(untitled) (Journal of Nonlinear Modeling and Analysis, 2025)
Contact tracing allows for more efficient quarantine and isolation, and is thus a key control measure in combating infectious diseases. Mathematical models that accurately describe the contact tracing process are important tools for studying the effectiveness of contact tracing. Recently, we developed a novel contact tracing SIR model based on pair dynamics, which uses pairs (two-individual) interactions to approximate triple (three-individual) interactions to close the model. However, the pair approximation used in the model is only a crude estimate. We extend this model to improve the approximation. Specifically, the new model tracks infectious individuals who have or have not infected others, as they play different roles in triples. We conduct a theoretical analysis to calculate the control reproduction number. The results of the new model are compared with those of the original model by numerical analysis. We find that the two models yield a similar epidemic final size. However, the original model yields a larger control reproduction number and thus underestimates the effect of contact tracing. This discrepancy increases as contact tracing is strengthened.