Model equations

Model equations associated with Hogan et al

Vaccination group v0 - unvaccinated

dS(t,a,v0)dt=2ρR2(t,a,v0)−βS(t,a,v0)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−κ(a)S(t,a,v0)\frac{dS(t,a,v_0)}{dt}=2\rho R_2(t,a,v_0)-\beta\frac{S(t,a,v_0)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-κ(a)S(t,a,v0)dE1(t,a,v0)dt=βS(t,a,v0)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2αE1(t,a,v0)−κ(a)E1(t,a,v0)\frac{dE_1(t,a,v_0)}{dt}=\beta\frac{S(t,a,v_0)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-2αE1(t,a,v0)-κ(a)E1(t,a,v0)dE2(t,a,v0)dt=2αE1(t,a,v0)−2αE2(t,a,v0)−κ(a)E2(t,a,v0)dIMILD(t,a,v0)dt=(1−ϕ1(a))(2αE2(t,a,v0))−γ1IMILD(t,a,v0)\frac{dE_2(t,a,v_0)}{dt}=2\alpha E_1(t,a,v_0)-2\alpha E_2(t,a,v_0)-\kappa(a)E_2(t,a,v_0)\frac{dI_{MILD}(t,a,v_0)}{dt}=(1-\phi_1(a))(2\alpha E_2(t,a,v_0))-\gamma_1I_{MILD}(t,a,v_0)dICASE,0(t,a,v0)dt=ϕ1(a)(2αE2(t,a,v0))−2γ2ICASE,0(t,a,v0)\frac{dI_{CASE,0}(t,a,v_0)}{dt}=\phi_1(a)(2\alpha E_2(t,a,v_0))-2\gamma_2I_{CASE,0}(t,a,v_0)dICASE,1(t,a,v0)dt=2γ2ICASE,0(t,a,v0)−2γ2ICASE,1(t,a,v0)dIHOSPITAL,0(t,a,v0,0,0)dt=(1−δ(H))μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v0)−2γ3,0IHOSPITAL,0(t,a,v0,0,0)\frac{dI_{CASE,1}(t,a,v_0)}{dt}=2\gamma_2I_{CASE,0}(t,a,v_0)-2\gamma_2I_{CASE,1}(t,a,v_0)\frac{dI_{HOSPITAL,0}(t,a,v_0,0,0)}{dt}=(1-\delta(H))\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_0,0,0)dIHOSPITAL,1(t,a,v0,0,0)dt=2γ3,0IHOSPITAL,0(t,a,v0,0,0)−2γ3,0IHOSPITAL,1(t,a,v0,0,0)\frac{dI_{HOSPITAL,1}(t,a,v_0,0,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_0,0,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_0,0,0)dIHOSPITAL,0(t,a,v0,1,0)dt=δ(H)μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v0)−2γ3,0IHOSPITAL,0(t,a,v0,1,0)\frac{dI_{HOSPITAL,0}(t,a,v_0,1,0)}{dt}=\delta(H)\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_0,1,0)dIHOSPITAL,1(t,a,v0,1,0)dt=2γ3,0IHOSPITAL,0(t,a,v0,1,0)−2γ3,0IHOSPITAL,1(t,a,v0,1,0)\frac{dI_{HOSPITAL,1}(t,a,v_0,1,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_0,1,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_0,1,0)dIHOSPITAL,0(t,a,v0,0,1)dt=(1−δ(H))(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v0)−2γ3,1IHOSPITAL,0(t,a,v0,0,1)\frac{dI_{HOSPITAL,0}(t,a,v_0,0,1)}{dt}=(1-\delta(H))(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_0,0,1)dIHOSPITAL,1(t,a,v0,0,1)dt=2γ3,1IHOSPITAL,0(t,a,v0,0,1)−2γ3,1IHOSPITAL,1(t,a,v0,0,1)\frac{dI_{HOSPITAL,1}(t,a,v_0,0,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_0,0,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_0,0,1)dIHOSPITAL,0(t,a,v0,1,1)dt=δ(H)(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v0)−2γ3,1IHOSPITAL,0(t,a,v0,1,1)\frac{dI_{HOSPITAL,0}(t,a,v_0,1,1)}{dt}=\delta(H)(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_0,1,1)dIHOSPITAL,1(t,a,v0,1,1)dt=2γ3,1IHOSPITAL,0(t,a,v0,1,1)−2γ3,1IHOSPITAL,1(t,a,v0,1,1)\frac{dI_{HOSPITAL,1}(t,a,v_0,1,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_0,1,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_0,1,1)dIICU,0(t,a,v0,0,0)dt=(1−δ(ICU))μ(a)ϕ2(a)2γ2ICASE,1(t,a,v0)−2γ4,0IICU,0(t,a,v0,0,0)\frac{dI_{ICU,0}(t,a,v_0,0,0)}{dt}=(1-\delta(ICU))\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{4,0}I_{ICU,0}(t,a,v_0,0,0)dIICU,1(t,a,v0,0,0)dt=2γ4,0IICU,0(t,a,v0,0,0)−2γ4,0IICU,1(t,a,v0,0,0)\frac{dI_{ICU,1}(t,a,v_0,0,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_0,0,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_0,0,0)dIICU,0(t,a,v0,1,0)dt=δ(ICU)μ(a)ϕ2(a)2γ2ICASE,1(t,a,v0)−2γ4,0IICU,0(t,a,v0,1,0)\frac{dI_{ICU,0}(t,a,v_0,1,0)}{dt}=\delta(ICU)\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{4,0}I_{ICU,0}(t,a,v_0,1,0)dIICU,1(t,a,v0,1,0)dt=2γ4,0IICU,0(t,a,v0,1,0)−2γ4,0IICU,1(t,a,v0,1,0)\frac{dI_{ICU,1}(t,a,v_0,1,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_0,1,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_0,1,0)dIICU,0(t,a,v0,0,1)dt=(1−δ(ICU))(1−μ(a))ϕ2(a)2γ2ICASE,1(t,v0,a)−2γ4,1IICU,0(t,a,v0,0,1)\frac{dI_{ICU,0}(t,a,v_0,0,1)}{dt}=(1-\delta(ICU))(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,v_0,a)-2\gamma_{4,1}I_{ICU,0}(t,a,v_0,0,1)dIICU,1(t,a,v0,0,1)dt=2γ4,1IICU,0(t,a,v0,0,1)−2γ4,1IICU,1(t,a,v0,0,1)\frac{dI_{ICU,1}(t,a,v_0,0,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_0,0,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_0,0,1)dIICU,0(t,a,v0,1,1)dt=δ(ICU)(1−μ(a))ϕ2(a)2γ2ICASE,1(t,a,v0)−2γ4,1IICU,0(t,a,v0,1,1)\frac{dI_{ICU,0}(t,a,v_0,1,1)}{dt}=\delta(ICU)(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_0)-2\gamma_{4,1}I_{ICU,0}(t,a,v_0,1,1)dIICU,1(t,a,v0,1,1)dt=2γ4,1IICU,0(t,a,v0,1,1)−2γ4,1IICU,1(t,a,v0,1,1)\frac{dI_{ICU,1}(t,a,v_0,1,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_0,1,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_0,1,1)dIREC,0(t,a,v0)dt=2γ4,1IICU,1(t,a,v0,0,1)+2γ4,1IICU,1(t,a,v0,1,1)−2γ5IREC,0(t,a,v0)\frac{dI_{REC,0}(t,a,v_0)}{dt}=2\gamma_{4,1}I_{ICU,1}(t,a,v_0,0,1)+2γ4,1IICU,1(t,a,v0,1,1)-2γ5IREC,0(t,a,v0)dIREC,1(t,a,v0)dt=2γ5IREC,0(t,a,v0)−2γ5IREC,1(t,a,v0)\frac{dI_{REC,1}(t,a,v_0)}{dt}=2\gamma_5I_{REC,0}(t,a,v_0)-2\gamma_5I_{REC,1}(t,a,v_0)dR1(t,a,v0)dt=γ1IMILD(t,a,v0)+2γ3,1IHOSPITAL,1(t,a,v0,0,1)+2γ3,1IHOSPITAL,1(t,a,v0,1,1)+2γ5IREC,1(t,a,v0)+2γ4,1IICU,1(t,a,v0,0,1)+2γ4,1IICU,1(t,a,v0,1,1)−2ρR1(t,a,v0)−κ(a)R1(t,a,v0)\frac{dR_1(t,a,v_0)}{dt}=\gamma_1I_{MILD}(t,a,v_0)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_0,0,1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_0,1,1)+2\gamma_5I_{REC,1}(t,a,v_0)+2\gamma_{4,1}I_{ICU,1}(t,a,v_0,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_0,1,1)-2\rho R_1(t,a,v_0)-\kappa(a)R_1(t,a,v_0)dR2(t,a,v0)dt=2ρR1(t,a,v0)−2ρR2(t,a,v0)−κ(a)R2(t,a,v0)\frac{dR_2(t,a,v_0)}{dt}=2\rho R_1(t,a,v_0)-2\rho R_2(t,a,v_0)-\kappa(a)R_2(t,a,v_0)dD(t,a,v0)dt=2γ3,0IHOSPITAL,1(t,a,v0,0,0)+2γ3,0IHOSPITAL,1(t,a,v0,1,0)+2γ4,0IICU,1(t,a,v0,0,0)+2γ4,0IICU,1(t,a,v0,1,0)\frac{dD(t,a,v_0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_0,0,0)+2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_0,1,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_0,0,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_0,1,0)

