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Convergence Analysis of an Endemic Time Delay Model Using Dirac and Radon Measures

Tin Nwe Aye, Linus Carlsson

TL;DR

Addresses convergence of discrete-delay approximations for the $SLIR^{T}R^{P}D$ endemic model with distributed delays implemented via Dirac and Radon measures. The authors formulate the continuous-time kernel-based model with compact-support kernels $\Phi$ and $\Psi$ and extend to positive Radon measures, proving that the discrete-delay solutions converge uniformly on compact time intervals to the continuous solution. They provide numerical demonstrations using exponential kernels, showing that the discrete $(N_{\tau},N_{\rho})$ models converge to the exact solution of the continuous system, with substantial computational speed-ups. The work strengthens the theoretical link between discrete and continuous delay representations in epidemic modeling and offers a practical framework for efficient simulations of latency and immune dynamics.

Abstract

This article explores the convergence properties of an $SLIR^\text{T}R^\text{P}D$ endemic model, incorporating Dirac and Radon measures, alongside distributed delays to represent latency and temporary immunity. A class of delays is defined for both continuous and discrete endemic models using continuous integral kernels with compact support and discrete terms expressed through Dirac and Radon measures. Numerical results show that the continuous model can be approximated by a discrete lag endemic model. Furthermore, the simulation time for the numerical solution is significantly shorter than that for the exact solution.

Convergence Analysis of an Endemic Time Delay Model Using Dirac and Radon Measures

TL;DR

Addresses convergence of discrete-delay approximations for the endemic model with distributed delays implemented via Dirac and Radon measures. The authors formulate the continuous-time kernel-based model with compact-support kernels and and extend to positive Radon measures, proving that the discrete-delay solutions converge uniformly on compact time intervals to the continuous solution. They provide numerical demonstrations using exponential kernels, showing that the discrete models converge to the exact solution of the continuous system, with substantial computational speed-ups. The work strengthens the theoretical link between discrete and continuous delay representations in epidemic modeling and offers a practical framework for efficient simulations of latency and immune dynamics.

Abstract

This article explores the convergence properties of an endemic model, incorporating Dirac and Radon measures, alongside distributed delays to represent latency and temporary immunity. A class of delays is defined for both continuous and discrete endemic models using continuous integral kernels with compact support and discrete terms expressed through Dirac and Radon measures. Numerical results show that the continuous model can be approximated by a discrete lag endemic model. Furthermore, the simulation time for the numerical solution is significantly shorter than that for the exact solution.

Paper Structure

This paper contains 10 sections, 4 theorems, 67 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Modeling Endemic Dynamics with Continuous Delays
  3. Convergence analysis for delay differential equations
  4. Endemic model with generalized measure approach
  5. Convergence analysis of endemic model with generalized measure
  6. Numerical example
  7. Discrete finite time-delay approximation using DDEs
  8. Continuous time-delay approximation using ODEs
  9. Simulation results
  10. Conclusions and Remarks

Key Result

Lemma 1

Let $H$ be uniformly continuous function and let $\Phi$ be a probability density function with $\mathrm{supp}(\Phi)\subset([\sigma,M])$ then In measure theoritical arguments, we prove that $d\nu_{j}\underset {j\rightarrow\infty}{\longrightarrow}\Phi d\lambda$ on $C(\mathbb{R}_{+}).$

Figures (7)

  • Figure 1: The number of infectious individuals is presented under one year by setting different values of discrete ($N_{\tau}$,$N_{\rho}$) model and the continuous endemic model. The solid curve is shown by using the exact solution of System \ref{['model:mainSystemOfEquation']} equipped with the kernal function equations \ref{['eq:Rho']} and \ref{['eq:Tau']}. The rest of dotted curves are simulated utilizing discrete ($N_{\tau}$,$N_{\rho}$) model by using the values $(1,2)$, $(10,20)$ and $(100,200)$.
  • Figure 2: The number of individuals for each compartment is presented under one year by setting $N_{\tau}=100$ and $N_{\rho}=200$ in the discrete ($N_{\tau}$,$N_{\rho}$) model, equipped with the kernal function equations \ref{['eq:Rho']} and \ref{['eq:Tau']}.
  • Figure 3: The number of susceptible individuals is presented under 1 year by setting different values of discrete ($N_{\tau}$,$N_{\rho}$) model and the continuous endemic model. The solid curve is shown by using the exact solution of System \ref{['model:mainSystemOfEquation']}. The rest of dotted curves are simulated utilizing discrete ($N_{\tau}$,$N_{\rho}$) model by setting (1,2), (10,20) and (100,200).
  • Figure 4: The number of latent individuals is presented under 1 year by setting different values of discrete ($N_{\tau}$,$N_{\rho}$) model and the continuous endemic model. The solid curve is shown by using the exact solution of System \ref{['model:mainSystemOfEquation']}. The rest of dotted curves are simulated utilizing discrete ($N_{\tau}$,$N_{\rho}$) model by setting (1,2), (10,20) and (100,200).
  • Figure 5: The number of temporary recovery individuals is presented under 1 year by setting different values of discrete ($N_{\tau}$,$N_{\rho}$) model and the continuous endemic model. The solid curve is shown by using the exact solution of System \ref{['model:mainSystemOfEquation']}. The rest of dotted curves are simulated utilizing discrete ($N_{\tau}$,$N_{\rho}$) model by setting (1,2), (10,20) and (100,200).
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 2
  • Lemma 3
  • Theorem 4