Effects of contact tracing and non-Markovian awareness process on the spread of air-borne diseases

Contact tracing is an important tool to contain the spread of many airborne diseases. We develop an approximated pairwise model to investigate the impact of non-Markovian awareness process among infectious individuals represented by pairwise endemic models. We derive the basic reproduction number and the final epidemic size, which are dependent on the tracing rate and the distribution of awareness process. The model analysis provides the threshold of contact tracing rate. When the contact tracing rate is greater than the threshold, the basic reproduction number will be less than one, then the epidemic will eventually die out. The analysis further shows that, higher variance in the awareness process generates smaller basic reproduction number, lower tracing threshold, and larger final epidemic size, when the mean awareness period is fixed. Extensive numerical analysis show the comprehensive effects of tracing rates and non-Markovian awareness processes on human behavior and the transmissibility ability of epidemic. It turns out that large tracing rates and high variances in awareness process lead to obvious reductions in contacts between susceptible and infectious individuals, so as to curb the transmission of infectious diseases. Moreover, contact tracing is more effective in reducing the number of infected individuals and the contacts between susceptible and infected individuals when the awareness process has a larger variance.