Fractional modelling of COVID-19 transmission incorporating asymptomatic and super-spreader individuals

TL;DR

The paper develops an eight-compartment epidemiological framework for COVID-19 that jointly models asymptomatic and super-spreader transmission using incommensurate Caputo fractional derivatives to capture memory effects. Through qualitative analysis, it derives a basic reproduction number $\mathcal{R}_0$ via the next-generation matrix and proves global stability of the disease-free equilibrium when $\mathcal{R}_0<1$, along with generalized Ulam–Hyers stability. Numerical results calibrated to Portugal data show that fractional-order formulations (FM1–FM3) outperform integer-order variants, with the full fractional model FM3 providing the best fit and offering detailed sensitivity insights for policy design. The work demonstrates that incorporating supplementary transmission pathways and fractional calculus improves model fidelity and interpretability, with publicly available code and data for reproducibility. Overall, the approach advances fractional epidemiology by integrating asymptomatic and super-spreader dynamics and by quantifying memory effects in disease spread.

Abstract

The COVID-19 pandemic has presented unprecedented challenges worldwide, necessitating effective modelling approaches to understand and control its transmission dynamics. In this study, we propose a novel approach that integrates asymptomatic and super-spreader individuals in a single compartmental model. We highlight the advantages of utilizing incommensurate fractional order derivatives in ordinary differential equations, including increased flexibility in capturing disease dynamics and refined memory effects in the transmission process. We conduct a qualitative analysis of our proposed model, which involves determining the basic reproduction number and analysing the disease-free equilibrium's stability. By fitting the proposed model with real data from Portugal and comparing it with existing models, we demonstrate that the incorporation of supplementary population classes and fractional derivatives significantly improves the model's goodness of fit. Sensitivity analysis further provides valuable insights for designing effective strategies to mitigate the spread of the virus.

Figures (2)

• Figure 1: Illustration of the accuracy of the studied models in fitting (a) daily new confirmed cases and (b) cumulative death cases, as retrieved from CSSE DataCSSE. The estimation of variables $I+P+H$ and $F$ is shown. Data points are represented by black circles. Errors are evaluated using the root mean square deviation (RMSD). Model M1, which lacks super-spreaders and uses integer orders, is shown to have an inferior performance with an error of 123.30. The fractional version of this model, FM1 (solid purple line), improves accuracy significantly, reducing the error to 107.45. Model M2, which lacks asymptomatic individuals (dashed orange line), performs better than M1 with an error of 116.41. However, its fractional version, FM2 (solid red line), shows a smaller improvement compared to FM1, with an error of 109.81. Our proposed model \ref{['model']} with integer orders, M3 (dashed green line), which includes both super-spreader and asymptomatic compartments, performs better than both M1 and M2, with an error of 108.21. The fractional version of this model, FM3 (solid green line), shows the best performance, reducing the error to 99.40.
• Figure 2: Comparison of individual dynamics from simulations of six studied models: Model M1 excludes super-spreaders (with parameters $\beta'$, $\rho_2$, $\delta_p = 0$), and uses integer-order derivatives. Model M2 excludes asymptomatic individuals (with parameters $\beta"$, $\delta_a = 0$, $\rho_2 = 1-\rho_1$), and also uses integer-order derivatives. Model M3 is the proposed model \ref{['model']} that includes both super-spreader and asymptomatic compartments, using integer-order derivatives. The fractional-order models are as follows: FM1 is the fractional-order version of M1, FM2 is the fractional-order version of M2, and FM3 is the fractional-order version of M3. The values for parameters and orders of the derivatives used in simulations are provided in Tables \ref{['tab:parCovid']} and \ref{['tab:FittedPara']}.

Theorems & Definitions (17)

• Definition 2.1
• Lemma 2.2: Gronwall inequality gronwall
• Theorem 2.3: Existence and uniqueness of solutions
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Remark 1
• Theorem 4.3