Modeling Public Health Campaigns for Sexually Transmitted Infections via Optimal and Feedback Control

Abstract

Control of sexually transmitted infections (STIs) poses important challenges to public health authorities. Obstacles for STIs’ control include low priority in public health programs and disease transmission mechanisms. This work uses a compartmental pair model to explore different public health strategies on the evolution of STIs. Optimal control and feedback control are used to model realistic strategies for reducing the prevalence of these infections. Feedback control is proposed to model the reaction of public health authorities relative to an alert level. Optimal control is used to model the optimization of available resources for implementing strategies. Numerical simulations are performed using trichomoniasis, gonorrhea, chlamydia and human papillomavirus (HPV) as study cases. HPV is non-curable, and it is analyzed only under transmission control such as condom promotion campaigns. Trichomoniasis, gonorrhea and chlamydia are curable STIs that are modeled here additionally under treatment control. Increased cost-effectiveness ratio is employed as a criterion to measure control strategies performance. The features and drawbacks of control strategies under the pair formation process are discussed.

Appendices

Appendix A. Positively Invariant Sets

Consider the pair model (1):

$$\begin{aligned} \begin{aligned} X'_{0}&= \nu + (\sigma + \mu )(2P_{00} + P_{01}) - (\mu + \rho ) X_{0} + (\gamma + u_T) X_{1},\\ X'_{1}&= (\sigma + \mu )(2P_{11} + P_{01}) - (\mu + \rho ) X_{1} - (\gamma + u_T) X_{1},\\ P'_{00}&= \dfrac{1}{2}\rho \dfrac{X_{0}^{2}}{X} - (\sigma + 2\mu ) P_{00} + (\gamma + u_T) P_{01},\\ P'_{01}&= \rho \left( 1 - h (1 - u_C)\right) \dfrac{X_{0} X_{1}}{X} - \left( \sigma + \phi h (1 - u_C) + 2\mu \right) P_{01} \\&\quad - (\gamma + u_T) P_{01} + 2(\gamma + u_T) P_{11},\\ P'_{11}&= \dfrac{1}{2}\rho \dfrac{X_{1}^{2}}{X} + \rho h (1 - u_C) \dfrac{X_{0} X_{1}}{X} + \phi h (1 - u_C) P_{01} - (\sigma + 2\mu ) P_{11} \\&\quad - 2(\gamma + u_T) P_{11}. \end{aligned} \end{aligned}$$

By adding the equations, note that the total population size $N = X_0 + X_1 + 2(P_{00} + P_{01} + P_{11})$ satisfies $N' = -\mu N + \nu$, and thus,

$$\begin{aligned} N(t) \le N(0) e^{-\mu t} + \frac{\nu }{\mu }\left( 1 - e^{-\mu t} \right) . \end{aligned}$$

(20)

If we consider the set

$$\begin{aligned} \Omega = \{ (X_0, X_1, P_{00}, P_{01}, P_{11}) \in \mathbb {R}^5_+\ |\ X_0 + X_1 + 2(P_{00} + P_{01} + P_{11}) \le \nu /\mu \}, \end{aligned}$$

then, from (20), we get that if $N(0) \in \Omega$ then $N(t) \in \Omega$ for all $t > 0$. We say that $\Omega$ is a positively invariant set under (1).

Appendix B. Existence of Solutions to the Optimal Control Problem

To prove existence of solutions for the only-treatment optimal control problem, we use Theorem 4.1 and Corollary 4.1 from Fleming and Rishel (1975, Chapter III, Section 4). Such result requires the following:

• 1. The set of solutions of system (6) (called admissible pairs) is not empty.

• 2. The set of admissible controls, i.e., functions u satisfying the control conditions (2), is closed and convex.

• 3. The right-hand side of system (6) is continuous, bounded from above by a sum of the states and the control, and it can be written as a linear function of the control.

• 4. Finally, the integrand of (13) is convex in the control, and it is bounded below by $c_1|u|^{g} - c_2$ with $c_1 > 0$ and $g > 1$.

We can see that $u_T \equiv 0$ is an admissible solution, so the set of admissible pairs is not empty. Since $u_T \in D(t_f)$ (see Sect. 3 to recall the definition of the set D), the set of admissible controls is closed and convex. Note that the supersolutions of (6), which we are going to denote by $\hat{X}_1$, $\hat{P}_{01}$ and $\hat{I}$, satisfy the following ODE system:

$$\begin{aligned} \begin{aligned} \widehat{X_{1}}'&= (\sigma +\mu )\widehat{I},\\ \widehat{P_{01}}'&= \rho \widehat{X_{1}}+M \widehat{I},\\ \widehat{I}'&= \rho \widehat{X_{1}}+\phi \widehat{P_{01}}, \end{aligned} \end{aligned}$$

(21)

which is a linear system. Thus, the solutions of system (21) are uniformly bounded for any finite time interval $[0,t_f]$. Let us define

$$\begin{aligned} f(X_1, P_{01}, I) = \begin{pmatrix} (\sigma +\mu )I-(2\mu +\rho +\sigma )X_{1}-(\gamma +u_T) X_{1}\\ \rho (1-h)X_{1}\left( 1-\dfrac{X_{1}}{X^{*}}\right) -(\sigma +\phi h+2\mu )P_{01}+(\gamma +u_T) (I-X_{1}-2P_{01})\\ \rho hX_{1}\left( 1-\dfrac{X_{1}}{X^{*}}\right) +\phi hP_{01}-\mu I-(\gamma +u_T) I \end{pmatrix}. \end{aligned}$$

