Optimizing an HIV testing program using a system dynamics model of the continuum of care

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1 Health Care Manag Sc (15) 18: DOI 1.7/s Optmzng an HIV testng program usng a system dynamcs model of the contnuum of care Sarah Kok Alexander R. Rutherford Reka Gustafson Rolando Barros Julo S. G. Montaner Krsztna Vasarhely, on behalf of the Vancouver HIV Testng Program Modellng Group Receved: 22 November 13 / Accepted: 26 November 14 / Publshed onlne: 17 January 15 Sprnger Scence+Busness Meda New York 15 Abstract Realzng the full ndvdual and populaton-wde benefts of antretrovral therapy for human mmunodefcency vrus (HIV) nfecton requres an effcent mechansm of HIV-related health servce delvery. We developed a system dynamcs model of the contnuum of HIV care n Vancouver, Canada, whch reflects key actvtes and S. Kok The IRMACS Centre, Smon Fraser Unversty, Burnaby, Brtsh Columba, Canada e-mal: sarah kok@sfu.ca A. R. Rutherford The IRMACS Centre and Department of Mathematcs, Smon Fraser Unversty, Burnaby, Brtsh Columba, Canada e-mal: sandyr@rmacs.sfu.ca R. Gustafson Vancouver Coastal Health, Vancouver, Brtsh Columba, Canada e-mal: reka.gustafson@vch.ca R. Barros Brtsh Columba Centre for Excellence n HIV/AIDS and Vancouver Coastal Health, Vancouver, Brtsh Columba, Canada e-mal: rbarros@cfenet.ubc.ca J. S. G. Montaner Brtsh Columba Centre for Excellence n HIV/AIDS and Faculty of Medcne, Unversty of Brtsh Columba, Vancouver, Brtsh Columba, Canada e-mal: jmontaner@cfenet.ubc.ca K. Vasarhely ( ) Faculty of Health Scences and The IRMACS Centre, Smon Fraser Unversty, Burnaby, Brtsh Columba, Canada e-mal: kvasarhe@sfu.ca decsons n the delvery of antretrovral therapy, ncludng HIV testng, lnkage to care, and long-term retenton n care and treatment. To measure the nfluence of operatonal nterventons on populaton health outcomes, we ncorporated an HIV transmsson component nto the model. We determned optmal resource allocatons among targeted and routne testng programs to mnmze new HIV nfectons over fve years n Vancouver. Smulaton scenaros assumed varous constrants nformed by the local health polcy. The project was conducted n close collaboraton wth the local health care provders, Vancouver Coastal Health Authorty and Provdence Health Care. Keywords HIV/AIDS Health servce delvery Publc health Treatment as Preventon System dynamcs HIV testng Optmzaton Resource allocaton 1 Introducton 1.1 Antretrovral therapy and the HIV epdemc Human mmunodefcency vrus (HIV) nfecton s a major contrbutor to dsease burden [38, 5, 62] and a leadng cause of death [88]. Globally, 35 mllon people are lvng wth HIV. Treatment has advanced enormously n recent years, although a vaccne or a cure s stll lkely to be years away. Combnatons of drugs known as hghly actve antretrovral therapy (HAART) 1 have the capacty to dramatcally alter the course of HIV dsease, mprovng health outcomes and extendng lfe expectancy. 1 Hghly actve antretrovral therapy (HAART) and antretrovral therapy (ART) are used nterchangeably n the lterature.

2 System dynamcs modellng to optmze an HIV testng program 335 HIV brngs on a progressve falure of the mmune system. Untreated nfecton generally proceeds to acqured mmunodefcency syndrome (AIDS) wthn eght to twelve years. The frst stage of dsease s the hghly nfectous acute phase that lasts sx to eght weeks, followed by a long asymptomatc latent phase precedng the onset of AIDS. Ths natural dsease progresson may be nterrupted through treatment wth HAART, whch restrcts vral replcaton and thereby slows mmune system deteroraton. The rsk of HIV transmsson s nfluenced by vral load the amount of vrus crculatng n the bloodstream. HAART has the potental to reduce HIV transmsson by as much as 96 % [18]. Therefore, antretrovral treatment also functons as a method of preventon [55]. Several studes found a declne n HIV dagnoses after mproved access to HAART, provdng emprcal support for a preventon effect at the populaton level [19, 54, 56, 79]. These observatons form the ratonale for the Treatment as Preventon (TasP) publc health strategy, whch ams to combat the HIV epdemc through mproved access to HAART. 1.2 Challenges n treatment access Access to HAART s nequtable around the world. Margnalzed groups wth hgh HIV prevalence often face the greatest barrers to health care access. Only about 3 % of people lvng wth HIV n low and mddle ncome countres receve treatment [38]. HAART coverage s nadequate n resource-rch countres as well. For example, only 25 % of people lvng wth HIV n the USA have suppressed vral load [26]. As of 11, an estmated 65 % of HIV-postve ndvduals were potentally nfectous due to unsuppressed vral load n Brtsh Columba (BC), Canada [59]. The World Health Organzaton (WHO), other nternatonal bodes, and a growng number of natonal governments are ntegratng TasP nto gudelnes and polces to reduce HIV transmsson, as well as HIV-related morbdty and mortalty. In 14, the Jont Unted Natons Programme on HIV/AIDS (UNAIDS) announced ambtous new testng and treatment coverage targets n order to acheve vral suppresson for 73 % of all people lvng wth HIV/AIDS by the year [39]. Expandng treatment coverage poses sgnfcant operatonal challenges [, 83]. Treatment transforms HIV nfecton nto a chronc condton; therefore, HIV management demands a shft to a chronc dsease servce delvery model [24]. HAART must be taken for lfe to mantan vral load suppresson, mproved health, and low nfectousness. Ths requres an effectve mechansm for the sustaned delvery of a contnuum of health care servces that range from HIV testng and lnkage to care, through to long-term montorng and retenton n treatment. Numerous gaps and neffcences n the HIV care contnuum have been documented [26, 33, 59]. Implementaton studes and ongong projects worldwde are evaluatng the effectveness of TasP strateges n overcomng these obstacles to treatment delvery [3, 83]. TasP was evaluated n Brtsh Columba durng 1 13 through the Seek and Treat for the Optmal Preventon of HIV/AIDS (STOP HIV/AIDS) plot project [36]. 