Global stability and uniform persistence in an epidemic model with saturating fomite-mediated transmission
We analyse the global dynamics of a Susceptible--Vaccinated--Exposed--Infected--Recovered (SVEIR) epidemic model with demographic turnover, imperfect vaccination, and two transmission routes: direct host-to-host contagion and indirect transmission via contaminated fomites. Indirect transmission is described through an environmental pathogen concentration and a Holling-type dose--response function, accounting for nonlinear incidence at high contamination levels. Threshold conditions separating disease elimination from long-term persistence are expressed in terms of the control reproduction number $\mathcal R_c$, and the classical threshold condition $\mathcal R_c<1$ is derived for the local asymptotic stability of the disease-free equilibrium. For the Holling type~II case, we further obtain an explicit closed-form sufficient condition for the global asymptotic stability of the disease-free equilibrium by applying the Kamgang--Sallet approach for monotone systems with a Metzler infected subsystem. In the absence of vaccination, this criterion recovers the sharp threshold $\mathcal R_0\le 1$ for the global asymptotic stability of the disease-free equilibrium, where $\mathcal R_0$ denotes the basic reproduction number. Conversely, when $\mathcal R_c>1$, we establish uniform persistence of the infection and the existence of at least one endemic equilibrium using persistence theory for semiflows and an acyclicity analysis of the boundary dynamics. Overall, our results quantify the combined impact of vaccination and saturating fomite-mediated transmission on the global behaviour of the model.