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Epidemic Transmission Modeling with Fractional Derivatives and Environmental Pathogens

Moein Khalighi, Faïçal Ndaïrou, Leo Lahti

TL;DR

This work develops a fractional-order epidemiological model for COVID-19 that explicitly includes environmental pathogen shedding. Using Caputo derivatives with compartmental states S, E, I_A, I_S, R, D, and W, the authors derive existence, uniqueness, boundedness, and both local and global stability results via a next-generation approach and Lyapunov methods. Numerical experiments fit the model to South Africa data, showing that fractional orders improve data alignment and highlighting the significance of environmental transmission in driving $R_0$ and informing interventions. The study demonstrates that environmental sanitation and control of shedding channels can meaningfully reduce transmission, with implications for public health policy and future research in memory-affected epidemic modeling.

Abstract

This research presents an advanced fractional-order compartmental model designed to delve into the complexities of COVID-19 transmission dynamics, specifically accounting for the influence of environmental pathogens on disease spread. By enhancing the classical compartmental framework, our model distinctively incorporates the effects of order derivatives and environmental shedding mechanisms on the basic reproduction numbers, thus offering a holistic perspective on transmission dynamics. Leveraging fractional calculus, the model adeptly captures the memory effect associated with disease spread, providing an authentic depiction of the virus's real-world propagation patterns. A thorough mathematical analysis confirming the existence, uniqueness, and stability of the model's solutions emphasizes its robustness. Furthermore, the numerical simulations, meticulously calibrated with real COVID-19 case data, affirm the model's capacity to emulate observed transmission trends, demonstrating the pivotal role of environmental transmission vectors in shaping public health strategies. The study highlights the critical role of environmental sanitation and targeted interventions in controlling the pandemic's spread, suggesting new insights for research and policy-making in infectious disease management.

Epidemic Transmission Modeling with Fractional Derivatives and Environmental Pathogens

TL;DR

This work develops a fractional-order epidemiological model for COVID-19 that explicitly includes environmental pathogen shedding. Using Caputo derivatives with compartmental states S, E, I_A, I_S, R, D, and W, the authors derive existence, uniqueness, boundedness, and both local and global stability results via a next-generation approach and Lyapunov methods. Numerical experiments fit the model to South Africa data, showing that fractional orders improve data alignment and highlighting the significance of environmental transmission in driving and informing interventions. The study demonstrates that environmental sanitation and control of shedding channels can meaningfully reduce transmission, with implications for public health policy and future research in memory-affected epidemic modeling.

Abstract

This research presents an advanced fractional-order compartmental model designed to delve into the complexities of COVID-19 transmission dynamics, specifically accounting for the influence of environmental pathogens on disease spread. By enhancing the classical compartmental framework, our model distinctively incorporates the effects of order derivatives and environmental shedding mechanisms on the basic reproduction numbers, thus offering a holistic perspective on transmission dynamics. Leveraging fractional calculus, the model adeptly captures the memory effect associated with disease spread, providing an authentic depiction of the virus's real-world propagation patterns. A thorough mathematical analysis confirming the existence, uniqueness, and stability of the model's solutions emphasizes its robustness. Furthermore, the numerical simulations, meticulously calibrated with real COVID-19 case data, affirm the model's capacity to emulate observed transmission trends, demonstrating the pivotal role of environmental transmission vectors in shaping public health strategies. The study highlights the critical role of environmental sanitation and targeted interventions in controlling the pandemic's spread, suggesting new insights for research and policy-making in infectious disease management.
Paper Structure (14 sections, 9 theorems, 62 equations, 4 figures, 2 tables)

This paper contains 14 sections, 9 theorems, 62 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The fractional-order model
  3. Mathematical analysis of the model
  4. Preliminaries
  5. Existence and Uniqueness
  6. Boundedness of the Solution
  7. Basic reproduction number
  8. Local stability analysis
  9. Global stability analysis
  10. Numerical results
  11. Sensitivity analysis of the basic reproduction number
  12. Numerical methods and implementation
  13. Data and code availability
  14. Conclusion

Key Result

Lemma 1

Let $f(t)$ be a function such that $f(t) \in C([0,T])$ and its Caputo fractional derivative $^{C}D^{\alpha}_{0+}f(t) \in C((0,T])$, with $0 < \alpha \leq 1$. Then, the following relation holds: where $0 \leq \xi \leq t$ for all $t \in (0,T]$. This relation represents the fractional generalization of the Mean Value Theorem, as established in ODIBAT2007286.

Figures (4)

  • Figure 1: Model Accuracy Assessment: It demonstrates the accuracy of the model \ref{['model']}, with integer order derivatives (left panels) and fractional order derivatives (right panels), in fitting the daily new confirmed cases, recovered individuals, and deceased individuals, obtained from CSSE DataCSSE. The circles represent the data points. The grey curves show the model's results with estimated parameter values and initial conditions, and the optimal fit across all three categories, evaluated via root mean square deviation (RMSD), is depicted with black curves.
  • Figure 2: Comparative Error Score Distribution: It illustrates the distribution of root mean square deviation (RMSD) comparing real data with the fitted models using integer (left) and fractional (right) orders.
  • Figure 3: Sensitivity Analysis of $R_0$ for Model \ref{['model']} with Integer Order Derivatives. (a) The boxplots of the sensitivity indices of $R_0$ to the parameter values, and (b) the distribution of $R_0$ for these values. (c) The sensitivity indices of $R_0$ to parameters when the environmental pathogen-related coefficients $\beta_1$, $\eta_S$, and $\eta_A$ are constrained to zero. (d) Illustration of the resulting influence on $R_0$ when the contributions of environmental transmission factors are eliminated.
  • Figure 4: Sensitivity Analysis of $R_0$ for Model \ref{['model']} with Fractional Order Derivatives. (a) The boxplots of the sensitivity indices of $R_0$ to the parameter values and (b) order derivatives, and (c) the distribution of $R_0$ for these values. (d) The sensitivity indices of $R_0$ to parameters and (e) order derivatives when the environmental pathogen-related coefficients $\beta_1$, $\eta_S$, and $\eta_A$ are constrained to zero. (f) Illustration of the resulting influence on $R_0$ when the contributions of environmental transmission factors are eliminated.

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 6 more