January  2017, 14(1): 321-337. doi: 10.3934/mbe.2017021

Optimal control of a Tuberculosis model with state and control delays

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: maurer@math.uni-muenster.de

Received  October 31, 2015 Accepted  June 26, 2016 Published  October 2016

Fund Project: The first author is supported by the FCT post-doc grant SFRH/BPD/72061/2010.

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.

Citation: Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
References:
[1]

R. Bellmann and K. L. Cooke, Differential-Difference Equations Academic Press, New York, 1963.  Google Scholar

[2]

B. Buonomo and M. Cerasuolo, The effect of time delay in plant-pathogen interactions with host demography, Math. Biosci. Eng., 12 (2015), 473-490.  doi: 10.3934/mbe.2015.12.473.  Google Scholar

[3]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen PhD thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. Google Scholar

[4]

C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control, J. Comput. Appl. Math., 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[5]

C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656.  doi: 10.1007/s002850050069.  Google Scholar

[6]

T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nat. Med., 10 (2004), 1117-1121.  doi: 10.1038/nm1110.  Google Scholar

[7]

R. V. 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

[8]

J. Dieudonné, Foundations of Modern Analysis Academic Press, New York, 1960.  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, Special Issue on Computational Methods for Optimization and Control, J. Ind. Manag. Optim., 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.  Google Scholar

[11]

M. G. M. GomesP. RodriguesF. M. HilkerN. B. Mantilla-BeniersM. MuehlenA. C. Paulo and G. F. Medley, Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions, J. Theoret. Biol., 248 (2007), 608-617.  doi: 10.1016/j.jtbi.2007.06.005.  Google Scholar

[12]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[13]

H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[14]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Academic Press, San Diego, 1993.  Google Scholar

[15]

M. L. Lambert and P. Van der Stuyft, Delays to tuberculosis treatment: Shall we continue to blame the victim?, Trop. Med. Int. Health, 10 (2005), 945-946.  doi: 10.1111/j.1365-3156.2005.01485.x.  Google Scholar

[16]

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

[17]

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 SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.  Google Scholar

[18]

P. RodriguesC. Rebelo and M. G. M. Gomes, Drug resistance in tuberculosis: A reinfection model, Theor. Popul. Biol., 71 (2007), 196-212.  doi: 10.1016/j.tpb.2006.10.004.  Google Scholar

[19]

P. RodriguesC. J. Silva and D. F. M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis, Bull. Math. Biol., 76 (2014), 2627-2645.  doi: 10.1007/s11538-014-0028-6.  Google Scholar

[20]

H. SchättlerU. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L^2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657.  Google Scholar

[21]

L. F. Shampine and S. Thompson, Solving DDEs in MATLAB, Appl. Numer. Math., 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.  Google Scholar

[22]

C. J. Silva and D. F. M. Torres, Optimal control strategies for tuberculosis treatment: A case study in Angola, Numer. Algebra Control Optim., 2 (2012), 601-617.  doi: 10.3934/naco.2012.2.601.  Google Scholar

[23]

C. J. Silva and D. F. M. Torres, Optimal Control of Tuberculosis: A Review, Dynamics, Games and Science, CIM Series in Mathematical Sciences, 1 (2015), 701-722.  doi: 10.1007/978-3-319-16118-1_37.  Google Scholar

[24]

C. T. Sreeramareddy, K. V. Panduru, J. Menten and J. Van den Ende, Time delays in diagnosis of pulmonary tuberculosis: A systematic review of literature BMC Infectious Diseases 9 (2009), p91. doi: 10.1186/1471-2334-9-91.  Google Scholar

[25]

D. G. Storla, S. Yimer and G. A. Bjune, A systematic review of delay in the diagnosis and treatment of tuberculosis BMC Public Health 8 (2008), p15. doi: 10.1186/1471-2458-8-15.  Google Scholar

[26]

K. Toman, Tuberculosis case-finding and chemotherapy: Questions and answers, WHO Geneva, 1979. Google Scholar

[27]

P. W. Uys, M. Warren and P. D. van Helden, A threshold value for the time delay to TB diagnosis PLoS ONE 2(2007), e757. doi: 10.1371/journal.pone.0000757.  Google Scholar

[28]

P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[29]

H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1573-1582.  doi: 10.3934/dcdsb.2015.20.1573.  Google Scholar

[30]

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

[31]

Systematic Screening for Active Tuberculosis --Principles and Recommendations Geneva, World Health Organization, 2013, http://www.who.int/tb/tbscreening/en/. Google Scholar

[32]

Global Tuberculosis Report 2014 Geneva, World Health Organization, 2014, http://www.who.int/tb/publications/global_report/en/. Google Scholar

[33]

Centers for Disease and Control Prevention http://www.cdc.gov/tb/topic/treatment/ltbi.htm Google Scholar

show all references

References:
[1]

R. Bellmann and K. L. Cooke, Differential-Difference Equations Academic Press, New York, 1963.  Google Scholar

[2]

