Discrete state-structured epidemic model with varying susceptibility and infectivity and its application to the measles outbreak in the United States

Host individuals with vaccinations for infectious diseases often transition among different states of susceptibility and infectivity. Thus, we derive a class of epidemic models in which susceptible and infected individuals have a discrete set of susceptibility and infectivity states, respectively. Our model is based on double-dose vaccination and accounts for all possible state transitions between states. A comprehensive mathematical analysis of the proposed model is conducted, including assessing the control reproduction number and the global dynamical behaviors of the solutions. Using an improved affine-invariant ensemble Markov Chain Monte Carlo method, we also fit the model with measles case data in the United States. The results indicate that despite high vaccination coverage and efficacy rates, sporadic or small-scale measles outbreaks still occur. This suggests that reliance solely on vaccination policies is insufficient to fully interrupt its transmission chain. Consequently, enhancing the diagnosis rate of measles cases is another crucial measure for controlling measles transmission and outbreaks, building upon existing vaccination efforts.