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Math Biosci. Author manuscript; available in PMC 2011 December 9.

Published in final edited form as:

Published online 2010 January 4. doi: 10.1016/j.mbs.2009.12.006

PMCID: PMC3235175

NIHMSID: NIHMS168176

Benjamin Armbruster: moc.liamg@bretsurbmra; Margaret L. Brandeau: ude.drofnats@uaednarb

The publisher's final edited version of this article is available at Math Biosci

See other articles in PMC that cite the published article.

Chronic viral diseases such as human immunodeficiency virus (HIV) and hepatitis B virus (HBV) afflict millions of people worldwide. A key public health challenge in managing such diseases is identifying infected, asymptomatic individuals so that they can receive antiviral treatment. Such treatment can benefit both the treated individual (by improving quality and length of life) and the population as a whole (through reduced transmission). We develop a compartmental model of a chronic, treatable infectious disease and use it to evaluate the cost and effectiveness of different levels of screening and contact tracing. We show that: 1) the optimal strategy is to get infected individuals into treatment at the maximal rate until the incremental health benefits balance the incremental cost of controlling the disease; 2) as one reduces the disease prevalence by moving people into treatment (which decreases the chance that they will infect others), one should increase the level of contact tracing to compensate for the decreased effectiveness of screening; 3) as the disease becomes less prevalent, it is optimal to spend more per case identified; and 4) the relative mix of screening and contact tracing at any level of disease prevalence is such that the marginal efficiency of contact tracing (cost per infected person found) equals that of screening if possible (e.g., when capacity limitations are not binding). We also show how to determine the cost-effective equilibrium level of disease prevalence (among untreated individuals), and we develop an approximation of the path of the optimal prevalence over time. Using this, one can obtain a close approximation of the optimal solution without having to solve an optimal control problem. We apply our methods to an example of hepatitis B virus.

Chronic viral diseases such as hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV) infect millions of people worldwide each year (CDC, 2008c; UNAIDS, 2006). Many such infections have no cure and, if untreated, can lead to disease, disability, and death. However, for HBV, HCV, HIV, and other chronic viral infections, antiviral treatments can often extend life, improve quality of life, and in some cases reduce a treated individual's infectivity. Such treatments thus can provide benefit both to the treated individual and to the population as a whole (through reduced transmission). In recent years, the World Health Organization and other health organizations have devoted significant resources to scaling up HIV treatment worldwide (UNAIDS, 2005). Similar efforts are underway, on a smaller scale, to increase treatment of individuals infected with HBV and HCV (AB 158, 2008).

A key public health challenge in managing chronic viral diseases is identifying infected, asymptomatic individuals so that they can receive treatment. Individuals identified before symptoms develop typically derive greater benefit from treatment than individuals who receive treatment only in an advanced stage of disease. For example, highly active antiretroviral therapy for HIV, when given before the development of AIDS, significantly extends length and quality of life (Freedberg et al., 2001). As another example, early management and treatment of chronic HBV infection can provide significant health benefits for treated individuals (Hutton et al., 2007; Brooks et al., 2001; Kanwal et al., 2005; Shepherd et al., 2006).

Two key means of identifying chronically infected, asymptomatic individuals are screening and contact tracing. Screening (via a blood test) may take place during routine care, or may be part of a targeted campaign. In contact tracing (also known as partner notification), the contacts of a newly identified infected individual (a so-called “index case”) are located and then screened for the infection. Contact tracing is typically more expensive per case found than screening, but can be an effective means of identifying infected individuals.

In this paper we consider the optimal mix of screening and contact tracing for a chronic viral disease. The goal is to maximize net health benefit in the population, which we define as the value of (quality-adjusted) life years experienced minus the cost of screening and contact tracing.

Analyses of optimal resource allocation for disease control date back to the late 1960s. Taylor (1968) focused on bovine virus diarrhea and Revelle et al. (1969) focused on tuberculosis control measures. Sanders (1971) and Sethi (1974) applied optimal control to curable diseases. Their models bear some similarity to our model except that our model is complicated by the presence of additional states, the lack of a cure, and contact tracing as a second control in addition to screening. Armbruster and Brandeau (2007b) used an optimal control approach to find the best combination of screening and contact tracing to control an infectious disease. They examined a curable endemic disease (instead of a chronic one as we do here) and assumed that contact tracing has a fixed efficacy, if it is performed at all. Here we allow the investment in contact tracing (and thus the resulting efficacy) to vary, similar to Armbruster and Brandeau (2007a) who performed a simulation study of contact tracing for a curable endemic disease.