B. Buonomo and M. Cerasuolo, The effect of time delay in plant-pathogen interactions with host demography, Math. Biosci. Eng., 12 (2015), 473-490.  doi: 10.3934/mbe.2015.12.473.  Google Scholar

[3]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen PhD thesis, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. Google Scholar

[4]

C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control, J. Comput. Appl. Math., 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[5]

C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656.  doi: 10.1007/s002850050069.  Google Scholar

[6]

T. Cohen and M. Murray, Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nat. Med., 10 (2004), 1117-1121.  doi: 10.1038/nm1110.  Google Scholar

[7]

R. V. 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

[8]

J. Dieudonné, Foundations of Modern Analysis Academic Press, New York, 1960.  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, Special Issue on Computational Methods for Optimization and Control, J. Ind. Manag. Optim., 10 (2014), 413-441.  doi: 10.3934/jimo.2014.10.413.  Google Scholar

[11]

M. G. M. GomesP. RodriguesF. M. HilkerN. B. Mantilla-BeniersM. MuehlenA. C. Paulo and G. F. Medley, Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions, J. Theoret. Biol., 248 (2007), 608-617.  doi: 10.1016/j.jtbi.2007.06.005.  Google Scholar

[12]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[13]

H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[14]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics Academic Press, San Diego, 1993.  Google Scholar

[15]

M. L. Lambert and P. Van der Stuyft, Delays to tuberculosis treatment: Shall we continue to blame the victim?, Trop. Med. Int. Health, 10 (2005), 945-946.  doi: 10.1111/j.1365-3156.2005.01485.x.  Google Scholar

[16]

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

[17]

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 SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.  Google Scholar

[18]

P. RodriguesC. Rebelo and M. G. M. Gomes, Drug resistance in tuberculosis: A reinfection model, Theor. Popul. Biol., 71 (2007), 196-212.  doi: 10.1016/j.tpb.2006.10.004.  Google Scholar

[19]

P. RodriguesC. J. Silva and D. F. M. Torres, Cost-effectiveness analysis of optimal control measures for tuberculosis, Bull. Math. Biol., 76 (2014), 2627-2645.  doi: 10.1007/s11538-014-0028-6.  Google Scholar

[20]

H. SchättlerU. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with $L^2$-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657.  Google Scholar

[21]

L. F. Shampine and S. Thompson, Solving DDEs in MATLAB, Appl. Numer. Math., 37 (2001), 441-458.  doi: 10.1016/S0168-9274(00)00055-6.  Google Scholar

[22]

C. J. Silva and D. F. M. Torres, Optimal control strategies for tuberculosis treatment: A case study in Angola, Numer. Algebra Control Optim., 2 (2012), 601-617.  doi: 10.3934/naco.2012.2.601.  Google Scholar

[23]

C. J. Silva and D. F. M. Torres, Optimal Control of Tuberculosis: A Review, Dynamics, Games and Science, CIM Series in Mathematical Sciences, 1 (2015), 701-722.  doi: 10.1007/978-3-319-16118-1_37.  Google Scholar

[24]

C. T. Sreeramareddy, K. V. Panduru, J. Menten and J. Van den Ende, Time delays in diagnosis of pulmonary tuberculosis: A systematic review of literature BMC Infectious Diseases 9 (2009), p91. doi: 10.1186/1471-2334-9-91.  Google Scholar

[25]

D. G. Storla, S. Yimer and G. A. Bjune, A systematic review of delay in the diagnosis and treatment of tuberculosis BMC Public Health 8 (2008), p15. doi: 10.1186/1471-2458-8-15.  Google Scholar

[26]

K. Toman, Tuberculosis case-finding and chemotherapy: Questions and answers, WHO Geneva, 1979. Google Scholar

[27]

P. W. Uys, M. Warren and P. D. van Helden, A threshold value for the time delay to TB diagnosis PLoS ONE 2(2007), e757. doi: 10.1371/journal.pone.0000757.  Google Scholar

[28]

P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosc., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[29]

H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1573-1582.  doi: 10.3934/dcdsb.2015.20.1573.  Google Scholar

[30]

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

[31]

Systematic Screening for Active Tuberculosis --Principles and Recommendations Geneva, World Health Organization, 2013, http://www.who.int/tb/tbscreening/en/. Google Scholar

[32]

Global Tuberculosis Report 2014 Geneva, World Health Organization, 2014, http://www.who.int/tb/publications/global_report/en/. Google Scholar

[33]

Centers for Disease and Control Prevention http://www.cdc.gov/tb/topic/treatment/ltbi.htm Google Scholar

