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.  Google Scholar

[2]

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

[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.   Google Scholar
[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[7]

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

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

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.   Google Scholar
[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[7]

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

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

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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