Section 2 presents our model and describes its solution. In section 2.1, we determine the optimal rate of individuals entering treatment and then in section 2.2, we determine the optimal mix of screening and contact tracing. In section 2.3, we describe how the optimal solution, including the optimal equilibrium, can be calculated numerically. In section 2.4, we develop a useful approximation which we use to analytically characterize the optimal trajectory of control and generate insight. We present a numerical example in section 3 and conclude with discussion in section 4.

Consider a population in which a chronic viral disease is spreading. Susceptible individuals acquire the infection through contact with infected individuals. Once infected, individuals are initially asymptomatic. Their immune system may resolve the infection, in which case they are immune from further infection. Otherwise, the infection becomes chronic (i.e., not resolved). Chronically infected individuals learn of their infection either through the development of symptoms or via a screening test. Individuals whose infection has been identified enter treatment. All infected individuals are infectious; those in treatment may be less infectious. We allow for the possibility that some chronically infected individuals in treatment can recover from the infection.

We model the disease using an epidemic model, figure 1, with four compartments: *S*(*t*) susceptible individuals, *IU*(*t*) infected but untreated individuals, *IT*(*t*) infected and treated individuals, and *R*(*t*) recovered individuals who are now immune from infection. We define ** s**(

We consider two interventions: a screening program that screens individuals not in treatment at a rate *λ*(*t*), and a contact tracing program implemented at a rate that finds an average of *κ*(*t*) secondary infections per otherwise reported index case. We let *u*(*t*) be the rate at which people enter treatment (i.e., the rate of movement from state *IU* to state *IT*). This is the rate at which we identify new index cases (found through screening and symptom development) and their contacts (found through contact tracing): thus,
$u(t)=\left(\frac{\lambda (t)}{S(t)+IU(t)+R(t)}+\gamma \right)IU(t)(1+\kappa (t))$. In solving the problem, we find it convenient to first determine the optimal total level of control *u**(*t*), and then the optimal values of *λ*(*t*) and *κ*(*t*) given *u**(*t*). For simplicity in notation we drop the dependence on *t* in the state and control variables (** s**,

The dynamics of the model are as follows:

$$\begin{array}{cc}\dot{S}={f}_{1}(\mathit{\text{s}},u):={\eta}_{S}-{\mu}_{S}S& \phantom{\rule{3em}{0ex}}-S\frac{{\beta}_{IU}IU+{\beta}_{IT}IT}{S+IU+IT+R}\end{array}$$

(1)

$$\begin{array}{cc}\stackrel{.}{IU}={f}_{2}(\mathit{\text{s}},u):={\eta}_{IU}-{\mu}_{IU}IU-\rho IU& \phantom{\rule{3em}{0ex}}+S\frac{{\beta}_{IU}IU+{\beta}_{IT}IT}{S+IU+IT+R}-u\end{array}$$

(2)

$$\begin{array}{ccc}\stackrel{\cdot}{IT}={f}_{3}(\mathit{\text{s}},u):={\eta}_{IT}-{\mu}_{IT}IT& \phantom{\rule{1em}{0ex}}-\omega IT& \phantom{\rule{2em}{0ex}}+u\end{array}$$

(3)

$$\dot{R}={f}_{4}(\mathit{\text{s}},u):={\eta}_{R}-{\mu}_{R}R+\rho IU+\omega IT$$

(4)

$$u=\left(\frac{\lambda}{S+IU+R}+\gamma \right)IU(1+\kappa ).$$

(5)

We impose three constraints on this model: the size of each intervention must be nonnegative, *λ* ≥ 0 and *κ* ≥ 0, and the rate at which people can be identified for treatment is bounded from above, *u* ≤ *ū*. Here *ū* is the capacity limit for putting new patients on treatment. From the fact that ≥ 0 whenever *S* = 0 (and similarly for *IU, IT* and *R*), it follows that *s* ≥ 0.

We now develop the objective function for the problem. We use a standard cost-effectiveness framework (Gold et al., 1996) and consider both the costs and benefits of disease control. The costs comprise the costs of screening, contact tracing, and health care (including both normal health care costs and the costs of treatment for the disease); the benefits are a function of quality-adjusted life years lived. We define *c _{s}* as the cost of screening one person;

$$J(\mathit{\text{s}},\kappa ,\lambda ,\mathit{\text{u}}):=\mathit{\text{s}}\cdot \mathit{\text{v}}-\lambda {c}_{s}-\left(\frac{\lambda}{S+IU+R}+\gamma \right)IU{c}_{c}(\kappa )-u{c}_{t}.$$

(6)

The first cost term is the cost of screening (at rate *λ*); the second cost term is the cost of contact tracing (the cost to trace, at intensity *κ*, all contacts found by screening and by symptom development); and the third cost term is the cost of treating all newly identified index cases and their contacts, who are found at total rate *u* through screening, symptom development, and contact tracing. Our goal is to determine the mix of screening and contact tracing that will maximize this net benefit over time.

