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<p>Understanding the dynamics of infectious diseases requires sophisticated modeling approaches that can account for various demographic and epidemiological factors. This study explores a biological disease model incorporating four population classes: susceptible adults, susceptible children, infected adults, and infected children. To analyze the system, we employ equilibrium analysis and bifurcation theory, uncovering both co-dimension-one (Fold, Hopf) and co-dimension-two (Bautin, Fold-Hopf, Bogdanov–Takens, and Cusp) bifurcations, including several degenerate cases. Dynamical tools such as Lyapunov exponents, bifurcation diagrams, Poincaré sections, and phase space reconstruction characterize chaotic and hyper-chaotic behavior. We identify hidden attractors, suggesting multi-stability and complex transitions that may evade traditional stability analysis. Long short-term memory (LSTM) neural networks are implemented to forecast the temporal evolution of the system and demonstrate high predictive accuracy with low error rates. These findings illustrate the power of combining mathematical modeling with machine learning to enhance our understanding of disease spread. The integration of bifurcation theory and LSTM-based forecasting provides a comprehensive framework for anticipating dynamic shifts in epidemiological systems and supports the development of data-driven public health interventions.</p>

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Dynamical analysis and bifurcation exploration in a biological model with neural network forecasting

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Muhammad Waseem Akhtar,
  • Zia Bashir,
  • M. G. Abbas Malik

摘要

Understanding the dynamics of infectious diseases requires sophisticated modeling approaches that can account for various demographic and epidemiological factors. This study explores a biological disease model incorporating four population classes: susceptible adults, susceptible children, infected adults, and infected children. To analyze the system, we employ equilibrium analysis and bifurcation theory, uncovering both co-dimension-one (Fold, Hopf) and co-dimension-two (Bautin, Fold-Hopf, Bogdanov–Takens, and Cusp) bifurcations, including several degenerate cases. Dynamical tools such as Lyapunov exponents, bifurcation diagrams, Poincaré sections, and phase space reconstruction characterize chaotic and hyper-chaotic behavior. We identify hidden attractors, suggesting multi-stability and complex transitions that may evade traditional stability analysis. Long short-term memory (LSTM) neural networks are implemented to forecast the temporal evolution of the system and demonstrate high predictive accuracy with low error rates. These findings illustrate the power of combining mathematical modeling with machine learning to enhance our understanding of disease spread. The integration of bifurcation theory and LSTM-based forecasting provides a comprehensive framework for anticipating dynamic shifts in epidemiological systems and supports the development of data-driven public health interventions.