An SEIR epidemic model with a nonconstant vaccination strategy is
studied. This SEIR model has two disease transmission rates β1 and β2 which imitate
the fact that, for some infectious diseases, a latent person can pass the disease into
a susceptible one. Here we study the spread of some childhood infectious diseases as
good examples of diseases with infectious latent. We found that our SEIR model has a
unique disease free solution (DFS). A lower bound R0inf and an upper bound R0sup of
the basic reproductive number, R0 are estimated. We show that, the DFS is globally
asymptotically stable when R0sup < 1 and unstable if R0inf > 1. Computer simulations
have been conducted to show that non trivial periodic solutions are possible. Moreover
the impact of the contact rate between the latent and the susceptibles is simulated.
Different periodic solutions with different periods including one, two and three years,
are obtained. These results give a clearer view for the decision makers to know how
and when they should take action against a possible new wave of these infectious
diseases. This action is mainly, applying a suitable dose of vaccination just before a
severe peak of infection occurs