1.3 The role of HIV testng Clearly, dagnoss of HIV nfecton s a prerequste for treatment. Untl recently, the WHO has recommended routne voluntary HIV testng wth counsellng n generalzed epdemcs and targeted testng to ndvduals presentng wth the sgns and symptoms of HIV nfecton n concentrated epdemcs [86]. The HIV epdemc n most countres outsde of sub-saharan Afrca s concentrated n key populatons, ncludng men who have sex wth men (MSM), njecton drug users (IDU), and female sex workers (FSW). Targeted HIV testng s usually employed n these countres. In sub-saharan Afrca, routne voluntary testng s recommended because HIV prevalence s hgh n the general populaton [27]. Recently updated WHO gudelnes nclude provderntated testng at health care facltes servng populatons most at rsk for HIV. These gudelnes also go a step further, recommendng the mplementaton of an approprate mx of routne and targeted testng delvery models based on the local epdemologcal context and avalable resources to ensure equtable access to HIV testng and counsellng [87]. A re-examnaton of the tradtonal dchotomzed approach to HIV testng s now necessary to develop nuanced strateges that better address the need to dagnose HIV as early n the course of the nfecton as possble [9, 84, 87]. Over the past decade, some jursdctons wth concentrated epdemcs have already begun to challenge conventonal testng recommendatons. Routne testng n nontradtonal settngs was found acceptable and effectve n several studes, ncludng n Argentna [75], Span [], and the UK [67, 78]. On the other hand, a study n the USA found only modest gans n new dagnoses most of these late n the course of nfecton through non-targeted HIV screenng n an emergency department [34]. Nevertheless, economc modellng studes consstently fnd routne screenng to be cost-effectve n terms of ncreased qualty-adjusted-lfe-years (QALYs) [47, 48, 64, 65, 71, 89], although the mpact vares by the state of the epdemc and model assumptons. Routne testng gudelnes and polces already exst n a few countres wth concentrated epdemcs. For example, the U. S. Preventve Servces Task Force has recently recommended screenng for the general populaton, renforcng the routne testng polcy launched by the Center for Dsease Control already n 6 [57]. Routne testng s recommended

3 336 S. Kok et al. n the Unted Kngdom for any settng wth HIV prevalence above.2 % [35]. Targeted testng has been the default polcy n Canada. In Vancouver, Brtsh Columba, Vancouver Coastal Health (VCH) and Provdence Health Care (PHC) local organzatons that oversee HIV testng have undertaken a plot project wthn the context of the STOP HIV/AIDS ntatve to evaluate routne HIV testng n hosptals [32]. In parallel, we conducted a resource allocaton analyss of Vancouver s HIV testng program to dentfy strateges to acheve optmal TasP outcomes. 1.4 Mathematcal models of HIV testng programs Resource allocaton models for HIV health servces have been under-utlzed n the past [3, 11, 44, 87]. Exstng models focus manly on preventon and treatment rather than testng. Alstar and Brandeau provde an overvew of the lterature on resource allocaton models [2], n whch they descrbe three man methodologes: lnear models, determnstc nonlnear epdemc models, and stochastc agent-based models. In contrast to lnear models, epdemc models capture non-lnear epdemc growth and can account for changes n epdemc outcomes due to nterventons. Stochastc agent-based models are more complex and have been used to study mxed nterventons. Modellng analyses have largely approached HIV testng from a health economcs perspectve to evaluate the cost effectveness of HIV testng programs [4, 5, 6, 31, 37, 48, 41]. These studes, whch we revew below, dffer from our operatonal approach, whch seeks ways to utlze a prevously allocated budget for the best possble effect on populaton health. Cost-effectveness models have been used to determne optmal HIV testng frequences for hgh-rsk groups [31, 48]. Gray et al. [31] used a stochastc agent-based model to assess the mpact on HIV ncdence of ncreasng testng coverage and frequency for gay men n New South Wales, Australa. They found that n the current testng and treatment envronment, further ncreases n the testng rate would only modestly reduce HIV ncdence. However, ncreasng testng coverage and frequency s lkely to have a greater mpact on ncdence n jursdctons wth lower testng levels. Lucas and Armbruster [48] used a smple model wthout dsease dynamcs to assess the cost-effectveness of ncreasng testng rates n screenng hgh-rsk groups n the Unted States. They found that the testng rates recommended by the Centers for Dsease Control are too conservatve and that ncreasng the rate of testng s cost-effectve for all rsk groups. Katz et al. [41] used a nonlnear determnstc model to determne potental changes n HIV prevalence assocated wth promotng home-use testng as an alternatve to testng n clncs for MSM n Seattle, USA. They found that HIV prevalence would ncrease n ths scenaro even wth an overall greater testng rate, because the length of the HIV nfecton-to-detecton wndow perod for homeuse testng s 9 days, as opposed to 15 days for clncbased testng. They also determned the requred wndowperod for home-use testng that would decrease HIV prevalence. The relatve effectveness of random HIV screenng and contact tracng was analyzed by Hyman et al. [37] usng two dfferental equaton models: the dfferental nfectvty model and the staged-progresson model. In the former, newly nfected ndvduals enter one of several compartments wth specfc nfectvtes. Ths captures rsk behavour varaton n the populaton. The epdemc n ths model s drven by superspreaders who frequently engage n hgh-rsk behavours. In the staged-progresson model, ndvduals progress through dsease stages wth dfferent nfectvtes, whch represent the acute, latent, and AIDS stages of HIV nfecton. The dfferental nfectvty model captures dfferences n nfectvty between ndvduals, whle the staged-progresson model captures varaton n nfectvty over tme for a sngle ndvdual. The authors fnd that contact tracng s more effectve n controllng the spread of HIV f epdemc dynamcs follow the dfferental nfectvty model. However, random screenng s more effectve f epdemc dynamcs follow the staged-progresson model. Armbruster and Brandeau analyzed the cost effectveness of mxed contact tracng and screenng strateges n a seres of papers [4 6]. They approach the problem usng both compartmental models [4, 6] anda networkmodel [5].Generally, they fnd that there s a threshold nvestment n testng programs requred to acheve cost-effectve control of an endemc dsease. Investment n testng programs beyond ths level leads to dmnshng returns and s not cost effectve. The cost-effectveness threshold depends on the mx of contact tracng and screenng programs and there s an optmal mx of testng programs, whch mnmzes total testng costs at threshold. The authors consder not only HIV, but also other chronc nfectous dseases, such as tuberculoss, hepatts B, and hepatts C. Addtonally, operatons research has been appled to determne the optmal procedure for dentfyng all postve blood samples n a routne screenng program [82]. Specfcally, the authors compared the effcency of drectly testng ndvdual samples to testng sub-groups of samples pror to ndvdual testng. Ths method to optmze screenng for nfectous dseases n large populatons was frst used durng World War II for syphls screenng [21, 82]. Optmal algorthms for testng large numbers of samples were also nvestgated by Abolnkov and Dukhovny usng queueng theory; they found that the optmal testng approach depends on HIV prevalence n the populaton [1].

