Unvaccinated	Vaccinated
$$\frac{d{S}_{10}}{dt}=-{\lambda }_1{S}_{10}$$	$$\frac{d{S}_{11}}{dt}=-{\lambda }_1\theta S_{11}$$ (1)
$$\frac{d{S}_{20}}{dt}=-{\lambda }_2{S}_{20}$$	$$\frac{d{S}_{21}}{dt}=-{\lambda }_2\theta S_{21}$$ (2)
$$\frac{d{A}_{10}}{dt}={\lambda }_1(1-\rho )S_{10}-{\gamma }_1{A}_{10}$$	$$\frac{d{A}_{11}}{dt}={\lambda }_1(1-\rho \psi )\theta S_{11}-{\gamma }_1{A}_{11}$$ (3)
$$\frac{d{A}_{20}}{dt}={\lambda }_2(1-\rho )S_{20}-{\gamma }_2{A}_{20}$$	$$\frac{d{A}_{21}}{dt}={\lambda }_2(1-\rho \psi )\theta S_{21}-{\gamma }_2{A}_{21}$$ (4)
$$\frac{d{I}_{10}}{dt}={\lambda }_1\rho S_{10}-{\lambda }_1{I}_{10}$$	$$\frac{d{I}_{11}}{dt}={\lambda }_1\rho \psi \theta S_{11}-{\gamma }_1{I}_{11}$$ (5)
$$\frac{d{I}_{20}}{dt}={\lambda }_2\rho S_{20}-{\lambda }_2{I}_{20}$$	$$\frac{d{I}_{21}}{dt}={\lambda }_2\rho \psi \theta S_{21}-{\gamma }_2{I}_{21}$$ (6)
$$\frac{d{R}_{10}}{dt}={\gamma }_1(A_{10}+I_{10})$$	$$\frac{d{R}_{11}}{dt}={\gamma }_1(A_{11}+I_{21)}$$ (7)
$$\frac{d{R}_{20}}{dt}={\gamma }_2(A_{20}+I_{20})$$	$$\frac{d{R}_{21}}{dt}={\gamma }_2(A_{21}+I_{21})$$ (8)