Nonlinear dynamical analysis of an SEIRP model with treatment saturation effects and public perception

Abstract Classical SEIR-type epidemic models often rely on idealised assumptions, such as unlimited healthcare capacity and static human behaviour, which can underestimate disease persistence and fail to reproduce recurrent endemic waves observed during outbreaks such as COVID-19. In this study, we propose and analyse a novel SEIRP (Susceptible-Exposed-Infectious-Recovered-Perception) model that explicitly couples treatment saturation effects with dynamic public perception feedback. Healthcare limitations are modelled through a nonlinear saturated treatment function, while behavioural adaptation is incorporated using a perception variable that modulates transmission intensity. We establish the positivity and boundedness of solutions, derive the basic reproduction number $R_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and perform a detailed bifurcation analysis. Our results reveal that treatment saturation can induce backward bifurcation, leading to bistability between disease-free and endemic equilibria even when $R_{0}&lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> . Moreover, we demonstrate that perception-driven feedback can destabilise the endemic equilibrium through Hopf bifurcation, generating sustained oscillatory dynamics that correspond to recurrent epidemic waves. These findings uncover a previously uncharacterised mechanism whereby the interaction between healthcare constraints and adaptive behavioural responses fundamentally alters endemic outcomes. The proposed model provides new theoretical insight into epidemiological phenomena and highlights that effective disease control requires coordinated management of healthcare capacity and public awareness, rather than reliance on transmission reduction alone.