Narrow Results

In this paper, an SIQS epidemic model with constant recruitment and standard incidence is investigated. Quarantine is taken into consideration on the basis of SIS model. The asymptotic stability of the equilibrium to a reaction–diffusion system with homogeneous Neumann boundary conditions is considered. Sufficient conditions for the local and global asymptotic stability are given by linearization and the method of upper and lower solutions and its associated monotone iterations. The result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.

The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.