Vaccination group v1 - vaccinated but not yet protected (state 1)

dS(t,a,v1)dt=κ(a)S(t,a,v0)+2ρR2(t,a,v1)−βS(t,a,v1)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2ωS(t,a,v1)\frac{dS(t,a,v_1)}{dt}=\kappa(a)S(t,a,v_0)+2\rho R_2(t,a,v_1)-\beta\frac{S(t,a,v_1)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-2ωS(t,a,v1)dE1(t,a,v1)dt=κ(a)E1(t,a,v0)+βS(t,a,v1)Na′c(a,a′)[v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2αE1(t,a,v1)−2ωE1(t,a,v1)\frac{dE_1(t,a,v_1)}{dt}=\kappa(a)E_1(t,a,v_0)+βS(t,a,v1)Na'c(a,a')[v(IMILD(t,a',v)+ICASE(t,a',v))]-2αE1(t,a,v1)-2ωE1(t,a,v1)dE2(t,a,v1)dt=κ(a)E2(t,a,v0)+2αE1(t,a,v1)−2αE2(t,a,v1)−2ωE2(t,a,v1)\frac{dE_2(t,a,v_1)}{dt}=\kappa(a)E_2(t,a,v_0)+2\alpha E_1(t,a,v_1)-2\alpha E_2(t,a,v_1)-2\omega E_2(t,a,v_1)
dIMILD(t,a,v1)dt=(1−ϕ1(a))(2αE2(t,a,v1))−γ1IMILD(t,a,v1)−2ωIMILD(t,a,v1)\frac{dI_{MILD}(t,a,v_1)}{dt}=(1-\phi_1(a))(2\alpha E_2(t,a,v_1))-\gamma_1I_{MILD}(t,a,v_1)-2\omega I_{MILD}(t,a,v_1)
dICASE,0(t,a,v1)dt=ϕ1(a)(2αE2(t,a,v1))−2γ2ICASE,0(t,a,v1)−2ωICASE,0(t,a,v1)\frac{dI_{CASE,0}(t,a,v_1)}{dt}=\phi_1(a)(2\alpha E_2(t,a,v_1))-2\gamma_2I_{CASE,0}(t,a,v_1)-2\omega I_{CASE,0}(t,a,v_1)
dICASE,1(t,a,v1)dt=2γ2ICASE,0(t,a,v1)−2γ2ICASE,1(t,a,v1)−2ωICASE,1(t,a,v1)\frac{dI_{CASE,1}(t,a,v_1)}{dt}=2\gamma_2I_{CASE,0}(t,a,v_1)-2\gamma_2I_{CASE,1}(t,a,v_1)-2\omega I_{CASE,1}(t,a,v_1)dIHOSPITAL,0(t,a,v1,0,0)dt=(1−δ(H))μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v1)−2γ3,0IHOSPITAL,0(t,a,v1,0,0)−2ωIHOSPITAL,0(t,a,v1,0,0)\frac{dI_{HOSPITAL,0}(t,a,v_1,0,0)}{dt}=(1-\delta(H))\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_1,0,0)-2\omega I_{HOSPITAL,0}(t,a,v_1,0,0)dIHOSPITAL,1(t,a,v1,0,0)dt=2γ3,0IHOSPITAL,0(t,a,v1,0,0)−2γ3,0IHOSPITAL,1(t,a,v1,0,0)−2ωIHOSPITAL,1(t,a,v1,0,0)\frac{dI_{HOSPITAL,1}(t,a,v_1,0,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_1,0,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_1,0,0)-2\omega I_{HOSPITAL,1}(t,a,v_1,0,0)dIHOSPITAL,0(t,a,v1,1,0)dt=δ(H)μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v1)−2γ3,0IHOSPITAL,0(t,a,v1,1,0)−2ωIHOSPITAL,0(t,a,v1,1,0)\frac{dI_{HOSPITAL,0}(t,a,v_1,1,0)}{dt}=\delta(H)\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_1,1,0)-2\omega I_{HOSPITAL,0}(t,a,v_1,1,0)dIHOSPITAL,1(t,a,v1,1,0)dt=2γ3,0IHOSPITAL,0(t,a,v1,1,0)−2γ3,0IHOSPITAL,1(t,a,v1,1,0)−2ωIHOSPITAL,1(t,a,v1,1,0)\frac{dI_{HOSPITAL,1}(t,a,v_1,1,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_1,1,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_1,1,0)-2\omega I_{HOSPITAL,1}(t,a,v_1,1,0)dIHOSPITAL,0(t,a,v1,0,1)dt=(1−δ(H))(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v1)−2γ3,1IHOSPITAL,0(t,a,v1,0,1)−2ωIHOSPITAL,0(t,a,v1,0,1)\frac{dI_{HOSPITAL,0}(t,a,v_1,0,1)}{dt}=(1-\delta(H))(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_1,0,1)-2\omega I_{HOSPITAL,0}(t,a,v_1,0,1)dIHOSPITAL,1(t,a,v1,0,1)dt=2γ3,1IHOSPITAL,0(t,a,v1,0,1)−2γ3,1IHOSPITAL,1(t,a,v1,0,1)−2ωIHOSPITAL,1(t,a,v1,0,1)\frac{dI_{HOSPITAL,1}(t,a,v_1,0,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_1,0,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_1,0,1)-2\omega I_{HOSPITAL,1}(t,a,v_1,0,1)dIHOSPITAL,0(t,a,v1,1,1)dt=δ(H)(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v1)−2γ3,1IHOSPITAL,0(t,a,v1,1,1)−2ωIHOSPITAL,0(t,a,v1,1,1)\frac{dI_{HOSPITAL,0}(t,a,v_1,1,1)}{dt}=\delta(H)(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_1,1,1)-2\omega I_{HOSPITAL,0}(t,a,v_1,1,1)dIHOSPITAL,1(t,a,v1,1,1)dt=2γ3,1IHOSPITAL,0(t,a,v1,1,1)−2γ3,1IHOSPITAL,1(t,a,v1,1,1)−2ωIHOSPITAL,1(t,a,v1,1,1)\frac{dI_{HOSPITAL,1}(t,a,v_1,1,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_1,1,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_1,1,1)-2\omega I_{HOSPITAL,1}(t,a,v_1,1,1)dIICU,0(t,a,v1,0,0)dt=(1−δ(ICU))μ(a)ϕ2(a)2γ2ICASE,1(t,a,v1)−2γ4,0IICU,0(t,a,v1,0,0)−2ωIICU,0(t,a,v1,0,0)\frac{dI_{ICU,0}(t,a,v_1,0,0)}{dt}=(1-\delta(ICU))\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{4,0}I_{ICU,0}(t,a,v_1,0,0)-2\omega I_{ICU,0}(t,a,v_1,0,0)dIICU,1(t,a,v1,0,0)dt=2γ4,0IICU,0(t,a,v1,0,0)−2γ4,0IICU,1(t,a,v1,0,0)−2ωIICU,1(t,a,v1,0,0)\frac{dI_{ICU,1}(t,a,v_1,0,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_1,0,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_1,0,0)-2\omega I_{ICU,1}(t,a,v_1,0,0)dIICU,0(t,a,v1,1,0)dt=δ(ICU)μ(a)ϕ2(a)2γ2ICASE,1(t,a,v1)−2γ4,0IICU,0(t,a,v1,1,0)−2ωIICU,0(t,a,v1,1,0)\frac{dI_{ICU,0}(t,a,v_1,1,0)}{dt}=\delta(ICU)\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{4,0}I_{ICU,0}(t,a,v_1,1,0)-2\omega I_{ICU,0}(t,a,v_1,1,0)dIICU,1(t,a,v1,1,0)dt=2γ4,0IICU,0(t,a,v1,1,0)−2γ4,0IICU,1(t,a,v1,1,0)−2ωIICU,1(t,a,v1,1,0)\frac{dI_{ICU,1}(t,a,v_1,1,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_1,1,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_1,1,0)-2\omega I_{ICU,1}(t,a,v_1,1,0)dIICU,0(t,a,v1,0,1)dt=(1−δ(ICU))(1−μ(a))ϕ2(a)2γ2ICASE,1(t,v1,a)−2γ4,1IICU,0(t,a,v1,0,1)−2ωIICU,0(t,a,v1,0,1)\frac{dI_{ICU,0}(t,a,v_1,0,1)}{dt}=(1-\delta(ICU))(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,v_1,a)-2\gamma_{4,1}I_{ICU,0}(t,a,v_1,0,1)-2\omega I_{ICU,0}(t,a,v_1,0,1)dIICU,1(t,a,v1,0,1)dt=2γ4,1IICU,0(t,a,v1,0,1)−2γ4,1IICU,1(t,a,v1,0,1)−2ωIICU,1(t,a,v1,0,1)\frac{dI_{ICU,1}(t,a,v_1,0,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_1,0,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_1,0,1)-2\omega I_{ICU,1}(t,a,v_1,0,1)dIICU,0(t,a,v1,1,1)dt=δ(ICU)(1−μ(a))ϕ2(a)2γ2ICASE,1(t,a,v1)−2γ4,1IICU,0(t,a,v1,1,1)−2ωIICU,0(t,a,v1,1,1)\frac{dI_{ICU,0}(t,a,v_1,1,1)}{dt}=\delta(ICU)(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_1)-2\gamma_{4,1}I_{ICU,0}(t,a,v_1,1,1)-2\omega