Cost-Effectiveness Analysis of Optimal Control Measures for Tuberculosis

Abstract

We propose and analyze an optimal control problem where the control system is a mathematical model for tuberculosis that considers reinfection. The control functions represent the fraction of early latent and persistent latent individuals that are treated. Our aim was to study how these control measures should be implemented, for a certain time period, in order to reduce the number of active infected individuals, while minimizing the interventions implementation costs. The optimal intervention is compared along different epidemiological scenarios, by varying the transmission coefficient. The impact of variation of the risk of reinfection, as a result of acquired immunity to a previous infection for treated individuals on the optimal controls and associated solutions, is analyzed. A cost-effectiveness analysis is done, to compare the application of each one of the control measures, separately or in combination.

Appendices

Appendix 1: Proof of Theorem 3.1

The Hamiltonian $H$ associated with the problem (1)–(3) is given by

$$\begin{aligned} H&= H(S(t), L_1(t), I(t), L_2(t), R(t), \lambda (t), u_1(t), u_2(t)) \\&= W_0 I(t) + \frac{W_1}{2}u_1^2(t) + \frac{W_2}{2}u_2^2(t) \\&+\, \lambda _1(t) \left( \mu N - \frac{\beta }{N} I(t) S(t) - \mu S(t) \right) \\&+\, \lambda _2(t) \left( \frac{\beta }{N} I(t)\left( S(t) + \sigma L_2(t) + \sigma _R R(t)\right) - (\delta + \tau _1 u_1(t) + \mu )L_1(t) \right) \\&+\, \lambda _3(t) \left( \phi \delta L_1(t) + \omega L_2(t) + \omega _R R(t) - (\tau _0 + \mu ) I(t) \right) \\&+ \,\lambda _4(t) \left( (1 - \phi ) \delta L_1(t) - \sigma \frac{\beta }{N} I(t) L_2(t) - (\omega + \tau _2 u_2(t) + \mu )L_2(t) \right) \\&+\, \lambda _5(t) \left( \tau _0 I(t) + \tau _1 u_1(t) L_1(t) + \tau _2 u_2(t) L_2(t) - \sigma _R \frac{\beta }{N} I(t) R(t) - (\omega _R + \mu )R(t) \right) , \end{aligned}$$

where $\lambda (t) = \left( \lambda _1(t), \lambda _2(t), \lambda _3(t), \lambda _4(t), \lambda _5(t)\right)$ is the adjoint vector. According to the Pontryagin maximum principle (Pontryagin et al. 1962), if $(u_1^*(\cdot ), u_2^*(\cdot )) \in \Omega$ is optimal for problem (1)–(3) with the initial conditions given in Table 2 and fixed final time $t_f$, then there exists a non-trivial absolutely continuous mapping $\lambda : [0, t_f] \rightarrow \mathbb {R}^5$, $\lambda (t) = \left( \lambda _1(t), \lambda _2(t), \lambda _3(t), \lambda _4(t), \lambda _5(t)\right)$, such that

$$\begin{aligned} \dot{S} = \frac{\partial H}{\partial \lambda _1}, \quad \dot{L}_1= \frac{\partial H}{\partial \lambda _2}, \quad \dot{I}= \frac{\partial H}{\partial \lambda _3}, \quad \dot{L}_2 = \frac{\partial H}{\partial \lambda _4}, \quad \dot{R} = \frac{\partial H}{\partial \lambda _5} \end{aligned}$$

and

$$\begin{aligned} \dot{\lambda }_1 = -\frac{\partial H}{\partial S}, \quad \dot{\lambda }_2 = -\frac{\partial H}{\partial L_1}, \quad \dot{\lambda }_3 = -\frac{\partial H}{\partial I}, \quad \dot{\lambda }_4 = -\frac{\partial H}{\partial L_2}, \quad \dot{\lambda }_5 = -\frac{\partial H}{\partial R}. \end{aligned}$$

(9)

The minimality condition

$$\begin{aligned}&H(S^*(t), L_1^*(t), I^*(t), L_2^*(t), R^*(t), \lambda ^*(t), u_1^*(t), u_2^*(t))\nonumber \\&\quad =\, \min _{0 \le u_1, u_2 \le 1} H(S^*(t), L_1^*(t), I^*(t), L_2^*(t), R^*(t), \lambda ^*(t), u_1, u_2) \end{aligned}$$

(10)

holds almost everywhere on $[0, t_f]$. Moreover, the transversality conditions

$$\begin{aligned} \lambda _i(t_f) = 0, \quad i =1,\ldots , 5, \end{aligned}$$

hold.

