The effectiveness of contact tracing in mitigating COVID-19 outbreak: A model-based analysis in the context of India

The ongoing pandemic situation due to COVID-19 originated from the Wuhan city, China affects the world in an unprecedented scale. Unavailability of totally effective vaccination and proper treatment regimen forces to employ a non-pharmaceutical way of disease mitigation. The world is in desperate demand of useful control intervention to combat the deadly virus. This manuscript introduces a new mathematical model that addresses two different diagnosis efforts and isolation of confirmed cases. The basic reproductive number, $R_0,$ is inspected, and the model's dynamical characteristics are also studied. We found that with the condition $R_0<1,$ the disease can be eliminated from the system. Further, we fit our proposed model system with cumulative confirmed cases of six Indian states, namely, Maharashtra, Tamil Nadu, Andhra Pradesh, Karnataka, Delhi and West Bengal. Sensitivity analysis carried out to scale the impact of different parameters in determining the size of the epidemic threshold of $R_0$ . It reveals that unidentified symptomatic cases result in an underestimation of $R_0$ whereas, diagnosis based on new contact made by confirmed cases can gradually reduce the size of $R_0$ and hence helps to mitigate the ongoing disease. An optimal control problem is framed using a control variable $u\left(t\right),$ projecting the effectiveness of diagnosis based on traced contacts made by a confirmed COVID patient. It is noticed that optimal contact tracing effort reduces $R_0$ effectively over time.