I_{ICU,0}(t,a,v_1,1,1)dIICU,1(t,a,v1,1,1)dt=2γ4,1IICU,0(t,a,v1,1,1)−2γ4,1IICU,1(t,a,v1,1,1)−2ωIICU,1(t,a,v1,1,1)\frac{dI_{ICU,1}(t,a,v_1,1,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_1,1,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_1,1,1)-2\omega I_{ICU,1}(t,a,v_1,1,1)dIREC,0(t,a,v1)dt=2γ4,1IICU,1(t,a,v1,0,1)+2γ4,1IICU,1(t,a,v1,1,1)−2γ5IREC,0(t,a,v1)−2ωIREC,0(t,a,v1)\frac{dI_{REC,0}(t,a,v_1)}{dt}=2\gamma_{4,1}I_{ICU,1}(t,a,v_1,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_1,1,1)-2\gamma_5I_{REC,0}(t,a,v_1)-2\omega I_{REC,0}(t,a,v_1)dIREC,1(t,a,v1)dt=2γ5IREC,0(t,a,v1)−2γ5IREC,1(t,a,v1)−2ωIREC,1(t,a,v1)\frac{dI_{REC,1}(t,a,v_1)}{dt}=2\gamma_5I_{REC,0}(t,a,v_1)-2\gamma_5I_{REC,1}(t,a,v_1)-2\omega I_{REC,1}(t,a,v_1)dR1(t,a,v1)dt=κ(a)R1(t,a,v0)+γ1IMILD(t,a,v1)+2γ3,1IHOSPITAL,1(t,a,v1,0,1)+2γ3,1IHOSPITAL,1(t,a,v1,1,1)+2γ5IREC,1(t,a,v1)+2γ4,1IICU,1(t,a,v1,0,1)+2γ4,1IICU,1(t,a,v1,1,1)−2ρR1(t,a,v1)−2ωR(t,a,v1)\frac{dR_1(t,a,v_1)}{dt}=\kappa(a)R_1(t,a,v_0)+\gamma_1I_{MILD}(t,a,v_1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_1,0,1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_1,1,1)+2\gamma_5I_{REC,1}(t,a,v_1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_1,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_1,1,1)-2\rho R_1(t,a,v_1)-2\omega R(t,a,v_1)
dR2(t,a,v1)dt=κ(a)R2(t,a,v0)+2ρR1(t,a,v1)−2ρR2(t,a,v1)\frac{dR_2(t,a,v_1)}{dt}=\kappa(a)R_2(t,a,v_0)+2\rho R_1(t,a,v_1)-2\rho R_2(t,a,v_1)dD(t,a,v1)dt=2γ3,0IHOSPITAL,1(t,a,v1,0,0)+2γ3,0IHOSPITAL,1(t,a,v1,1,0)+2γ4,0IICU,1(t,a,v1,0,0)+2γ4,0IICU,1(t,a,v1,1,0)\frac{dD(t,a,v_1)}{dt}=2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_1,0,0)+2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_1,1,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_1,0,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_1,1,0)

Vaccination group v2 - vaccinated but not yet protected (state 2)

dS(t,a,v2)dt=2ρR2(t,a,v2)−βS(t,a,v2)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]+2ωS(t,a,v1)−2ωS(t,a,v2)\frac{dS(t,a,v_2)}{dt}=2\rho R_2(t,a,v_2)-\beta\frac{S(t,a,v_2)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]+2ωS(t,a,v1)-2ωS(t,a,v2)dE1(t,a,v2)dt=βS(t,a,v2)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2αE1(t,a,v2)+2ωE1(t,a,v1)−2ωE1(t,a,v2)\frac{dE_1(t,a,v_2)}{dt}=\beta\frac{S(t,a,v_2)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-2αE1(t,a,v2)+2ωE1(t,a,v1)-2ωE1(t,a,v2)dE2(t,a,v2)dt=2αE1(t,a,v2)−2αE2(t,a,v2)+2ωE2(t,a,v1)−2ωE2(t,a,v2)\frac{dE_2(t,a,v_2)}{dt}=2\alpha E_1(t,a,v_2)-2\alpha E_2(t,a,v_2)+2\omega E_2(t,a,v_1)-2\omega E_2(t,a,v_2)
dIMILD(t,a,v2)dt=(1−ϕ1(a))(2αE2(t,a,v2))−γ1IMILD(t,a,v2)+2ωIMILD(t,a,v1)−2ωIMILD(t,a,v2)\frac{dI_{MILD}(t,a,v_2)}{dt}=(1-\phi_1(a))(2\alpha E_2(t,a,v_2))-\gamma_1I_{MILD}(t,a,v_2)+2\omega I_{MILD}(t,a,v_1)-2\omega I_{MILD}(t,a,v_2)
dICASE,0(t,a,v2)dt=ϕ1(a)(2αE2(t,a,v2))−2γ2ICASE,0(t,a,v2)+2ωICASE,0(t,a,v1)−2ωICASE,0(t,a,v2)\frac{dI_{CASE,0}(t,a,v_2)}{dt}=\phi_1(a)(2\alpha E_2(t,a,v_2))-2\gamma_2I_{CASE,0}(t,a,v_2)+2\omega I_{CASE,0}(t,a,v_1)-2\omega I_{CASE,0}(t,a,v_2)
dICASE,1(t,a,v2)dt=2γ2ICASE,0(t,a,v2)−2γ2ICASE,1(t,a,v2)+2ωICASE,1(t,a,v1)−2ωICASE,1(t,a,v2)\frac{dI_{CASE,1}(t,a,v_2)}{dt}=2\gamma_2I_{CASE,0}(t,a,v_2)-2\gamma_2I_{CASE,1}(t,a,v_2)+2\omega I_{CASE,1}(t,a,v_1)-2\omega I_{CASE,1}(t,a,v_2)dIHOSPITAL,0(t,a,v2,0,0)dt=(1−δ(H))μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v2)−2γ3,0IHOSPITAL,0(t,a,v2,0,0)+2ωIHOSPITAL,0(t,a,v1,0,0)−2ωIHOSPITAL,0(t,a,v2,0,0)\frac{dI_{HOSPITAL,0}(t,a,v_2,0,0)}{dt}=(1-\delta(H))\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_2,0,0)+2\omega I_{HOSPITAL,0}(t,a,v_1,0,0)-2\omega I_{HOSPITAL,0}(t,a,v_2,0,0)dIHOSPITAL,1(t,a,v2,0,0)dt=2γ3,0IHOSPITAL,0(t,a,v2,0,0)−2γ3,0IHOSPITAL,1(t,a,v2,0,0)+2ωIHOSPITAL,1(t,a,v1,0,0)−2ωIHOSPITAL,1(t,a,v2,0,0)\frac{dI_{HOSPITAL,1}(t,a,v_2,0,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_2,0,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_2,0,0)+2\omega I_{HOSPITAL,1}(t,a,v_1,0,0)-2\omega I_{HOSPITAL,1}(t,a,v_2,0,0)dIHOSPITAL,0(t,a,v2,1,0)dt=δ(H)μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v2)−2γ3,0IHOSPITAL,0(t,a,v2,1,0)+2ωIHOSPITAL,0(t,a,v1,1,0)−2ωIHOSPITAL,0(t,a,v2,1,0)\frac{dI_{HOSPITAL,0}(t,a,v_2,1,0)}{dt}=\delta(H)\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_2,1,0)+2\omega I_{HOSPITAL,0}(t,a,v_1,1,0)-2\omega I_{HOSPITAL,0}(t,a,v_2,1,0)dIHOSPITAL,1(t,a,v2,1,0)dt=2γ3,0IHOSPITAL,0(t,a,v2,1,0)−2γ3,0IHOSPITAL,1(t,a,v2,1,0)+2ωIHOSPITAL,1(t,a,v1,1,0)−2ωIHOSPITAL,1(t,a,v2,1,0)\frac{dI_{HOSPITAL,1}(t,a,v_2,1,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_2,1,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_2,1,0)+2\omega I_{HOSPITAL,1}(t,a,v_1,1,0)-2\omega I_{HOSPITAL,1}(t,a,v_2,1,0)dIHOSPITAL,0(t,a,v2,0,1)dt=(1−δ(H))(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v2)−2γ3,1IHOSPITAL,0(t,a,v2,0,1)+2ωIHOSPITAL,0(t,a,v1,0,1)−2ωIHOSPITAL,0(t,a,v2,0,1)\frac{dI_{HOSPITAL,0}(t,a,v_2,0,1)}{dt}=(1-\delta(H))(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_2,0,1)+2\omega I_{HOSPITAL,0}(t,a,v_1,0,1)-2\omega I_{HOSPITAL,0}(t,a,v_2,0,1)dIHOSPITAL,1(t,a,v2,0,1)dt=2γ3,1IHOSPITAL,0(t,a,v2,0,1)−2γ3,1IHOSPITAL,1(t,a,v2,0,1)+2ωIHOSPITAL,1(t,a,v1,0,1)−2ωIHOSPITAL,1(t,a,v2,0,1)\frac{dI_{HOSPITAL,1}(t,a,v_2,0,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_2,0,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_2,0,1)+2\omega I_{HOSPITAL,1}(t,a,v_1,0,1)-2\omega I_{HOSPITAL,1}(t,a,v_2,0,1)dIHOSPITAL,0(t,a,v2,1,1)dt=δ(H)(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v2)−2γ3,1IHOSPITAL,0(t,a,v2,1,1)+2ωIHOSPITAL,0(t,a,v1,1,1)−2ωIHOSPITAL,0(t,a,v2,1,1)\frac{dI_{HOSPITAL,0}(t,a,v_2,1,1)}{dt}=\delta(H)(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_2,1,1)+2\omega