynamics upon the introduction of nonpharmaceutical interventions.
Paper Structure (17 sections, 6 theorems, 63 equations, 5 figures, 5 tables)

This paper contains 17 sections, 6 theorems, 63 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. A novel SECIR-type IDE-based model
  3. A nonstandard discretization scheme
  4. Nonstandard discretization of compartments based on integral formulation
  5. Nonstandard discretization of compartments based on flows
  6. Connection between sum and update discretizations
  7. Initialization of the discretized system
  8. Theoretical properties of the numerical solution
  9. Numerical results
  10. Order of convergence
  11. Model behavior at change points
  12. A COVID-19 inspired real scenario
  13. Discussion
  14. Conclusion
  15. Relation between IDE and ODE model
  16. ...and 2 more sections

Key Result

Theorem 3.1

Let the derivative of $\gamma_{z_1}^{z_2}$ for appropriate combinations of $z_1,z_2\in\mathcal{Z}$ be approximated with a backwards difference scheme, i.e., let for $i\in\mathbb{Z}$. Then the sum discretization of the compartments, as defined in eq:diskret_sum, is equivalent to the update discretization, as described in eq:diskret_update.

Figures (5)

  • Figure 1: Structure of the IDE model. Schematic representation of the compartments and the transitions between the compartments in the IDE model. The states in which individuals are infectious are highlighted in red.
  • Figure 2: Convergence of IDE model for compartments (left) and transitions (right). Relative error of the simulation results of the IDE model with different time step sizes compared with numerical results of the ODE model with a step size $\Delta t=10^{-6}$ (ground truth) for the compartments (left) and the transitions (right). A linear function is plotted to compare the slope of the errors.
  • Figure 3: Daily new transmissions at change points. Comparison of the simulation results for the daily new transmissions of the ODE and the IDE model for a halving (left) or doubling (right) of the contact rate $\phi(t)$ after two simulation days.
  • Figure 4: Simulation results for COVID-19 in Germany from Oct 1, 2020, onwards. Comparison of extrapolated real data with the simulation results of the IDE and the ODE model. The results are shown for the number of daily new transmissions (top left), the number of mildly symptomatic individuals (top right), the number of patients in intensive care units (bottom left) and the number of deaths (bottom right).
  • Figure 5: Share of considered population in respective age groups. For dates Jun 1, 2020, and Oct 1, 2020, we consider the number of confirmed cases between $T_I^H + T_H^U + T_U$ and $T_I^H + T_H^U$ days before these dates. This corresponds to the individuals that we expect to be in compartment $U$ at time $t_0$. We use data on the confirmed cases of COVID-19 in Germany provided by RKI_data_2023. Above, the proportion per age group of these confirmed cases is depicted. For comparison, the proportion of each age group in the total population in Germany as reported in regionaldatenbank_deutschland_fortschreibung is shown.

Theorems & Definitions (13)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • ...and 3 more