Modelling cost-effectiveness of syphilis detection strategies in prisoners: exploratory exercise in a Chilean male prison

Background Syphilis, together with other sexually transmitted infections, remains a global public health problem that is far from controlled. People deprived of liberty are a vulnerable population. Control activities in prisons rely mostly on passive case detection, despite the existence of affordable alternatives that would allow switching to active case-finding strategies. Our objective was to develop a mathematical modelling framework for cost-effectiveness evaluation, from a health system perspective, of different approaches using rapid tests for the detection of syphilis in inmates' populations and to explore the results based on a Chilean male prison population. Methods A compartmental model was developed to characterize the transmission dynamics of syphilis inside a prison with the ongoing strategy (passive case detection, with VRDL + FTA-ABS), considering the entrance and exit of inmates over a 40 year period. The model allows simulation of the implementation of a reverse algorithm for the current situation (rapid test + VDRL), different screening strategies (entry point, massive periodically; both with rapid test + VDRL) and treatment of detected cases. The parameters for the exploratory exercise were obtained from systematic searches of indexed and grey literature and field work (EQ-5D questionnaire application and key actors interviews). Probabilistic sensitivity analysis was conducted to account for uncertainty in relevant parameters. Results The proposed framework allows the evaluation of different detection strategies. In this study, all the strategies were cost-effective in the baseline scenario when considering an ICER threshold of 1 Chilean GDP per capita (US$15,000). The strategies most likely to be cost-effective (over 80% probability) were: current situation with reverse algorithm, entry point screening and mass screening every two years; the latter was the most effective, achieving the lowest prevalence (0.7% and 1.7% over the period versus the 3% prevalence in the current situation). Conclusions Mathematical modelling that considers the performance of different tests and detection strategies could be a useful tool for decision making. The exploratory results show the efficiency of adopting both the use of the rapid tests and performing active case detection to significantly reduce the burden of syphilis in Chilean prisons in the near future.

