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Optimizing strategies for populationbased chlamydia infection screening among young women: an agestructured system dynamics approach
 Yu Teng^{1},
 Nan Kong^{2}Email author and
 Wanzhu Tu^{3}
Received: 29 December 2014
Accepted: 25 June 2015
Published: 11 July 2015
Abstract
Background
Chlamydia infection (CT) is one of the most commonly reported sexually transmitted diseases. It is often referred to as a “silent” disease with the majority of infected people having no symptoms. Without early detection, it can progress to serious reproductive and other health problems. Economical identification of asymptomatically infected is a key public health challenge. Increasing evidence suggests that CT infection risk varies over the range of adolescence. Hence, agedependent screening strategies with more frequent testing for certain age groups of higher risk may be costsaving in controlling the disease.
Methods
We study the optimization of agedependent screening strategies for populationbased chlamydia infection screening among young women. We develop an agestructured compartment model for CT natural progress, screening, and treatment. We apply parameter optimization on the resultant PDEbased system dynamical models with the objective of minimizing the total care spending, including screening and treatment costs during the program period and anticipated costs of treating the sequelae afterwards). For ease of practical implementation, we also search for the best screening initiation age for strategies with a constant screening frequency.
Results
The optimal agedependent strategies identified outperform the current CDC recommendations both in terms of total care spending and disease prevalence at the termination of the program. For example, the agedependent strategy that allows monthly screening rate changes can save about 5 % of the total spending. Our results suggest early initiation of CT screening is likely beneficial to the cost saving and prevalence reduction. Finally, our results imply that the strategy design may not be sensitive to accurate quantification of the agespecific CT infection risk if screening initiation age and screening rate are the only decisions to make.
Conclusions
Our research demonstrates the potential economic benefit of agedependent screening strategy design for populationbased screening programs. It also showcases the applicability of agestructured system dynamical modeling to infectious disease control with increasing evidence on the age differences in infection risk. The research can be further improved with consideration of the difference between firsttime infection and reinfection, as well as population heterogeneity in sexual partnership.
Keywords
Agestructured system dynamics Infectious disease modeling Disease screening Parameter optimizationBackground
Sexually transmitted infections with Chlamydia trachomatis (CT) are among the most commonly reported infectious diseases in the United States [10] and many other developed countries [38]. The infection is caused by bacterium C. trachomatis [7]. It is estimated that about 1 million individuals in the U.S. are infected with CT. Due to lack of specific symptoms in many CT infection cases [22], the infection may lead to major longterm morbidities such as pelvic inflammatory disease, ectopic pregnancy, and infertility [9, 36]. Together with other STDs, CT infection inflicts significant human and economic costs [26].
At present, CT infection can be accurately detected and easily treated with early detection. Thus, CT screening has emerged as a key public health intervention [6] and the disease control relies primarily on the cost and effectiveness of the screening. Several economic studies found CT screening to be costeffective, and even costsaving (e.g., [17–19, 21, 35]). For literature reviews on the economic studies, we refer to Low et al. [23, 24]; Roberts et al. [28]. However, most of the existing economic studies assumed a constant CT infection rate over the studied age range, which typically spans adolescence and early