$\mu$-Stability of Positive Homogeneous Differential-Difference Equations With Unbounded Time-Varying Delays

This note studies the stability problem of nonlinear positive differential-difference equations with unbounded time-varying delays. Under assumptions on the nonlinear vector fields, such as being homogeneous, cooperative, and order-preserving, conditions are derived for positivity and asymptotic stability of nonlinear differential-difference equations, which include the corresponding linear system as a special case. This work features three main contributions: first, a necessary and sufficient positivity condition is proposed for nonlinear differential-difference equations with delays. Then, utilizing the concept of homogeneity, a necessary and sufficient condition is provided for the global asymptotic stability of such positive nonlinear systems with unbounded delays. Finally, we analyze the global $\mu$-stability of the systems with unbounded time-varying delays to estimate the convergence rates of the system dynamics. The effectiveness of the obtained theoretical results is illustrated by numerical examples, including an analysis of nonlinear epidemic-spreading processes.