Dynamics of a Symmetric Seasonal Influenza Model with Variable Recovery, Treatment, and Fear Effects

This study proposes and examines the dynamics of a susceptible–exposed–infectious–recovered (SEIR) model for the spread of seasonal influenza. The population is categorized into four distinct groups: susceptible (S), exposed (E), infectious (I), and recovered (R) individuals. The symmetric model integrates a bilinear incidence rate alongside a nonlinear recovery rate that depends on the quality of healthcare services. Additionally, it accounts for the impact of fear related to the disease and includes a constant vaccination rate as well as a nonlinear treatment function. The model advances current epidemiological frameworks by simultaneously accounting for these interrelated mechanisms, which are typically studied in isolation. We derive the expression for the basic reproduction number and analyze the essential stability properties of the model. Key analytical results demonstrate that the system exhibits rich dynamic behavior, including backward bifurcation (where stable endemic equilibria persist even when the basic reproduction number is less than one) and Hopf bifurcation. These phenomena emerge from the interplay between fear-induced suppression of transmission, treatment saturation, and healthcare quality. Numerical simulations using Saudi Arabian demographic and epidemiological data quantify how increased fear perception shrinks the bistability region, facilitating eradication. Healthcare capacity improvements, on the other hand, reduce the critical reproduction number threshold while treatment accessibility suppresses infection loads. The model’s practical significance lies in its ability to identify intervention points where small parameter changes yield disproportionate control benefits and evaluate trade-offs between pharmaceutical (vaccination/treatment) and non-pharmaceutical (fear-driven distancing) strategies. This work establishes a versatile framework for public health decision making and the integrated approach offers policymakers a tool to simulate combined intervention scenarios and anticipate nonlinear system responses that simpler models cannot capture.