4 System dynamcs modellng to optmze an HIV testng program 337 We are not aware of prevous analyses that use operatons research to optmze exstng testng programs by reallocatng resources among testng actvtes. Ths paper descrbes an operatonal analyss n whch we assume an aprortestng budget and seek to optmally allocate resources between targeted and routne testng programs to acheve the best publc health outcomes. Our focus on testng resources was motvated by the prortes of the publc health stakeholders on the team, who manage Vancouver s HIV testng program. However, the nfluence of testng on the HIV epdemc depends strongly on the entre follow-up and treatment framework. Vancouver follows a test and treat polcy [29], n whch treatment s offered at no cost to patents, mmedately after an HIV dagnoss. Therefore, to assess the epdemc mpact of testng program nterventons, we consder HIV testng n the context of the entre contnuum of HIV care. We explore the TasP objectve of preventng nfectons by optmzng resource allocaton to mnmze new HIV nfectons over fve and ten years. 1.5 Collaboraton background From 1 13, Vancouver Coastal Health and Provdence Health Care were partcpants n the STOP HIV/ AIDS Treatment as Preventon mplementaton plot project n Vancouver. Algned wth TasP goals, VCH and PHC sought ways to boost case fndng and early HIV dagnoses. We assembled a team of operatons research and VCH/PHC publc health experts to carry out the mathematcal modellng analyss to mprove the performance of Vancouver s HIV testng program. Pror to the collaboraton, VCH/PHC partners developed a conceptual framework for categorzng HIV testng actvtes to facltate plannng and management of the testng program. Smlarly, the operatons research partcpants brought to the collaboraton ther prevously developed conceptual framework for modellng the HIV care contnuum n Vancouver. The team decded to ntegrate the VCH/PHC testng framework nto a contnuum of HIV care model to produce a hghly customzed modellng tool for answerng questons of nterest to the local publc health stakeholders. The model was developed jontly, wth frequent consultatons and meetngs of the entre team to ensure that the relevance of the model to the practcal publc health questons was retaned. The project concluded wth several meetngs wth the team dedcated to nterpretaton of the fndngs and ther relevance to programmng and polcy. 2 System dynamcs to model the HIV care contnuum Health servces n the contnuum of HIV care nclude testng programs; lnkage of the newly dagnosed to care; clncal assessment; montorng for clncal ndcators of treatment ntaton, adverse events and drug resstance; long-term retenton n treatment; and hosptal care. Components of the Fg. 1 A smple causal loop dagram of the contnuum of HIV care

5 338 S. Kok et al. care contnuum are nterdependent a change n one has the potental to trgger changes n others creatng a complex system wth outcomes that are often dffcult to predct or quantfy. Therefore, the full mpact of a sngle nterventon s most accurately assessed n the context of the entre system. System dynamcs modellng s well suted to capturng complex feedback structures wthn systems. The causal loop dagram n Fg. 1 s a hgh-level depcton of relatonshps among elements of the HIV care contnuum. In the dagram, arrows represent postve and negatve causal nfluences between elements of the system. The small, crcular arrows represent two feedback loops. For example, consder an nterventon that mproves the effcency of an HIV testng program. As people lvng wth HIV are dagnosed and referred to further care, the proporton of the nfected populaton engaged n care and recevng antretrovral therapy grows. Those who acheve vral suppresson and adhere to HAART have a reduced probablty of transmttng HIV. Fewer new nfectons eventually leads to a drop n new dagnoses. Fgure 1 shows how ths feedback loop spans the entre contnuum of HIV care. The system dynamcs model of the HIV care contnuum n Vancouver we descrbe n ths paper s coupled to a non-lnear compartmental HIV transmsson model. Mathematcally, both system dynamcs models and compartmental transmsson models are systems of coupled ordnary dfferental equatons. Therefore, the combned model s smply a large nonlnear system of ordnary dfferental equatons, whch relates testng strateges to HIV ncdence, the number of new nfectons per unt tme. Ths model s used to fnd optmal allocatons of testng resources, whch mnmze the total number of new nfectons over fve and ten years. 3 Developng a qualtatve model We developed a qualtatve model of Vancouver s contnuum of HIV-related health servces ncorporatng a detaled representaton of the testng program to gude constructon of the system dynamcs model. Through ntervews wth system experts, ncludng HIV physcans and publc health representatves, we dentfed key actvtes and decsons, whch were documented usng the Unfed ModellngLanguage (UML) [] actvty dagram shown n Fg. 2. UML was orgnally developed as a vsual modellng language for software desgn. The applcaton of UML n the context of smulaton model development s dscussed by Sonnessa [76]. UML actvty dagrams provde a graphcal depcton of system workflow. Ther vsually ntutve structure facltated our dscussons wth system experts, whle the precse UML syntax served as a blueprnt for constructng the system dynamcs model descrbed n Secton 4. Below, we explan the dagram n Fg. 2, startng at the black crcle at the top left and proceedng through the publc health, communty care and acute care sectors of the health care system, whch are shown as separate swm lanes. In the Publc Health swm lane, the event boxes Identfy Populatons at Rsk, Identfy Cases and Test for HIV show HIV testng actvtes n a smplfed form. If the test s negatve, further tests may follow. A postve test leads to the Dagnose Infecton event. Each new dagnoss trggers Partner Notfcaton, whch s a publc health servce to dentfy and offer HIV testng to partners of HIV cases. The newly dagnosed are ether lnked or lost to care, as shown by Lnk to Care from Publc Health Sector or Patent Lost to Follow-Up, respectvely. In the Communty Care swm lane, health servce provders Identfy Cases for testng and offer a Test for HIV. A negatve test result may lead to further testng, whereas a postve result leads to the event Dagnose Infecton. Each dagnoss trggers a Report to Publc Health and partner notfcaton. In parallel to partner notfcaton, a newly dagnosed ndvdual may be lnked to care by the test provder (Lnk to Care from Communty Care Sector) or through publc health. If nether avenue s successful, the patent s lost to follow-up. In the Acute Care swm lane, HIV testng may be admnstered to patents presentng wth symptoms suggestve of HIV nfecton, or through routne screenng for HIV (Identfy Cases va Dagnostcs or Screen). As