adulthood. Increasing evidence suggests that the CT infection risk decreases with age (e.g., [3, 13, 31]), mainly due to more stabilized sexual partnership and possibly also due to increased immunological response to CT over age. Hence, one would expect that a screening strategy with agedependent screening rate, i.e., treating screening proportion in the population as a function of age, would be more costsaving than the strategies assuming a constant rate. In this paper, we incorporate the age dependency of the infection risk into an economic study of CT screening with nucleic acid amplification testing [33]. We optimize agedependent screening strategies for a populationbased screening program, which offers tests systematically to all individuals in the target group within a framework of agreed policy, protocols, quality management, monitoring and evaluation [16].
To the best of our knowledge, only few simulationbased economic studies have taken the agedependency into account. For example, Hu et al. [18, 19], basing their studies on an earlier observational study in the Netherlands [8, 15], assumed that the probability of acquiring CT is constant for women from early ages and decreases with a constant annual rate after then. While the simulationbased analyses have compared tailored screening strategies that recommend different screening rates to different population subgroups based on some risk measure (e.g., [18, 19, 21]), we have not witnessed any optimization work on identifying agedependent CT screening strategies, which are, in some sense, a subset of riskbased strategies.
In this paper, we model the population dynamics, related to CT transmission, screening, and treatment, with a set of partial differential equations (PDE) that incorporate agedependency on the CT infection risk. We formulate a parameter optimization problem subject to the PDE model to identify the screening rates at different age points over a range (i.e., an agedependent parameter profile) such that some percapita cumulative cost is minimized. To summarize our contribution, we are among the first that conduct economic analyses of populationbased CT screening programs through agestructured systems modeling and optimization.
In this paper, we also reasonably specify the studied cohort so that we can reduce the PDE model to a set of ordinary differential equations (ODEs) for simplifying the numerical optimization. We next focus on the optimization over a set of more implementable strategies. In anticipation that the optimal agespecific screening strategy may be difficult to implement as optimal screening rates obtained from the above model may vary significantly between consecutive age points, we consider cases where a constant screening rate is applied to a truncated age range. Specifically, we consider optimizing the screening start age. Finally, we make a simplifying assumption on the agespecific infection risk, with which we remodel the system dynamics and explore the benefit in the numerical optimization. Through this simplification, we also check how robust the optimal strategy with a constant screening rate is to the estimate of the agespecific CT infection risk profile. After presenting the research methodology, we report our numerical studies and discuss their policy implications. At the end of the paper, we draw conclusions and outline future research.
Differential equation based systems dynamic modeling has been widely used in infectious disease control. For a general introduction, we refer to Keeling and Rohani [20]. For studies on CT transmission dynamics, we refer to Martin et al. [25]; Sharomi and Gumel [29]. Meanwhile, ODEbased models have been applied to economic studies of screening programs. For example, Althaus et al. [1] applied an SEIRS (susceptibleexposedinfectedrecoveredsusceptible) model, which is widely used in the infectious disease modeling literature (e.g., [2, 15]), to assess the impact of screening programs on CT prevalence reduction. Regan et al. [27] extended the SEIRS model to incorporate the additional state of receiving treatment. Note that the two studies above did not consider cost or costeffectiveness of the screening programs. Our work differs from previous in that we apply nonlinear optimization to design optimal strategies.