Test (FTA-Abs). Depending on the moment when the screening tests are applied (at entry point or inside the prison) and on the order in which they are applied, the strategies considered are the following.
-Strategy 0 (current situation, passive detection): Screening of inmates inside the prison with a Non Treponemal Test, based mainly on symptoms. This is followed by a confirmation Treponemal Test applied to detected cases; -Strategy 1 (entry screening): Screening of a large percentage of inmates entering the prison using the Rapid Test; -Strategy 2a, 2b, 2c, and 2d (mass screening inside the prison): Mass screening of inmates inside the prison using the Rapid Test, with three different frequencies: every 1, 2, 5, and 10 years, respectively; -Strategy 3 (current situation, reverse algorithm): Screening of inmates inside the prison using the Rapid Test based mainly on symptoms.
In strategies 1, 2, and 3, after the first test, a Non Treponemal Test is applied to confirm detected cases. In strategies 1 and 2 the procedure described in Strategy 0 continues being applied simultaneously. In every strategy, all confirmed cases are treated. In the model description, we will use the notation E ∈ {0, 1, 2, 3} for referring to a strategy in particular.
Concerning the model of the disease dynamics under the described intervention strategies, we adapt the model proposed in [4] to a prison context, considering the data available for this study. Thus, the model proposed consists of a compartmental model, where the inmates are distributed into 6 groups corresponding to different stages of the disease: Susceptible (S), primary (Y 1 ), secondary (Y 2 ), latent (L), tertiary (T ), and immune (I). As usual in controlled dynamical systems, these groups of stages are called state variables. The evolution of these state variables Modelling cost-effectiveness of syphilis detection strategies in prisoners 3 is described by the following system of ordinary differential equations: The above system is represented also in Figure A1.1 below. Since model (A1.1) is an adaptation of that introduced in, [4]  Susceptible individuals (S) can be infected by diseased individuals in primary (Y 1 ) or secondary (Y 2 ) stages (see Figure A1.1, transition 1). The transitions from different disease stages, following the natural history of Syphilis, are: from primary to secondary (transition 2), from secondary to latent (L) (transition 3), from latent to tertiary (T ) (transition 4), and from immune (I) to susceptible (transition 5).
Notation-wise, the transition from a disease state, say X, to the following stage is occurring at rate σ X , where σ −1 X represents the mean duration of stage X. In addition, a transition from latent state L to secondary state Y 2 is considered (transition 6), due to the relapse of latent cases to an infectious syphilitic state.
Treatments inside the prison, for individuals at stage X of the disease, are represented by the treatment rate τ X . These rates depend on the strategy E considered. Treated inmates at the primary stage of the disease reenter into the susceptible class (transition 7). Treated individuals at the latent or tertiary stages of the disease acquire immunity (transitions 10 and 11). Treated inmates at the secondary stage can reenter into the susceptible class (transition 8) or may acquire immunity (transition 9).
Structure of the mathematical model for the dynamics of syphilis in prison. Each circle represents a compartment. Susceptible individuals (S), and different disease states: primary (Y 1 ), secondary (Y 2 ), latent (L), tertiary (T ), and immune (I). Treatment rates from compartment X is represented by τ X . Discharge of the prison is represented by µ and entry to compartment X (depending on the strategy E considered, specifically on ξ E ) is denoted by Λ X (ξ E ).
Finally, we are assuming that the number of inmates in the prison is constant and equal to N . Therefore, entry and exit rates in and out of the prison are equal (denoted by µ). In Figure A1.1, at each compartment there is an exit flow (out of the prison) at rate µ (see flows [12][13][14][15][16][17]. Nevertheless, the entries to the prison have to be distributed in different compartments, and this distribution depends on the Modelling cost-effectiveness of syphilis detection strategies in prisoners 5 strategy E considered because it may involve the application of the Rapid Test and treatment before entering the prison. Thus, the entry flow of new inmates to a stage X is denoted by Λ X (ξ E ) (see flows [18][19][20][21][22][23]. Moreover, the sum of all the entry rates Λ X (ξ E ) over all the stages X is equal to µN .
The dynamics under strategies 0, 1 and 3 (E ∈ {0, 1, 3}) are represented by system (A1.1) with the corresponding choice of ξ E . Namely, if the Rapid Test is applied constantly to a fraction ξ E ∈ [0, 1] of new inmates, strategies 0 and 3 correspond to ξ E = 0. On the other hand, Strategy 1 is represented by having ξ E positive.
The main difference of Strategies 2a, 2b, 2c, and 2d concerns the application of the mass screening to a portion of the prison population, which is done in a very short period of time. Hence, it implies abrupt jumps in the values of state variables in (A1.1). For this reason, in Section 1.4 we describe separately these strategies (E = 2).
From the mathematical modeling viewpoint, we recall that the objective of this study is to implement a decision model for evaluating the cost-effectiveness of different interventions. The proposed model representing the disease dynamics does not consider all the disease stages as such published in [4] and it is valid under important assumptions. This is explained by the available data to which we had access. In this regard, we are not considering interactions between inmates and the outside population, the high variability of the number of sexual partners of inmates (represented by parameters c j ), and the high variability of inmates' conviction times, represented by 1/µ. While some degree of realism may be spared -as in any mathematical model-, these assumptions allow us to construct a fair and tractable mathematical representation of our object of study that successfully accounts for its complexity. Namely, the contribution of the proposed mathematical model is its ability to represent the different interventions and their effects in the disease dynamics (as explained in the following sections), taking into account the performance of the applied tests, which can be adapted to more realistic models when knowledge and data are available.
Comparing model (A1.1) with the proposed in [4], we do not consider three state variables: incubation, relapse, and remission stages. To omit the incubation stage implies to consider that a proportion of inmates in the primary stage are not infectious. For this reason we have introduced the parameter α in the first and second equation in (A1.1). Ifσ −1 E andσ −1 Y 1 represent the mean duration of incubation and primary stages in the model proposed in [4], we compute the proportion of infectious individuals in primary stage in our model as Considering the values ofσ −1 E andσ −1 Y 1 in [4], we obtain α = 0.622 as specified in Table A2.1 in Appendix S2. On the other hand, the value of σ Y 1 was also computed taking into account the valuesσ E andσ Y 1 in [4], because in our model σ Y 1 −1 would represent the time in the primary stage plus an incubation time. Thus, we have According to the values ofσ −1 The distribution between susceptible and immune inmates treated at secondary stage (φ(E) ∈ [0, 1]), explained in Section 1.2 below, is another parameter in (A1.1) not considered originally in [4]. This parameter is introduced because we are not considering relapse and remission stages. Although treatment rates are considered in [4], how these rates vary according each strategy is described in Section 1.1.
Finally, since in [4] all new individuals are entering to the susceptible stage and we are adapting this model to a prison context, we have to consider flows of new inmates entering eventually to all stages of the disease. These flows, represented by Λ X (ξ E ) in (A1.1) and in Figure A1.1, depend on the applied strategy. This modeling approach is described in Section 1.3.

