Dynamic Drug Policy: Optimally Varying the Mix of Treatment, Price-Raising Enforcement, and Primary Prevention Over Time

Abstract

A central question in drug policy is how control efforts should be divided among enforcement, treatment, and prevention. Of particular interest is how the mix should vary dynamically over the course of an epidemic. Recent work considered how various pairs of these interventions interact. This paper considers all three simultaneously in a dynamic optimal control framework, yielding some surprising results. Depending on epidemic parameters, one of three situations pertains. It may be optimal to eradicate the epidemic, to “accommodate” it by letting it grow, or to eradicate if control begins before drug use passes a DNSS threshold but accommodate if control begins later. Relatively modest changes in parameters such as the perceived social cost per unit of drug use can push the model from one regime to another, perhaps explaining why opinions concerning proper policy diverge so sharply. If eradication is pursued, then treatment and enforcement should be funded very aggressively to reduce use as quickly as possible. If accommodation is pursued then spending on all three controls should increase roughly linearly but less than proportionally with the size of the epidemic. With the current parameterization, optimal spending on prevention varies the least among the three types of control interventions.

Appendix: Optimality Conditions

The current value Hamiltonian H is given by

$$\displaystyle{H = -(\kappa \theta Ap^{-\omega } + u + v + w) +\lambda (kA^{\alpha }p^{-a}\varPsi - c\beta A -\mu p^{b}A),}$$

where λ describes the current-value costate variable.

Note that it is not necessary to formulate the maximum principle for the Lagrangian, which incorporates the non-negativity constraints for the controls, since u, v, and w all turn out to be positive in the analysis described in this paper.

According to Pontryagin’s maximum principle we have the following three necessary optimality conditions:

$$\displaystyle\begin{array}{rcl} & u =\mathrm{ arg}\max _{u}H,& {}\\ & v =\mathrm{ arg}\max _{v}H,& {}\\ \end{array}$$

and

$$\displaystyle{w =\mathrm{ arg}\max _{w}H.}$$

Due to the concavity of the Hamiltonian H with respect to (u, v, w), setting the first order partial derivatives equal to zero leads to the unrestricted extremum, and we get the following expressions for the costate λ:

$$\displaystyle\begin{array}{rcl} H_{u} = 0 \Rightarrow & \lambda = \frac{-1} {c\beta _{u}A},&{}\end{array}$$

(1)

$$\displaystyle\begin{array}{rcl} H_{w} = 0 \Rightarrow & \lambda = \frac{1} {kp^{-a}A^{\alpha }\varPsi _{w}},&{}\end{array}$$

(2)

$$\displaystyle\begin{array}{rcl} H_{v} = 0 \Rightarrow & \lambda = \frac{1-\kappa \theta \omega p^{-\omega -1}p_{ v}A} {-akp^{-a-1}p_{v}A^{\alpha }\varPsi -\mu bp^{b-1}p_{v}A},&{}\end{array}$$

(3)

where subscripts denote derivatives w.r.t. the corresponding variables.

The concavity of the maximized Hamiltonian with respect to the state variable, however, cannot be guaranteed, so the usual sufficiency conditions are not satisfied.

With Eqs. (1)–(3) we can describe u, w, and λ as functions of A and v as follows:

$$\displaystyle\begin{array}{rcl} & \lambda (A,v):= \frac{\frac{p_{v}} {p} \left ( \frac{a} {m}+\kappa \theta \omega p^{-\omega }A\right )-1} {ahkp^{-a-1}p_{v}A^{\alpha }+\mu bp^{b-1}p_{v}A}, & \\ & u(A,v):= \left ( \frac{-(A+\delta )^{z}} {czA\lambda (A,v)}\right )^{ \frac{1} {z-1} }, & \\ & w(A,v):= \frac{1} {m}\mathrm{ln}\left ((h - 1)kmp^{-a}A^{\alpha }\lambda (A,v)\right ).&{}\end{array}$$

(4)

Due to this simplification we can concentrate on the two variables A and v.

To gain an equation for $\dot{v}$ we differentiate λ(A, v) with respect to time:

$$\displaystyle\begin{array}{rcl} \dot{\lambda } =\lambda _{A}\dot{A} +\lambda _{v}\dot{v}.& &{}\end{array}$$

(5)

Setting (5) equal to the costate equation

$$\displaystyle{\dot{\lambda }= r\lambda - H_{A},}$$

yields:

$$\displaystyle{\dot{v} = \frac{r\lambda - H_{A} -\lambda _{A}\dot{A}} {\lambda _{v}},}$$

where we insert λ(A, v) from (4) and the corresponding derivatives λ _A and λ _v as well as H _A given by

$$\displaystyle{\begin{array}{rcl} H_{A}& =& -\kappa \theta p^{-\omega -1}(p -\omega p_{ A}A) +\lambda [kp^{-a-1}\varPsi (\alpha A^{\alpha -1}p - aA^{\alpha }p_{ A}) - \\ & &\quad - c(\beta _{A}A+\beta ) -\mu p^{b-1}(bp_{A}A + p)]. \end{array} }$$