The impact of information and saturated treatment with time delay in an infectious disease model

Abstract

In this paper, we propose a mathematical model with a saturated treatment rate in the presence of information. We consider that the information about the disease affects the transmission rate of infection and hence the transmission rate is corrected. We also assume that people are losing their immunity against disease and the model is of SIRS type. We analyse the stability of the model system and our analysis shows that the model possesses the existence of backward bifurcation and multiple endemic steady states. The saturation in treatment is an important factor causing backward bifurcation. Various situations of multiple endemic steady states are explored numerically. We observe that our model shows existence of bi-stability via backward bifurcation, oscillations and hysteresis. Further, we extend the model to include the time lag in information and we find that in presence of time delay, the endemic steady state destabilizes and oscillations are observed. Thus, we conclude that if information dissemination is delayed beyond a threshold time then the infection oscillates in population and it may lead to difficulty in controlling the disease. Also, nonlinear incidence rate and saturated treatment may cause the existence of multiple endemic steady states and hence leads to complex dynamics.