<p>Rabies remains a major public health concern, particularly in regions where dog-mediated transmission sustains human infection risk. Classical rabies models often rely on integer-order dynamics and overlook the long incubation periods and delayed behavioral responses that characterize the disease. Moreover, neural-network-based surrogate frameworks have rarely been developed for multi-host rabies systems incorporating biologically realistic incidence mechanisms and fractional memory effects.In this study, we propose an eight-compartment human–dog rabies transmission model governed by the Atangana–Baleanu–Caputo (ABC) fractional derivative, which captures nonlocal memory through a non-singular Mittag–Leffler kernel. Transmission between hosts is modeled using a harmonic-mean incidence rate to reflect saturation effects in dog–human and dog–dog contacts. To efficiently approximate the resulting fractional dynamics, we develop a Levenberg–Marquardt-based deep neural network (LMB–DNN) surrogate trained on reference solutions generated by a fractional Adams–Bashforth–Moulton (ABM) predictor–corrector scheme. The neural surrogate accurately reproduces the fractional model dynamics across all eight state variables, achieving mean squared errors in the range <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(10^{-8}\)</EquationSource> </InlineEquation>–<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(10^{-6}\)</EquationSource> </InlineEquation>, absolute errors below <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(10^{-5}\)</EquationSource> </InlineEquation>, and regression coefficients close to unity. A systematic investigation of the fractional order <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha \in (0.85,0.95]\)</EquationSource> </InlineEquation> reveals that memory effects significantly influence outbreak timing, peak magnitude, and persistence, with moderate memory (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha \approx 0.90\)</EquationSource> </InlineEquation>–0.95) producing smoother trajectories consistent with rabies incubation delays. Results from the sensitivity analysis indicate that parameters associated with dog-to-dog transmission and the incubation process exert the strongest influence on the basic reproduction number, suggesting that effective rabies control should primarily focus on dogs. The proposed ABC fractional LMB–DNN framework provides a computationally efficient and biologically meaningful tool for analyzing rabies dynamics with memory effects. The approach offers a practical alternative to repeated fractional simulations, supports rapid scenario exploration, and establishes a foundation for future integration of epidemiological data, uncertainty quantification, and adaptive control strategies in rabies modeling.</p>

Enhancing rabies epidemic modeling with neural networks and fractional calculus

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2026年2月24日

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Ramsha Shafqat,
Imran,
Ashraf Al-Quran,
Abdelhamid Mohammed Djaouti

摘要

Rabies remains a major public health concern, particularly in regions where dog-mediated transmission sustains human infection risk. Classical rabies models often rely on integer-order dynamics and overlook the long incubation periods and delayed behavioral responses that characterize the disease. Moreover, neural-network-based surrogate frameworks have rarely been developed for multi-host rabies systems incorporating biologically realistic incidence mechanisms and fractional memory effects.In this study, we propose an eight-compartment human–dog rabies transmission model governed by the Atangana–Baleanu–Caputo (ABC) fractional derivative, which captures nonlocal memory through a non-singular Mittag–Leffler kernel. Transmission between hosts is modeled using a harmonic-mean incidence rate to reflect saturation effects in dog–human and dog–dog contacts. To efficiently approximate the resulting fractional dynamics, we develop a Levenberg–Marquardt-based deep neural network (LMB–DNN) surrogate trained on reference solutions generated by a fractional Adams–Bashforth–Moulton (ABM) predictor–corrector scheme. The neural surrogate accurately reproduces the fractional model dynamics across all eight state variables, achieving mean squared errors in the range \(10^{-8}\) – \(10^{-6}\) , absolute errors below \(10^{-5}\) , and regression coefficients close to unity. A systematic investigation of the fractional order \(\alpha \in (0.85,0.95]\) reveals that memory effects significantly influence outbreak timing, peak magnitude, and persistence, with moderate memory ( \(\alpha \approx 0.90\) –0.95) producing smoother trajectories consistent with rabies incubation delays. Results from the sensitivity analysis indicate that parameters associated with dog-to-dog transmission and the incubation process exert the strongest influence on the basic reproduction number, suggesting that effective rabies control should primarily focus on dogs. The proposed ABC fractional LMB–DNN framework provides a computationally efficient and biologically meaningful tool for analyzing rabies dynamics with memory effects. The approach offers a practical alternative to repeated fractional simulations, supports rapid scenario exploration, and establishes a foundation for future integration of epidemiological data, uncertainty quantification, and adaptive control strategies in rabies modeling.