Dynamical analysis of a spatially heterogeneous disease model with degenerate structure

This study analyzes a reaction-diffusion model for cocoa black pod disease (CBPD) with space-dependent parameters and partial degeneracy, which assumes that susceptible and infected cocoa pods remain immobile within the region. Spores produced by infected pods, which drive both direct and indirect infections, are assumed to disperse at distinct rates depending on their mode of transmission. Firstly, we establish the model's well-posedness, including the boundedness and global existence of solutions. Furthermore, to address the non-compactness of solution semiflow, we demonstrate the asymptotic smoothness of the semiflow generated by the model. Subsequently, the basic reproduction number (BRN) is defined via the next generation operator approach, following which we derive and discuss its some properties. The BRN, R₀, is shown as a threshold parameter for disease eradication versus uniform persistence: if R₀≤1, the disease will become extinct; if R₀>1, the infection will persist. Moreover, when R₀>1, we further show that under some conditions, by constructing delicate Lyapunov functions, the model admits a unique globally attractive positive steady state.