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Analyzing Smoothness and Dynamics in an SEIR$^{\text{T}}$R$^{\text{P}}$D Endemic Model with Distributed Delays

Tin Nwe Aye, Linus Carlsson

TL;DR

The paper analyzes a six-compartment endemic model, SEIR^T R^P D, with distributed delays for latency and temporary immunity implemented via compact-support density kernels. It establishes well-posedness, nonnegativity, and boundedness of the continuous-time system, and proves that the solution becomes increasingly smooth over time through an r-increasingly smooth framework. It further demonstrates that discrete-lag endemic models provide accurate approximations to the continuous model, with convergence improving as the number of lag points grows, validated by numerical simulations using Ebola-like kernels. The results enhance understanding of infectious-disease dynamics under realistic delay structures and offer practical guidance for numerically approximating exact delayed solutions using discrete kernels. The work also delineates a clear route for future rigorous convergence proofs of the discrete schemes to the continuous model.

Abstract

This article explores the properties of an SEIR$^{\text{T}}$R$^{\text{P}}$D endemic model expressed through delay-differential equations with distributed delays for latency and temporary immunity. Our research delves into the variability of latent periods and immunity durations across diseases, in particular, we introduce a class of delays defined by continuous integral kernels with compact support. The main result of the paper is a kind of smoothening property which the solution function posesses under mild conditions of the system parameter functions. Also, boundedness and non-negativity is proved. Numerical simulations indicates that the continuous model can be approximated with a discrete lag endemic models. The study contributes to understanding infectious disease dynamics and provides insights into the numerical approximation of exact solution for different delay scenarios.

Analyzing Smoothness and Dynamics in an SEIR$^{\text{T}}$R$^{\text{P}}$D Endemic Model with Distributed Delays

TL;DR

The paper analyzes a six-compartment endemic model, SEIR^T R^P D, with distributed delays for latency and temporary immunity implemented via compact-support density kernels. It establishes well-posedness, nonnegativity, and boundedness of the continuous-time system, and proves that the solution becomes increasingly smooth over time through an r-increasingly smooth framework. It further demonstrates that discrete-lag endemic models provide accurate approximations to the continuous model, with convergence improving as the number of lag points grows, validated by numerical simulations using Ebola-like kernels. The results enhance understanding of infectious-disease dynamics under realistic delay structures and offer practical guidance for numerically approximating exact delayed solutions using discrete kernels. The work also delineates a clear route for future rigorous convergence proofs of the discrete schemes to the continuous model.

Abstract

This article explores the properties of an SEIRRD endemic model expressed through delay-differential equations with distributed delays for latency and temporary immunity. Our research delves into the variability of latent periods and immunity durations across diseases, in particular, we introduce a class of delays defined by continuous integral kernels with compact support. The main result of the paper is a kind of smoothening property which the solution function posesses under mild conditions of the system parameter functions. Also, boundedness and non-negativity is proved. Numerical simulations indicates that the continuous model can be approximated with a discrete lag endemic models. The study contributes to understanding infectious disease dynamics and provides insights into the numerical approximation of exact solution for different delay scenarios.

Paper Structure

This paper contains 16 sections, 5 theorems, 93 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. An endemic dynamics model with distributed delays
  3. Mathematical analysis of the model
  4. Positivity and boundedness of solution
  5. Some results on continuity and differentaibility
  6. Enhancing Solution Smoothness Over Time
  7. Numerical simulation
  8. Discrete lag endemic models
  9. Kernel functions in simulations
  10. Exact solution of System 1
  11. Simulation results
  12. Final remarks
  13. Acknowledgements
  14. Appendix
  15. Proof of Theorem \ref{['Thm:Non-negativity']}
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that the parameters $\mu$ and $\gamma$ in the System model:mainGenNoBirthOrNaturalDeath are non-negative constants. The contact rate, $\beta(t),$ is non-negative for all $t$. Let the history data for be given by equations eq:HS and eq:HI and the initial data be equations eq:Es--eq:RPs. Then,

Figures (6)

  • Figure 1: The number of susceptible individuals is represented for four different models under 10 years. The blue solid curve uses continuous time delay kernel functions. The red dotted curve is simulated utilizing discrete (60,60) model, the green dotted curve is plotted with discrete (3,3) model, and the yellow dashed curve is produced with the discrete (1,1) model.
  • Figure 2: The number of exposed individuals is represented for four different models under 10 years. The blue solid curve uses continuous time delay kernel functions. The red dotted curve is simulated utilizing discrete (60,60) model, the green dotted curve is plotted with discrete (3,3) model, and the yellow dashed curve is produced with the discrete (1,1) model.
  • Figure 3: The number of infectious individuals is represented for four different models under 10 years. The blue solid curve uses continuous time delay kernel functions. The red dotted curve is simulated utilizing discrete (60,60) model, the green dotted curve is plotted with discrete (3,3) model, and the yellow dashed curve is produced with the discrete (1,1) model.
  • Figure 4: The number of temporary recovery individuals is represented for four different models under 10 years. The blue solid curve uses continuous time delay kernel functions. The red dotted curve is simulated utilizing discrete (60,60) model, the green dotted curve is plotted with discrete (3,3) model, and the yellow dashed curve is produced with the discrete (1,1) model.
  • Figure 5: The number of permanent recovery individuals is represented for four different models under 10 years. The blue solid curve uses continuous time delay kernel functions. The red dotted curve is simulated utilizing discrete (60,60) model, the green dotted curve is plotted with discrete (3,3) model, and the yellow dashed curve is produced with the discrete (1,1) model.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1: Non-negativity
  • Theorem 2: Boundedness
  • Theorem 3
  • Corollary 4
  • Definition 5
  • Theorem 6