Treatment rates inside the prison
Treatment rates inside the prison, denoted by τ X , correspond to flow rates from a stage X of the disease to susceptible or immune stages, due to treatments of inmates presenting symptoms. These rates depend on the distribution of inmates in each stage of the disease, which in turn depend on the applied strategy E ∈ {0, 1, 2, 3}.
Regarding Figure A1.1 or system (A1.1), the treatment rates inside the prison Then, in order to compute the rates τ j (E) (with j ∈ {Y 1 , Y 2 , L, T }), we need to consider the following equations: where -s j (d) and e j (d) are the sensitivity and specificity of the test denoted by d ∈ {VDRL, RT, FTA-Abs} (values indicated in Table A2.2 in Appendix S2) respectively.
-γ j is the fraction of people in the stage j to whom the screening process is applied. We suppose γ j does not depend on the strategy since it is mainly related to symptoms of stage j. In principle, these values are unknown, but we describe the assumptions that allow their computation.
-p j (E) is the number of treated individuals after being detected at stage j (including false positive cases), divided by the total number of people at this stage. These values are assumed to be known for strategy E = 0 (representing the current situation at steady state) and are shown in Table A2.2 (Appendix S2).
-P j (E) is the prevalence of the disease at stage j inside the prison, which depends on the chosen strategy E. These values are assumed to be known for strategy E = 0 (representing the current situation at steady state) and are shown in Table A2.2 (Appendix S2).
-The number η ∈ [0, 1] represents the proportion of inmates undergoing diagnosis inside the prison. We assume this number does not depend on the strategy considered. The value of this parameter is η = 0.0049, as indicated in Table   A2.2 in Appendix S2. Nevertheless, we will allow parameter η (and others) to vary in the sensitivity analysis described in the Section Results (see also

Figures 3 in the paper).
Sincep j (E) and P j (E) are known for the strategy E = 0 (at steady state), consisting in d 1 = VDRL and d 2 = FTA-Abs then, the values of γ j (independent on the strategy), according to equation (A1.3), are given by With these values of γ j one can compute τ j (E) using equation (A1.2).  Table A2.2 in Appendix S2), and values ofp j (E) and P j (E) indicated in Table A2.2 (Appendix S2), we obtain the values for γ j shown in Table A1.1 below, which can then be used to compute treatment rates for strategies 0, 1, 2 and 3 using equation (A1.2), thus obtaining  Table A1.1 Values of γ j representing the fraction of people in the stage j to whom the screening process is applied.
Notation  In order to compute φ(E) we have considered the original model published in [4]. For that model, we compute the steady states at secondary, latency, remission and relapse stages. Notice that in our model remission and relapse stages are Modelling cost-effectiveness of syphilis detection strategies in prisoners 9 not considered, so we are associating the remission stage with the latency, and the relapse stage to the secondary stage. Thus, in the model proposed in [4] we compute the number of individuals (at equilibrium) in the relapse stage divided by the sum of individuals in secondary and relapse stages. This fraction is considered as the proportion of individuals at secondary stage (in our model) that have previously been in the latency stage.
This procedure gives the following expression for φ(E): where σ Y 2 and µ are described in Table A2.2 (see Appendix S2) and τ Y 2 (E) and τ L (E) are given by to [4] is p = 0.25 (see Table A2.1 in Appendix S2). Finally,ρ is the inverse of the mean duration of the remission stage. This time, according to [4], is 6 months, so the value considered isρ = 2 (year) −1 (see Table A2.1 in Appendix S2).
To obtain (A1.5) we have assumed that there are no individuals entering at remission or relapse stages from outside the prison. New inmates presenting the disease in any of its stages and who were not detected at the entry point, will enter to the corresponding stage in prison.
We denote the inflow rate of individuals at stage j of the disease by Λ j (ξ E ) (see system (A1.1) or transitions 18-23 in Figure A1.1) according to the Rapid Test coverage ξ E . More explicitely, we use the following notations: immunes entering the prison where d 1 = RT and d 2 = VDRL.
Observe that if the Rapid Test coverage is total (ξ E = 1) and the applied tests The value used for the coverage of the mass screening isξ 2 = 0.8. On the other hand, the time instants where the mass screening is applied are considered to be periodic, that is t i = t 0 + i∆t for i = 1, 2, . . . , f , where ∆t is the period between two mass screening with the Rapid Test. In this study, we consider ∆t equal to 1, 2, 5, and 10 years.
Modelling cost-effectiveness of syphilis detection strategies in prisoners

