Modelling and optimal control of immune response of renal transplant recipients

Figures & data

Table 1. Description of state variables.

States	Description	Units
H S	Concentration of susceptible cells	cells/μl-blood
H I	Concentration of infected cells	cells/μl-blood
V	Concentration of free HCMV	copies/μl-blood
E V	Concentration of HCMV-specific CD8+T cells	cells/μl-blood
E K	Concentration of allospecific CD8+T cells that targets kidney	cells/μl-blood
C	Serum creatine concentration	mg/dl

Figure 1. Model schematic.

Table 2. Parameter values used in the simulations.

Parameters	Value	Parameters	Value
λ HS	10^−3 per day	κ HS	400 cells/μl-blood
β	l-blood/(copies day)	δ HI	0.2 per day
δ EH	l-blood/(cells day)	ρ V	12 copies/cells
δ V	0.2 per day	λ EV	0.5 cells/(μl-blood day)
ρ¯ EV	1 per day	κ V	20 copies/μl-blood
δ EV	0.05 per day	λ EK	0.5 cells/(μl-blood day)
δ EK	0.1 per day	λ C	0.2 mg/(dL day)
δ C0	0.2 per day	κ EK	8 cells/μl-blood

Figure 2. Simulations of primary infection with different efficacy levels (ε _I =0, 0.4, 0.8) of immunosuppression in the case of no antiviral drug treatment (ε _V ≡ 0).

Figure 3. Simulations of primary infection with different numbers of days of prophylactic antiviral treatment (ε _V is set at 0.8 during antiviral treatment) during immunosuppression (ε _I ≡ 0.8).

Figure 4. Simulations of reactivation following immunosuppression (ε _I ≡ 0.8) with different numbers of days of prophylactic antiviral treatment (ε _V is set at 0.8 during antiviral treatment).

Figure 5. Simulation results obtained with no treatment, i.e., ϵ _I =ϵ _V ≡ 0 (dashed line) and with full treatment, i.e., ϵ _I ≡ 0.8 and ϵ _V ≡ 0.2 (dash-dot lines). The solid lines indicate the threshold quantities, V* and C*.

Figure 6. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=0.90, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =15.

Figure 7. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=0.95, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =15.

Figure 8. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=1.05, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =15.

Figure 9. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=1.10, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =15.

Figure 10. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=0.90, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =20.

Figure 11. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=0.95, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =20.

Figure 12. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=1.05, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =20.

Figure 13. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=1.10, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =20.

Figure 14. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=0.90, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =30.

Figure 15. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=0.95, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =30.

Figure 16. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=1.05, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =30.

Figure 17. OBSE-MPC controls and solutions (solid line), and simple MPC controls and solutions (dashed line) obtained with [Pcirc]=1.10, |w _i |≤1, Q=1, R=100, [Qcirc]=[Rcirc]=10⁴, , , and t _i+1−t _i =30.