Assuming that the epidemic is initially in state *s*_{0}, we want to optimize the infinite-horizon net present value of our policy with a discount rate of *r*. This can be written as an optimal control problem with value function *V*(** s**):

$$\begin{array}{ll}\multicolumn{2}{c}{V({\mathit{\text{s}}}_{0}):=\underset{\kappa (t),\lambda (t),u(t)}{max}{\int}_{0}^{\infty}{e}^{-rt}J(\mathit{\text{s}}(t),\kappa (t),\lambda (t),u(t))\text{d}t}\\ \hfill & \phantom{\rule{7em}{0ex}}\dot{\mathit{\text{s}}}(t)=\mathit{\text{f}}(\mathit{\text{s}}(t),u(t)),\phantom{\rule{1em}{0ex}}\mathit{\text{s}}(0)={\mathit{\text{s}}}_{0},\hfill \\ \hfill & \phantom{\rule{7em}{0ex}}u(t)=\left(\frac{\lambda (t)}{S(t)+IU(t)+R(t)}+\gamma \right)IU(t)(1+\kappa (t)),\hfill \\ \hfill & \phantom{\rule{7em}{0ex}}0\le \kappa (t),0\le \lambda (t),u(t)\le \overline{u}.\hfill \end{array}$$

(7)

We solve this problem by first determining the optimal rate *u**(*t*) at which people enter treatment (section 2.1) and then we determine the cost-minimizing mix of screening and contact tracing, *λ**(*t*) and *κ**(*t*) to achieve *u**(*t*) (section 2.2).

Let *ϕ* := (*ϕ _{S}, ϕ_{IU}, ϕ_{IT}, ϕ_{R}*) be our vector of adjoint variables and

$$\begin{array}{l}{\kappa}^{\ast}(t),{\lambda}^{\ast}(t),{u}^{\ast}(t)\in arg\underset{\kappa ,\lambda ,u}{max}H({s}^{\ast}(t),{\phi}^{\ast}(t),\kappa ,\lambda ,u)\hfill \\ \phantom{\rule{9em}{0ex}}s.t.\phantom{\rule{0.3em}{0ex}}u=\left(\frac{\lambda}{{S}^{\ast}(t)+I{U}^{\ast}(t)+{R}^{\ast}(t)}+\gamma \right)I{U}^{\ast}(t)(1+\kappa )\hfill \\ \phantom{\rule{11em}{0ex}}0\le \kappa ,0\le \lambda ,u\le \overline{u}.\hfill \end{array}$$

(8)

Rewriting the above equality constraint, we have

$$\lambda =({S}^{\ast}(t)+I{U}^{\ast}(t)+{R}^{\ast}(t))\left(\frac{u}{I{U}^{\ast}(t)(\kappa +1)}-\gamma \right).$$

(9)

Substituting (9) into (8) we obtain an optimization problem with only *κ* and *u* as the decision variables. Then dropping the terms that do not depend on *u* and *κ*, and simplifying, we obtain

$$\begin{array}{l}{\kappa}^{\ast}(t),{u}^{\ast}(t)\in arg\underset{\kappa ,u}{max}u\cdot h({s}^{\ast}(t),{\phi}^{\ast}(t),\kappa )\\ \phantom{\rule{8em}{0ex}}s.t.\phantom{\rule{0.3em}{0ex}}0\le \kappa ,\phantom{\rule{0.3em}{0ex}}I{U}^{\ast}(t)\gamma (\kappa +1)\le u\le \overline{u},\end{array}$$

(10)

where

$$h(\mathit{\text{s}},\phi ,\kappa ):={\phi}_{IT}-{\phi}_{IU}-{c}_{t}-\frac{{c}_{c}(\kappa )+{c}_{s}/p(\mathit{\text{s}})}{\kappa +1}\phantom{\rule{0.3em}{0ex}}\text{and}$$

(11)

$$p(\mathit{\text{s}}):=IU/(S+IU+R).$$

(12)

The term *h*(* s, ϕ, κ*) represents the marginal benefit of moving an infected individual into treatment including the costs of finding and treating the individual while holding the contact tracing effort constant. The term

