The impact of vaccine failure rate on epidemic dynamics in responsive networks

An SIS model based on the microscopic Markov-chain approximation is considered in this paper. It is assumed that the individual vaccination behavior depends on the contact awareness, local and global information of an epidemic. To better simulate the real situation, the vaccine failure rate is also taken into consideration. Our main conclusions are given in the following. First, we show that if the vaccine failure rate α is zero, then the epidemic eventually dies out regardless of what the network structure is or how large the effective spreading rate and the immunization response rates of an epidemic are. Second, we show that for any positive α, there exists a positive epidemic threshold depending on an adjusted network structure, which is only determined by the structure of the original network, the positive vaccine failure rate and the immunization response rate for contact awareness. Moreover, the epidemic threshold increases with respect to the strength of the immunization response rate for contact awareness. Finally, if the vaccine failure rate and the immunization response rate for contact awareness are positive, then there exists a critical vaccine failure rate α_c > 0 so that the disease free equilibrium (DFE) is stable (resp., unstable) if α < α_c (resp., α > α_c). Numerical simulations to see the effectiveness of our theoretical results are also provided.