Elderly Care Sensor System

Patients, Primary Care, and Policy: Simulation Modeling for

Health Care Decision Support

Martin Comis · Catherine Cleophas · Christina B¨using

Creasing Due To The Ongoing Demographic Change,

while at the same time health service providers face difficulties in finding skilled personnel. This cre-

Ates Pressure On Health Care Systems Around The

world, such that the efficient, nationwide provision of primary health care has become one of society’s greatest challenges. Due to the complexity of health care systems, unforeseen future events, and a fre-

Performance Of Health Care Systems Means Tackling

a wicked problem. To support this task for primary care, this paper introduces the hybrid agent-based

Simulation Model Sim-Care. Sim-Care Models The

interactions of patients and primary care physicians on an individual level. By tracking agent interac- tions, it enables modelers to assess multiple key in- dicators such as patient waiting times and physician utilization. Based on these indicators, primary care

Systems Can Be Assessed And Compared. Moreover,

changes in the infrastructure, patient behavior, and service design can be directly evaluated. To show- case the opportunities offered by SiM-Care and aid model validation, we present a case study for a pri-

Mary Care System In Germany. Specifically, We In-

This work is supported by the Freigeist-Fellowship of the Volkswagen Stiftung. This work is supported by the German research council (DFG) Research Training Group 2236 UnRAVeL.

Catherine Cleophas

Working Group Service Analytics, Christian-Albrechts- Universit¨at zu Kiel, Westring 425, 24118 Kiel, Germany vestigate the effects of an aging population, a de- crease in the number of physicians, as well as the combined effects.

1 Introduction

Health is one of the most important factors for the prosperity and well-being of a society. Therefore,

Eryone’S Access To Health Services . The Foun-

dation of universally accessible health services is

Physicians , Primary Care Systems “Serve As The

patient’s first point of entry into the health care sys-

Tem And The Continuing Focal Point For All Needed

health services”. To that end, they feature a set of

Mary Care Services To A Defined Population Of Pa-

tients”. Such primary care services include “health promotion, disease prevention, health maintenance, counseling, patient education, diagnosis and treat- ment of acute and chronic illnesses”.

Progress Paired With Improved Living Conditions

and reduced birth rates lead to a rise in the share of elderly citizens. In the United States, the percent- age of individuals aged 65 and older is predicted to exceed 21% of the total population by 2030 .

2

Martin Comis et al.

Age, Their Demand For Primary Care Services Tends

to increase. This is primarily due to the prevalence of chronic illnesses which disproportionately affect

Older Adults . On The Supply Side, Primary

care physicians are also aging, e.g., 34.1% of all

Or Older By The End Of 2017 And Thus About To

retire. Moreover, fewer medical students are will-

Private Primary Care Practice . This Leads To Re-

duced treatment capacities and exacerbates the risk for supply disruptions. In the United States, the “confluence of a rising de-

Mand For Primary Care Services And A Decreasing

supply of professionals providing these services” is

Considered A “Crisis In Primary Care” . In Order

to manage this crisis, existing primary care systems

Have To Be Fundamentally Adjusted . Various

new concepts and policies to maintain the standard of health care provision are discussed by the statu-

Cians . However, All Such Concepts Call For

validation and evaluation prior to their potentially

Costly Implementation . Naturally, This Leads To

the pressing question: How can the quality of pri- mary care systems and the effect of changes to them

Which Defines Adequate Health Care Supply On The

basis of profession-specific ratios. The law subdi-

Ical Specialization. For Example, The Predefined

nominal ratio of primary care physicians is one PCP per 1,671 inhabitants [24, §11(4)]. This base indi-

Dividual Demographic And Geographic Characteris-

tics [24, §2]. If the actual ratio of a zone is signif- icantly higher than the nominal ratio, closing prac- tices will not be replaced. If it is significantly lower, new practices are permitted to be opened.

Services Administration (Hrsa) Defines Adequate

health care supply based on profession- and region- specific population-to-provider ratios. If the pre-

Needs ) Of A Geographic Area Is Exceeded, Hrsa

designates it a health professional shortage area to

Which National Health Service Corps Personnel Is

directed with priority. Obviously all ratio-based assessments have several

Justment Rules Are Highly Dependent On The Defini-

tion of the underlying zones or geographic areas. Factors such as the accessibility of practices and

The Pcp’S Individual Workload Are Completely Ne-

glected. Finally, ratio-based assessments cannot ac-

Count For New Health Care Concepts Such As Tele-

medicine, mobile medical units, or centralized ap- pointment scheduling.