11
A mass screening at time t i ∈ {t 1 , t 2 , . . . , t f } ⊂ [t 0 , T ], with the respective confirmation test and the corresponding treatment to positive individuals, will produce an instantaneous variation in the state variables. Let us then consider the following notation: If X j denotes one of the six state variables, then the value of this variable at time t i is X j (t i ); moreover, we denote X j (t + i ) the value of X j just after the screening, confirmation test, and the corresponding treatments. Values X j (t i ) are obtained from system (A1.1). In order to determine the values X j (t + i ) we follow an approach similar to the one described in Section 1.3, obtaining: immunes before mass screening where d 1 = RT and d 2 = VDRL.
Thus, after the procedure is finished at time t i , the evolution of the disease follows the dynamics given by (A1.1) -but considering the new initial conditions X j (t + i ) described above-until the next instant t i+1 , when the mass screening (and the corresponding procedure) is applied again.

Costs of interventions
In this section we present the models of costs associated with each type of intervention involved in the different strategies described in the Section Materials and methods. In all the interventions considered, infected individuals are detected and treated in one of the following stages of the disease: primary (Y 1 ), secondary (Y 2 ), latency (L) or tertiary (T ).
We distinguish two types of interventions and their respective costs: detection and treatment inside the prison (mainly for inmates presenting symptoms), detection and treatment for individuals about to enter the prison (only valid for Strategy 1, i.e., E = 1). However, since Strategies 2a, 2b, 2c, and 2d (i.e., E = 2) involve jumps (or discontinuities) in the evolution of state variables due to mass interventions, we should consider the costs of detection and treatment for inmates inside the prison, as consequence of such mass screenings, in some specific instants of time. Hence, the expressions for the costs of detection and treatment inside the prison for inmates presenting symptoms will be different for Strategies 2a, 2b, 2c, and 2d. (strategy E = 1); and costs of strategy E = 2. In some parts we write X E j , for j ∈ {Y 1 , Y 2 , L, T } to denote the respective state variables when strategy E ∈ {0, 1, 2, 3} is applied.
All the costs are computed for a given interval of time [t 0 , T ], where t 0 corresponds to present time. In order to bring costs at a future time t ≥ t 0 to present time we need to multiply by 1/(1 + r) (t−t 0 ) , where r > 0 is a given discount rate.
For this purpose, if we denote then the discount factor is ε (t−t 0 ) . {VDRL, FTA-Abs}. This procedure is applied constantly to a group of individuals.
The expression for detection costs is given by On the other hand, the cost of treatment of individuals with positive results from the confirmation test d 2 (true and false) is given by

Costs of detection and treatment at the prison entry
The costs of detection and treatment at the entry point are valid only for strategy E = 1. Nevertheless, we establish these costs for a general coverage ξ E ∈ [0, 1] of the Rapid Test at the prison entry, recalling that ξ E = 0 when E ∈ {0, 2, 3}.
The first test applied to a fraction ξ E ∈ [0, 1] of newly admitted inmates is Then, a confirmation test d 2 = VDRL is applied to individuals giving (true and false) positive results. Therefore, the detection cost of this procedure is given by where d 1 = RT and d 2 = VDRL.
The cost of treatment of individuals resulting positive in the confirmation test is given by Notice that if ξ E = 0 (as in strategies E ∈ {0, 2, 3}) the above costs are equal to zero.
With expressions (A1.6), (A1.7), (A1.8) and (A1.9), we can write the cost of Strategy 1 (E = 1) as follows Observe that C D entrance (E) and C T entrance (E) are linear with respect to the coverage ξ E of the Rapid Test at entry. Other costs C D inside (E) and C T inside (E) are non linear because they depend on the state variables X E j (t).

Costs of Strategies 2a, 2b, 2c, and 2d
The costs of detection and treatment of Strategies 2a, 2b, 2c, and 2d (i.e., E = 2) have the same structure of those presented in Sections 2.1 and 2.2 taking into account that the state variables have to be updated after mass interventions at Costs of detection of inmates inside the prison -procedure which is applied constantly and based mainly on symptoms-, is given by true positives of d 1 (A1.10) The cost associated with treatments of inmates due to the detection procedure indicated above, is given by (A1.11) In the above expressions, one has t f +1 = T . Notice also that inside the integrals in (A1.10) and (A1.11), initial conditions of state variables have to be updated to X E j (t + i ), according to the dynamics of Strategies 2a, 2b, 2c, and 2d (i.e., E = 2) described in Section 1.4, in order to integrate in intervals [t i , t i+1 ].
The cost of mass interventions at times t i ∈ {t 1 , t 2 , . . . , t f } ⊂ [t 0 , T ], for the detection part is given by On the other hand, for the treatment part, i.e., to those positive results in the confirmation test d 2 , we have where d 1 = RT and d 2 = VDRL.

