Occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: A case study on COVID-19

The outbreak of COVID-19 caused by SARS-CoV-2 is spreading rapidly around the world, which is causing a major public health concerns. The outbreaks started in India on March 2, 2020. As of April 30, 2020, 34,864 confirmed cases and 1154 deaths are reported in India and more than 30,90,445 confirmed cases and 2,17,769 deaths are reported worldwide. Mathematical models may help to explore the transmission dynamics, prediction and control of COVID-19 in the absence of an appropriate medication or vaccine. In this study, we consider a mathematical model on COVID-19 transmission with the imperfect lockdown effect. The basic reproduction number, R(0), is calculated using the next generation matrix method. The system has a disease-free equilibrium (DFE) which is locally asymptotically stable whenever R(0) < 1. Moreover, the model exhibits the backward bifurcation phenomenon, where the stable DFE coexists with a stable endemic equilibrium when R(0) < 1. The epidemiological implications of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is only a necessary, but not sufficient for effectively controlling the spread of COVID-19 outbreak. It is observed that the system undergoes backward bifurcation which is a new observation for COVID-19 disease transmission model. We also noticed that under the perfect lockdown scenario, there is no possibility of having backward bifurcation. Using Lyapunov function theory and LaSalle Invariance Principle, the DFE is shown globally asymptotically stable for perfect lockdown model. We have calibrated our proposed model parameters to fit daily data from India, Mexico, South Africa and Argentina. We have provided a short-term prediction for India, Mexico, South Africa and Argentina of future cases of COVID-19. We calculate the basic reproduction number from the estimated parameters. We further assess the impact of lockdown during the outbreak. Furthermore, we find that effective lockdown is very necessary to reduce the burden of diseases.