Methods
Optimization of agedependent screening strategies
An agestructured SEIRS model
We adapt a widely used SEIRS compartment model [1] to illustrate the system dynamics associated with CT transmission, screening, and treatment. We then capture the system dynamics with a multicompartment model and mathematically formalize the agestructured population heterogeneity with a set of PDEs.
Compartment modeling has been widely used in modeling infectious disease transmission [2, 4, 20, 34]. In recent years, it has been used to model various specific screening, vaccinating, pharmaceutical, and therapeutic interventions for dealing with relevant public health problems (e.g., [12]). To many infectious diseases, age has a deep influence on the rate of disease spread in a population, especially the contact rate [2, 20]. To sexually transmitted diseases, the contact rate is affected by the sexual behavior, which is often age dependent.
Notation in the agestructured compartment model and corresponding PDEs
S(t,τ)  Susceptible  f  Fraction of asymptomatic infections 
E(t,τ)  Exposed  1/γ  Incubation time 
I_{ a }(t,τ)  Asymptomatically infected  1/r_{ a }  Duration of the asymptomatic period 
I_{ s }(t,τ)  Symptomatically infected  1/r_{ s }  Duration of the symptomatic period 
R(t,τ)  Recovered  1/μ  Duration of the temporary immunity 
β(τ)  Agedependent infection rate  after natural clearance of asymptomatic infection  
λ(τ)  Agedependent screening rate  1/r_{ PID }  Duration of acute PID onset 
\( \begin{array}{l}\left(\frac{\partial }{\partial t}+\frac{\partial }{\partial \tau}\right)S\left(t,\tau \right)=\beta \left(\tau \right)S\left(t,\tau \right){\displaystyle \underset{0}{\overset{A}{\int }}\left({I}_a\left(t,{\tau}^{\hbox{'}}\right)+{I}_s\left(t,{\tau}^{\hbox{'}}\right)\right)}d{\tau}^{\hbox{'}}+\left({r}_{PID}+\lambda \left(\tau \right)\right){I}_a\left(t,\tau \right)+\left({r}_s+\lambda \left(\tau \right)\right){I}_s\left(t,\tau \right)+\mu R\left(t,\tau \right);\hfill \\ {}\left(\frac{\partial }{\partial t}+\frac{\partial }{\partial \tau}\right)E\left(t,\tau \right)=\beta \left(\tau \right)S\left(t,\tau \right){\displaystyle \underset{0}{\overset{A}{\int }}\left({I}_a\left(t,{\tau}^{\hbox{'}}\right)+{I}_s\left(t,{\tau}^{\hbox{'}}\right)\right)}d{\tau}^{\hbox{'}}\gamma E\left(t,\tau \right);\hfill \\ {}\left(\frac{\partial }{\partial t}+\frac{\partial }{\partial \tau}\right){I}_a\left(t,\tau \right)=f\gamma E\left(t,\tau \right)\left({r}_a+{r}_{PID}+\lambda \left(\tau \right)\right){I}_a\left(t,\tau \right);\hfill \\ {}\left(\frac{\partial }{\partial t}+\frac{\partial }{\partial \tau}\right){I}_s\left(t,\tau \right)=\left(1f\right)\gamma E\left(t,\tau \right)\left({r}_s+\lambda \left(\tau \right)\right){I}_s\left(t,\tau \right);\hfill \\ {}\left(\frac{\partial }{\partial t}+\frac{\partial }{\partial \tau}\right)R\left(t,\tau \right)={r}_a{I}_a\left(t,\tau \right)\mu R\left(t,\tau \right).\hfill \end{array} \) Typically, a screening program estimates in advance the size of the cohort it can deal with based on its capacity and keeps its size relatively constant by synchronizing the recruitment and exit processes. Without loss of generality, we set the cohort size to be 1 at any time point, i.e., \( {\displaystyle \underset{0}{\overset{A}{\int }}\left(S\left(t,\tau \right)+E\left(t,\tau \right)+{I}_a\left(t,\tau \right)+{I}_s\left(t,\tau \right)+R\left(t,\tau \right)\right)}d\tau =1,\forall t. \)
Once the screening rate profile, as well as the boundary and initial conditions, are given, the state of the system can be determined for any given time point with the above PDEs. The screening and treatment costs are cumulated accordingly over the program duration. An optimal screening rate profile can then be identified to minimize the percapita cumulative cost. We next present a parameter optimization problem subject to the PDE constraints.
A parameter optimization problem

