Dynamical analysis of deterministic and stochastic epidemic models with contact tracing

Contact tracing is an important non-pharmaceutical intervention for interrupting disease transmission chains. In this study, we formulate a deterministic epidemic model and its corresponding continuous-time Markov chain model to investigate the transmission and extinction dynamics of an emerging infectious disease with symptomatic and asymptomatic infections. For the deterministic model, the control reproduction number \(R_{c}\) is derived, and its biological interpretation is provided. We prove that the disease-free equilibrium is globally asymptotically stable when \(R_{c}<1\) , while the disease is uniformly persistent and a unique endemic equilibrium exists when \(R_{c}>1\) . In the early stage of disease transmission, the nonlinear Markov chain is approximated by a multitype branching process. This approximation allows us to derive the ultimate extinction probability and the mean of the extinction time. When \(\rho (\mathbf {M})<1\) , disease extinction occurs with probability one, whereas when \(\rho (\mathbf {M})>1\) , the disease still has a positive probability of extinction. Moreover, the deterministic threshold \(R_{c}\) and the stochastic threshold \(\rho (\mathbf {M})\) are shown to yield equivalent threshold conditions. Numerical simulations show that increasing the contact tracing proportion and hospitalization rate reduces epidemic peaks and increases the probability of extinction, with their joint implementation being more effective than either intervention alone. The mean of extinction time exhibits a pronounced maximum near the critical threshold, indicating that epidemics in the critical regime may persist for a prolonged period. These results highlight the importance of contact tracing in accelerating disease extinction.