Global analysis of a network-based SIR epidemic model with a saturated treatment function

Abstract

In this paper, we study a network-based SIR epidemic model with a saturated treatment function in which a parameter [Formula: see text] is introduced to measure the extent of the effect of the infected being delayed for treatment. Our aim is to present a global analysis and to investigate how the parameter [Formula: see text] affects the spreading of diseases. Our main results are as follows: (1) In the case of the threshold value [Formula: see text], there exist two values of [Formula: see text]: [Formula: see text] and [Formula: see text], such that the disease-free equilibrium is globally asymptotically stable when [Formula: see text] and multiple endemic equilibria exist when [Formula: see text]. This means that the parameter [Formula: see text] has an essential influence on the spreading of the disease. (2) In the case of the threshold value [Formula: see text], if the model has only one endemic equilibrium, then the unique endemic equilibrium is globally attractive. In this case, it is also proved that if [Formula: see text], then the endemic equilibrium has only one, so is globally attractive. In addition, numerical simulation is performed to illustrate our theoretical results.