In this paper, we consider an efficient way to investigate infectious diseases by constructing and analyzing differential equations. Some infectious diseases such as COVID-19, AIDS, and hepatitis B, involve both symptomatic patients and asymptomatic carriers. These asymptomatic carriers are contagious. However, only a small amount of works consider asymptomatic carriers in their models. Other factors such as vaccination and ages of infection are also important to the spread of infectious diseases. Therefore, we incorporate these factors together into an SVEIR model that includes continuous ages of infection for both symptomatic patients and asymptomatic carriers in this work. The conditions for existence and local stability of disease-free and endemic steady states together with the basic reproduction number $R_B$ are presented. Furthermore, the global stability of disease-free and endemic steady states is considered. Several examples by simulations are presented to demonstrate the obtained theoretical results. The importance of asymptomatic carriers in the infected population is also shown in simulations.