Two patterns of recruitment in an epidemic model with difference in immunity of individuals

In this paper, two SIR epidemic models with different patterns of recruitment and difference in immunity are investigated. When the recruitment rate is less than some threshold value, the disease will be eradicated. Furthermore, for the continuous recruitment model, according to the Poincare–Bendixson theorem, the global asymptotical stability of a unique positive equilibrium is obtained. For the pulse recruitment model, we investigated the existence of nontrivial periodic solutions via a supercritical (subcritical) bifurcation. From a biological point of view, our results indicate that (1) the disease can be eradicated if the recruitment rate is controlled under some threshold; (2) the number of the infected increases as the difference in immunity increases; (3) fewer individuals are infected as the pulse recruitment is taken, displaying its effect on the control of the disease.