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Qualitative and Numerical Simulation of a Time-Fractional SEIR Mpox Model Arising in Population Epidemiology

Gaurav Saini, Bappa Ghosh, Sunita Chand, Jugal Mohapatra

TL;DR

This paper addresses Mpox transmission modeling using a time-fractional SEIR framework based on the Caputo derivative of order $\rho\in(0,1)$ to capture memory in disease dynamics. The authors establish existence, uniqueness and Hyers-Ulam stability of the solution and develop an efficient L1 finite difference scheme (coupled with Newton–Raphson) to discretize and solve the nonlinear system, with rigorous error analysis showing algebraic convergence. They compare the L1 scheme against the Fractional Modified Euler Method (FMEM) and demonstrate superior accuracy and stability, plus numerical experiments revealing how the fractional order and vaccination rate affect outbreak trajectories. The work highlights memory effects in Mpox spread and positions the L1 method as a robust tool for simulating fractional-order epidemiological models.

Abstract

Epidemiological modeling is vital in understanding disease dynamics and guiding public health interventions. This study presents a time-fractional SEIR model to describe the transmission dynamics of Mpox, incorporating memory effects via the fractional derivative. We perform an extensive qualitative investigation, proving that there is a unique solution and that the solutions are Hyers-Ulam stable. To approximate the model numerically, we implement the L1 finite difference scheme for the Caputo derivative and solve the resulting nonlinear system using the Newton-Raphson technique. A detailed error analysis is provided, demonstrating that the scheme achieves algebraic convergence. Comparative results with the Fractional Modified Euler method (FMEM) confirm the superior accuracy and stability of the proposed approach. Numerical simulations under biologically relevant parameters illustrate the impact of the non-integer order and vaccination rate on disease progression. The study underscores the effectiveness of fractional order models in capturing epidemic memory effects and positions the L1 scheme as a robust numerical tool for simulating such dynamics.

Qualitative and Numerical Simulation of a Time-Fractional SEIR Mpox Model Arising in Population Epidemiology

TL;DR

This paper addresses Mpox transmission modeling using a time-fractional SEIR framework based on the Caputo derivative of order to capture memory in disease dynamics. The authors establish existence, uniqueness and Hyers-Ulam stability of the solution and develop an efficient L1 finite difference scheme (coupled with Newton–Raphson) to discretize and solve the nonlinear system, with rigorous error analysis showing algebraic convergence. They compare the L1 scheme against the Fractional Modified Euler Method (FMEM) and demonstrate superior accuracy and stability, plus numerical experiments revealing how the fractional order and vaccination rate affect outbreak trajectories. The work highlights memory effects in Mpox spread and positions the L1 method as a robust tool for simulating fractional-order epidemiological models.

Abstract

Epidemiological modeling is vital in understanding disease dynamics and guiding public health interventions. This study presents a time-fractional SEIR model to describe the transmission dynamics of Mpox, incorporating memory effects via the fractional derivative. We perform an extensive qualitative investigation, proving that there is a unique solution and that the solutions are Hyers-Ulam stable. To approximate the model numerically, we implement the L1 finite difference scheme for the Caputo derivative and solve the resulting nonlinear system using the Newton-Raphson technique. A detailed error analysis is provided, demonstrating that the scheme achieves algebraic convergence. Comparative results with the Fractional Modified Euler method (FMEM) confirm the superior accuracy and stability of the proposed approach. Numerical simulations under biologically relevant parameters illustrate the impact of the non-integer order and vaccination rate on disease progression. The study underscores the effectiveness of fractional order models in capturing epidemic memory effects and positions the L1 scheme as a robust numerical tool for simulating such dynamics.
Paper Structure (7 sections, 7 theorems, 50 equations, 4 figures, 6 tables)

This paper contains 7 sections, 7 theorems, 50 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Existence and Uniqueness of Solutions
  3. Stability Analysis of the Solution
  4. The Discretized Problem
  5. Error Analysis
  6. Numerical illustration and discussions
  7. Conclusion

Key Result

Theorem 2.1

The kernel functions $F_1, F_2, F_3, F_4$ satisfy the Lipschitz condition under assumption $(C)$, provided that $0 \leq \kappa_i < 1$ for all $i = 1, 2, 3, 4$.

Figures (4)

  • Figure 1: Comparison of $\widehat{S}(t)$ and $\widehat{E}(t)$ for different values of $\rho$
  • Figure 2: Comparison of $\widehat{I}(t)$ and $\widehat{R}(t)$ for different values of $\rho$
  • Figure 3: Effect of vaccination rate on $\widehat{E}(t)$ and $\widehat{I}(t)$
  • Figure 4: Log-log plots for Example \ref{['example']}

Theorems & Definitions (16)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3: Uniqueness of solution
  • proof
  • Definition 3.1: Hyers--Ulam stability Jung2004
  • Theorem 3.2: Hyers-Ulam Stability
  • proof
  • Theorem 5.1: Local Truncation Error
  • ...and 6 more