CT screening cost: \( {C}_s\left(\lambda \left(\tau \right)\right)={c}_sT{\displaystyle \underset{0}{\overset{A}{\int }}\lambda \left(\tau \right)}, \)

CT Treatment cost: \( {C}_t\left(\lambda \left(\tau \right)\right)={\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{0}{\overset{A}{\int }}{c}_t\left[\left({r}_s+\lambda \left(\tau \right)\right){I}_s\left(t,\tau \right)+\lambda \left(\tau \right){I}_a\left(t,\tau \right)\right]}} d\tau dt, \)

Acute PID treatment cost: \( {C}_{PID}\left(\lambda \left(\tau \right)\right)={\displaystyle \underset{0}{\overset{T}{\int }}{\displaystyle \underset{0}{\overset{A}{\int }}{c}_{PID}}}{r}_{PID}{I}_a\left(t,\tau \right) d\tau dt, \)

PID sequelae treatment cost: \( {C}_{end}\left(\lambda \left(\tau \right)\right)={\displaystyle \underset{0}{\overset{T}{\int }}{c}_{end}{I}_a\left(t,A\right)dt}. \)
Note that the screening cost applies to the entire cohort, which is assumed to be 1. We define the cumulative cost as C_{ total }(λ(τ)) = C_{ s }(λ(τ)) + C_{ t }(λ(τ)) + C_{ PID }(λ(τ)) + C_{ end }(λ(τ)). The optimization problem is then formulated as \( \underset{\lambda \left(\tau \right)}{ \min }{C}_{total}\left(\lambda \left(\tau \right)\right) \) subject to the PDEs introduced above and the boundary and initial conditions. While attempting to minimize the percapita cumulative cost, we also compare different strategies in terms of the terminal CT prevalence at time t, defined as \( {\displaystyle \underset{0}{\overset{A}{\int }}\left({I}_a\left(t,\tau \right)+{I}_s\left(t,\tau \right)\right)d\tau } \).
The objective function is discretized as: \( {C}_{total}\left({\lambda}^0,{\lambda}^1,\dots, {\lambda}^{N_{\tau }1}\right)={c}_sT{\displaystyle \sum_{j=0}^{N_{\tau }1}{\lambda}^j}+{\displaystyle \sum_{i=0}^{N_T1}{\displaystyle \sum_{j=0}^{N_{\tau }1}{c}_t\left[\left({r}_s+{\lambda}^{i,j}\right){I}_s^{i,j}+{\lambda}^{i,j}{I}_a^{i,j}\right]}+}{\displaystyle \sum_{i=0}^{N_T1}{\displaystyle \sum_{j=0}^{N_{\tau }1}{c}_{PID}{r}_{PID}{I}_a^{i,j}}+}{\displaystyle \sum_{i=0}^{N_T1}{c}_{end}{I}_a^{i,{N}_{\tau }}}. \)
Given the two subinterval counts (N_{ t } and N_{ τ }), the boundary and initial conditions, and the estimated CT infection risk for j = 1,…, N_{ τ }, we obtain a nonlinear optimization model with finitely many decision variables, linear objective function, and quadratic constraints. We use standard constrained nonlinear optimization solvers (e.g., activeset and interior point) available in the MATLAB Optimization Toolbox [4].
A special case for cohorts with uniform age distribution
In this section, we consider a special case of the above PDE model, which is more suitable to the real practice of a screening program. In real practice, a screening program often only targets those of age 0 (i.e., the smallest age to be concerned for CT infection) for recruitment and terminates CT screening for those who reach A (i.e., the largest age to be concerned for CT infection). A general belief is that the number of infected individuals at age 0 is negligible. That is, for any t, we have S(t,0) = p, where p is denoted as the rate with which new participants enter the cohort, and E(t,0) = I_{ a }(t,0) = I_{ s }(t,0) = R(t,0) = 0. We further assume that the age of the studied open cohort follows a uniform distribution and term such a cohort uniformly aged cohort. That is, for any t, we have S(t,τ) + E(t,τ) + I_{ a }(t,τ) + I_{ s }(t,τ) + R(t,τ) = p = 1/A for τ ∈ (0, A]. Hence, we can align the age domain with the time domain and thus reduce the agestructured PDE model to a timeinvariant ODE model with agespecific CT infection risks. We term this model ODE_1. Since the screening strategy design is only considered up to age A, β(τ) for τ ≥ A can be arbitrarily specified. To solve the parameter optimization problem for ODE_1, we again resort to discretization. In the following, we further study this special case with a smaller set of ageindependent screening strategies, which are more implementable in practice.
Optimization of ageindependent screening policies
Our study in this section was inspired by the current CT screening recommendations. The CDC guideline recommends annual CT screening for women under age 25 but does not specify the initial screening age [34]. We consider policies similar to the current CDC recommendations structurewise. The considered policies recommend to start CT screening for women at some age between 0 and A, and continue the screening until A with a constant frequency. Hence, the optimization problem is intended to determine an optimal screening initiation age and optimal screening rate. Note that Teng et al. [12] studied the problem with fixed screening initiation age and only optimized the screening rate over a fixed age range. Their problem is a parameter optimization problem with only one decision variable and assumes a constant infection risk. For each screening initiation age \( \widehat{\tau} \), we have a similar parameter optimization problem, but with agedependent infection risk. We use a standard line search algorithm without derivative information in MATLAB to solve the inner problem for each given screening initiation age. We apply onedimensional explicit enumeration to select the optimal screening initiation age.

