Gamma–distributed transmission memory in a reaction–diffusion epidemic model with nonlinear incidence

We extend the delayed reaction–diffusion SI model with nonlinear incidence proposed by Song et al. by replacing the fixed transmission lag with weak and strong gamma-distributed memory kernels. Because the model has no exposed compartment, the memory term is interpreted phenomenologically as a distributed timing of transmission-related effects of past infectious states rather than as an explicit latent-stage mechanism. Using the same parameter set adopted by Song et al., we show that the distributed-memory models have a stability structure that is qualitatively different from that of the discrete-delay model. For the spatially homogeneous mode, both gamma kernels exhibit a small stabilising spectral Hopf crossing at a mean memory time near 0.08. The discrete-delay model has a comparable small stabilising crossing but, in addition, a much larger destabilising crossing near τ = 1.335 . Consequently, at the nominal values 1.2 and 1.6 used in the reference simulations, the gamma models converge to the selected endemic equilibrium, whereas the discrete-delay model crosses into an oscillatory regime. The first spatial mode is different: for the gamma formulations its modal constant term is negative for every positive memory rate, so this mode is unstable for every positive mean-memory value. We therefore interpret this behavior as a memory-independent spatial-mode instability, distinct from both a memory-induced Hopf crossing and a classical Turing instability.