Model Dynamic Effects In Primary Care Systems Is

required. However, analyzing and evaluating health

Due To The Large Number Of Involved Individuals

and uncertain nature of health care processes, e.g., fluctuating demand, arrival time of patients, emer-

Modeling (Abm), Which Account For Individual

agents and their interactions on the micro-level. A general introduction to the concept of ABM is pro-

Numerous Applications Of Abm In Field Of Health

care , e.g., for accountable care organizations [5,

44], For Medical Workforce Forecasting , And In

epidemiology .

Care Systems. Sim-Care Models Patients And Pcps

on an individual level as illustrated by Figure 1:

On Their State Of Health And Pcps Handle Appoint-

ment requests, manage patient admission, and treat patients. By tracking the resulting interactions in Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

Patients

Fig. 1 Geo-social system of patients and physicians.1

Different Subproblems, Evaluate New Planning Ap-

proaches, and quantify the effects of interventions

Measurements. From Empirical Data, We Develop Re-

alistic test scenarios including a controllable degree of uncertainty realized via stochastic simulation ex- periments.

Model Sim-Care, Which Provides Decision Makers

with a versatile decision support tool for primary

Be Easily Modified And Extended To Meet Each Mod-

eler’s needs. Patients and physicians are modeled as individuals who follow their own objectives, learn, and adapt. To ensure computational tractability, the model incorporates a global event queue at its core.

From Agent-Based Modeling And Discrete Event Sim-

ulation. Based on empirical data for a German pri-

Mary Care System, We Illustrate How Scenarios For

the simulation model can be generated. Finally, we

Showcase The Opportunities Of Sim-Care Through A

case study. To the best of our knowledge, SiM-Care is the first model of its kind that captures entire pri-

Mary Care System With All Physicians And Patients

as individual agents and allows for the simultane-

Ous Consideration Of Microsystem Improvements As

well as macrosystem reforms. 1 Map tiles by Stamen Design, under CC BY 3.0. Data by OpenStreetMap, under ODbL.

Based Simulations With A Focus On Health Care Ap-

plications. Section 3 introduces SiM-Care based on

Aid Model Validation And Showcase How Sim-Care

can be applied to support health care planning. Sec- tion 5 discusses the potential applications and entry

Requirements Of Sim-Care. Finally, Section 6 Con-

cludes and provides directions for future work.

2 Related Work

Decision support for health care planning is an area

Care Systems, Decision Support Tools Have To Deal

with the detail complexity that is inherent to the

Health Care Sector : Patients Schedule Appoint-

ments based on their preferences and state of health, while PCPs offer appointments and treat patients. Thereby, micro-interactions affect macro-level in- dicators as agents observe, learn and adapt, decide and act, and - as a group - determine the system’s

Behavior . When A System’S Status Depends On

such micro-interactions, its behavior becomes diffi- cult to predict and a “wide range of possible out- comes may arise from any policy change” .

Simulation Modeling Can Deal With This Kind Of

complexity by “simulating the life histories of in- dividuals and then estimating the population effect

Such, Simulation Models Represent A Powerful Tool

to inform policy makers: They can provide valuable

Insights Into The Dependencies Within Health Care

systems and allow for the prediction of the outcome of a change in strategy ahead of a potentially costly and risky real-world intervention .

Given these potentials, the use of computer simula- tion to study health care delivery systems has sig-

Nificantly Increased Over The Recent Years . The

resulting body of literature on applications of simu- lation in health care is rich and can, from a method- ological point of view, be classified into four main

More Of The Former Paradigms . To Catego-

rize related work on the study of primary care sys- tems even further, we distinguish in the following

Models That Focus On Microsystem Improvements

and models that aim at macrosystem reforms.

4

Martin Comis et al.

Improvements In Primary Care Systems Mostly In-

clude a detailed model of a single (specific) outpa-

Potential Improvements. Zong Et Al. Present A

discrete event simulation for a pediatric clinic at includes a very detailed representation of the se- quential stages during a patient’s visit. In a set of

The Overall Performance Of The Clinic Is Impacted

by different scheduling templates, a change in the medical assistant to physician ratio, and the pairing of resident doctors with clinicians. Shi et al.