n communty care, all newly dagnosed nfectons are reported to publc health offcals and are ether lnked to care by the test provder, lnked to care from publc health, or lost to follow-up. Patents dagnosed wth HIV whle n a hosptal may undergo Clncal Assessment n Acute Care. All others lnked to care undergo ntal Clncal Assessment n communty care, ncludng vral load testng. Clncal assessment may have varous outcomes. Patents n the acute or latent phase of nfecton may proceed to montorng wth or wthout treatment ntaton, as ndcated by the Acute and Latent events, or Acute on HAART and Latent on HAART events, respectvely. Patents wth AIDS trgger the event Dagnose AIDS n Acute Care. Fnally, patents may become dsengaged from the health care system after ntal assessment n the Patent Lost to Follow-Up event. Once n care, t s possble to ntate treatment, acheve vral suppresson, experence treatment nterruptons, or become non-adherent to treatment regmens, as shown by the vertcal arrows between the Latent or AIDS events, Latent on HAART or AIDS on HAART events, and Latent Suppressed or AIDS Suppressed events. The events labelled Suppressed refer to patents who are vrally suppressed on HAART. The events labelled on HAART refer to treatment falure, non-adherent cases, and those who have not yet acheved vral suppresson. After the

6 System dynamcs modellng to optmze an HIV testng program 339 Fg. 2 A Unfed Modellng Language (UML) dagram of the contnuum of HIV care n Vancouver. Actvtes take place n the publc health, communty care, and acute care sectors of the health care system, depcted as separate swm lanes. Boxes wth rounded corners are events. Purple and pnk damonds represent decson and merge ponts, respectvely. Vertcal bars wth multple out arrows are logcal or statements for events that may occur smultaneously, whereas multple n arrows represent logcal and statements, whch means that all pror events must occur before movng forward. Pnk boxes represent epdemc outcomes of the HIV care contnuum. Patents may de and leave the system at any stage onset of AIDS, hosptalzaton may result for patents who are not vrologcally suppressed, whch s ndcated by arrows from AIDS and AIDS on HAART down to Hosptalzaton ICU or Hosptalzaton General Medcal Bed. Sustaned vral suppresson reduces morbdty, mortalty, and rsk of HIV transmsson, whch s ndcated by the arrows from Latent Suppressed and AIDS Suppressed to the outcomes Sustaned Suppressed Vral Load, Reduced Morbdty and Mortalty and HIV Incdence. Fewer new nfectons lead to a reducton n HIV Prevalence, whch n turn nfluences the yeld of postve dagnoses through testng n the three health care sectors. These populaton health outcomes reflect the combned effects of nterventons and are measures of the effcency of the HIV care contnuum. We ncorporated a detaled representaton of Vancouver s HIV testng program nto the qualtatve model n Fg. 2 by adaptng a conceptual framework developed by VCH for categorzng ther HIV testng actvtes. In Vancouver, HIV dagnoses may occur through targeted testng based on an dentfed HIV rsk; routnely offered voluntary testng n hgh HIV-prevalence settngs, for example detox centres; or routne testng programs n acute care settngs, such as hosptal departments for nternal medcne, renal or cardac servces. Expanson of the routne testng program to famly practces, prmary care clncs, and other general health care settngs was under consderaton durng our collaboraton and has subsequently moved forward. However, the only general health care settng ncluded n our model s acute care n hosptals because data were not avalable for other general health care settngs at the tme of the analyss. The VCH testng framework consders the key populatons of men who have sex wth men, njecton drug users and street-based female sex workers. In addton to reachng key populatons, routne testng n general health care settngs also reaches the general populaton. A summary of the VCH testng framework as ncorporated nto the model s gvenntable1. 4 System dynamcs model The actvtes and decson ponts n the UML actvty dagram of Fg. 2 were translated nto the stocks and flows of a

7 3 S. Kok et al. Table 1 HIV testng categores and subpopulatons Type of testng Targeted Routne n hgh prevalence settngs Routne n general health care settngs (Acute care only) Subpopulaton reached Key populatons Key populatons Key populatons and General populaton system dynamcs model. In the model, we combne the IDU and street-based FSW nto one key populaton, because they are closely lnked n Vancouver. A 6 cohort study of street-based FSW found that % were engaged n njecton drug use [74]. Therefore, there s substantal crosstransmsson between these groups. Furthermore, VCH confrmed that the two groups have smlar rsks for acqurng HIV. We denote ths combned key populaton n the model as IDU-FSW. Independent system dynamcs models were constructed for each of the subpopulatons MSM, IDU-FSW, and the general populaton.as shown n Fg. 3, each model conssts of 17 compartments representng stocks of undagnosed and dagnosed groups. Susceptble, HIV-negatve ndvduals S may become nfected and subsequently progress through the acute E, latent L, and AIDS A phases of nfecton. Dagnoses can take place durng any nfecton phase, ether through targeted testng, routne testng n hgh prevalence settngs or routne testng n acute care. Followng testng and dagnoss (flows a, b, andc n Fg. 3), there s a perod of watng to be lnked to care, represented by the fve compartments denoted by W, wth subscrpts ndcatng dsease state and method of dagnoss. Due to the short sx to eght week duraton of the acute phase [14] and the delay n engagement n care after dagnoss, we make the smplfyng assumpton that patents can only be lnked to care n the latent or AIDS phase of nfecton. Once n care, patents may be: () off treatment (L C and A C ), () recevng treatment yet not have acheved vral suppresson (L T and A T ), or () recevng treatment and vrologcally suppressed (L S and A S ). Patents may also leave care after dagnoss and become lost to follow up (L O and A O ). 4.1 Model assumptons The subpopulatons MSM, IDU-FSW, and the general populaton are treated as ndependent and non-nteractng n the model. These groups are socally, and to some extent, Fg. 3 System dynamcs model of the HIV care contnuum n Vancouver. Labels for flows of new dagnoses from each stage of nfecton are a: targeted testng and routne testng n hgh prevalence settngs; b: routne testng n acute care; c: dagnoses through symptom-based testng. Labels for stocks are S: HIV-negatve, susceptble to nfecton; E: undagnosed HIV-postve n acute phase; L: undagnosed HIV-postve n latent phase; A: undagnosed HIV-postve wth AIDS; W: watng to be lnked to care. Subscrpts for L and A are O: out of care; C: n care, off treatment; T : treated, not vrologcally suppressed; S: treated, vrologcally suppressed. Death from each compartment s not shown but taken nto account. The flow 1, s used to lnk dfferent portons of the dagram