Optimizing (10) first over *u*, we determine that *IU**(*t*)*γ*(*κ* + 1) ≤ *ū* is required for feasibility and that a bang-bang solution,

$${u}^{\ast}(t)=\{\begin{array}{ll}\overline{u}& \text{if}\phantom{\rule{0.3em}{0ex}}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )>0,\\ I{U}^{\ast}(t)\gamma (\kappa +1)& \text{if}\phantom{\rule{0.3em}{0ex}}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )<0,\end{array}$$

(13)

is optimal when *h*(** s***(

$$\begin{array}{l}{\kappa}^{\ast}(t)\in arg\underset{\kappa}{max}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )\cdot \{\begin{array}{ll}\overline{u}& \text{if}\phantom{\rule{0.3em}{0ex}}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )>0,\\ I{U}^{\ast}(t)\gamma (\kappa +1)& \text{if}\phantom{\rule{0.3em}{0ex}}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )<0,\end{array}\\ \phantom{\rule{5em}{0ex}}s.t.\phantom{\rule{0.2em}{0ex}}0\le \kappa \le \frac{\overline{u}}{I{U}^{\ast}(t)\gamma}-1.\end{array}$$

(14)

The key to solving (14) is to determine the sign of

$$\begin{array}{l}\underset{\kappa}{max}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )\\ \phantom{\rule{0.2em}{0ex}}s.t.\phantom{\rule{0.2em}{0ex}}0\le \kappa \le \frac{\overline{u}}{I{U}^{\ast}(t)\gamma}-1,\end{array}$$

(15)

that is, to determine whether a feasible contact tracing level *κ* exists where the marginal benefit of finding and treating a case is positive. If (15) is negative, then *u**(*t*) = *IU**(*t*)*γ*(*κ**(*t*) + 1) and

$$\begin{array}{c}{\kappa}^{\ast}(t)\in arg\underset{\kappa}{max}h({\mathit{\text{s}}}^{\ast}(t),{\phi}^{\ast}(t),\kappa )\cdot I{U}^{\ast}(t)\gamma (\kappa +1)\\ s.t.\phantom{\rule{0.2em}{0ex}}0\le \kappa \le \frac{\overline{u}}{I{U}^{\ast}(t)\gamma}-1.\end{array}$$

(16)

Substituting (11) into (16) and dropping the terms that do not depend on *κ*, we obtain

$$\begin{array}{c}{\kappa}^{\ast}(t)\in arg\underset{\kappa}{max}\kappa ({\phi}_{IT}^{\ast}(t)-{\phi}_{IU}^{\ast}(t)-{c}_{t})I{U}^{\ast}(t)\gamma -{c}_{c}(\kappa )I{U}^{\ast}(t)\gamma \\ s.t.\phantom{\rule{0.2em}{0ex}}0\le \kappa \le \frac{\overline{u}}{I{U}^{\ast}(t)\gamma}-1.\end{array}$$

(17)

Since *c _{c}*(

$${\kappa}^{\ast}(t)=\{\begin{array}{ll}\overline{u}/(I{U}^{\ast}(t)\gamma )-1\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}{c}_{c}^{\prime}(\overline{u}/(I{U}^{\ast}(t)\gamma )-1)<{\phi}_{IT}^{\ast}(t)-{\phi}_{IU}^{\ast}(t)-{c}_{t},\hfill \\ {{c}_{c}^{\prime}}^{-1}({\phi}_{IT}^{\ast}(t)-{\phi}_{IU}^{\ast}(t)-{c}_{t})\hfill & \text{otherwise}\hfill \\ 0\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}{\phi}_{IT}^{\ast}(t)-{\phi}_{IU}^{\ast}(t)-{c}_{t}<{c}_{c}^{\prime}(0).\phantom{\rule{0.3em}{0ex}}\hfill \end{array}$$

(18)

Thus if (15) is negative (i.e., *λ**(*t*) = 0), then *κ* is at a level such that the marginal benefit of getting an additional person into treatment equals the marginal cost of treating the person and finding them using an increase in the contact tracing effort,
${\phi}_{IT}^{\ast}(t)-{\phi}_{IU}^{\ast}(t)={c}_{t}+{c}_{c}^{\prime}(\kappa )$. However, when the number of untreated infections is large enough, then the treatment capacity *ū* becomes binding and limits our contact tracing effort.

If on the other hand, (15) is nonnegative, then *κ**(*t*) is the optimizer of (15) (and *λ**(*t*) is given by (9)), and if in addition (15) is strictly positive, then *u**(*t*) = *ū*. In the next section we study (15) more closely.