I_{HOSPITAL,0}(t,a,v_1,1,1)-2\omega I_{HOSPITAL,0}(t,a,v_2,1,1)dIHOSPITAL,1(t,a,v2,1,1)dt=2γ3,1IHOSPITAL,0(t,a,v2,1,1)−2γ3,1IHOSPITAL,1(t,a,v2,1,1)+2ωIHOSPITAL,1(t,a,v1,1,1)−2ωIHOSPITAL,1(t,a,v2,1,1)\frac{dI_{HOSPITAL,1}(t,a,v_2,1,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_2,1,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_2,1,1)+2\omega I_{HOSPITAL,1}(t,a,v_1,1,1)-2\omega I_{HOSPITAL,1}(t,a,v_2,1,1)dIICU,0(t,a,v2,0,0)dt=(1−δ(ICU))μ(a)ϕ2(a)2γ2ICASE,1(t,a,v2)−2γ4,0IICU,0(t,a,v2,0,0)+2ωIICU,0(t,a,v1,0,0)−2ωIICU,0(t,a,v2,0,0)\frac{dI_{ICU,0}(t,a,v_2,0,0)}{dt}=(1-\delta(ICU))\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{4,0}I_{ICU,0}(t,a,v_2,0,0)+2\omega I_{ICU,0}(t,a,v_1,0,0)-2\omega I_{ICU,0}(t,a,v_2,0,0)dIICU,1(t,a,v2,0,0)dt=2γ4,0IICU,0(t,a,v2,0,0)−2γ4,0IICU,1(t,a,v2,0,0)+2ωIICU,1(t,a,v1,0,0)−2ωIICU,1(t,a,v2,0,0)\frac{dI_{ICU,1}(t,a,v_2,0,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_2,0,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_2,0,0)+2\omega I_{ICU,1}(t,a,v_1,0,0)-2\omega I_{ICU,1}(t,a,v_2,0,0)dIICU,0(t,a,v2,1,0)dt=δ(ICU)μ(a)ϕ2(a)2γ2ICASE,1(t,a,v2)−2γ4,0IICU,0(t,a,v2,1,0)+2ωIICU,0(t,a,v1,1,0)−2ωIICU,0(t,a,v2,1,0)\frac{dI_{ICU,0}(t,a,v_2,1,0)}{dt}=\delta(ICU)\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{4,0}I_{ICU,0}(t,a,v_2,1,0)+2\omega I_{ICU,0}(t,a,v_1,1,0)-2\omega I_{ICU,0}(t,a,v_2,1,0)dIICU,1(t,a,v2,1,0)dt=2γ4,0IICU,0(t,a,v2,1,0)−2γ4,0IICU,1(t,a,v2,1,0)+2ωIICU,1(t,a,v1,1,0)−2ωIICU,1(t,a,v2,1,0)\frac{dI_{ICU,1}(t,a,v_2,1,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_2,1,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_2,1,0)+2\omega I_{ICU,1}(t,a,v_1,1,0)-2\omega I_{ICU,1}(t,a,v_2,1,0)dIICU,0(t,a,v2,0,1)dt=(1−δ(ICU))(1−μ(a))ϕ2(a)2γ2ICASE,1(t,v2,a)−2γ4,1IICU,0(t,a,v2,0,1)+2ωIICU,0(t,a,v1,0,1)−2ωIICU,0(t,a,v2,0,1)\frac{dI_{ICU,0}(t,a,v_2,0,1)}{dt}=(1-\delta(ICU))(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,v_2,a)-2\gamma_{4,1}I_{ICU,0}(t,a,v_2,0,1)+2\omega I_{ICU,0}(t,a,v_1,0,1)-2\omega I_{ICU,0}(t,a,v_2,0,1)dIICU,1(t,a,v2,0,1)dt=2γ4,1IICU,0(t,a,v2,0,1)−2γ4,1IICU,1(t,a,v2,0,1)+2ωIICU,1(t,a,v1,0,1)−2ωIICU,1(t,a,v2,0,1)\frac{dI_{ICU,1}(t,a,v_2,0,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_2,0,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_2,0,1)+2\omega I_{ICU,1}(t,a,v_1,0,1)-2\omega I_{ICU,1}(t,a,v_2,0,1)dIICU,0(t,a,v2,1,1)dt=δ(ICU)(1−μ(a))ϕ2(a)2γ2ICASE,1(t,a,v2)−2γ4,1IICU,0(t,a,v2,1,1)+2ωIICU,0(t,a,v1,1,1)−2ωIICU,0(t,a,v2,1,1)\frac{dI_{ICU,0}(t,a,v_2,1,1)}{dt}=\delta(ICU)(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_2)-2\gamma_{4,1}I_{ICU,0}(t,a,v_2,1,1)+2\omega I_{ICU,0}(t,a,v_1,1,1)-2\omega I_{ICU,0}(t,a,v_2,1,1)dIICU,1(t,a,v2,1,1)dt=2γ4,1IICU,0(t,a,v2,1,1)−2γ4,1IICU,1(t,a,v2,1,1)+2ωIICU,1(t,a,v1,1,1)−2ωIICU,1(t,a,v2,1,1)\frac{dI_{ICU,1}(t,a,v_2,1,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_2,1,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_2,1,1)+2\omega I_{ICU,1}(t,a,v_1,1,1)-2\omega I_{ICU,1}(t,a,v_2,1,1)dIREC,0(t,a,v2)dt=2γ4,1IICU,1(t,a,v2,0,1)+2γ4,1IICU,1(t,a,v2,1,1)−2γ5IREC,0(t,a,v2)+2ωIREC,0(t,a,v1)−2ωIREC,0(t,a,v2)\frac{dI_{REC,0}(t,a,v_2)}{dt}=2\gamma_{4,1}I_{ICU,1}(t,a,v_2,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_2,1,1)-2\gamma_5I_{REC,0}(t,a,v_2)+2\omega I_{REC,0}(t,a,v_1)-2\omega I_{REC,0}(t,a,v_2)dIREC,1(t,a,v2)dt=2γ5IREC,0(t,a,v2)−2γ5IREC,1(t,a,v2)+2ωIREC,1(t,a,v1)−2ωIREC,1(t,a,v2)\frac{dI_{REC,1}(t,a,v_2)}{dt}=2\gamma_5I_{REC,0}(t,a,v_2)-2\gamma_5I_{REC,1}(t,a,v_2)+2\omega I_{REC,1}(t,a,v_1)-2\omega I_{REC,1}(t,a,v_2)dR1(t,a,v2)dt=γ1IMILD(t,a,v2)+2γ3,1IHOSPITAL,1(t,a,v2,0,1)+2γ3,1IHOSPITAL,1(t,a,v2,1,1)+2γ5IREC,1(t,a,v2)+2γ4,1IICU,1(t,a,v2,0,1)+2γ4,1IICU,1(t,a,v2,1,1)−2ρR1(t,a,v2)+2ωR1(t,a,v1)−2ωR1(t,a,v2)\frac{dR_1(t,a,v_2)}{dt}=\gamma_1I_{MILD}(t,a,v_2)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_2,0,1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_2,1,1)+2\gamma_5I_{REC,1}(t,a,v_2)+2\gamma_{4,1}I_{ICU,1}(t,a,v_2,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_2,1,1)-2\rho R_1(t,a,v_2)+2\omega R_1(t,a,v_1)-2\omega R_1(t,a,v_2)
dR2(t,a,v2)dt=2ρR1(t,a,v2)−2ρR2(t,a,v2)+2ωR2(t,a,v1)−2ωR2(t,a,v2)\frac{dR_2(t,a,v_2)}{dt}=2\rho R_1(t,a,v_2)-2\rho R_2(t,a,v_2)+2\omega R_2(t,a,v_1)-2\omega R_2(t,a,v_2)dD(t,a,v2)dt=2γ3,0IHOSPITAL,1(t,a,v2,0,0)+2γ3,0IHOSPITAL,1(t,a,v2,1,0)+2γ4,0IICU,1(t,a,v2,0,0)+2γ4,0IICU,1(t,a,v2,1,0)\frac{dD(t,a,v_2)}{dt}=2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_2,0,0)+2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_2,1,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_2,0,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_2,1,0)

Vaccination group v3 - vaccinated and protected (state 1)

dS(t,a,v3)dt=2ρR2(t,a,v3)−υinf(a)βS(t,a,v3)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]+2ωS(t,a,v2)−2ψS(t,a,v3)\frac{dS(t,a,v_3)}{dt}=2\rho R_2(t,a,v_3)-\upsilon_{inf}\left(a\right)\beta\frac{S\left(t,a,v_3\right)}{N}\sum_{a^\prime}{c(a,a\prime)[\sum_{v}\left(I_{MILD}\left(t,a^\prime,v\right)+I_{CASE}\left(t,a^\prime,v\right)\right)}]+2ωS(t,a,v2)-2ψS(t,a,v3)dE1(t,a,v3)dt=υinf(a)βS(t,a,v3)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2αE1(t,a,v2)+2ωE1(t,a,v2)−2ψE1(t,a,v3)\frac{dE_1(t,a,v_3)}{dt}=\upsilon_{inf}\left(a\right)\beta\frac{S(t,a,v_3)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-2αE1(t,a,v2)+2ωE1(t,a,v2)-2ψE1(t,a,v3)dE2(t,a,v3)dt=2αE1(t,a,v3)−2αE2(t,a,v3)+2ωE2(t,a,v2)−2ψE2(t,a,v3)\frac{dE_2(t,a,v_3)}{dt}=2\alpha E_1(t,a,v_3)-2\alpha E_2(t,a,v_3)+2\omega E_2(t,a,v_2)-2\psi E_2(t,a,v_3)
dIMILD(t,a,v3)dt=(1−υdis(a)ϕ1(a))(2αE2(t,a,v3))−γ1IMILD(t,a,v3)+2ωIMILD(t,a,v2)−2ψIMILD(t,a,v3)\frac{dI_{MILD}(t,a,v_3)}{dt}=(1-\upsilon_{dis}(a)\phi_1(a))(2\alpha E_2(t,a,v_3))-\gamma_1I_{MILD}(t,a,v_3)+2\omega I_{MILD}(t,a,v_2)-2\psi I_{MILD}(t,a,v_3)
