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Mathematical Analysis and Modeling of Ebola Virus Dynamics via Optimal Control and Neural Network Paradigms

Noor Muhammad, Md. Nur Alam, Zhang Shiqing

TL;DR

This study addresses Ebola transmission by integrating memory-aware (Caputo fractional) eight-compartment dynamics with rigorous mathematical analysis, an optimal-control framework, and a disease-informed neural network (DINN). It derives the basic reproduction number via a next-generation approach, proves global stability results, and demonstrates a forward bifurcation at $\mathcal{R}_0=1$, highlighting memory effects on outbreak behavior. The control analysis shows that combining treatment with safe burial yields the largest mortality reduction (up to 86.5%), while DINN provides near-perfect predictive accuracy and reliable parameter estimation. Collectively, the work offers a principled, memory-inclusive toolset for forecasting, intervention design, and real-time inference in Ebola epidemics, with potential extensions to spatial and stochastic settings.

Abstract

Ebola virus disease is a severe hemorrhagic fever with rapid transmission through infected fluids and surfaces. We develop a fractional-order model using Caputo derivatives to capture memory effects in disease dynamics. An eight-compartment structure distinguishes symptomatic, asymptomatic, and post-mortem transmission pathways. We prove global well-posedness, derive the basic reproduction number $\mathcal{R}_0$, and establish stability theorems. Sensitivity analysis shows $\mathcal{R}_0$ is most sensitive to transmission rate, incubation period, and deceased infectivity. Treatment-safe burial synergy achieves 86.5\% morbidity-mortality control, with safe burial being most effective. Our disease-informed neural network achieves near-perfect predictive accuracy ($R^2$: 0.991-0.999, 99.1-99.9\% accuracy), closely matching real epidemic behavior.

Mathematical Analysis and Modeling of Ebola Virus Dynamics via Optimal Control and Neural Network Paradigms

TL;DR

This study addresses Ebola transmission by integrating memory-aware (Caputo fractional) eight-compartment dynamics with rigorous mathematical analysis, an optimal-control framework, and a disease-informed neural network (DINN). It derives the basic reproduction number via a next-generation approach, proves global stability results, and demonstrates a forward bifurcation at , highlighting memory effects on outbreak behavior. The control analysis shows that combining treatment with safe burial yields the largest mortality reduction (up to 86.5%), while DINN provides near-perfect predictive accuracy and reliable parameter estimation. Collectively, the work offers a principled, memory-inclusive toolset for forecasting, intervention design, and real-time inference in Ebola epidemics, with potential extensions to spatial and stochastic settings.

Abstract

Ebola virus disease is a severe hemorrhagic fever with rapid transmission through infected fluids and surfaces. We develop a fractional-order model using Caputo derivatives to capture memory effects in disease dynamics. An eight-compartment structure distinguishes symptomatic, asymptomatic, and post-mortem transmission pathways. We prove global well-posedness, derive the basic reproduction number , and establish stability theorems. Sensitivity analysis shows is most sensitive to transmission rate, incubation period, and deceased infectivity. Treatment-safe burial synergy achieves 86.5\% morbidity-mortality control, with safe burial being most effective. Our disease-informed neural network achieves near-perfect predictive accuracy (: 0.991-0.999, 99.1-99.9\% accuracy), closely matching real epidemic behavior.

Paper Structure

This paper contains 31 sections, 13 theorems, 92 equations, 20 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional-Order Ebola Transmission Dynamics
  4. Mathematical Analysis
  5. Well-Posedness of the Model
  6. Invariant Region and Attractivity
  7. EVD-Free Equilibrium
  8. Basic Reproduction NumberR0
  9. Local Stability Analysis of the EVD-Free Equilibrium
  10. Global Stability of EVD-Free Equilibrium
  11. Endemic Equilibrium Analysis
  12. Local Stability Analysis of the Endemic Equilibrium
  13. Global Stability of the Endemic Equilibrium
  14. Sensitivity Analysis
  15. Analytical Sensitivity Framework
  16. ...and 16 more sections

Key Result

Theorem 2.1

Let $\alpha \in (0,1]$. An equilibrium point $y^*$ of the autonomous system ${}^{C}D_{t}^{\alpha} y(t) = f(y(t))$ is asymptotically stable if all eigenvalues $\lambda_i$ of the Jacobian matrix $J = \partial f / \partial y |_{y^*}$ satisfy the condition li2010stability: This condition generalizes the classical Routh-Hurwitz criterion to fractional-order systems.

Figures (20)

  • Figure 1: Transmission architecture of the fractional Ebola model.
  • Figure 2: Ebola Virus Disease outbreak analysis during the 2014-2016 West Africa outbreak showing: (a) cumulative confirmed cases, (b) cumulative deaths, (c) active cases, and (d) comparative analysis of all outbreak metrics.
  • Figure 3: PRCC ranking identifies transmission parameters ($\beta$, $\sigma$, $\eta_d$) as dominant.
  • Figure 4: Contour plots reveal nonlinear parameter interactions and threshold behavior at $\mathcal{R}_0=1$.
  • Figure 5: Optimal Control Analysis : (a) Algorithm Convergence, (b) Control Path, (c) Cost Control Effectiveness, (d) Control of Epidemics and Containment
  • ...and 15 more figures

Theorems & Definitions (16)

  • Definition 2.1: Caputo Fractional Derivative
  • Definition 2.2: Fractional Lyapunov Stability
  • Theorem 2.1: Stability Condition for Fractional Systems
  • Theorem 4.1: Existence and Uniqueness
  • Theorem 4.2: Invariance and Attractivity
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5: Existence of Endemic Equilibrium
  • Theorem 4.6
  • Theorem 4.7
  • ...and 6 more