Impact of awareness and time-delayed saturated treatment on the transmission of infectious diseases

We have proposed a mathematical model with saturated incidence and treatment rates along with awareness and delay in treatment. We analyze the model and find the equilibrium points and their stability. We also find the basic reproduction number $R_0$ to understand the disease dynamics. We performed the sensitivity analysis and found that treatment along with awareness plays a significant role in controlling infectious disease. We deduce that awareness about the disease affects the transmission rate of infection. As people become aware of aspects of the infectious disease, they amend their behavior so that they fend themselves from catching the disease. We have introduced a time lag in treatment and found the threshold value of the time delay. It is observed that when the value of the time delay crosses the threshold value, we get a Hopf bifurcation i.e. endemic steady state becomes unstable above the threshold value and it may become difficult to control the disease beyond the threshold value of delay in treatment.