dICASE,0(t,a,v3)dt=υdis(a)ϕ1(a)(2αE2(t,a,v3))−2γ2ICASE,0(t,a,v3)+2ωICASE,0(t,a,v2)−2ψICASE,0(t,a,v3)\frac{dI_{CASE,0}(t,a,v_3)}{dt}=\upsilon_{dis}(a)\phi_1(a)(2\alpha E_2(t,a,v_3))-2\gamma_2I_{CASE,0}(t,a,v_3)+2\omega I_{CASE,0}(t,a,v_2)-2\psi I_{CASE,0}(t,a,v_3)dICASE,1(t,a,v3)dt=2γ2ICASE,0(t,a,v3)−2γ2ICASE,1(t,a,v3)+2ωICASE,1(t,a,v2)−2ψICASE,1(t,a,v3)\frac{dI_{CASE,1}(t,a,v_3)}{dt}=2\gamma_2I_{CASE,0}(t,a,v_3)-2\gamma_2I_{CASE,1}(t,a,v_3)+2\omega I_{CASE,1}(t,a,v_2)-2\psi I_{CASE,1}(t,a,v_3)dIHOSPITAL,0(t,a,v3,0,0)dt=(1−δ(H))μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v3)−2γ3,0IHOSPITAL,0(t,a,v3,0,0)+2ωIHOSPITAL,0(t,a,v2,0,0)−2ψIHOSPITAL,0(t,a,v3,0,0)\frac{dI_{HOSPITAL,0}(t,a,v_3,0,0)}{dt}=(1-\delta(H))\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_3,0,0)+2\omega I_{HOSPITAL,0}(t,a,v_2,0,0)-2\psi I_{HOSPITAL,0}(t,a,v_3,0,0)dIHOSPITAL,1(t,a,v3,0,0)dt=2γ3,0IHOSPITAL,0(t,a,v3,0,0)−2γ3,0IHOSPITAL,1(t,a,v3,0,0)+2ωIHOSPITAL,1(t,a,v2,0,0)−2ψIHOSPITAL,1(t,a,v3,0,0)\frac{dI_{HOSPITAL,1}(t,a,v_3,0,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_3,0,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_3,0,0)+2\omega I_{HOSPITAL,1}(t,a,v_2,0,0)-2\psi I_{HOSPITAL,1}(t,a,v_3,0,0)dIHOSPITAL,0(t,a,v3,1,0)dt=δ(H)μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v3)−2γ3,0IHOSPITAL,0(t,a,v3,1,0)+2ωIHOSPITAL,0(t,a,v2,1,0)−2ψIHOSPITAL,0(t,a,v3,1,0)\frac{dI_{HOSPITAL,0}(t,a,v_3,1,0)}{dt}=\delta(H)\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_3,1,0)+2\omega I_{HOSPITAL,0}(t,a,v_2,1,0)-2\psi I_{HOSPITAL,0}(t,a,v_3,1,0)dIHOSPITAL,1(t,a,v3,1,0)dt=2γ3,0IHOSPITAL,0(t,a,v3,1,0)−2γ3,0IHOSPITAL,1(t,a,v3,1,0)+2ωIHOSPITAL,1(t,a,v2,1,0)−2ψIHOSPITAL,1(t,a,v3,1,0)\frac{dI_{HOSPITAL,1}(t,a,v_3,1,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_3,1,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_3,1,0)+2\omega I_{HOSPITAL,1}(t,a,v_2,1,0)-2\psi I_{HOSPITAL,1}(t,a,v_3,1,0)dIHOSPITAL,0(t,a,v3,0,1)dt=(1−δ(H))(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v3)−2γ3,1IHOSPITAL,0(t,a,v3,0,1)+2ωIHOSPITAL,0(t,a,v2,0,1)−2ψIHOSPITAL,0(t,a,v3,0,1)\frac{dI_{HOSPITAL,0}(t,a,v_3,0,1)}{dt}=(1-\delta(H))(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_3,0,1)+2\omega I_{HOSPITAL,0}(t,a,v_2,0,1)-2\psi I_{HOSPITAL,0}(t,a,v_3,0,1)dIHOSPITAL,1(t,a,v3,0,1)dt=2γ3,1IHOSPITAL,0(t,a,v3,0,1)−2γ3,1IHOSPITAL,1(t,a,v3,0,1)+2ωIHOSPITAL,1(t,a,v2,0,1)−2ψIHOSPITAL,1(t,a,v3,0,1)\frac{dI_{HOSPITAL,1}(t,a,v_3,0,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_3,0,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_3,0,1)+2\omega I_{HOSPITAL,1}(t,a,v_2,0,1)-2\psi I_{HOSPITAL,1}(t,a,v_3,0,1)dIHOSPITAL,0(t,a,v3,1,1)dt=δ(H)(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v3)−2γ3,1IHOSPITAL,0(t,a,v3,1,1)+2ωIHOSPITAL,0(t,a,v2,1,1)−2ψIHOSPITAL,0(t,a,v3,1,1)\frac{dI_{HOSPITAL,0}(t,a,v_3,1,1)}{dt}=\delta(H)(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_3,1,1)+2\omega I_{HOSPITAL,0}(t,a,v_2,1,1)-2\psi I_{HOSPITAL,0}(t,a,v_3,1,1)dIHOSPITAL,1(t,a,v3,1,1)dt=2γ3,1IHOSPITAL,0(t,a,v3,1,1)−2γ3,1IHOSPITAL,1(t,a,v3,1,1)+2ωIHOSPITAL,1(t,a,v2,1,1)−2ψIHOSPITAL,1(t,a,v3,1,1)\frac{dI_{HOSPITAL,1}(t,a,v_3,1,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_3,1,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_3,1,1)+2\omega I_{HOSPITAL,1}(t,a,v_2,1,1)-2\psi I_{HOSPITAL,1}(t,a,v_3,1,1)dIICU,0(t,a,v3,0,0)dt=(1−δ(ICU))μ(a)ϕ2(a)2γ2ICASE,1(t,a,v3)−2γ4,0IICU,0(t,a,v3,0,0)+2ωIICU,0(t,a,v2,0,0)−2ψIICU,0(t,a,v3,0,0)\frac{dI_{ICU,0}(t,a,v_3,0,0)}{dt}=(1-\delta(ICU))\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{4,0}I_{ICU,0}(t,a,v_3,0,0)+2\omega I_{ICU,0}(t,a,v_2,0,0)-2\psi I_{ICU,0}(t,a,v_3,0,0)dIICU,1(t,a,v3,0,0)dt=2γ4,0IICU,0(t,a,v3,0,0)−2γ4,0IICU,1(t,a,v3,0,0)+2ωIICU,1(t,a,v2,0,0)−2ψIICU,1(t,a,v3,0,0)\frac{dI_{ICU,1}(t,a,v_3,0,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_3,0,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_3,0,0)+2\omega I_{ICU,1}(t,a,v_2,0,0)-2\psi I_{ICU,1}(t,a,v_3,0,0)dIICU,0(t,a,v3,1,0)dt=δ(ICU)μ(a)ϕ2(a)2γ2ICASE,1(t,a,v3)−2γ4,0IICU,0(t,a,v3,1,0)+2ωIICU,0(t,a,v2,1,0)−2ψIICU,0(t,a,v3,1,0)\frac{dI_{ICU,0}(t,a,v_3,1,0)}{dt}=\delta(ICU)\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{4,0}I_{ICU,0}(t,a,v_3,1,0)+2\omega I_{ICU,0}(t,a,v_2,1,0)-2\psi I_{ICU,0}(t,a,v_3,1,0)dIICU,1(t,a,v3,1,0)dt=2γ4,0IICU,0(t,a,v3,1,0)−2γ4,0IICU,1(t,a,v3,1,0)+2ωIICU,1(t,a,v2,1,0)−2ψIICU,1(t,a,v3,1,0)\frac{dI_{ICU,1}(t,a,v_3,1,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_3,1,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_3,1,0)+2\omega I_{ICU,1}(t,a,v_2,1,0)-2\psi I_{ICU,1}(t,a,v_3,1,0)dIICU,0(t,a,v3,0,1)dt=(1−δ(ICU))(1−μ(a))ϕ2(a)2γ2ICASE,1(t,v3,a)−2γ4,1IICU,0(t,a,v3,0,1)+2ωIICU,0(t,a,v2,0,1)−2ψIICU,0(t,a,v3,0,1)\frac{dI_{ICU,0}(t,a,v_3,0,1)}{dt}=(1-\delta(ICU))(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,v_3,a)-2\gamma_{4,1}I_{ICU,0}(t,a,v_3,0,1)+2\omega I_{ICU,0}(t,a,v_2,0,1)-2\psi I_{ICU,0}(t,a,v_3,0,1)dIICU,1(t,a,v3,0,1)dt=2γ4,1IICU,0(t,a,v3,0,1)−2γ4,1IICU,1(t,a,v3,0,1)+2ωIICU,1(t,a,v2,0,1)−2ψIICU,1(t,a,v3,0,1)\frac{dI_{ICU,1}(t,a,v_3,0,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_3,0,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_3,0,1)+2\omega I_{ICU,1}(t,a,v_2,0,1)-2\psi I_{ICU,1}(t,a,v_3,0,1)dIICU,0(t,a,v3,1,1)dt=δ(ICU)(1−μ(a))ϕ2(a)2γ2ICASE,1(t,a,v3)−2γ4,1IICU,0(t,a,v3,1,1)+2ωIICU,0(t,a,v2,1,1)−2ψIICU,0(t,a,v3,1,1)\frac{dI_{ICU,0}(t,a,v_3,1,1)}{dt}=\delta(ICU)(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_3)-2\gamma_{4,1}I_{ICU,0}(t,a,v_3,1,1)+2\omega I_{ICU,0}(t,a,v_2,1,1)-2\psi I_{ICU,0}(t,a,v_3,1,1)dIICU,1(t,a,v3,1,1)dt=2γ4,1IICU,0(t,a,v3,1,1)−2γ4,1IICU,1(t,a,v3,1,1)+2ωIICU,1(t,a,v2,1,1)−2ψIICU,1(t,a,v3,1,1)\frac{dI_{ICU,1}(t,a,v_3,1,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_3,1,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_3,1,1)+2\omega