8 System dynamcs modellng to optmze an HIV testng program 341 geographcally dstnct n Vancouver. The sze of any brdge groups have not been precsely estmated. A mathematcal model developed to study HAART expanson n Brtsh Columba assumed that 5 % of the MSM populaton and 6.5 % of the IDU populaton belong to both key populatons [46]. The assumpton of mnmal overlap was supported by survellance data from the Brtsh Columba Centre for Excellence n HIV/AIDS (BC-CfE) [46] and expert opnon of our publc health partners. No equvalent estmates for nteractons of key populatons wth the general populaton are currently avalable. Whle we mght expect some nteracton between IDU-FSW and the general populaton through clents of FSW n the general populaton, t s mportant to note that the IDU-FSW key populaton does not nclude brothel-based sex workers or sex work through escort agences. Based on the expert optnon of our publc health partners, HIV prevalence among these sex workers s qute low, because they receve frequent HIV testng. Clents of street-based sex workers are often engaged n njecton drug use and, therefore, are also members of the IDU- FSW key populaton n the model. We wll nvestgate the potental mpact of brdge groups n future analyss usng a network model. Brtsh Columba practces a test and treat polcy for HIV management [29]. Local gudelnes recommend treatment for everyone lvng wth HIV, except for elte controllers and long-term non-progressors, who are nfected but have the bologcal capacty to mantan low or undetectable vral load wthout treatment [16]. Antretrovral therapy s suppled by the provnce at no cost to patents. Ths mmedate treatment polcy n BC dffers from nternatonal recommendatons: WHO gudelnes for treatment ntaton are based on CD4 cell count, whch s an ndcator of the health of the mmune system. To reflect ths culture of wllngness to treat and be treated, treatment n our model may be ntated upon lnkage to care. Although physcans usually offer mmedate treatment, there s a mnorty of patents who choose to reman n care and begn treatment only when symptoms arse. To smplfy the model, we assume that these patents wll not be treated untl hosptalzaton due to an AIDS-related llness or the onset of AIDS. In the model, progresson from beng on HAART to beng on HAART wth vral suppresson occurs at a rate correspondng to the mean tme requred to acheve vral suppresson. In practce, ndvduals may move n and out of a vrologcally suppressed state due to short treatment nterruptons or low adherence to the treatment regmen. We address ths by usng data on the proporton of the populaton on treatment, but not vrologcally suppressed, to calbrate an effectve suppresson level for the L S and A S compartments. We assume that death rate and nfectousness are not affected by treatment status untl an ndvdual has acheved vral suppresson and those who do acheve vral suppresson are assumed to have the same lfe expectancy [7]. Retenton rates n Vancouver are hgh, wth approxmately 88 % of people dagnosed wth HIV retaned n care at the end of 13 [15]. We assume that patents not retaned n care are those who never engaged n care after a possble ntal clncal assessment. Although some ndvduals not retaned n care may have become dsengaged from the health care system at a later tme, there s no data avalable on when ths occurs. For ndvduals not n care, entry to care may occur after the onset of AIDS, when medcal attenton becomes necessary. In practce, AIDS patents who are hosptalzed may start treatment ether durng ther stay or upon dscharge. To capture ths opportunty for ntatng treatment, we model hosptalzaton as a trgger for treatment ntaton. Snce the average length of stay n the hosptal due to an AIDS-related llness s less than 11 days [8], we do not consder nhosptal dynamcs. Therefore, to smplfy the model, there s no hosptal compartment and patents who are hosptalzed move to ether the treatment compartment A T or to the n care off treatment compartment A C. We assume that hosptalzaton occurs only due to AIDS-related llnesses and vrally suppressed patents are not hosptalzed. We also assume that the HIV epdemc n Vancouver s approxmately n equlbrum at the tme of model ntaton. Ths assumpton smplfed calbraton and valdaton of the model. Valdty of ths approxmaton does not requre the state of the epdemc to be constant n recent years. Rather, f the epdemc s evolvng at a rate much slower than the response tme of the model, then we can assume adabatc or movng statonary evoluton of the dynamcal system over recent years. Ths assumpton s supported by data on the number of new dagnoses, whch s a commonly used, albet not necessarly accurate, proxy for the number of new nfectons. The estmated number of ncdent HIV nfectons n Brtsh Columba has been relatvely stable, fallng from 418 cases n 7 to 368 n 12 [56]. Furthermore, n the VCH jursdcton the number of new dagnoses has also been relatvely stable n recent years, decreasng from 164 n 9 to 151 n 13 [15]. The most sgnfcant perturbaton to the HIV epdemc was the ntroducton of HAART, whch occurred n 1996, well before the tme of model ntaton. 4.2 Model equatons The tme evoluton of each subpopulaton n the system dynamcs model s governed by the system of ordnary dfferental Eqs Each equaton corresponds to a model compartment n Fg. 3. The superscrpt denotes the subpopulatons, where s 1 for MSM, 2 for IDU-FSW, and 3 for the general populaton. Therefore, the total number

9 342 S. Kok et al. of equatons s 51. A complete lst of model parameters s gven n Tables 2 and 3. ds ( = d S + L S + A S + E + L + WEa + W Eb + W La ) +WLb + L C + L T + L O )( ) + (d + d a A + Wc + A C + A T + A O S β ( δ a E + L + δ b A + δ a δ d (W Ea + δ pweb ) +δ d (L C + L T + L O + W La + δ pwlb ) ( )) +δ h (L S + A S + δ b δ d A C + A T + A O + δ pwc d S (1) ) dl C dl T dl S ( ) = γ (1 l h 1 ) WEa + W La ( ) +γ (1 l h 2 ) WEb + W Lb ( ) +ζ L T + L S = h 1 γ ( τ d + d ) L C (11) ( ) ( ) WEa + W La + h 2 γ WEb + W Lb ζl T νl T (τ d + d ) L T (12) = νl T (ζ + d ) L S (13) de = S β ( δ a E + L + δ b A + δ a δ d (WEa + δ pweb ) +δ d (L C + L T + L O + W La + δ pwlb ) +δ h (L S + A S )) +δ b δ d (A C + A T + A O + δ pwc ) ( E ) ) da O da C = lγ W c + τ dl O ( ρ u + d + d a ) A O (14) = γ (1 l h 2 ) Wc (A + ζ T + A S ( ) +(1 h 3 )τ d L C ρ v + d + d a A C ( ) ) + (1 h 4 ) (ρ v A C + A T + ρ u A O ) (15) dl ( a λ 1 b 1 + λ 2 b 2 + λ 3 b 3 S + E + L ( τ a + d ) E (2) ( ) ( = τ a E λ 1 b 1 + λ 2 b 2 + L ) λ 3 b 3 S + E + L ( τ l + d ) L (3) da T = h 2 γwc + h 3τ d L C + τ dl T ( ) ) +h 4 ρ v (A C + A T + ρ u A O ) (ζ + ν + ρ v + d + d a A T (16) da dw Ea dw