Developed A Discrete Event Simulation Model For

eran Affairs. Within the model, the different patient flow routes for appointment patients, walk-ins, and nurse-only patients are distinguished. In a scenario analysis, the authors investigate how the clinic’s performance is affected by six distinct factors that

Include Walk-In And No-Show Rates As Well As The

double booking of appointments. Cayirli et al.

Used Empirical Data Collected At A Primary Care

clinic in New York to devise a discrete event simu- lation of a generic single-server primary care prac- tice. The model distinguishes new and returning pa- tients and accounts for walk-ins, no-shows, patient punctuality, and service time variations. In a sim- ulation study, the authors evaluate 42 appointment

Systems That Vary In The Implemented Sequencing-

and appointment rules. A similar discrete event sim- ulation of a generic single-server primary care prac-

Tice Is Introduced By Schacht . In His Model,

all arriving patients have a stochastic willingness

To Wait And Always Request An Appointment. If The

indirect waiting time for this appointment exceeds a patient’s willingness to wait, they become walk- ins. The arrival rate of patients depends on the ses-

Sion, Day, And Month To Model Seasonality. In A

case study, the author evaluates a class of appoint-

Make Up A Primary Care System. Moreover, All Of

these models adopt a different approach to the rep-

Resentation Of Patients: While Sim-Care Models A

persistent patient population that is shared by all providers, the previous models perceive patients as non-persistent, i.e., patients are generated as they arrive at the practice and cease to exist as soon as they are discharged. As a result, the previous mod- els cannot account for the effects of individual mi-

Crosytem Improvements On The Entire Primary Care

system itself. Simulation models aimed at investigating macrosys- tem reforms of primary care systems mostly include

An Entire Primary Care System, However They Are

usually much more high level. Matchar et al.

Use The Methodology Of System Dynamics To De-

velop a simulation model to aid primary care plan-

Work. The Authors Evaluate Three Policy Changes

that constitute in reducing the service gap, reduc- ing the out-of-pocket costs, and increasing the num-

Sim-Care And Does Not Model Patients Or Physi-

cians as individuals. Consequently, the model can- not account for the objectives and satisfaction of in- dividual stakeholders which limits the possibilities

Based Model To Investigate The Paradox Of Primary

care. Their model features patients, PCPs, and spe- cialists as individual agents. Every patient has a health status that changes over time: The contrac- tion of illnesses leads to a (temporary) decrease in the patients’ health; the treatment of acute illnesses

By Pcps And Specialists As Well As Regular Check-

ups (performed exclusively by PCPs) lead to an in- crease in the patients’ health. Tracking the evolu- tion of the patients’ average health status over time, the authors investigate how public health is posi- tively affected by the interplay of different mecha-

Nisms In Primary Care. As Such, The Model Of Homa

et. al has a different objective than SiM-Care: While Homa et al. investigate the external effects of treat- ments in primary care on the entire health care sys- tem, SiM-Care focuses on the processes within pri-

Mary Care Systems. To That End, Sim-Care Models

the scheduling of appointments, the patients’ actual practice visits that result in waiting times through the interaction of patients, and the physicians’ treat- ments of patients with variable service times which are not part of the model by Homa et al.

Ous Work On Simulation Models For The Evaluation

of primary care systems that allow for the simulta-

Neous Consideration Of Microsystem Improvements

and macrosystem reforms as in SiM-Care. Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

Sim-Care Models The Interaction Of Patients And

primary care physicians in a restricted geographical area over a given period of time; compare Figure 1.

The Model Features Two Types Of Agents: A Popu-

lation of potential patients P and a population of

Primary Care Physicians G. Every Patient Ρ ∈P Re-

sides at a specific location, belongs to a certain age group, has an individual health status, as well as in- dividual treatment preferences; compare Figure 2.

Patients continuously develop acute illnesses that

Depend On Their Age And Health Status And Require

treatment until they subside. Additionally, patients

Need To Be Monitored By A Physician On A Regular

basis. To receive medical attention, patients either

Schedule An Appointment Or Visit A Pcp’S Practice

without prior notice. Thereby, the patients’ decision making process depends on their individual prefer- ences and health status which determine the choice of physician, the type of the visit (walk-in/ appoint- ment), as well as the time of the visit. Physicians φ ∈G practice at a certain location and have weekly opening hours during which they admit patients for treatment; see Figure 2. Moreover, every physician φ ∈G follows individual strategies that govern how they manage appointments, admit patients, and per- form treatments. As patients and physicians inter- act, they influence each other and adapt by adjust- ing their preferences and strategies.