I_{ICU,1}(t,a,v_2,1,1)-2\psi I_{ICU,1}(t,a,v_3,1,1)dIREC,0(t,a,v3)dt=2γ4,1IICU,1(t,a,v3,0,1)+2γ4,1IICU,1(t,a,v3,1,1)−2γ5IREC,0(t,a,v3)+2ωIREC,0(t,a,v2)−2ψIREC,0(t,a,v3)\frac{dI_{REC,0}(t,a,v_3)}{dt}=2\gamma_{4,1}I_{ICU,1}(t,a,v_3,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_3,1,1)-2\gamma_5I_{REC,0}(t,a,v_3)+2\omega I_{REC,0}(t,a,v_2)-2\psi I_{REC,0}(t,a,v_3)dIREC,1(t,a,v3)dt=2γ5IREC,0(t,a,v3)−2γ5IREC,1(t,a,v3)+2ωIREC,1(t,a,v2)−2ψIREC,1(t,a,v3)\frac{dI_{REC,1}(t,a,v_3)}{dt}=2\gamma_5I_{REC,0}(t,a,v_3)-2\gamma_5I_{REC,1}(t,a,v_3)+2\omega I_{REC,1}(t,a,v_2)-2\psi I_{REC,1}(t,a,v_3)dR1(t,a,v3)dt=γ1IMILD(t,a,v3)+2γ3,1IHOSPITAL,1(t,a,v3,0,1)+2γ3,1IHOSPITAL,1(t,a,v3,1,1)+2γ5IREC,1(t,a,v3)+2γ4,1IICU,1(t,a,v3,0,1)+2γ4,1IICU,1(t,a,v3,1,1)−2ρR1(t,a,v3)+2ωR1(t,a,v2)−2ψR1(t,a,v3)\frac{dR_1(t,a,v_3)}{dt}=\gamma_1I_{MILD}(t,a,v_3)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_3,0,1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_3,1,1)+2\gamma_5I_{REC,1}(t,a,v_3)+2\gamma_{4,1}I_{ICU,1}(t,a,v_3,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_3,1,1)-2\rho R_1(t,a,v_3)+2\omega R_1(t,a,v_2)-2\psi R_1(t,a,v_3)
dR2(t,a,v3)dt=2ρR1(t,a,v3)−2ρR2(t,a,v3)+2ωR2(t,a,v2)−2ψR2(t,a,v3)\frac{dR_2(t,a,v_3)}{dt}=2\rho R_1(t,a,v_3)-2\rho R_2(t,a,v_3)+2\omega R_2(t,a,v_2)-2\psi R_2(t,a,v_3)dD(t,a,v3)dt=2γ3,0IHOSPITAL,1(t,a,v3,0,0)+2γ3,0IHOSPITAL,1(t,a,v3,1,0)+2γ4,0IICU,1(t,a,v3,0,0)+2γ4,0IICU,1(t,a,v3,1,0)\frac{dD(t,a,v_3)}{dt}=2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_3,0,0)+2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_3,1,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_3,0,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_3,1,0)

Vaccination group v4 - vaccinated and protected (state 2)

dS(t,a,v4)dt=2ρR2(t,a,v4)−υinf(a)βS(t,a,v4)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]+2ψS(t,a,v3)−2ψS(t,a,v4)\frac{dS(t,a,v_4)}{dt}=2\rho R_2(t,a,v_4)-\upsilon_{inf}\left(a\right)\beta\frac{S\left(t,a,v_4\right)}{N}\sum_{a^\prime}{c(a,a\prime)[\sum_{v}\left(I_{MILD}\left(t,a^\prime,v\right)+I_{CASE}\left(t,a^\prime,v\right)\right)}]+2ψS(t,a,v3)-2ψS(t,a,v4)dE1(t,a,v4)dt=υinf(a)βS(t,a,v4)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2αE1(t,a,v4)+2ψE1(t,a,v3)−2ψE1(t,a,v4)\frac{dE_1(t,a,v_4)}{dt}=\upsilon_{inf}\left(a\right)\beta\frac{S(t,a,v_4)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-2αE1(t,a,v4)+2ψE1(t,a,v3)-2ψE1(t,a,v4)
dE2(t,a,v4)dt=2αE1(t,a,v4)−2αE2(t,a,v4)+2ψE2(t,a,v3)−2ψE2(t,a,v4)\frac{dE_2(t,a,v_4)}{dt}=2\alpha E_1(t,a,v_4)-2\alpha E_2(t,a,v_4)+2\psi E_2(t,a,v_3)-2\psi E_2(t,a,v_4)
dIMILD(t,a,v4)dt=(1−υdis(a)ϕ1(a))(2αE2(t,a,v4))−γ1IMILD(t,a,v4)+2ψIMILD(t,a,v3)−2ψIMILD(t,a,v4)\frac{dI_{MILD}(t,a,v_4)}{dt}=(1-\upsilon_{dis}(a)\phi_1(a))(2\alpha E_2(t,a,v_4))-\gamma_1I_{MILD}(t,a,v_4)+2\psi I_{MILD}(t,a,v_3)-2\psi I_{MILD}(t,a,v_4)
dICASE,0(t,a,v4)dt=υdis(a)ϕ1(a)(2αE2(t,a,v4))−2γ2ICASE,0(t,a,v4)+2ψICASE,0(t,a,v3)−2ψICASE,0(t,a,v4)\frac{dI_{CASE,0}(t,a,v_4)}{dt}=\upsilon_{dis}(a)\phi_1(a)(2\alpha E_2(t,a,v_4))-2\gamma_2I_{CASE,0}(t,a,v_4)+2\psi I_{CASE,0}(t,a,v_3)-2\psi I_{CASE,0}(t,a,v_4)dICASE,1(t,a,v4)dt=2γ2ICASE,0(t,a,v4)−2γ2ICASE,1(t,a,v4)+2ψICASE,1(t,a,v3)−2ψICASE,1(t,a,v4)\frac{dI_{CASE,1}(t,a,v_4)}{dt}=2\gamma_2I_{CASE,0}(t,a,v_4)-2\gamma_2I_{CASE,1}(t,a,v_4)+2\psi I_{CASE,1}(t,a,v_3)-2\psi I_{CASE,1}(t,a,v_4)dIHOSPITAL,0(t,a,v4,0,0)dt=(1−δ(H))μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v4)−2γ3,0IHOSPITAL,0(t,a,v4,0,0)+2ψIHOSPITAL,0(t,a,v3,0,0)−2ψIHOSPITAL,0(t,a,v4,0,0)\frac{dI_{HOSPITAL,0}(t,a,v_4,0,0)}{dt}=(1-\delta(H))\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_4,0,0)+2\psi I_{HOSPITAL,0}(t,a,v_3,0,0)-2\psi I_{HOSPITAL,0}(t,a,v_4,0,0)dIHOSPITAL,1(t,a,v4,0,0)dt=2γ3,0IHOSPITAL,0(t,a,v4,0,0)−2γ3,0IHOSPITAL,1(t,a,v4,0,0)+2ψIHOSPITAL,1(t,a,v3,0,0)−2ψIHOSPITAL,1(t,a,v4,0,0)\frac{dI_{HOSPITAL,1}(t,a,v_4,0,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_4,0,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_4,0,0)+2\psi I_{HOSPITAL,1}(t,a,v_3,0,0)-2\psi I_{HOSPITAL,1}(t,a,v_4,0,0)dIHOSPITAL,0(t,a,v4,1,0)dt=δ(H)μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v4)−2γ3,0IHOSPITAL,0(t,a,v4,1,0)+2ψIHOSPITAL,0(t,a,v3,1,0)−2ψIHOSPITAL,0(t,a,v3,1,0)\frac{dI_{HOSPITAL,0}(t,a,v_4,1,0)}{dt}=\delta(H)\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_4,1,0)+2\psi I_{HOSPITAL,0}(t,a,v_3,1,0)-2\psi I_{HOSPITAL,0}(t,a,v_3,1,0)dIHOSPITAL,1(t,a,v4,1,0)dt=2γ3,0IHOSPITAL,0(t,a,v4,1,0)−2γ3,0IHOSPITAL,1(t,a,v4,1,0)+2ψIHOSPITAL,1(t,a,v3,1,0)−2ψIHOSPITAL,1(t,a,v4,1,0)\frac{dI_{HOSPITAL,1}(t,a,v_4,1,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_4,1,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_4,1,0)+2\psi I_{HOSPITAL,1}(t,a,v_3,1,0)-2\psi I_{HOSPITAL,1}(t,a,v_4,1,0)dIHOSPITAL,0(t,a,v4,0,1)dt=(1−δ(H))(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v4)−2γ3,1IHOSPITAL,0(t,a,v4,0,1)+2ψIHOSPITAL,0(t,a,v3,0,1)−2ψIHOSPITAL,0(t,a,v4,0,1)\frac{dI_{HOSPITAL,0}(t,a,v_4,0,1)}{dt}=(1-\delta(H))(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_4,0,1)+2\psi I_{HOSPITAL,0}(t,a,v_3,0,1)-2\psi I_{HOSPITAL,0}(t,a,v_4,0,1)dIHOSPITAL,1(t,a,v4,0,1)dt=2γ3,1IHOSPITAL,0(t,a,v4,0,1)−2γ3,1IHOSPITAL,1(t,a,v4,0,1)+2ψIHOSPITAL,1(t,a,v3,0,1)−2ψIHOSPITAL,1(t,a,v4,0,1)\frac{dI_{HOSPITAL,1}(t,a,v_4,0,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_4,0,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_4,0,1)+2\psi I_{HOSPITAL,1}(t,a,v_3,0,1)-2\psi I_{HOSPITAL,1}(t,a,v_4,0,1)dIHOSPITAL,0(t,a,v4,1,1)dt=δ(H)(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v4)−2γ3,1IHOSPITAL,0(t,a,v4,1,1)+2ψIHOSPITAL,0(t,a,v3,1,1)−2ψIHOSPITAL,0(t,a,v4,1,1)\frac{dI_{HOSPITAL,0}(t,a,v_4,1,1)}{dt}=\delta(H)(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_4,1,1)+2\psi I_{HOSPITAL,0}(t,a,v_3,1,1)-2\psi