We need to solve the optimization problem (15) to determine *u**(*t*), and thus whether we need to treat as many or as few as possible. If it is optimal to treat as many people as possible, *u**(*t*) = *ū*, the solution *κ ^{o}* of (15) will also determine the optimal level of contact tracing (

Dropping the terms in (15) that do not depend on *κ* eliminates the dependence on *ϕ**(*t*). We therefore focus on the equivalent problem,

$$\begin{array}{l}{\kappa}^{o}(\mathit{\text{s}})\in arg\underset{\kappa}{max}-\frac{{c}_{c}(\kappa )+{c}_{s}/p(\mathit{\text{s}})}{\kappa +1}\\ \phantom{\rule{5em}{0ex}}s.t.\phantom{\rule{0.2em}{0ex}}0\le \kappa \le \frac{\overline{u}}{IU\gamma}-1,\end{array}$$

(19)

whose solution only depends on the state. Thus the value of (15) is *h*(** s***(

$$0={c}_{c}(\kappa )+{c}_{s}/p-(\kappa +1){c}_{c}^{\prime}(\kappa ).$$

(20)

Since
${c}_{c}(\kappa )-(\kappa +1){c}_{c}^{\prime}(\kappa )$ is decreasing in *κ* (due to the strict convexity of *c _{c}*(·)), the solution to the first-order condition (20) is unique and corresponds to a maximum:

$${\kappa}^{o}(\mathit{\text{s}})=\{\begin{array}{ll}\overline{u}/(IU\gamma )-1\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}\overline{u}/(IU\gamma )-1\le g(p(\mathit{\text{s}})),\phantom{\rule{0.3em}{0ex}}\hfill \\ g(p(\mathit{\text{s}}))\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}g(p(\mathit{\text{s}}))<\overline{u}/(IU\gamma )-1\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}p(\mathit{\text{s}})\le {c}_{s}/{c}_{c}^{\prime}(0),\hfill \\ 0\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}{c}_{s}/{c}_{c}^{\prime}(0)<p(\mathbf{s}).\hfill \end{array}$$

(21)

Thus, *κ ^{o}*(

Note that when *κ ^{o}* is not at its upper bound, then

When the first-order condition (20) holds and the treatment capacity is not a binding limitation, then the optimal level of contact tracing is such that the incremental cost of finding an additional infected person via screening and tracing his or her contacts (yielding a total of *κ* + 1 additional cases) equals the cost of finding an additional infected person via contact tracing for each of *κ* + 1 otherwise reported index cases. Hence the relative mix of screening and contact tracing at any point in time is such that the marginal efficiency (cost per infected person found) of contact tracing equals that of screening if possible. This makes intuitive sense: in an optimal resource allocation, the marginal efficiency of competing interventions is balanced (Mas-Collel et al., 1995).

Since the right-hand side of (20) is decreasing in *κ* (due to the strict convexity of *c _{c}*(·)), it follows that

Using (21), we can write the optimum value of (19), the cost of finding an infected individual with the optimal mix of contact tracing and screening:

$$\begin{array}{l}\frac{{c}_{c}({\kappa}^{o})+{c}_{\mathit{\text{s}}}/p(\mathit{\text{s}})}{{\kappa}^{o}+1}\\ =\{\begin{array}{ll}(IU\gamma /\overline{u})({c}_{c}(\overline{u}/(IU\gamma )-1)+{c}_{\mathit{\text{s}}}/p(\mathit{\text{s}}))\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}\overline{u}/(IU\gamma )-1\le g(p(\mathit{\text{s}})),\hfill \\ {c}_{c}^{\prime}(g(p(\mathit{\text{s}})))\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}g(p(\mathit{\text{s}}))<\overline{u}/(IU\gamma )-1\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}p(\mathit{\text{s}})\le {c}_{s}/{c}_{c}^{\prime}(0),\hfill \\ {c}_{s}/p(\mathbf{s})\hfill & \text{if}\phantom{\rule{0.3em}{0ex}}{c}_{s}/{c}_{c}^{\prime}(0)<p(\mathbf{s}).\hfill \end{array}\end{array}$$

(22)

In the first case of (22), the amount of contact tracing is limited by the treatment rate. In the second (interior) case, where *κ ^{o}* is neither at its minimal nor maximal level, the per person cost of finding an untreated infection equals the marginal cost of additional contact tracing
${c}_{c}^{\prime}({\kappa}^{o})$. In the third case (

We have two approaches to further characterize the solution to the optimal control problem (7). In section 2.3 we describe how the solution can be calculated numerically, and in section 2.4 we describe an approximation that yields further analytic insight.