Eb dw La dw Lb dw c dl O ) = τ l L (ρ u + d + d a A (4) = a ( λ 1 b 1 + λ 2 b 2 = a λ 3 b 3 ( ) ( E S + E + L E S + E + L ) ( γ + d ) WEa (5) ) ( γ + d ) WEb (6) ( ) ( = λ 1 b 1 + L ) ( λ 2 b 2 S + E + L γ + d ) WLa (7) = λ 3 b 3 ( L S + E + L ) ( γ + d ) WLb (8) ) = ρ u A (γ + d + d a Wc (9) = lγ ( ) ( WEa + W Eb + W La + W Lb τ d + d ) L O (1) da S = νa T (ζ + d ) A S (17) Equaton 1 determnes the tme evoluton of the susceptble compartment S. The rate of flow nto compartment S s set equal to the total populaton death rate n order to mantan a constant populaton. The natural death rate for subpopulaton s denoted d. There s an addtonal HIVrelated death rate d a for the AIDS phase of nfecton. Ths addtonal death rate s the same for all subpopulatons. Susceptbles n each subpopulaton are nfected at a rate gven by a blnear contact term that s proportonal to the number of nfected ndvduals tmes the number of susceptble ndvduals. The proportonalty constant, whch s termed the nfectvty, s the probablty per unt tme of dsease transmsson from a sngle nfected ndvdual to a sngle susceptble ndvdual. It depends on both behavoural and bologcal factors, such as the vral load of the nfected ndvdual. Behavoural factors whch nfluence nfectvty nclude condom use and needle exchange programs. The prevalence of male crcumcson n the populaton wll also

10 System dynamcs modellng to optmze an HIV testng program 343 Table 2 Model parameters Parameter Defnton Value Reference N 1 Sze of MSM populaton, [51] N 2 Sze of IDU-FSW populaton 6,5 [51, 85] N 3 Sze of general populaton 47, [51] λ 1 1 Number of targeted tests per month (MSM) 43 Unpublshed a λ 1 2 Number of routne tests n hgh prevalence settngs per month (MSM) 3 Unpublshed a λ 1 3 Number of routne tests n acute care per month (MSM) λ 3 (N 1 /N 1 + N 2 + N 3 ) λ 2 1 Number of targeted tests per month (IDU-FSW) Unpublshed a λ 2 2 Number of routne tests n hgh prevalence settngs per month (IDU-FSW) 61 Unpublshed a λ 2 3 Number of routne tests n acute care per month (Key populatons) λ 3 (N 2 /N 1 + N 2 + N 3 ) λ 3 1 Number of targeted tests per month (General populaton) λ 3 2 Number of routne tests n hgh prevalence settngs per month (General populaton) λ 3 3 Number of routne tests n acute care per month (General populaton) λ 3 (N 3 /N 1 + N 2 + N 3 ) λ 3 Number of routne tests n acute care per month (Total) 681 Unpublshed a μ Proporton of patents on treatment who acheve vral suppresson.77 [15] δ a Infectvty multpler n acute phase 9.2 [1] δ b Infectvty multpler n AIDS phase 7.3 [1] δ d Infectvty multpler after dagnoses 1.68 [49] δ v Infectvty multpler for patents wth suppressed vral load 1.96 [18] δ h Infectvty multpler for patents n treatment μδ v + (1 μ) δ p Infectvty multpler for patents n hosptal Assumpton b 1/τ a Length of acute phase 7 weeks [14] 1/τ l Length of latent phase 1 years [52] 1/τ d Length of tme from acute or latent dagnoss to onset of AIDS 7 years Assumpton c d 1 Natural death rate (MSM) 68 years 1 [77] d 2 Natural death rate (IDU-FSW) 58.8 years 1 [] d 3 Natural death rate (General populaton) 68 years 1 [77] d a HIV-related death rate (AIDS phase) 2 years 1 [52] 1/γ Mean tme to lnkage to care 11 days [81] l Proporton of patents not retaned n care.12 [15] k Proporton of patents dagnosed n acute care or n AIDS phase Model dynamcs α Proporton of patents ntatng treatment wthn 1 month of dagnoss.38 Unpublshed d h 1 Probablty of non-aids patents ntatng treatment after dagnoss out of acute care See Eq. h 2 Probablty of patents ntatng treatment after dagnoss n acute care or AIDS phase.9 (1 l) Unpublshed e h 3 Proporton of patents ntatng treatment mmedately after AIDS dagnoss.9 Unpublshed e h 4 Proporton of patents ntatng treatment after dscharge from hosptal.9 Unpublshed e 1/ν Mean tme to vral suppresson 4.1 months Unpublshed d ζ Rate of treatment nterruptons.55/person-month [28] 1/ρ u Mean tme to dagnoss after onset of AIDS (Undagnosed HIV nfecton) 6 months Assumpton ρ v AIDS-related hosptalzaton rate.345/person-month [22] The superscrpt determnes the subpopulatons, where s 1 for MSM, 2 for IDU-FSW and 3 for the general populaton a Vancouver Coastal Health data b Assume hosptalzed patents do not nteract wth susceptble subpopulaton c Based on stage of dsease at dagnoses data from [15] d Drug Treatment Program at the Brtsh Columba Centre for Excellence n HIV/AIDS data e Expert opnon, Brtsh Columba Centre for Excellence n HIV/AIDS

11 344 S. Kok et al. Table 3 Model calbraton parameters Parameter Defnton Value HIV nfectvty β 1 MSM β 2 IDU-FSW β 3 General populaton Modfer for probablty of acute phase dagnoses a 1 MSM 9. a 2 IDU-FSW 6.32 a 3 General populaton Modfer for probablty of latent phase dagnoses Targeted testng b1 1 MSM.212 b1 2 IDU-FSW Modfer for probablty of latent phase dagnoses Routne testng n hgh prevalence settngs b2 1 MSM b2 2 IDU-FSW Modfer for probablty of latent phase dagnoses Routne testng n acute care b3 1 MSM b3 2 IDU-FSW b3 3 General populaton.7429 nfluence the nfectvty for female to male sexual transmsson. The force of nfecton, defned as the rate at whch susceptble ndvduals acqure HIV, s the nfectvty tmes the number of nfected ndvduals. The baselne nfectvty β for subpopulaton s defned to be the nfectvty when the nfected ndvdual s n the latent stage of the dsease. When the nfected ndvdual s n the acute or AIDS stages, the nfectvty s ncreased through multplyng by δ a or δ b, respectvely. Furthermore, β s decreased through multplyng by δ d or δ h when ndvduals are dagnosed or treated, respectvely. Hosptalzed patents have ther nfectvty further reduced by the factor δ p. The nfectvty modfers δ a, δ b,andδ h are the same for all subpopulatons, because they account for the mpact of changes n vral load on nfectvty. The nfectvty modfer δ d arses from changes n the rsk behavour of patents after dagnoss. Although ths could be dfferent for the subpopulatons, there s no data avalable to calbrate ths dfference. Therefore, we made the smplfyng assumpton that δ d s the same for all subpopulatons. The nfectvty modfer δ p for hosptalzed patents s also assumed to be the same for all subpopulatons. Followng nfecton, ndvduals progress through the acute phase E, latent phase L, and AIDS phase A of undagnosed HIV, as descrbed n Eqs The mean duraton of the acute phase and latent phase are 1/τ a and 1/τ l, respectvely. Undagnosed ndvduals n states E and L are typcally asymptomatc. Therefore, dagnoss n these compartments occurs at a rate proportonal to the undagnosed prevalence n the populaton beng tested wthn each subpopulaton. The number of tests per unt tme n each testng stream for subpopulaton s λ n,wherens 1 for targeted testng, 2 for routne testng n hgh prevalence settngs, or 3 for routne testng n acute care. Conversely, the AIDS phase of HIV nfecton s