An analytical insight into the impact of disease-induced mortality on spatiotemporal patterns on directed networks

Directed population movement and disease-induced mortality critically influence the spread of infectious diseases. Network reaction diffusion equations offer a powerful tool for characterizing spatiotemporal pattern dynamics. Nevertheless, systematic analytical studies on epidemic patterns in directed networks remain notably scarce. To address this gap, this research aims to develop a theoretical framework for analytically characterizing spatiotemporal patterns in reaction–diffusion epidemic models defined on directed networks, with particular emphasis on disease-induced mortality as a key epidemiological parameter. The primary methodological innovation consists in deriving explicit analytical expressions for the spatiotemporal period and amplitude of emerging patterns through weakly nonlinear analysis conducted near the Turing instability threshold. This analytical approach enables quantitative prediction of pattern characteristics directly from model parameters, overcoming the limitations of purely numerical investigations. Numerical simulations validate the accuracy of these solutions and demonstrate their ability to capture key dynamical features in parameter regimes where mortality significantly influences pattern formation. The results provide a theoretical basis for interpreting periodic transmission patterns of pathogens such as the highly pathogenic avian influenza H5N1 virus in spatially structured populations with directional movement.