To numerically calculate the solution to the optimal control problem (7), we use an indirect method that solves a boundary value problem for the optimal trajectories of the state and adjoint variables (** s**(

$$\dot{\mathit{\text{s}}}(t)=\mathit{\text{f}}(\mathbf{s}(t),{u}^{\ast}(t))$$

(23)

$$\dot{\phi}(t)=r\phi (t)-\frac{\partial}{\partial \mathit{\text{s}}}H(\mathit{\text{s}}(t),\phi (t),{\kappa}^{\ast}(t),{\lambda}^{\ast}(t),{u}^{\ast}(t))$$

(24)

$$\mathit{\text{s}}(0)={\mathit{\text{s}}}_{0},\phantom{\rule{0.2em}{0ex}}\mathit{\text{s}}(T)={\mathit{\text{s}}}_{eq},\phantom{\rule{0.2em}{0ex}}\phi (T)={\phi}_{eq},$$

(25)

where *u**(*t*), *κ**(*t*), and *λ**(*t*) can be determined from the optimal trajectories ** s***(

The steady state is the combination of state, adjoint, and control variables (** s**,

In practice, the bang-bang nature of *u* (switching between two levels, *ū* and *IUγ*, depending on the sign of *h*(** s**,

In this section we approximate the dynamics of the optimal policy (specifically, the path along which the prevalence decreases) in order to gain analytical insight into its dependence on the parameters. We will assume that the initial prevalence is above the level that is cost-effective. On the optimal path, then, we start by getting as many people into treatment as possible, *u**(0) = *ū*. Hence the system of differential equations, **(***t*) = ** f**(

Since this system of differential equations does not have an analytic solution, we make several approximations in order to obtain an analytically solvable problem. We focus on the number of untreated infections, which is given by (2), and substitute in *u* = *ū*. This yields

$$\stackrel{.}{IU}={f}_{2}(\mathit{\text{s}},\overline{u})={\eta}_{IU}-{\mu}_{IU}IU-\rho IU+\frac{S}{N}({\beta}_{IU}IU+{\beta}_{IT}IT)-\overline{u},$$

(26)

where *N*(*t*) := *S*(*t*) + *IU*(*t*) + *IT*(*t*) + *R*(*t*) is the total population size. Assuming the demographics are stable, *S*/*N* is roughly constant and we obtain a linear approximation of (26):

$$\stackrel{.}{IU}\approx {\eta}_{IU}-{\mu}_{IU}IU-\rho IU+\frac{S(0)}{N(0)}({\beta}_{IU}IU+{\beta}_{IT}IT)-\overline{u}.$$

(27)

We assume in addition that people in treatment are not infectious; thus, *β _{IT}* ≈ 0. Then (27) reduces to

$$\stackrel{.}{IU}\approx {\eta}_{IU}-\overline{u}-({\mu}_{IU}+\rho -{\beta}_{IU}S(0)/N(0))IU.$$

(28)

We can write (28) as a differential equation of the untreated prevalence *p*(** s**(

$$\dot{p}\approx -\frac{\overline{u}-{\eta}_{IU}}{N(0)-IT(0)}-({\mu}_{IU}+\rho -{\beta}_{IU}S(0)/N(0))p=-\alpha -\sigma p,$$

(29)

where we define *α* := (*ū* − *η _{IU}*)/(

We now take a qualitative look at the long-term dynamics of the approximate solution. Let *p*_{0} := *p*(*s*_{0}) be the initial untreated prevalence and *p _{eq}* :=

From (29) we can gain insight into the factors that affect the (approximate) change in prevalence among people not in treatment when the optimal control is applied. The decay rate *σ* is the sum of the death rate from chronic untreated infection plus the spontaneous recovery rate from chronic infection (these represent exits from the infected state) minus the rate of new infections per untreated infection (this represents entry into the infected state). If the death rate is higher or the recovery rate is faster, then the prevalence associated with the optimal control will decrease more quickly; conversely, if the sufficient contact rate is higher, then the prevalence associated with the optimal control will decrease more slowly. These relationships are only approximate due to the above approximations; for example, the size of the total population and the fraction of susceptible individuals are actually changing over time (and individuals in treatment may infect a few people).

We observe that, using the methods in section 2.3, we can calculate the optimal equilibrium prevalence (among people not in treatment). We can then use (29) to estimate the optimal trajectory. This allows us to approximately determine the optimal trajectory without having to solve the full optimal control problem (7). We demonstrate the quality of this approximation in the next section.

In this section we apply our model to a numerical example. We consider the case of hepatitis B virus (HBV) infection among Asian and Pacific Island adults in the US. Parameter values are shown in table 1.

