SIR model with time-varying contact rate

The contact rate is defined as the average number of contacts adequate for disease transmission by an individual per unit time and it is usually assumed to be constant in time. However, in reality, the contact rate is not always constant throughout the year due to different factors such as population behavior, environmental factors and many others. In the case of serious diseases with a high level of infection, the population tends to reduce their contacts in the hope of reducing the risk of infection. Therefore, it is more realistic to consider it to be a function of time. In particular, the study of models with contact rates decreasing in time is well worth exploring. In this paper, an SIR model with a time-varying contact rate is considered. A new form of a contact rate that decreases in time from its initial value till it reaches a certain level and then remains constant is proposed. The proposed form includes two important parameters, which represent how far and how fast the contact rate is reduced. These two parameters are found to play important roles in disease dynamics. The existence and local stability of the equilibria of the model are analyzed. Results on the global stability of disease-free equilibrium and transcritical bifurcation are proved. Numerical simulations are presented to illustrate the theoretical results and to demonstrate the effect of the model parameters related to the behavior of the contact rate on the model dynamics. Finally, comparisons between the constant, variable contact rate and variable contact rate with delay in response cases are presented.