CT screening cost: \( {C}_s\left(\widehat{\tau},\lambda \right)={c}_s\lambda M\left(A\widehat{\tau}\right), \)

CT treatment cost: \( {C}_t\left(\widehat{\tau},\lambda \right)={\displaystyle \underset{0}{\overset{A}{\int }}{c}_t\left[{r}_s\left({I}_{s0}+{I}_s\right)+\lambda \left({I}_s+{I}_a\right)\right]d\tau }, \)

Acute PID treatment cost: \( {C}_{PID}\left(\widehat{\tau},\lambda \right)={\displaystyle \underset{0}{\overset{A}{\int }}{c}_{PID}{r}_{PID}\left({I}_{a0}+{I}_a\right)d\tau }, \)

PID sequelae treatment cost: \( {C}_{end}\left(\widehat{\tau},\lambda \right)={\displaystyle \underset{0}{\overset{A}{\int }}{c}_{end}p\frac{I_a}{M}d\tau }, \)

Percapita cumulative cost: \( {C}_{total}\left(\widehat{\tau},\lambda \right)={C}_s\left(\widehat{\tau},\lambda \right)+{C}_t\left(\widehat{\tau},\lambda \right)+{C}_{PID}\left(\widehat{\tau},\lambda \right)+{C}_{end}\left(\widehat{\tau},\lambda \right). \)
The optimization problem is thus presented as \( \underset{\widehat{\tau},\lambda }{ \min }{C}_{total}\left(\widehat{\tau},\lambda \right) \) subject to ODE_2.
With any given screening initiation age \( \widehat{\tau}\in \left[0,A\right) \), β_{0} and β become known. Hence, we can uniquely set the initial condition on S(\( \widehat{\tau} \)), E(\( \widehat{\tau} \)), I_{ a }(\( \widehat{\tau} \)), I_{ s }(\( \widehat{\tau} \)), and R(\( \widehat{\tau} \)). We also determine the cost accumulated from 0 to \( \widehat{\tau} \). Then we can reduce the optimization problem to a parameter optimization problem based on the 5compartment ODE model for \( \tau \in \left[\widehat{\tau},A\right) \), for which we can adapt the optimization method proposed in Teng et al. [12]. That is, for any \( \widehat{\tau} \), the gradient of the objective function, i.e., \( \frac{d{C}_{total}\left(\widehat{\tau},\lambda \right)}{d\lambda } \), can be derived with a cubic interpolation method. We apply a standard linear search algorithm with derivatives in MATLAB to solve the inner problem given each screening initiation age. We then apply onedimensional explicit enumeration to select the optimal screening initiation age.
Results and discussion
Parameters pertaining to costs and disease transition rates
Parameter value  

f  0.625 
1/γ  14 days 
1/r_{ a }  433 days 
1/r_{ s }  35 days 
1/μ  90 days 
1/r_{ PID }  1000 days 
c _{ s }  $13 
c _{ t }  $36 
c _{ PID }  $1898 
c _{ end }  $192 

For S1, the screening rate profile is represented as a multistep function with identical step size depending on the maximal allowable frequency of strategy update. We chose to update the screening strategy either yearly or monthly. We report the optimal strategies in Fig. 4.

For S2, we present the optimal screening rate with all possible screening initiation ages (every month between 0 and A), as well as the associated percapita cumulative cost and terminal prevalence in Fig. 5. The smallest unit for the screening initiation age is one month. The strategy with the minimum cost is the one that starts the screening for every individual when she reaches the 6^{th} month after the 14^{th} birthday. The screening rate is 1.511 times per year, which implies that an individual should test for CT roughly every 8 months.

For S3, we present the optimal screening rates with all possible screening initiation ages, as well as the associated percapita cumulative cost and terminal prevalence in Fig. 6. The strategy with the minimum cost is the one that starts the screening for every individual when she reaches the 4^{th} month after the 14^{th} birthday. The screening rate is 1.499 times per year.