There are approximately 12 million Asian Americans and Native Hawaiian and Pacific Islanders above the age of 18 in the US (U.S. Census Bureau, 2007). Approximately 50% of this population is immune to hepatitis B virus (either from vaccination or previous infection which resolved itself) and 10% is chronically infected (Hutton et al., 2007). Only one-third of chronically infected individuals in the US know that they are infected (Lin et al., 2007); we assume that these individuals are receiving treatment. Thus our initial state is *S* = 4.8 million, *IU* = 800, 000, *IT* = 400, 000, *R* = 6 million.

We assume that the remaining life expectancy is 40 years for susceptible individuals, immune individuals, and chronically infected individuals in treatment. Approximately 25% of chronically infected individuals not in treatment develop liver disease (Hutton et al., 2007). We assume that these individuals have a remaining life expectancy of 25 years (5 years to develop the disease and 20 years with the disease). Thus *μS* = *μ _{IT}* =

We assume that entry of infected individuals into the population is due to immigration and that these individuals are not yet in treatment. Each year approximately 22,000 legal immigrants above the age of 18 with chronic hepatitis B infection emigrate to the US from Oceania and East and South Asia (we take country level HBV prevalence data from table 3 in annex 4 of WHO (2007) and figure 37 of WHO (2001) and immigration statistics from (DHS, 2008)). We assume that the aggregate rate of entry into the population, Σ* _{i} η_{i}*, matches the exit rate from the initial state,

Since our focus is on hepatitis B infections that do not resolve themselves, we set *ρ* = 0. We set *ω* = 0 because treatment does not effect a cure. In 2006 there were an estimated 46,000 new (acute) hepatitis B infections in the US (CDC, 2008a). Since almost all acute infections in the US occur in adults (see figure 14 of CDC (2008d)) and only 5% of adults develop chronic infection after acute infection (the remaining 95% recover and become immune) (CDC, 2008b), we estimate that approximately 2,300 new cases of adult chronic infection occurred in the US in 2006. Since there were 226 million adults in the US in 2006 (US Census, 2006) and the incidence of hepatitis B infection in Asian Americans and Pacific Islanders is similar to that of whites (figure 15 of CDC (2008d)), this translates to a 1.0 in 100,000 annual rate of new chronic infection among Asian and Pacific Islander adults. We assume that only chronically infected individuals can infect others and that treatment reduces infectiousness by 50% (i.e., *β _{IT}* =

We assume that the average quality multiplier for an untreated infection *q _{IU}* is 0.985 (we assume that 3/4 of those individuals have no symptoms and the remaining have a quality multiplier of 0.7 in the last 5 years of a 25 year life-span, c.f., the quality multipliers in Hutton et al. (2007)). We assume that the quality multiplier for the remaining states (

We use a discount factor of 3% (Gold et al., 1996), a screening cost *c _{s}* of $27 (the cost of an HBsAg test and the blood test administration in Hutton et al. (2007)), and no cost for the initial treatment,

The solution to the optimal control problem (7) is illustrated in figure 2. The maximum level of control *ū* is applied until the cost-effective equilibrium is reached. The steady state is *S* = 5.2 million, *IU* = 26, 000, *IT* = 850, 000, and *R* = 6.06 million. The cost-effective steady-state untreated disease prevalence,
$\frac{IU}{S+IU+R}$, is 0.2%, a level comparable to the prevalence of HBV in the general US population (CDC, 2008a). This suggests that efforts should be made to reduce the untreated prevalence of hepatitis B among Asian and Pacific Islander adults in the US to a level comparable to that for the rest of the population. Using the condition in (20), the threshold prevalence *p* for contact tracing (this is the prevalence among those not in treatment) is
${c}_{s}/{c}_{c}^{\prime}(0)=3\%$. If this prevalence is above 3%, we perform no contact tracing; if this prevalence is below 3%, we do perform contact tracing. In addition, *κ _{eq}* = 4.3: at the cost-effective steady state it is optimal to trace contacts at a rate that finds an average of 4.3 untreated cases per identified index case. Finally, we note that at equilibrium, the marginal overall benefit of moving a person out of the infected untreated state and into the infected treated state exactly equals the cost of finding and treating this person,

Solutions of the exact and approximate models: Prevalence among untreated individuals over time for the hepatitis B example

Figure 2 compares the optimal policy and the approximation of its dynamics that we developed in section 2.4. The results are remarkably similar. This suggests that, if the untreated disease prevalence is higher than the optimal equilibrium level (section 2.3 describes how it can be calculated), and thus the maximum level of control must be applied until the cost-effective equilibrium prevalence is reached, one can estimate the optimal trajectory for prevalence reduction using the simple exponential decay formula given by (29). Instead of solving the full optimal control problem, one can easily calculate the equilibrium prevalence and the rate of decay using the simple formulas in sections 2.3 and 2.4.