Health outcomes of strategies
The health outcomes associated with uninfected and infected inmates, for a given strategy E ∈ {0, 1, 2, 3}, were assessed using QALY. For this purpose, the quality of life coefficients indicated in Table A2.2 (Appendix S2), were considered. We denote by Q j the quality of life coefficients associated with inmates in the stage j of the disease, with j ∈ {S, Y 1 , Y 2 , L, T, I}. Thus, the health outcomes of strategies E ∈ {0, 1, 3} are given by where X E (t) indicates the corresponding value of the state variable X at time t under the strategy E.
For Strategies 2a, 2b, 2c, and 2d (i.e., E = 2) the health outcome expression is different due to the actualization procedure of states variables described in Section 1.4. So, for E = 2, the health outcome expression is given by where t f +1 = T . In the above expression (specifically inside the integrals) initial conditions of state variables have to be updated to X E j (t + i ), according to the dynamics of Strategies 2a, 2b, 2c, and 2d (i.e., E = 2) described in Section 1.4, in order to integrate in intervals [t i , t i+1 ].

Calibration of key parameters
Most of the parameters, mainly those related to the disease dynamic and test performance, were obtained from systematic searches of indexed (PubMed, Cochrane, Scielo) and grey literature, complemented with epidemiological textbooks. We proceed with a sensitivity analysis for almost all of these parameters (see Table A2.2 in Appendix S2) according to the probability distribution indicated in the literature.
On the other hand, taking the reference values of parameters in Table A2.2 (Appendix S2) and based on a previous work that measured the syphilis prevalence in two Chilean prisons using rapid tests, we calibrate other (key) parameters in order to obtain a stationary 3% prevalence as an outcome for the Strategy 0 (current situation) (see initial conditions or the distribution by stage at the beginning of simulations in Table A2.1 in Appendix S2). We implemented this procedure in absence of historical data which has allowed an accurate model fit.
The calibrated parameters were c 1 and c 2 (number of sexual partners per year for inmates in primary and secondary stages); β 1 and β 2 (probabilities of transmission of inmates in primary and secondary stages); and µ (inmates turnover, that is, µ −1 is the average imprisonment time). Parameters c j and µ have an important influence in the model outcomes and they have a great variability among inmates.
For this reason we decided to calibrate them. We also included parameters β j in this calibration, because they appear multiplying c j in the model (A1.1). While our deterministic modeling approach may not deal with the entire variability of the number of sexual partners and imprisonment times, our model is still able to present qualitatively realistic and insightful results. Indeed, the main goal of our work has been to implement a decision model for evaluating the cost-effectiveness of different interventions rather than to represent the evolution of the disease with full accuracy. In spite of the lack of available data needed to implement a more realistic version, the simplicity of the proposed model successfully represents the desired amount of complexity observed in the disease dynamics.

Number of inmates in
Discount rate Discount rate for cost effectiveness analysis r % 3% Based on [22] i Parameters obtained from a calibration process described in Section 4 in Appendix S1.
ii Stage 1 includes incubation (a separate stage in Garnett et al. [4]) and there is no transmission at this part of the stage.
iii Estimation based on the population incarcerated in the Santiago Sur Penitentiary Detention Center in December 2016 [6].
iv Based on the average distributions observed in the reported cases of syphilis in Chilean general population during the period 2007-2012 [21].  i Primary stage duration considers merging primary and incubation stages included in Garnett et. al [4]. ii Weighted average prevalence of general syphilis in men from 20 to 49 years old in Chile 2015, according to estimated prevalence of the IHME [10] and estimated population [11]. iii Based on % of infected people who reported having a diagnosis in surveys conducted in Puente Alto and Arica's prisons. Stage distribution is based on the average distributions observed in the reported cases of syphilis in Chile during the period 2007-2012 [21]. iv A prevalence of 3% was assumed based on the results obtained in Arica prison [25] and exams applied in Puente Alto prison. Stage distribution is based on results found in Spain by Garriga et al. For the purposes of modeling, the existence of a case in tertiary syphilis was assumed (in the study by Garriga et al. there were no cases in this stage) [5]. v US$ 2017.
vi Includes costs for sample collection and medical consultation.
vii Includes costs for sample collection.