I_{HOSPITAL,0}(t,a,v_4,1,1)dIHOSPITAL,1(t,a,v4,1,1)dt=2γ3,1IHOSPITAL,0(t,a,v4,1,1)−2γ3,1IHOSPITAL,1(t,a,v4,1,1)+2ψIHOSPITAL,1(t,a,v3,1,1)−2ψIHOSPITAL,1(t,a,v4,1,1)\frac{dI_{HOSPITAL,1}(t,a,v_4,1,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_4,1,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_4,1,1)+2\psi I_{HOSPITAL,1}(t,a,v_3,1,1)-2\psi I_{HOSPITAL,1}(t,a,v_4,1,1)dIICU,0(t,a,v4,0,0)dt=(1−δ(ICU))μ(a)ϕ2(a)2γ2ICASE,1(t,a,v4)−2γ4,0IICU,0(t,a,v4,0,0)+2ψIICU,0(t,a,v3,0,0)−2ψIICU,0(t,a,v4,0,0)\frac{dI_{ICU,0}(t,a,v_4,0,0)}{dt}=(1-\delta(ICU))\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{4,0}I_{ICU,0}(t,a,v_4,0,0)+2\psi I_{ICU,0}(t,a,v_3,0,0)-2\psi I_{ICU,0}(t,a,v_4,0,0)dIICU,1(t,a,v4,0,0)dt=2γ4,0IICU,0(t,a,v4,0,0)−2γ4,0IICU,1(t,a,v4,0,0)+2ψIICU,1(t,a,v3,0,0)−2ψIICU,1(t,a,v4,0,0)\frac{dI_{ICU,1}(t,a,v_4,0,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_4,0,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_4,0,0)+2\psi I_{ICU,1}(t,a,v_3,0,0)-2\psi I_{ICU,1}(t,a,v_4,0,0)dIICU,0(t,a,v4,1,0)dt=δ(ICU)μ(a)ϕ2(a)2γ2ICASE,1(t,a,v4)−2γ4,0IICU,0(t,a,v4,1,0)+2ψIICU,0(t,a,v3,1,0)−2ψIICU,0(t,a,v4,1,0)\frac{dI_{ICU,0}(t,a,v_4,1,0)}{dt}=\delta(ICU)\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{4,0}I_{ICU,0}(t,a,v_4,1,0)+2\psi I_{ICU,0}(t,a,v_3,1,0)-2\psi I_{ICU,0}(t,a,v_4,1,0)dIICU,1(t,a,v4,1,0)dt=2γ4,0IICU,0(t,a,v4,1,0)−2γ4,0IICU,1(t,a,v4,1,0)+2ψIICU,1(t,a,v3,1,0)−2ψIICU,1(t,a,v4,1,0)\frac{dI_{ICU,1}(t,a,v_4,1,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_4,1,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_4,1,0)+2\psi I_{ICU,1}(t,a,v_3,1,0)-2\psi I_{ICU,1}(t,a,v_4,1,0)dIICU,0(t,a,v4,0,1)dt=(1−δ(ICU))(1−μ(a))ϕ2(a)2γ2ICASE,1(t,v4,a)−2γ4,1IICU,0(t,a,v4,0,1)+2ψIICU,0(t,a,v3,0,1)−2ψIICU,0(t,a,v4,0,1)\frac{dI_{ICU,0}(t,a,v_4,0,1)}{dt}=(1-\delta(ICU))(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,v_4,a)-2\gamma_{4,1}I_{ICU,0}(t,a,v_4,0,1)+2\psi I_{ICU,0}(t,a,v_3,0,1)-2\psi I_{ICU,0}(t,a,v_4,0,1)dIICU,1(t,a,v4,0,1)dt=2γ4,1IICU,0(t,a,v4,0,1)−2γ4,1IICU,1(t,a,v4,0,1)+2ψIICU,1(t,a,v3,0,1)−2ψIICU,1(t,a,v4,0,1)\frac{dI_{ICU,1}(t,a,v_4,0,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_4,0,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_4,0,1)+2\psi I_{ICU,1}(t,a,v_3,0,1)-2\psi I_{ICU,1}(t,a,v_4,0,1)dIICU,0(t,a,v4,1,1)dt=δ(ICU)(1−μ(a))ϕ2(a)2γ2ICASE,1(t,a,v4)−2γ4,1IICU,0(t,a,v4,1,1)+2ψIICU,0(t,a,v3,1,1)−2ψIICU,0(t,a,v4,1,1)\frac{dI_{ICU,0}(t,a,v_4,1,1)}{dt}=\delta(ICU)(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_4)-2\gamma_{4,1}I_{ICU,0}(t,a,v_4,1,1)+2\psi I_{ICU,0}(t,a,v_3,1,1)-2\psi I_{ICU,0}(t,a,v_4,1,1)dIICU,1(t,a,v4,1,1)dt=2γ4,1IICU,0(t,a,v4,1,1)−2γ4,1IICU,1(t,a,v4,1,1)+2ψIICU,1(t,a,v3,1,1)−2ψIICU,1(t,a,v4,1,1)\frac{dI_{ICU,1}(t,a,v_4,1,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_4,1,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_4,1,1)+2\psi I_{ICU,1}(t,a,v_3,1,1)-2\psi I_{ICU,1}(t,a,v_4,1,1)dIREC,0(t,a,v4)dt=2γ4,1IICU,1(t,a,v4,0,1)+2γ4,1IICU,1(t,a,v4,1,1)−2γ5IREC,0(t,a,v4)+2ψIREC,0(t,a,v3)−2ψIREC,0(t,a,v4)\frac{dI_{REC,0}(t,a,v_4)}{dt}=2\gamma_{4,1}I_{ICU,1}(t,a,v_4,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_4,1,1)-2\gamma_5I_{REC,0}(t,a,v_4)+2\psi I_{REC,0}(t,a,v_3)-2\psi I_{REC,0}(t,a,v_4)dIREC,1(t,a,v4)dt=2γ5IREC,0(t,a,v4)−2γ5IREC,1(t,a,v4)+2ψIREC,1(t,a,v3)−2ψIREC,1(t,a,v4)\frac{dI_{REC,1}(t,a,v_4)}{dt}=2\gamma_5I_{REC,0}(t,a,v_4)-2\gamma_5I_{REC,1}(t,a,v_4)+2\psi I_{REC,1}(t,a,v_3)-2\psi I_{REC,1}(t,a,v_4)dR1(t,a,v4)dt=γ1IMILD(t,a,v4)+2γ3,1IHOSPITAL,1(t,a,v4,0,1)+2γ3,1IHOSPITAL,1(t,a,v4,1,1)+2γ5IREC,1(t,a,v4)+2γ4,1IICU,1(t,a,v4,0,1)+2γ4,1IICU,1(t,a,v4,1,1)−2ρR1(t,a,v4)+2ψR1(t,a,v3)−2ψR1(t,a,v4)\frac{dR_1(t,a,v_4)}{dt}=\gamma_1I_{MILD}(t,a,v_4)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_4,0,1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_4,1,1)+2\gamma_5I_{REC,1}(t,a,v_4)+2\gamma_{4,1}I_{ICU,1}(t,a,v_4,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_4,1,1)-2\rho R_1(t,a,v_4)+2\psi R_1(t,a,v_3)-2\psi R_1(t,a,v_4)
dR2(t,a,v4)dt=2ρR1(t,a,v4)−2ρR2(t,a,v4)+2ψR2(t,a,v3)−2ψR2(t,a,v4)\frac{dR_2(t,a,v_4)}{dt}=2\rho R_1(t,a,v_4)-2\rho R_2(t,a,v_4)+2\psi R_2(t,a,v_3)-2\psi R_2(t,a,v_4)dD(t,a,v4)dt=2γ3,0IHOSPITAL,1(t,a,v4,0,0)+2γ3,0IHOSPITAL,1(t,a,v4,1,0)+2γ4,0IICU,1(t,a,v4,0,0)+2γ4,0IICU,1(t,a,v4,1,0)\frac{dD(t,a,v_4)}{dt}=2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_4,0,0)+2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_4,1,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_4,0,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_4,1,0)

Vaccination group v5 - previously vaccinated but no longer protected

dS(t,a,v5)dt=2ρR2(t,a,v5)−βS(t,a,v5)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]+2ψS(t,a,v4)\frac{dS(t,a,v_5)}{dt}=2\rho R_2(t,a,v_5)-\beta\frac{S(t,a,v_5)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]+2ψS(t,a,v4)dE1(t,a,v5)dt=βS(t,a,v5)N∑a′c(a,a′)[∑v(IMILD(t,a′,v)+ICASE(t,a′,v))]−2αE1(t,a,v5)+2ψE1(t,a,v4)\frac{dE_1(t,a,v_5)}{dt}=\beta\frac{S(t,a,v_5)}{N}\sum_{a\prime}{c(a,a\prime)[\sum_{v}{(I_{MILD}(t,a\prime,v)+I_{CASE}(t,a\prime,v))}}]-2αE1(t,a,v5)+2ψE1(t,a,v4)dE2(t,a,v5)dt=2αE1(t,a,v5)−2αE2(t,a,v5)+2ψE2(t,a,v4)\frac{dE_2(t,a,v_5)}{dt}=2\alpha E_1(t,a,v_5)-2\alpha E_2(t,a,v_5)+2\psi E_2(t,a,v_4)
dIMILD(t,a,v5)dt=(1−ϕ1(a))(2αE2(t,a,v5))−γ1IMILD(t,a,v5)+2ψIMILD(t,a,v4)\frac{dI_{MILD}(t,a,v_5)}{dt}=(1-\phi_1(a))(2\alpha E_2(t,a,v_5))-\gamma_1I_{MILD}(t,a,v_5)+2\psi I_{MILD}(t,a,v_4)
dICASE,0(t,a,v5)dt=ϕ1(a)(2αE2(t,a,v5))−2γ2ICASE,0(t,a,v5)+2ψICASE,0(t,a,v4)\frac{dI_{CASE,0}(t,a,v_5)}{dt}=\phi_1(a)(2\alpha E_2(t,a,v_5))-2\gamma_2I_{CASE,0}(t,a,v_5)+2\psi I_{CASE,0}(t,a,v_4)