In Table 3, we compare the three strategies. First, the three studied strategies all outperform the strategy of no screening and the current CRC recommendations in both percapita cumulative cost and terminal CT prevalence. Second, the comparison indicates the superiority of agedependent CT screening strategy (S1 vs. S2) and quantifies its potential impact to the screening practice. Finally, the comparison shows comparable solution qualities between S2 and S3, suggesting the strategy design may not be sensitive to the quantification of agedependent CT infection risks. In terms of computation time, on a PC with a 2.33GHz Intel Core 2 Duo Processor and 2GB RAM, the computation time is about 3.5 s for identifying S3, compared to 13 s for S2. This is mainly due to the fact that the gradient is available to the onedimensional linear search for S3 but not for S2.Table 3
Comparison between the screening strategies
Screening strategies
Percapita cumulative cost
Terminal prevalence
No screening
$874
13.12 %
CDC Recommendations
$706
8.15 %
S1 w/ yearly update
$675.4
6.75 %
S1 w/ monthly update
$673.0
6.69 %
S2
$691.1
6.94 %
S3
$691.7
6.92 %
Discussion
Overall, our numerical studies suggest that considering agedependency in the screening strategy design is more costsaving than currently recommended strategies. Our results further offer insights into various aspects of the design. With the study on S1, the results suggest that the agedependency on the screening rate in an optimal screening policy roughly coincides with the agedependency on the CT infection risk. That is, the screening rate should be intensified around age 16 – 18, which is the age range where the infection risk is highest. Compared to the current recommendations, biannual screening or screening every 8 months is more likely to be optimal from the societal costsaving viewpoint. With the study on S2, the results suggest that it may be beneficial to initiate the screening earlier at least for the tested intercity cohort, which has relatively high CT prevalence. This also suggests that it is important to consider the potential costs incurred by the PID sequelae. Thus, it is important to provide accurate estimate on the probabilities of developing the sequelae in any strategy design activities.
Comparing S2 to S1 suggests that constant rate screening is likely to be acceptable given the small increase in both outcomes. Comparing S3 to S2 suggests that accurate quantification of agespecific CT infection risks may not be essential to the design of strategies with constant screening rate. Note that almost all the existing work largely relies on relatively crude estimates due to data scarcity and ethical concerns [31]. Finally, the fast computations suggest that it may be appealing to expand our models to incorporate highlevel population heterogeneities.
Conclusions
In this research, we present a series of parameter optimization models to investigate agedependent screening strategies for controlling chlamydia infection among young women. Through our modeling research, we attempt to inform the design of optimal populationbased CT screening strategies from a societal costsaving perspective while ensuring a sufficient level of practicality. For the analysis, we extend a widely used SEIRS model to incorporate agedependent screening rate profile and apply a gradientbased line search algorithm for ease of numerical optimization.
Our future research will mainly be focused on detailed model development. For example, it is evident that risks of firsttime infection and subsequent reinfection differ due to partial protective immunity against CT [11, 37]. We will formulate the parameter optimization models that differentiate individuals with firsttime infection and reinfection. We will also consider different patterns in ongoing sexual partnership. We plan to adapt the pair compartment model in Heijne et al. [30], which captures sexual partnership duration and reinfection. The investigation on sexual partnership and effective management of sex partners motivates us to explore the use of stochastic network models (e.g., [5, 32]), which provides added flexibility in modeling sexual partnership networks of complex structure. We will thereby develop optimization models based on stochastic network models for CT transmission among heterogeneous sex partners. In addition, we will model programmatic adherence and testing accuracy to make our strategy design more suitable in realworld CT infection control. Other future research directions include design of more efficient parameter optimization solution methods, systematic literature review for model parameter estimation, and sensitivity analyses on the model parameters.
Declarations
Acknowledgements
The data for CT agedependent risk estimation was originally collected through the Young Women project which was supported by grant R01 HD042404 from the US National Institutes of Health.
Authors’ Affiliations
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