January  2018, 23(1): 443-458. doi: 10.3934/dcdsb.2018030

Optimal control of a delayed HIV model

1. 

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

2. 

Institute of Computational and Applied Mathematics, University of Münster, D-48149 Münster, Germany

* Corresponding author: delfim@ua.pt

Received  July 2016 Published  January 2018

We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function representing the efficiency of reverse transcriptase inhibitors and consider the pharmacological delay associated to the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.

Citation: Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
References:
[1]

R. ArnaoutM. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 267 (2000), 1347-1354. doi: 10.1098/rspb.2000.1149.

[2]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer -und Zustands-Beschränkungen, Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.

[3] C. Büskens and H. Maurer, Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods (M. Gr"otschel, S. O. Krumke, J. Rambau, eds.), 57-68, Springer, Berlin, 2001.
[4] L. Cesari, Optimization — Theory and Applications. Problems with Ordinary Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.
[5]

R. Culshaw and S. Ruan, A delay differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.

[6]

R. CulshawS. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.

[7]

A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming Mathematics in Science and Engineering, 165, Academic Press, Orlando, FL, 1983.

[8] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975.
[9] R. FourerD. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.
[10]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413.

[11] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[12]

K. Hattaf and N. Yousfi, Optimal control of a delayed hiv infection model with immune response using an efficient numerical method ISRN Biomathematics 2012 (2012), Art. ID 215124, 7 pp. doi: 10.5402/2012/215124.

[13]

A. V. M. HerzS. BonhoeerR. M. AndersonR. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Nat. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247.

[14]

D. KirschnerS. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Mathematical Biology, 35 (1996), 775-792. doi: 10.1007/s002850050076.

[15]

J. KlamkaH. Maurer and A. Swierniak, Local controllability and optimal control for a model of combined anticancer therapy with control delays, Math. Biosci. Eng., 14 (2017), 195-216. doi: 10.3934/mbe.2017013.

[16] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, 1993.
[17]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Appl. Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.

[18]

J. E. MittlerM. MarkowitzD. D. Ho and A. S. Perelson, Improved estimates for HIV-1 clearance rate and intracellular delay, AIDS, 13 (1999), 1415-1417.

[19]

J. E. MittlerB. SulzerA. U. Neumann and A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5.

[20]

P. W. NelsonJ. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3.

[21]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8.

[22]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[23] M. A. Nowak and R. M. May, Virus Dynamics, Oxford Univ. Press, Oxford, 2000.
[24]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control Advances in Design and Control, 24, SIAM, Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368.

[25]

K. A. PawelekS. LiuF. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002.

[26]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[27]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.

[28]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes Interscience Publishers John Wiley & Sons, Inc., New York, 1962.

[29]

J. PrüssR. Schnaubelt and R. Zacher, Global asymptotic stability of equilibria in models for virus dynamics, Math. Model. Nat. Phenom., 3 (2008), 126-142. doi: 10.1051/mmnp:2008045.

[30]

D. Rocha, C. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model Math. Methods Appl. Sci. in press. doi: 10.1002/mma.4207.

[31]

C. J. SilvaH. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337. doi: 10.3934/mbe.2017021.

[32]

C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663. doi: 10.3934/dcds.2015.35.4639.

[33]

A. Świerniak and J. Klamka, Local controllability of models of combined anticancer therapy with delays in control, Math. Model. Nat. Phenom., 9 (2014), 216-226. doi: 10.1051/mmnp/20149413.

[34]

J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16 (1999), 29-37. doi: 10.1093/imammb/16.1.29.

[35]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

[36]

K. WangW. Wang and X. Liu, Global Stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593-1610. doi: 10.1016/j.camwa.2005.07.020.

[37]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524. doi: 10.3934/dcdsb.2009.12.511.

show all references

References:
[1]

R. ArnaoutM. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 267 (2000), 1347-1354. doi: 10.1098/rspb.2000.1149.

[2]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer -und Zustands-Beschränkungen, Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998.

[3] C. Büskens and H. Maurer, Sensitivity analysis and real-time control of parametric optimal control problems using nonlinear programming methods (M. Gr"otschel, S. O. Krumke, J. Rambau, eds.), 57-68, Springer, Berlin, 2001.
[4] L. Cesari, Optimization — Theory and Applications. Problems with Ordinary Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.
[5]

R. Culshaw and S. Ruan, A delay differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.

[6]

R. CulshawS. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3.

[7]

A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming Mathematics in Science and Engineering, 165, Academic Press, Orlando, FL, 1983.

[8] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975.
[9] R. FourerD. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks-Cole Publishing Company, 1993.
[10]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441. doi: 10.3934/jimo.2014.10.413.