And Policy Makers. These Objectives Are Assumed

by the model as follows: While PCPs strive to effi- ciently utilize their time, patients strive for a quick

Cost Of Health Care To Maximizing The Degree Of

patient-centered care. Policy makers are not repre-

Sented By Agents Within Sim-Care. Instead, Policy

decisions set relevant model parameters such as the

Number Of Physicians In The Area, Treatment Stan-

dards, and the financial reimbursement system.

Scribed By Grimm Et Al. . To Concisely High-

light the interaction of relevant components in that

Scales Within Sim-Care. Second, Section 3.2 De-

scribes the representation of the relevant entities and state variables, including patients and physi- cians with their sensing, predicting, adapting, in- teracting, and learning actions. Third, Section 3.3

Of Health Care Systems Through Stochastic Parame-

ters. Fifth, Section 3.5 briefly reviews the indica- tors that result from running the simulation and ex- plains their emergent properties. Sixth, Section 3.6 discusses the initialization of a simulation experi-

Ment. Seventh, Section 3.7 Documents The Submod-

els that implement, e.g., the PCP’s strategies to han- dle appointments. Finally, Section 3.8 describes our structural validation as well as our approach to ver- ification taken when implementing the model.

Sim-Care’S Environment Entails The Geographical

and temporal structure as well as policy effects.

Coordinates Latitude And Longitude Indicating The

north-south and east-west position, respectively.

The Modeled Time Period Is Considered As A Con-

tinuum structured by points in time and durations. For any time object (point in time or duration) t =

Same Encoding For Points In Time And Durations As

context uniquely defines which of the former a time object refers to. For example, (38,0.55) ∈T corre- sponds to day 38 and 24 · 60 · 0.55 = 792 minutes, i.e., 1:12 p.m. as a point in time or, analogously, to a duration of 38 days, 13 hours, and 12 minutes.

To ease notation, we associate every point in time

R≥0.

In addition to the continuous representation of time, we structure each day into a morning and an after-

Is Uniquely Defined By A Day Δ ∈N And A Binary

indicator γ ∈{0,1}. Thereby, the binary indicator γ

6

Martin Comis et al.

Illnesses

Fig. 2 Concept of SiM-Care showing both types of agents with their main attributes as well as interactions between agents. defines whether it is the morning (γ = 0) or the after- noon (γ = 1) session. Sessions reoccur on a weekly basis which yields an equivalence relation ∼on the

Set Of Sessions Λ Via

(δ1,γ1) ∼(δ2,γ2) :⇔δ1 ≡δ2 mod7 ∧γ1 = γ2. The resulting equivalence class for a session λ ∈Λ defined as [λ] := {λ′ ∈Λ : λ′ ∼λ} contains all ses-

Sions Sharing The Same Day Of The Week And Time

of the day, e.g., all Thursday afternoon sessions.

Classes Λ/∼:= {[Λ] : Λ ∈Λ} With The 14 Sessions

of the week, i.e., Monday to Sunday with a respec- tive morning and afternoon session.

Pcps Φ ∈G Are The Active Entities In The Simula-

tion. Their interaction is motivated by patients’ suf- fering from illnesses and therefore seeking treat-

Its. Both Patients And Pcps Are Complex Individu-

als featuring characteristics that represent entities

Themselves. Going From Simple To More Elaborated,

we begin by describing the self-containing entities

Of Sim-Care And End With The Description Of The

agents representing patients and physicians.

When Patients Suffer From An Acute Illness, They

want to be treated as soon as possible, ideally by their preferred physician. For the continuous treat-

Ment Of Chronic Illnesses And The Follow-Up Care

of acute illnesses, patients prefer treatment by the same physician through appointments in regular in- tervals. Physicians, on the other hand, aim at effi- ciently utilizing their available time while minimiz- ing overtime. Thus, patients’ and physicians’ objec- tives are in conflict as it is ineffective for physicians

Ments In Strict Intervals Would Prevent Pcps From

reacting to demand fluctuations.

