Effect of a novel generalized incidence rate function in SIR model: Stability switches and bifurcations

When a disease outbreaks in a population, the incidence rate function may increase initially with an increase of infective and then decrease due to inhibitory effects before getting saturated. For this purpose, we propose a new generalized incidence rate function and demonstrate the impact of this incidence rate function’s non-monotonic feature on the stability of endemic equilibria. In this paper, an SIR compartmental model with this new nonlinear incidence rate function and saturated treatment rate function is proposed. We obtain a disease free equilibrium which is locally stable for R0<1 and unstable otherwise. The system may have multiple endemic equilibria which can be either anti-saddle (node, focus or center) or saddle. We discover that when the incidence rate function is increasing at an anti-saddle endemic equilibrium, the endemic equilibrium may become either stable, unstable or switch stability via a parameter change. However, when the incidence rate function is non-monotonic and if the incidence rate decreases at an anti-saddle endemic equilibrium, the endemic equilibrium is always stable. Due to the limited treatment, we find forward (transcritical) and backward bifurcations. We also establish Hopf bifurcation and saddle–node bifurcation in the model system. Further, we incorporate the incubation delay in the model system. We obtain the same number of equilibria in the delay model. The disease free equilibrium is locally asymptotically stable for all delay lengths for R0<1. The endemic equilibrium switches its stability due to change in delay. Due to delay variation, the increasing incidence rate at an anti-saddle endemic equilibrium causes the multiple stability switches of the equilibrium from both stable and unstable initial states. Further, when the incidence rate decreases at the stable anti-saddle endemic equilibrium, it switches its stability once or multiple times and finally becomes unstable due to delay variation. We explicitly, show multiple stability switches of an endemic equilibrium due to existence of multiple Hopf bifurcations. When we further analyzed model system numerically, we could establish the existence of Hopf–Hopf bifurcation of codimension 2. In this case, the increase in the delay parameter causes the existence of two oscillatory solutions around the unstable endemic equilibrium. Hence, our results show that the delay system either destabilize developing the same frequency oscillations via Hopf bifurcation or two different frequency oscillations via Hopf–Hopf bifurcation of codimension 2. The existence and exhibition of these two frequency oscillations are novel and have not been explored much in epidemiological models.