typcally symptomatc. Therefore, HIV dagnoss n the AIDS phase occurs at a constant rate ρ u, whch represents the average tme from the onset of AIDS to the tme of seekng health care. A porton of these dagnoses are attrbuted to routne testng n acute care based on hstorcal numbers and the remanng AIDS dagnoses are equally attrbuted to targeted testng and routne testng n hgh prevalence settngs. An mportant lmtaton of compartmental models s that they assume perfect mxng of the populaton. In ths context, the probablty of a test yeldng a postve dagnoss s smply the fracton of the undagnosed populaton that s HIV postve. However, ths fals to capture the mportant concept of testng reach, wheren HIV-postve ndvduals have varyng probabltes of beng tested. Therefore, modfyng factors for each subpopulaton and testng program were ntroduced. For dagnoss n the acute stage of nfecton and subpopulaton, the modfyng factors are denoted by a. The modfyng factors for the latent stage of nfecton are bj, for subpopulaton and testng program j.inthe model, the probablty of a test beng postve s the nave compartmental model probablty, gven by the HIV prevalence n the undagnosed segment of the subpopulaton, tmes the modfyng factor. Data s not avalable for these modfyng factors and they are treated as calbraton parameters n the model. Equatons 5 through 9 correspond to compartments W Ea through W c, whch represent patents watng to be lnked to care, dependng on the method of testng and stage of nfecton at dagnoss. Compartment W Ea represents ndvduals n the acute phase of nfecton, who were dagnosed va targeted testng or routne testng n hgh prevalence settngs. Smlarly, compartment W La represents those n the latent phase of nfecton who were dagnosed va targeted testng or routne testng n hgh prevalence settngs. Compartments W Eb and W Lb represent ndvduals dagnosed n the acute phase and latent phase va routne testng

12 System dynamcs modellng to optmze an HIV testng program 345 n acute care settngs, respectvely. Compartment W c represents those dagnosed va symptom-based testng durng the AIDS phase of nfecton. We assume that all of these patents are hosptalzed after dagnoss to treat an AIDS defnng llnesses. Equatons 1 13 descrbe the populaton dynamcs of ndvduals n the latent phase of HIV who have been dagnosed. The subpopulaton of dagnosed but not retaned n care s denoted by L O ; the subpopulaton n care but not on treatment s denoted by L C ; the subpopulaton on treatment but not suppressed s denoted by L T ; and the subpopulaton wth suppressed vral load s denoted by L S. Equatons descrbe the dynamcs of correspondng compartments for ndvduals n the AIDS phase of nfecton. Lnkage to care occurs at rate γ. A fracton l of patents are not retaned n care and enter compartment L O or A O. The probablty of startng treatment after beng lnked to care from compartments W Ea or W La s h 1. The probablty of startng treatment after beng lnked to care from compartments W Eb, W Lb or W c s h 2. Patents may also ntate treatment wth the onset of AIDS or after an HIV-related hosptalzaton, whch occur wth probabltes h 3,andh 4, respectvely. Hosptalzaton occurs at rate ρ ν. Treatment nterruptons occur at rate ζ and ndvduals on treatment acheve vral suppresson at rate ν. The mean tme from dagnoss n the acute or latent phase of nfecton to AIDS s 1/τ d. The system of Eqs. 1 17was solved numercally n MAT- LAB usng the functon ode45, a4 th and 5 th order Runge- Kutta method wth varable tme step [72]. No numercal nstablty problems were encountered. 4.3 Parameters and data sources Parameter estmaton was based on data from our publc health partners (VCH and PHC), the Brtsh Columba Centre for Excellence n HIV/AIDS (BC-CfE), publc health reports [14, 15, 77,, 81] and publshed lterature [1, 22, 49, 51, 52, 85]. A complete lst of model parameters, values, and references s gven n Table 2. The model parameters for the monthly rate of targeted tests and routne tests n hgh prevalence settngs were calbrated from the average testng rate from January 1 through October 12. The number λ 3 of routne tests per month n acute care were calbrated from the acute care testng rate durng the sustaned mplementaton of the STOP HIV/AIDS acute care strategy, from July 12 through June 13. Data from the provncal Drug Treatment Program was provded by the BC-CfE, whch s responsble for dspensng all HIV antretrovral treatment n the provnce. Data for the year 11 were used to estmate model parameters for the proporton of patents startng treatment wthn one month of dagnoss and the mean tme to acheve vral suppresson. The mean tme for lnkage to care, 1/γ,was calbrated from the VCH report [81]. The proporton of dagnosed ndvduals l who are not retaned n care was calbrated from a VCH 13 publc health report [15]. Hosptalzaton rates n the model are based on Brtsh Columba data from the BC-CfE Drug Treatment Program and the BC Mnstry of Health admnstratve databases [22]. The rate at whch patents start treatment n the model was calbrated as follows. Let α be the probablty of startng treatment wthn one month of dagnoss, where the category of dagnoss s for all new dagnoses, 1 for acute and latent phase dagnoses outsde of acute care, and 2 for AIDS dagnoses and all dagnoses n acute care. Smlarly, let h be the probablty of startng treatment after beng lnked to care, where denotes the category of dagnoss as above. Compartmental models are contnuous or large populaton approxmatons to stochastc agent-based models n whch the length of tme that agents spend n a state s exponentally dstrbuted. Therefore, t follows from standard propertes of the exponental dstrbuton that α s gven by α = Pr{ndvdual s lnked to care wthn 1 month} h = (1 exp( γ 1)) h, (18) where γ s the rate per month of lnkage to care. Furthermore,wehavethat α = α 1 (1 k) + α 2 k and h = h 1 (1 k) + h 2 k, (19) where k s the fracton of all patents lnked to care after dagnoses n acute care or n the AIDS stage of nfecton. The parameter h 2 s estmated from data and the fracton k s determned through model dynamcs. Therefore, we solve for h 1 usng Eqs. 18 and 19 to obtan h 1 = α h 2 k ( 1 exp( γ 1) ) (1 k) ( 1 exp( γ 1) ). () Ths expresson for h 1 s substtuted nto Eqs. 11 and 12. The modfyng factors for the nfectvty n the acute and AIDS phases of nfecton are not specfc to Vancouver and estmates were taken from the lterature [1]. Estmates of the average length of the latent and AIDS phase of HIV nfecton were also also taken from the lterature [52]. Vancouver data on the change n nfectvty after dagnoss were not avalable and t was necessary to estmate ths modfer from the lterature [49]. However, the change n nfectvty after dagnoss results from behavour change and ths s lkely to depend on the counsellng and followup programs n the health care jursdcton.

