Brucellosis threshold dynamics: a time-fractional-order diffusion model with generalized incidence rate functions

Abstract This study proposes a new Caputo fractional-order brucellosis model that incorporates diffusion terms and generalized incidence rate functions, enabling more flexible simulation of various disease transmission patterns. Compared to traditional integer-order models, the fractional-order model has significant advantages in capturing the memory effect and long-term dynamics of disease spread. We rigorously prove that when $R_{0} &gt; 1 $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> , the endemic equilibrium is globally asymptotically stable, while when $R_{0} \leq 1 $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:math> , the disease-free equilibrium is globally stable. Numerical simulations validate the impact of diffusion and memory effects on the dynamic behavior of the model. Our model not only provides a more accurate description of transmission mechanisms but also offers more reliable theoretical support for the development of public health intervention strategies. This approach is highly flexible and can be applied to a broader range of epidemiological modeling studies.