Figure 3 shows the optimal equilibrium prevalence among untreated individuals as a function of key model parameters. As contact tracing cost increases (decreases), optimal untreated prevalence increases (decreases), but only by a relatively small amount. Equilibrium prevalence is quite sensitive to the net value of treatment (*v _{IT}* =

We have considered the optimal mix of screening and contact tracing for a chronic viral disease. We found that when we are above the equilibrium (and moving towards the long-term target steady state) the optimal strategy is to get as many people as possible into treatment per unit time until the incremental health benefits balance the cost of controlling the disease. We further found that the optimal contact tracing effort *κ**(*t*) is a simple function of the current prevalence among untreated individuals, and under the optimal strategy, the marginal effectiveness of contact tracing equals that of screening. Additionally, as the disease becomes less prevalent, it is optimal to spend more per case identified. We showed how to calculate the equilibrium cost-effective state of the system (e.g., number of people in treatment) and developed a simple approximation of the rate at which the system moves toward this equilibrium. Together, the equilibrium calculation and the approximation allow us to obtain a close characterization of the optimal solution without having to solve the full optimal control problem.

Simple compartmental models such as the one we use are standard and analytically tractable but are not completely realistic. Two well-known shortcomings of such models are the assumptions of random mixing and instantaneous transitions. One could ameliorate these shortcomings by constructing a more complicated model with more compartments: for example, one could include compartments for different risk groups (with different mixing rates), different genders in the case of diseases that can be sexually transmitted, and different disease states (such as an infected asymptomatic state). To calculate numerical solutions to optimal control problems arising from more complicated models it may be helpful to use specialized software such as the OCMat toolbox for MATLAB (Grass et al., 2008). Another alternative is to use a stochastic network simulation model to capture more details of mixing and infection transmission (e.g., Armbruster and Brandeau (2007a); Morris and Kretzschmar (1997); Enns et al. (2009); Halloran et al. (2008)). However, in both cases, analytical results may be difficult to develop. Simple models such as ours also ignore co-infection dynamics: in some cases, infection with one disease (e.g., HIV) may facilitate infection with another (e.g., tuberculosis) (Long et al., 2008; Porco et al., 2001). Additionally, our model assumes that the rate of infected individuals developing immunity (and entering compartment *R)* is proportional to the number of treated and untreated infections, *IT* and *IU*. A more accurate model could capture the possibility that a certain fraction of infections will quickly resolve themselves.

We have considered disease control via treatment but for some chronic infectious diseases, such as hepatitis B, a vaccine exists. A worthwhile extension of this work would consider the optimal allocation of resources between vaccination and treatment. We have modeled contact tracing and screening as two means of finding infected individuals. One could extend our model to also consider health-awareness campaigns as an additional intervention (such campaigns are common for HIV and hepatitis B, for example). The effect of including additional interventions is not additive because any particular infection can only be prevented, detected, or treated once. Thus, one would ideally examine the allocation of resources not only across time but also across a portfolio of possible interventions (e.g., Brandeau et al. (2003); Zaric (2003); Zaric and Brandeau (2001, 2002)).

Our analysis assumed dynamic control: we determined the optimal levels of screening and contact tracing over time to control a chronic viral disease. Dynamic control of this type could be extended to behavior change interventions. Typical models assume static levels of behavior change (e.g., Brandeau et al. (2003); Zaric (2003); Zaric and Brandeau (2001, 2002)). However, studies of risky behavior related to HIV suggest that risky behavior decreases as perceived risk increases (Hethcote et al., 1991; Velasco-Hernández et al., 1996; Zeiler, 2008). It may be possible to extend such models using our ideas to the case in which behavior change depends on prevalence.

Although our model is simple and the mathematics at times complicated, our analysis yields four insights. First, the optimal strategy is to get infected individuals into treatment as fast as possible when the disease prevalence is above the cost-effective equilibrium disease prevalence. Second, as one reduces the disease prevalence by moving people into treatment (which decreases the chance that they will infect others), one should increase the level of contact tracing to compensate for the decreased effectiveness of screening. Third, as disease prevalence decreases, it is optimal to spend more per case identified. Fourth, the relative mix of screening and contact tracing at any level of disease prevalence is such that the marginal efficiency of contact tracing (cost per infected person found) equals that of screening if possible (e.g., when capacity limitations are not binding).

The authors were supported by grant number R01-DA15612 from the National Institute on Drug Abuse.

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