Lemma

For problem (1)–(3) with fixed initial conditions $S(0)$, $L_1(0)$, $I(0)$, $L_2(0)$ and $R(0)$ and fixed final time $t_f$, there exists adjoint functions $\lambda _1^*(\cdot )$, $\lambda _2^*(\cdot )$, $\lambda _3^*(\cdot )$, $\lambda _4^*(\cdot )$ and $\lambda _5^*(\cdot )$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\lambda ^*_1}(t) = \lambda ^*_1(t) \left( \frac{\beta }{N} I^*(t) + \mu \right) - \lambda ^*_2(t) \frac{\beta }{N} I^*(t)\\ \dot{\lambda ^*_2}(t) = \lambda ^*_2(t)\left( \delta + \tau _1 + \mu \right) - \lambda ^*_3(t) \phi \delta - \lambda ^*_4(t) (1 - \phi ) \delta - \lambda ^*_5(t)\tau _1 u^*_1(t) \\ \dot{\lambda ^*_3}(t) = -W_0 + \lambda ^*_1(t) \frac{\beta }{N} S^*(t) - \lambda ^*_2(t) \frac{\beta }{N}(S^*(t) + \sigma L_2^*(t) + \sigma _R R^*(t)) \\ \qquad \qquad \quad +\,\lambda ^*_3(t) \left( \tau _0 + \mu \right) +\lambda ^*_4(t)\sigma \frac{\beta }{N} L_2^*(t) - \lambda ^*_5(t)\left( \tau _0 - \sigma _R \frac{\beta }{N} R^*(t) \right) \\ \dot{\lambda ^*_4}(t) = - \lambda ^*_2(t) \frac{\beta }{N}I^*(t) \sigma - \lambda ^*_3(t) \omega + \lambda ^*_4(t)\left( \sigma \frac{\beta }{N} I^*(t) + \omega + \tau _2 u^*_2(t) + \mu \right) \\ \qquad \qquad \quad -\,\lambda ^*_5(t)\left( \tau _2 u^*_2(t) \right) \\ \dot{\lambda ^*_5}(t) = -\lambda ^*_2(t) \sigma _R \frac{\beta }{N}I^*(t) - \lambda ^*_3(t) \omega _R + \lambda ^*_5(t)\left( \sigma _R \frac{\beta }{N} I^*(t) +\omega _R + \mu \right) , \end{array}\right. } \end{aligned}$$

(11)

with transversality conditions

$$\begin{aligned} \lambda ^*_i(t_f) = 0, \quad i=1, \ldots , 5. \end{aligned}$$

Furthermore,

$$\begin{aligned} u_1^*(t)&= \min \left\{ \max \left\{ 0, \frac{\tau _1 L_1^* \left( \lambda ^*_2 - \lambda ^*_5\right) }{W_1}\right\} , 1 \right\} ,\nonumber \\ u_2^*(t)&= \min \left\{ \max \left\{ 0, \frac{\tau _2 L^*_2 \left( \lambda ^*_4 - \lambda ^*_5\right) }{W_2}\right\} , 1 \right\} . \end{aligned}$$

(12)

Proof

System (11) is derived from the Pontryagin maximum principle (see (9), Pontryagin et al. 1962) and the optimal controls (12) come from the minimality condition (10). For small final time $t_f$, the optimal control pair given by (12) is unique due to the boundedness of the state and adjoint functions and the Lipschitz property of systems (1) and (11) (see Jung et al. 2002 and references cited therein). $\square$

Proof of Theorem 3.1

Existence of an optimal solution $\left( S^*, L_1^*, I^*, L_2^*, R^*\right)$ associated with an optimal control pair $\left( u_1^*, u_2^*\right)$ comes from the convexity of the integrand of the cost functional $\mathcal {J}$ with respect to the controls $(u_1, u_2)$ and the Lipschitz property of the state system with respect to state variables $\left( S, L_1, I, L_2, R\right)$ (see, e.g., Cesari 1983; Fleming and Rishel 1975). For small final time $t_f$, the optimal control pair is given by (12) that is unique by the Lemma above. Because the problem (1)–(3) is autonomous, uniqueness is valid for any time $t_f$ and not only for small time $t_f$. $\square$

Appendix 2: Sensitivity Analysis to the Duration of Intervention $t_f$

We fix $\beta =100$ and $\sigma _R=\sigma$ and the remaining parameters according to Table 1 and vary $t_f$. Results for the proportion of infectious individuals are shown in the Fig. 6. The general behavior do not change significantly with $t_f$. The proportion of infected individuals slightly increases toward the end of the intervention for $t_f>7$. This tendency is more pronounced for higher $t_f$.

Fig. 6

Proportion of infectious individuals for the optimal solution $I(t)$ with $t_f \in \{5, 7, 10, 12, 15, 17, 20, 22, 25 \}$. Parameters according to Table 1, $\beta =100$ and $\sigma _R=\sigma$

Appendix 3: Sensitivity Analysis to the Weight Constants on the Objective Functional $\mathcal {J}$

Figure 7 shows the results for different combination of the weight constants on the objective functional $\mathcal {J}$. We fix $\beta =100$ and $\sigma _R=\sigma$ and the remaining parameters according to Table 1 and vary $W_0$, $W_1$ and $W_2$. Efficacy decreases when the costs $W_1$ and $W_2$ increase, corresponding to an earlier relaxation of the intensity of treatment $(u_1(t), u_2(t))$ in the optimal solution due to cost restrictions. The change in efficacy is more pronounced for the cases where the weight associated with infectious individuals $W_0$ change in comparison with the weights associated with the controls $W_1=W_2$ (Fig. 7a, b). Results are less sensitive to the variation between the weight controls $W_1$ and $W_2$ (Fig. 7c, d).

Fig. 7

Sensitivity analysis to the weight constants on the objective functional $\mathcal {J}$. a $W_0=50$ and $W_1=W_2=5, 25, 50, 100, 200, 500$. b $W_1=W_2=50$ and $W_0=5, 25, 50, 100, 200, 500$. c $W_0=W_2=50$ and $W_1=5, 25, 50, 100, 200, 500$. d $W_0=W_1=50$ and $W_2=5, 25, 50, 100, 200, 500$