The author studies an SVIS epidemic model with infective immigrants
\[ \begin{aligned} S^{\prime}=\Lambda+(1-p)A- {\beta S I}{N}-(\mu+\phi)S+\gamma I+\theta V,\\ V^{\prime}=\phi S- {\sigma \beta V I}{N}-(\mu+\theta)V,\\ I^{\prime}=pA+ {\beta S I}{N}+ {\sigma \beta V I}{N}-(\mu+\gamma)I\end{aligned} \]
with \(S(t)+V(t)+I(t)=N=K:=\frac{A+\Lambda}{\mu}\) and an SVIR epidemic model with infective immigrants
\[ \begin{aligned} S^{\prime}=\Lambda+(1-p)A- {\beta S I}{N}-(\mu+\phi)S+\theta V,\\ V^{\prime}=\phi S- {\sigma \beta V I}{N}-(\mu+\theta)V,\\ I^{\prime}=pA+ {\beta S I}{N}+ {\sigma \beta V I}{N}-(\mu+\gamma)I,\\ R^{\prime}=\gamma I-\mu R \end{aligned} \]
with \(S(t)+V(t)+I(t)+R(t)=K\), where \(K=\frac{A+\Lambda}{\mu}\), \( S,I,R, V\) are susceptibles, infectives, removed and vaccinated, respectively. The effects of immigration and vaccination on disease dynamics of the above two models are explored. Conditions for the existence of multiple endemic steady states and a fold bifurcation are also addressed.