Optimal control of a Tuberculosis model with state and control delays

Cristiana J. Silva; Helmut Maurer; Delfim F. M. Torres

doi:10.3934/mbe.2017021

Article Contents

2017, Volume 14, Issue 1: 321-337. Doi: 10.3934/mbe.2017021

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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
* Corresponding author: maurer@math.uni-muenster.de

Received: October 30, 2015

Accepted: June 25, 2016

Published: September 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34D30, 92D30; Secondary: 49M05, 93A30.

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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)

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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 $L_2$

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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 $W_1=W_2 =50$

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Figure 4. Optimal controls $u_1$ and $u_2$ for the $L^1$-type objective (18) with weights $W_1=W_2 =150$

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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 $L_2$

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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 $k=2$

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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 $d_I=0.1, d_{u_1}= d_{u_2}=0.2$

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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 $S(T)$, $R(T)$, $I(T)$, $L_1(T)$, $L_2(T)$

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Table 1. Parameter values

Symbol	Description	Value
$\beta$	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 patients	0.25
$\tau_0$	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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