dICASE,1(t,a,v5)dt=2γ2ICASE,0(t,a,v5)−2γ2ICASE,1(t,a,v5)+2ψICASE,1(t,a,v4)\frac{dI_{CASE,1}(t,a,v_5)}{dt}=2\gamma_2I_{CASE,0}(t,a,v_5)-2\gamma_2I_{CASE,1}(t,a,v_5)+2\psi I_{CASE,1}(t,a,v_4)dIHOSPITAL,0(t,a,v5,0,0)dt=(1−δ(H))μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v5)−2γ3,0IHOSPITAL,0(t,a,v5,0,0)+2ψIHOSPITAL,0(t,a,v4,0,0)\frac{dI_{HOSPITAL,0}(t,a,v_5,0,0)}{dt}=(1-\delta(H))\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_5,0,0)+2\psi I_{HOSPITAL,0}(t,a,v_4,0,0)dIHOSPITAL,1(t,a,v5,0,0)dt=2γ3,0IHOSPITAL,0(t,a,v5,0,0)−2γ3,0IHOSPITAL,1(t,a,v5,0,0)+2ψIHOSPITAL,1(t,a,v4,0,0)\frac{dI_{HOSPITAL,1}(t,a,v_5,0,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_5,0,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_5,0,0)+2\psi I_{HOSPITAL,1}(t,a,v_4,0,0)dIHOSPITAL,0(t,a,v5,1,0)dt=δ(H)μ(a)(1−ϕ2(a))2γ2ICASE,1(t,a,v5)−2γ3,0IHOSPITAL,0(t,a,v5,1,0)+2ψIHOSPITAL,0(t,a,v3,1,0)\frac{dI_{HOSPITAL,0}(t,a,v_5,1,0)}{dt}=\delta(H)\mu(a)(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_5,1,0)+2\psi I_{HOSPITAL,0}(t,a,v_3,1,0)dIHOSPITAL,1(t,a,v5,1,0)dt=2γ3,0IHOSPITAL,0(t,a,v5,1,0)−2γ3,0IHOSPITAL,1(t,a,v5,1,0)+2ψIHOSPITAL,1(t,a,v4,1,0)\frac{dI_{HOSPITAL,1}(t,a,v_5,1,0)}{dt}=2\gamma_{3,0}I_{HOSPITAL,0}(t,a,v_5,1,0)-2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_5,1,0)+2\psi I_{HOSPITAL,1}(t,a,v_4,1,0)dIHOSPITAL,0(t,a,v5,0,1)dt=(1−δ(H))(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v5)−2γ3,1IHOSPITAL,0(t,a,v5,0,1)+2ψIHOSPITAL,0(t,a,v4,0,1)\frac{dI_{HOSPITAL,0}(t,a,v_5,0,1)}{dt}=(1-\delta(H))(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_5,0,1)+2\psi I_{HOSPITAL,0}(t,a,v_4,0,1)dIHOSPITAL,1(t,a,v5,0,1)dt=2γ3,1IHOSPITAL,0(t,a,v5,0,1)−2γ3,1IHOSPITAL,1(t,a,v5,0,1)+2ψIHOSPITAL,1(t,a,v4,0,1)\frac{dI_{HOSPITAL,1}(t,a,v_5,0,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_5,0,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_5,0,1)+2\psi I_{HOSPITAL,1}(t,a,v_4,0,1)dIHOSPITAL,0(t,a,v5,1,1)dt=δ(H)(1−μ(a))(1−ϕ2(a))2γ2ICASE,1(t,a,v5)−2γ3,1IHOSPITAL,0(t,a,v5,1,1)+2ψIHOSPITAL,0(t,a,v4,1,1)\frac{dI_{HOSPITAL,0}(t,a,v_5,1,1)}{dt}=\delta(H)(1-\mu(a))(1-\phi_2(a))2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_5,1,1)+2\psi I_{HOSPITAL,0}(t,a,v_4,1,1)dIHOSPITAL,1(t,a,v5,1,1)dt=2γ3,1IHOSPITAL,0(t,a,v5,1,1)−2γ3,1IHOSPITAL,1(t,a,v5,1,1)+2ψIHOSPITAL,1(t,a,v4,1,1)\frac{dI_{HOSPITAL,1}(t,a,v_5,1,1)}{dt}=2\gamma_{3,1}I_{HOSPITAL,0}(t,a,v_5,1,1)-2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_5,1,1)+2\psi I_{HOSPITAL,1}(t,a,v_4,1,1)dIICU,0(t,a,v5,0,0)dt=(1−δ(ICU))μ(a)ϕ2(a)2γ2ICASE,1(t,a,v5)−2γ4,0IICU,0(t,a,v5,0,0)+2ψIICU,0(t,a,v4,0,0)\frac{dI_{ICU,0}(t,a,v_5,0,0)}{dt}=(1-\delta(ICU))\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{4,0}I_{ICU,0}(t,a,v_5,0,0)+2\psi I_{ICU,0}(t,a,v_4,0,0)dIICU,1(t,a,v5,0,0)dt=2γ4,0IICU,0(t,a,v5,0,0)−2γ4,0IICU,1(t,a,v5,0,0)+2ψIICU,1(t,a,v4,0,0)\frac{dI_{ICU,1}(t,a,v_5,0,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_5,0,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_5,0,0)+2\psi I_{ICU,1}(t,a,v_4,0,0)dIICU,0(t,a,v5,1,0)dt=δ(ICU)μ(a)ϕ2(a)2γ2ICASE,1(t,a,v5)−2γ4,0IICU,0(t,a,v5,1,0)+2ψIICU,0(t,a,v4,1,0)\frac{dI_{ICU,0}(t,a,v_5,1,0)}{dt}=\delta(ICU)\mu(a)\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{4,0}I_{ICU,0}(t,a,v_5,1,0)+2\psi I_{ICU,0}(t,a,v_4,1,0)dIICU,1(t,a,v5,1,0)dt=2γ4,0IICU,0(t,a,v5,1,0)−2γ4,0IICU,1(t,a,v5,1,0)+2ψIICU,1(t,a,v4,1,0)\frac{dI_{ICU,1}(t,a,v_5,1,0)}{dt}=2\gamma_{4,0}I_{ICU,0}(t,a,v_5,1,0)-2\gamma_{4,0}I_{ICU,1}(t,a,v_5,1,0)+2\psi I_{ICU,1}(t,a,v_4,1,0)dIICU,0(t,a,v5,0,1)dt=(1−δ(ICU))(1−μ(a))ϕ2(a)2γ2ICASE,1(t,v5,a)−2γ4,1IICU,0(t,a,v5,0,1)+2ψIICU,0(t,a,v4,0,1)\frac{dI_{ICU,0}(t,a,v_5,0,1)}{dt}=(1-\delta(ICU))(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,v_5,a)-2\gamma_{4,1}I_{ICU,0}(t,a,v_5,0,1)+2\psi I_{ICU,0}(t,a,v_4,0,1)dIICU,1(t,a,v5,0,1)dt=2γ4,1IICU,0(t,a,v5,0,1)−2γ4,1IICU,1(t,a,v5,0,1)+2ψIICU,1(t,a,v4,0,1)\frac{dI_{ICU,1}(t,a,v_5,0,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_5,0,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_5,0,1)+2\psi I_{ICU,1}(t,a,v_4,0,1)dIICU,0(t,a,v5,1,1)dt=δ(ICU)(1−μ(a))ϕ2(a)2γ2ICASE,1(t,a,v5)−2γ4,1IICU,0(t,a,v5,1,1)+2ψIICU,0(t,a,v4,1,1)\frac{dI_{ICU,0}(t,a,v_5,1,1)}{dt}=\delta(ICU)(1-\mu(a))\phi_2(a)2\gamma_2I_{CASE,1}(t,a,v_5)-2\gamma_{4,1}I_{ICU,0}(t,a,v_5,1,1)+2\psi I_{ICU,0}(t,a,v_4,1,1)dIICU,1(t,a,v5,1,1)dt=2γ4,1IICU,0(t,a,v5,1,1)−2γ4,1IICU,1(t,a,v5,1,1)+2ψIICU,1(t,a,v4,1,1)\frac{dI_{ICU,1}(t,a,v_5,1,1)}{dt}=2\gamma_{4,1}I_{ICU,0}(t,a,v_5,1,1)-2\gamma_{4,1}I_{ICU,1}(t,a,v_5,1,1)+2\psi I_{ICU,1}(t,a,v_4,1,1)dIREC,0(t,a,v5)dt=2γ4,1IICU,1(t,a,v5,0,1)+2γ4,1IICU,1(t,a,v5,1,1)−2γ5IREC,0(t,a,v5)+2ψIREC,0(t,a,v4)\frac{dI_{REC,0}(t,a,v_5)}{dt}=2\gamma_{4,1}I_{ICU,1}(t,a,v_5,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_5,1,1)-2\gamma_5I_{REC,0}(t,a,v_5)+2\psi I_{REC,0}(t,a,v_4)
dIREC,1(t,a,v5)dt=2γ5IREC,0(t,a,v5)−2γ5IREC,1(t,a,v5)+2ψIREC,1(t,a,v4)\frac{dI_{REC,1}(t,a,v_5)}{dt}=2\gamma_5I_{REC,0}(t,a,v_5)-2\gamma_5I_{REC,1}(t,a,v_5)+2\psi I_{REC,1}(t,a,v_4)dR1(t,a,v5)dt=γ1IMILD(t,a,v5)+2γ3,1IHOSPITAL,1(t,a,v5,0,1)+2γ3,1IHOSPITAL,1(t,a,v5,1,1)+2γ5IREC,1(t,a,v5)+2γ4,1IICU,1(t,a,v5,0,1)+2γ4,1IICU,1(t,a,v5,1,1)−2ρR1(t,a,v5)+2ψR1(t,a,v4)\frac{dR_1(t,a,v_5)}{dt}=\gamma_1I_{MILD}(t,a,v_5)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_5,0,1)+2\gamma_{3,1}I_{HOSPITAL,1}(t,a,v_5,1,1)+2\gamma_5I_{REC,1}(t,a,v_5)+2\gamma_{4,1}I_{ICU,1}(t,a,v_5,0,1)+2\gamma_{4,1}I_{ICU,1}(t,a,v_5,1,1)-2\rho R_1(t,a,v_5)+2\psi R_1(t,a,v_4)
dR2(t,a,v5)dt=2ρR1(t,a,v5)−2ρR2(t,a,v5)+2ψR2(t,a,v4)\frac{dR_2(t,a,v_5)}{dt}=2\rho R_1(t,a,v_5)-2\rho R_2(t,a,v_5)+2\psi R_2(t,a,v_4)dD(t,a,v5)dt=2γ3,0IHOSPITAL,1(t,a,v5,0,0)+2γ3,0IHOSPITAL,1(t,a,v5,1,0)+2γ4,0IICU,1(t,a,v5,0,0)+2γ4,0IICU,1(t,a,v5,1,0)\frac{dD(t,a,v_5)}{dt}=2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_5,0,0)+2\gamma_{3,0}I_{HOSPITAL,1}(t,a,v_5,1,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_5,0,0)+2\gamma_{4,0}I_{ICU,1}(t,a,v_5,1,0)