A time-delayed SVEIR model for imperfect vaccine with a generalized nonmonotone incidence and application to measles.

In this paper, we investigate the effects of the latent period on the dynamics of infectious disease with an imperfect vaccine. We assume a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population when the number of infectious individuals increases. After we propose the model, we provide the well-posedness property by verifying the non-negativity and boundedness of the models solutions. Then, we calculate the effective reproduction number R E . The threshold dynamics of the system is obtained with respect to R E . We discuss the global stability of the disease-free equilibrium when R E < 1 and explore the system persistence when R E > 1 . Moreover, we prove the coexistence of an endemic equilibrium when the system persists. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to: (i) implement a case study regarding the measles disease transmission in the United States from 1963 to 2016; (ii) study the local and global sensitivity of R E with respect to the model parameters; (iii) discuss the stability of endemic equilibrium; and (iv) explore the sensitivity of the proposed model solutions with respect to the main parameters.