13 346 S. Kok et al. 4.4 Model calbraton and valdaton The model has fve free parameters each for the MSM and IDU-FSW populatons and three free parameters for the general populaton, for a total of 13 free parameters that are determned through model calbraton. These parameters are lsted n Table 3. Model calbraton was done at equlbrum, because we have assumed n Secton 4.1 that the HIV epdemc s close to equlbrum at the tme of model ntaton. There are three nfectvty parameters, one for each of the subpopulatons. These nfectvtes are dffcult to calculate from data because ths would requre accurate estmates of the frequency of potental transmsson acts and varous factors whch modfy the probablty of HIV transmsson per act, ncludng condom use, prevalence of other sexually transmtted dseases, and male crcumcson. Therefore, these nfectvtes were treated as free parameters and were determned through model calbraton. The modfyng factors a and bj for the probablty of a dagnoss n the acute phase and latent phase of nfecton, respectvely, were ntroduced n Secton 4.2 to account for testng reach. There are a total of ten dagnoss modfyng factors. MSM and IDU-FSW each have an acute phase modfer and one for each of the testng programs. The general populaton only has two dagnoss modfers, because routne testng n hgh prevalence settngs and targeted testng do not apply to the general populaton. Data were not avalable for the dagnoss modfyng factors and they were treated as free parameters, whch were determned through calbraton. Data used for model calbraton and valdaton are lsted n Table 4. Data on the number of new dagnoses were avalable from multple sources. VCH provded the number of new dagnoses stratfed by testng program for the years The Brtsh Columba Centre for Dsease Control (BCCDC) and the Drug Treatment Program of the BC-CfE both provded data on the total number of new dagnoses; however, they were unable to stratfy new dagnoses by testng program. The total number of new dagnoses n 11 reported by the BCCDC [14] s smlar to the total number reported by VCH. However, the annual numbers of new dagnoses reported by BCCDC and VCH are hstorcally lower than the annual number from the Drug Treatment Program. Data from the BC-CfE Drug Treatment Program are consdered to be more accurate [36]. The average value of the rato φ of the total number of new dagnoss from the Drug Treatment Program to the total number of new dagnoss from VCH s 1.2 for the years 7 9, whch are the most recent three years of common data. Data from VCH on the number of new dagnoses by testng program were corrected by multplyng by φ = 1.2. Vancouver Coastal Health Authorty provded survellance data from the STOP HIV/AIDS project on the proporton of dagnoses durng the acute phase of nfecton, determned through vral load testng; however they were only able to stratfy these data by testng program and not by subpopulaton. Therefore, we had to assume that wthn each testng program, the proporton of dagnoses n the acute phase of nfecton s the same for all subpopulatons. Table 4 lsts the 19 data values that were used for model calbraton and valdaton. We used 18 of these data ponts to calbrate the 13 free parameters, employng a nonlnear least squares ft. Therefore, calbraton s over-determned by 5 degrees of freedom, whch s suffcent for model valdaton. An addtonal data value, the percentage of the dagnosed populaton on treatment, was used for ndependent out-of-sample valdaton. The model for each of the three subpopulatons were calbrated ndependently, because they are treated as nonnteractng subpopulatons Calbraton of the MSM and IDU-FSW populatons We ft fve model parameters for each of the MSM and IDU-FSW subpopulatons. Denotng the MSM subpopulaton by superscrpt = 1 and the IDU-FSW subpopulaton by superscrpt = 2, these parameters are the HIV nfectvty β, the modfer a for the probablty of a postve dagnoss n the acute phase of nfecton, the modfer b1 for the probablty of a postve dagnoss n the latent phase of nfecton for targeted testng, the modfer b2 for routne testng n hgh prevalence settngs, and the modfer b3 for routne testng n acute care. Data used for optmzaton were HIV prevalence; the number of new dagnoses from targeted testng, routne testng n hgh prevalence settngs, and routne testng n acute care; and the proporton of dagnoses n the acute phase of nfecton for each of targeted testng, routne testng n hgh prevalence settngs, and routne testng n acute care. For each subpopulaton, these data ponts are denoted n correspondng order by yj,for j = 1, 2,...,7. All data values are lsted n Table 4. We calbrated the parameters for the MSM and IDU- FSW subpopulatons by weghted nonlnear least squares ft, defned by mnmzng the objectve functons ( ) F x,φ = ( 7 y j ỹj ( x ) ) 2, (21) j=1 y j where x = ( β,a,b1,b 2 3),b s the parameter vector and the model output correspondng to the data pont yj s ỹj ( x ). Mnmzng the objectve functon n Eq. 21 s a 5- dmensonal optmzaton problem. Each evaluaton of the objectve functon requres fndng the equlbrum soluton to the system of Eqs. 1 through 17. Hgh-dmensonal,