Asymptotic profiles of a diffusive SEIR epidemic model with underlying disease and mass action infection mechanism

We study a susceptible–exposed–infected–recovered (SEIR) reaction–diffusion epidemic model that includes susceptible individuals with underlying diseases, focusing on how these comorbidities, diffusion coefficients, and spatial heterogeneity affect disease's spread. The basic reproduction number $R_0$ is central to understanding and controlling infectious diseases' spread. We define $R_0$, analyze its behavior under low diffusion rates, and investigate the persistence of infection in relation to $R_0$. Our results show that underlying health conditions increase the value of $R_0$, enhancing the disease's transmission potential and persistence. In a homogeneous environment, if $R_0>1$, the system admits a constant endemic equilibrium that is globally asymptotically stable; if $R_0<1$, the disease-free equilibrium is globally attractive, implying eventual disease eradication. Furthermore, we analyze the asymptotic behavior of the endemic equilibrium as the diffusion rates approach zero. Our results indicate that limiting the mobility of susceptible, exposed, and infectious individuals alone is insufficient to eliminate the disease. By examining the influence of diffusion coefficients on the spatial dynamics and disease persistence, we conclude that effective control strategies must extend beyond diffusion control and incorporate interventions targeting additional transmission factors.