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Harmful information spreading and its impact on vaccination campaigns modeled through fractal-fractional operators

Ali Akgül, Auwalu Hamisu Usman, J. Alberto Conejero

TL;DR

The paper models the spread of harmful information and its impact on vaccination campaigns using fractal-fractional operators to capture memory and fractal effects. It presents a seven-compartment framework with $S_p, I, I_p, I_n, I_c, R, D$, analyzes existence, positivity, equilibria, and the basic reproduction number $\mathcal{R}_0$, and introduces a nonlinear strength number $\mathcal{SN}$ to assess amplification. A Lyapunov-based approach yields global stability results for the endemic state when $\mathcal{R}_0>1$, and a second-derivative analysis provides wave-detection criteria. The numerical scheme employs power-law, exponential, and Mittag-Leffler kernels to simulate fractal-fractional dynamics, demonstrating memory and kernel effects on wave-like patterns in misinformation spread and vaccination uptake.

Abstract

Despite the huge efforts to develop and administer vaccines worldwide to cope with the COVID-19 pandemic, misinformation spreading through fake news in media and social networks about vaccination safety, make that people refuse to be vaccinated, which harms not only these people but also the whole population. In this work, we model the effects of harmful information spreading in immunization acquisition through vaccination. Our model is posed for several fractional derivative operators. We have conducted a comprehensive foundation analysis of this model for the different fractional derivatives. Additionally, we have incorporated a strength parameter that shows the combined impact of nonlinear and linear components within an epidemiological model. We have used the second derivative of the Lyapunov function to ascertain the detection of wave patterns within the vaccination dynamics.

Harmful information spreading and its impact on vaccination campaigns modeled through fractal-fractional operators

TL;DR

The paper models the spread of harmful information and its impact on vaccination campaigns using fractal-fractional operators to capture memory and fractal effects. It presents a seven-compartment framework with , analyzes existence, positivity, equilibria, and the basic reproduction number , and introduces a nonlinear strength number to assess amplification. A Lyapunov-based approach yields global stability results for the endemic state when , and a second-derivative analysis provides wave-detection criteria. The numerical scheme employs power-law, exponential, and Mittag-Leffler kernels to simulate fractal-fractional dynamics, demonstrating memory and kernel effects on wave-like patterns in misinformation spread and vaccination uptake.

Abstract

Despite the huge efforts to develop and administer vaccines worldwide to cope with the COVID-19 pandemic, misinformation spreading through fake news in media and social networks about vaccination safety, make that people refuse to be vaccinated, which harms not only these people but also the whole population. In this work, we model the effects of harmful information spreading in immunization acquisition through vaccination. Our model is posed for several fractional derivative operators. We have conducted a comprehensive foundation analysis of this model for the different fractional derivatives. Additionally, we have incorporated a strength parameter that shows the combined impact of nonlinear and linear components within an epidemiological model. We have used the second derivative of the Lyapunov function to ascertain the detection of wave patterns within the vaccination dynamics.

Paper Structure

This paper contains 22 sections, 3 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation of harmful information impact on a vaccination campaign
  3. Existence and Uniqueness
  4. Proof:
  5. Positiveness and boundedness of the solution
  6. The Caputo fractional derivative case
  7. The Caputo-Fabrizio fractional derivative case
  8. The Atangana-Baleanu fractional derivative case
  9. The fractal fractional derivative case
  10. Analysis of equilibrium points
  11. Reproduction and strength numbers
  12. Stability analysis
  13. Proof:
  14. Proof:
  15. Second derivative of Lyapunov
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let us consider the model where the functions $G_i$ represent the left side of equations eqn1-eqn7. There are positive constants $\rho_i$ and $\bar{\rho_i}$ such that for all $(x_i,t)\in\mathbb{R}^7\times[0,T]$ and for all $1\le i\le 7$, we have

Figures (4)

  • Figure 1: Harmful information spreading model.
  • Figure 2: Evolution of the population classes modeled with FFP: susceptible $S_p$, impacted by media $I$, positive opinion $I_p$, negative opinion $I_n$, confused $I_c$, overcome the misinformation $R$, death or denials $D$.
  • Figure 3: Evolution of the population classes modeled with FFE: susceptible $S_p$, impacted by media $I$, positive opinion $I_p$, negative opinion $I_n$, confused $I_c$, overcome the misinformation $R$, death or denials $D$.
  • Figure 4: Evolution of the population classes modeled with FFM: susceptible $S_p$, impacted by media $I$, positive opinion $I_p$, negative opinion $I_n$, confused $I_c$, overcome the misinformation $R$, death or denials $D$.

Theorems & Definitions (7)

  • Definition 1.1: Caputo Fractional Derivative
  • Definition 1.2: Caputo–Fabrizio Fractional Derivative
  • Definition 1.3: Atangana–Baleanu Fractional Derivative in Caputo Sense
  • Definition 1.4: Fractal Fractional Derivatives
  • Theorem 3.1
  • Theorem 7.1
  • Theorem 7.2