It is straightforward to note that there exists a function $g(X_1, P_{01}, I)$ such that

$$\begin{aligned} f(X_1, P_{01}, I) = g(X_1, P_{01}, I) + u_T \begin{pmatrix} - X_1 \\ I - X_1 - 2 P_{01} \\ - I \end{pmatrix} \end{aligned}$$

and so the right-hand side of (6) can be written as a linear function of $u_T$. Also, we already have satisfied the continuity of the model. Finally, using the supersolutions system (21) we note that

$$\begin{aligned} \begin{aligned} \Vert f(X_1, P_{01}, I)\Vert&\le \left\| \begin{pmatrix} 0 &{} 0 &{} \sigma + \mu \\ \rho &{} 0 &{} M \\ \rho &{} \phi &{} P_{01} \end{pmatrix} \begin{pmatrix} X_1 \\ P_{01} \\ I \end{pmatrix} + u_T \begin{pmatrix} - X_1 \\ I - X_1 - 2 P_{01} \\ - I \end{pmatrix} \right\| \\&\le C \left( \Vert (X_1,P_{01},I)\Vert + \Vert u_T\Vert \right) \end{aligned} \end{aligned}$$

where C is a constant that depends on the model parameters. Thus, f is bounded from above by a sum of the states and the control. The integrand is $h(I,u_T) = I + B u_T^2$ and so $h(I,u) = I + B u_T^2 \ge B u_T^2$, so choosing $c_2 = 0$, $c_1 = B > 0$ and $g = 2$ we have the following:

Theorem 3

There exist an optimal control $u_T$ and state variables $(X_1, P_{01}, I)$ that minimize the objective functional (13) and satisfy system (6). $\square$

Appendix C. Optimality System for the Only-Treatment Model

The corresponding optimality system for the only-treatment control strategy ($u_C \equiv 0$) is given by:

$$\begin{aligned} u_T&= \min \left\{ M_T, \max \left\{ 0, \frac{X_1 \lambda _1 + \lambda _2 (2P_{01} - I + X_1) + \lambda _3 I}{2B_T} \right\} \right\} ,\\ X'_{1}&=(\sigma +\mu )I-(2\mu +\rho +\sigma )X_{1}-(\gamma + u_T) X_{1},\\ P'_{01}&=\rho (1-h)X_{1}\left( 1-\dfrac{X_{1}}{X^{*}}\right) -(\sigma +\phi h+2\mu )P_{01}+(\gamma + u_T) (I-X_{1}-2P_{01}),\\ I'&=\rho hX_{1}\left( 1-\dfrac{X_{1}}{X^{*}}\right) +\phi hP_{01}-\mu I-(\gamma + u_T) I,\\ \lambda '_1&= \left( 2 \mu +\rho +\sigma + \gamma + u_T \right) \lambda _{1}\\&\quad +\left( \gamma + u_T - \rho (1 - h) \left( 1 - \frac{2 X_{1} }{X^*}\right) \right) \lambda _{2} - h \rho \left( 1 - \frac{2 X_{1} }{X^*} \right) \lambda _{3},\\ \lambda '_2&= \left( 2 \mu + \sigma + 2 (\gamma + u_T) +h \phi \right) \lambda _{2}-h \phi \lambda _{3},\\ \lambda '_3&= - \left( \mu +\sigma \right) \lambda _{1} - (\gamma + u_T) \lambda _{2} + \left( \mu + \gamma + u_T \right) \lambda _{3} - 1,\\&\quad X_1(0),\ P_{01}(0),\ I(0) \ \text {given}, \ \lambda _1(t_f) = \lambda _2(t_f) = \lambda (t_f) = 0. \end{aligned}$$

Appendix D. Optimality System for the Only-Condom Promotion Model

The corresponding optimality system for the only-condom promotion control strategy ($u_T \equiv 0$) is given by:

$$\begin{aligned} u_C&= \min \left\{ M_C, \max \left\{ 0, \frac{ \lambda _2 \left( X_1 h \rho \left( \frac{X_1}{X^*} - 1\right) - P_{01} h \phi \right) + \lambda _3 \left( X_1 h \rho \left( \frac{X_1}{X^*} - 1\right) - P_{01} h \phi \right) }{2B_C} \right\} \right\} ,\\ X'_{1}&= (\sigma +\mu )I - (2\mu +\rho +\sigma )X_{1} - \gamma X_{1},\\ P'_{01}&= \rho \left( 1 - h(1 - u_C)\right) X_{1}\left( 1-\dfrac{X_{1}}{X^{*}}\right) -\left( \sigma +\phi h(1 - u_C)+2\mu \right) P_{01} + \gamma (I-X_{1}-2P_{01}),\\ I'&=\rho h(1 - u_C)X_{1}\left( 1-\dfrac{X_{1}}{X^{*}}\right) +\phi h(1 - u_C)P_{01}-\mu I - \gamma I,\\ \lambda '_1&= \left( 2 \mu +\rho +\sigma + \gamma \right) \lambda _{1} +\left( \gamma - \rho \left( 1 - h(1 - u_C)\right) \left( 1 - \frac{2 X_{1} }{X^*}\right) \right) \lambda _{2} \\&\quad - h (1 - u_C) \rho \left( 1 - \frac{2 X_{1} }{X^*} \right) \lambda _{3},\\ \lambda '_2&= \left( 2 \mu + \sigma + 2 \gamma + h (1 - u_C) \phi \right) \lambda _{2} - h (1 - u_C) \phi \lambda _{3},\\ \lambda '_3&= - \left( \mu +\sigma \right) \lambda _{1} - \gamma \lambda _{2} + \left( \mu + \gamma \right) \lambda _{3} - 1,\\&X_1(0),\ P_{01}(0),\ I(0) \ \text {given}, \ \lambda _1(t_f) = \lambda _2(t_f) = \lambda (t_f) = 0. \end{aligned}$$