Figure 1.  Disease free equilibrium with basic reproduction number $R_0 = 0.88$ ($\beta = 40$, $d_I = 0.1$ and the other values from Table 1)
Figure 2.  Optimal control and state variables of the non-delayed TB model with $L^1$ objective (18) and weights $W_1=W_2=50$. Top row: (a) control $u_1$ (23) and (scaled) switching function $\phi_1$ (21) satisfying the control law (22) for $k=1$, (b) susceptible individuals $S$ and recovered individuals $R$, (c) infectious individuals $I$. Bottom row: (a) control $u_2$ (23) and (scaled) switching function $\phi_2$ (21) satisfying the control law (22) for $k=2$, (b) early latent $L_1$, (c) persistent latent $L_2$
Figure 3.  Comparison of controls $u_1$ and $u_2$ for the $L^1$-type objective (18) and $L^2$-type objective (19) with weights $W_1=W_2 =50$
Figure 4.  Optimal controls $u_1$ and $u_2$ for the $L^1$-type objective (18) with weights $W_1=W_2 =150$
Figure 5.  Optimal control and state variables of the delayed TB model with $L^1$-objective (18), $W_1=W_2=50$ and delays $d_I=0.1, d_{u_1}= d_{u_2}=0.2$. Top row: (a) control $u_1$ (25) and (scaled) switching function $\phi_1$ (21) satisfying the control law (22) for $k=1$, (b) susceptible individuals $S$ and recovered individuals $R$, (c) infectious individuals $I$. Bottom row: (a) control $u_2$ (25) and (scaled) switching function $\phi_2$ (21) satisfying the control law (22) for $k=2$, (b) early latent $L_1$, (c) persistent latent $L_2$
Figure 6.  Extremal controls for the delayed TB model with $L^1$ objective (18), $W_1=W_2=150$ and delays $d_I=0.1$, $d_{u_1}= d_{u_2}=0.2$. (a) control $u_1$ (25) and (scaled) switching function $\phi_1$ (21) satisfying the control law (22) for $k=1$, (b) control $u_2$ (25) and (scaled) switching function $\phi_2$ (21) satisfying the control law (22) for $k=2$
Figure 7.  Comparison of extremal controls for parameters $\beta=50$ and $\beta=150$ in the delayed TB model with $L^1$ objective (18), weights $W_1=W_2=150$ and delays $d_I=0.1, d_{u_1}= d_{u_2}=0.2$
Figure 8.  Homotopic solutions of the delayed TB model with $L^1$ objective (18) and weights $W_1=W_2=50$ for parameters $\beta \in [50,150]$. Displayed are the objective value $J_1(x,u)$ and the terminal states $S(T)$, $R(T)$, $I(T)$, $L_1(T)$, $L_2(T)$
Table 1.  Parameter values
SymbolDescriptionValue
$\beta$Transmission coefficient$\in [50,150]$
$\mu$Death and birth rate$1/70 \, yr^{-1}$
$\delta$Rate at which individuals leave $L_1$$12 \, yr^{-1}$
$\phi$Proportion of individuals going to $I$$0.05$
$\omega$Endogenous reactivation rate for persistent latent infections$0.0002 \, yr^{-1}$
$\omega_R$Endogenous reactivation rate for treated individuals$0.00002 \, yr^{-1}$
$\sigma$Factor reducing the risk of infection as a result of acquired
immunity to a previous infection for $L_2$
$0.25$
$\sigma_R$Rate of exogenous reinfection of treated patients0.25
$\tau_0$Rate of recovery under treatment of active TB$2 \, yr^{-1}$
$\tau_1$Rate of recovery under treatment of early latent individuals $L_1$$2 \, yr^{-1}$
$\tau_2$Rate of recovery under treatment of persistent latent individuals $L_2$$1 \, yr^{-1}$
$N$Total population$30,000$
$T$Total simulation duration$5$ $yr$
$\epsilon_1$Efficacy of treatment of early latent $L_1$$0.5$
$\epsilon_2$Efficacy of treatment of persistent latent TB $L_2$$0.5$
SymbolDescriptionValue
$\beta$Transmission coefficient$\in [50,150]$
$\mu$Death and birth rate$1/70 \, yr^{-1}$
$\delta$Rate at which individuals leave $L_1$$12 \, yr^{-1}$
$\phi$Proportion of individuals going to $I$$0.05$
$\omega$Endogenous reactivation rate for persistent latent infections$0.0002 \, yr^{-1}$
$\omega_R$Endogenous reactivation rate for treated individuals$0.00002 \, yr^{-1}$
$\sigma$Factor reducing the risk of infection as a result of acquired
immunity to a previous infection for $L_2$
$0.25$
$\sigma_R$Rate of exogenous reinfection of treated patients0.25
$\tau_0$Rate of recovery under treatment of active TB$2 \, yr^{-1}$
$\tau_1$Rate of recovery under treatment of early latent individuals $L_1$$2 \, yr^{-1}$
$\tau_2$Rate of recovery under treatment of persistent latent individuals $L_2$$1 \, yr^{-1}$
$N$Total population$30,000$
$T$Total simulation duration$5$ $yr$
$\epsilon_1$Efficacy of treatment of early latent $L_1$$0.5$
$\epsilon_2$Efficacy of treatment of persistent latent TB $L_2$$0.5$
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