[11] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[12]

K. Hattaf and N. Yousfi, Optimal control of a delayed hiv infection model with immune response using an efficient numerical method ISRN Biomathematics 2012 (2012), Art. ID 215124, 7 pp. doi: 10.5402/2012/215124.

[13]

A. V. M. HerzS. BonhoeerR. M. AndersonR. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Nat. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247.

[14]

D. KirschnerS. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, J. Mathematical Biology, 35 (1996), 775-792. doi: 10.1007/s002850050076.

[15]

J. KlamkaH. Maurer and A. Swierniak, Local controllability and optimal control for a model of combined anticancer therapy with control delays, Math. Biosci. Eng., 14 (2017), 195-216. doi: 10.3934/mbe.2017013.

[16] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, 1993.
[17]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Appl. Methods, 26 (2005), 129-156. doi: 10.1002/oca.756.

[18]

J. E. MittlerM. MarkowitzD. D. Ho and A. S. Perelson, Improved estimates for HIV-1 clearance rate and intracellular delay, AIDS, 13 (1999), 1415-1417.

[19]

J. E. MittlerB. SulzerA. U. Neumann and A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5.

[20]

P. W. NelsonJ. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3.

[21]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8.

[22]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[23] M. A. Nowak and R. M. May, Virus Dynamics, Oxford Univ. Press, Oxford, 2000.
[24]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control Advances in Design and Control, 24, SIAM, Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368.

[25]

K. A. PawelekS. LiuF. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002.

[26]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[27]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.

[28]

L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mischenko, The Mathematical Theory of Optimal Processes Interscience Publishers John Wiley & Sons, Inc., New York, 1962.

[29]

J. PrüssR. Schnaubelt and R. Zacher, Global asymptotic stability of equilibria in models for virus dynamics, Math. Model. Nat. Phenom., 3 (2008), 126-142. doi: 10.1051/mmnp:2008045.

[30]

D. Rocha, C. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model Math. Methods Appl. Sci. in press. doi: 10.1002/mma.4207.

[31]

C. J. SilvaH. Maurer and D. F. M. Torres, Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337. doi: 10.3934/mbe.2017021.

[32]

C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663. doi: 10.3934/dcds.2015.35.4639.

[33]

A. Świerniak and J. Klamka, Local controllability of models of combined anticancer therapy with delays in control, Math. Model. Nat. Phenom., 9 (2014), 216-226. doi: 10.1051/mmnp/20149413.

[34]

J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16 (1999), 29-37. doi: 10.1093/imammb/16.1.29.

[35]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

[36]

K. WangW. Wang and X. Liu, Global Stability in a viral infection model with lytic and nonlytic immune response, Comput. Math. Appl., 51 (2006), 1593-1610. doi: 10.1016/j.camwa.2005.07.020.

[37]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524. doi: 10.3934/dcdsb.2009.12.511.

Figure 1.  Endemic equilibrium $E_2$ for the parameter values of Table 1 and time delay $\tau=0.5$
Figure 2.  State variables with time delay $\tau = 0.5$ (dashed curves) versus without delay (continuous curves)
Figure 3.  Bang-bang control $c(t)$ (20) (continuous curve) and switching function $\phi$ (18) matching the control law (19). Zoom into a neighborhood of the switching time $t_s$: (left) Case 1, (middle) Case 2, (right) Case 3
Figure 4.  A comparison of state trajectories in Case 1 (no delays, dashed line) and Case 3 (delays $\tau = 0.5$ and $\xi = 0.2$, continuous line). (left) zoom of infected cells $I(t)$ into $[0, 5]$, (middle) zoom of free virus particles $V(t)$ into $[0, 10]$, (right) zoom of CTL cells $T(t)$ into $[0, 10]$
Figure 5.  State variables in the case of an intracellular delay only ($\tau = 0.5$ and $\xi=0$): controlled (dashed lines) versus uncontrolled situations (continuous lines)
Table 1.  Parameter values
Parameter Value Units
λ 5 day-1mm-3
m 0.03 day-1
r 0.0014 mm3virion-1day-1
u 0.32 day-1
s 0.05 mm3day-1
k 153.6 day-1
v 1 day-1
a 0.2 mm3day-1
n 0.3 day-1
tf 50 day
τ 0.5 day
ξ 0.2 day
Parameter Value Units
λ 5 day-1mm-3
m 0.03 day-1
r 0.0014 mm3virion-1day-1
u 0.32 day-1
s 0.05 mm3day-1
k 153.6 day-1
v 1 day-1
a 0.2 mm3day-1
n 0.3 day-1
tf 50 day
τ 0.5 day
ξ 0.2 day
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