One Hand, They Need To Ensure A Certain Minimum

standard in health care quality to guarantee patients are treated when necessary. On the other hand, they

Cannot Afford To Subsidize An Excessive Number Of

physicians. Thus, policy makers necessarily aim at a trade-off: A purely patient-based system that dis- regards efficiency is likely to turn out to be unaf-

Cess Times. Sim-Care Represents Policy Decisions

through their resulting parameter values, e.g., the number of physicians and their distribution.

3.2.2 Illnesses And Families Of Illnesses

Illnesses are health concerns that cause discomfort to patients and require treatment. They belong to a certain illness family (e.g. cold or heartburn), have a certain seriousness (e.g. mild or severe), persist over a certain period of time, and require an initial

Treatment Within An Acceptable Time Frame As Well

as subsequent follow-up visits in regular time inter- vals. In SiM-Care, we formalize illnesses as tuples Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

7

i = (si, fi,di,ωi,νi) ∈I with attributes as shown in

Table 1. Thereby, Si ∈ Defines The Seriousness

of the illness, fi ∈F defines the illness family of

Illness. The Parameter Ωi ∈T Defines The Illness’

willingness to wait, which is the patient’s maximum accepted waiting time for the initial treatment. The

Parameter Νi ∈T Defines The Illness’ Follow-Up In-

terval, which specifies the frequency of the required aftercare that follows the initial treatment. When we use this representation to model health concerns that are not strictly illnesses like the need for vac- cination, the characteristics duration and follow-up interval may not apply. In such cases, setting pa- rameter values di = /0 and νi = /0 indicates that the respective characteristic is not applicable for i ∈I .

Families of illnesses serve as the classification sys-

Tem Of Illnesses Within Sim-Care. While Emerg-

ing illnesses vary in their manifestation, families of

Illnesses Define The Common Constant Traits Of All

illnesses belonging to the same illness family. In

Ing To Illness Family Fi ∈F Are The Expected Du-

ration D fi(si) ∈T , the expected willingness to wait Wfi(si) ∈T , and the follow-up interval Nfi(si) ∈T . The expected duration D fi(si) and expected willing- ness to wait Wfi(si) are exclusively used during the

The Means For Distributions From Which We Sample

each illness’ stochastic duration di and stochastic

Willingness To Wait Ωi. Thus For All Emerged Ill-

nesses i ∈I , it generally holds that di̸ = D fi(si)

Emerged Illnesses I ∈I Derives From The Illness

family in a deterministic way, i.e., νi = Nfi(si).

In Order To Define The Common Traits Of Emerg-

ing illnesses, families of illnesses f ∈F are for-

Tion D F : →T That Defines The Expected Dura-

tion D f (s) in days for all emerging illnesses with

And Nf : →T That Analogously Define The Ex-

pected willingness to wait in days and follow-up in- terval in days; see Table 2. As above, we indicate the inapplicability of the characteristics duration or follow-up interval to families of illnesses by setting D f = /0 and Nf = /0, respectively.

To illustrate the concept of illnesses and families

Cold” With Expected Illness Duration Defined By

Table 1 Attributes of illnesses i ∈I .

[Days]

Table 2 Attributes of families of illnesses f ∈F .

Κf ∈{0,1}

D f (s) = 10s + 3, expected willingness to wait de-

Ness Family “Common Cold” Defines The Expected

duration, expected willingness to wait, and follow- up interval of the mild cold as D f (si) = 5 days, Wf (si) = 2.4 days, and Nf (si) = 6.6 days. The ac- tual duration and willingness to wait of the devel-

Oped Mild “Common Cold” Are Stochastic And Vary

around their expected counterparts, e.g., di = 5.5 days and ωi = 2.7 days. The illness’ follow-up in- terval is deterministic and derives from the illness family via νi = Nfi(si) = 6.6 days.

Chronic Attribute Κf ∈{0,1} Identifies Families Of

chronic illnesses. Thereby, κf partitions F into the set of acute families of illnesses F act := { f ∈F : κf = 0} and the set of chronic families of illnesses

F Chro := { F ∈F : Κf = 1}. This Directly Induces

a partition of the set of illnesses I into the set of acute I act and the set of chronic illnesses I chro.

Time And Patients Can Simultaneously Suffer From

an arbitrary number of acute illnesses I act ⊆I act. Chronic illnesses ς ∈I chro are conceived as static

By The Model – They Neither Develop Nor Heal In

the modeled time period. Instead, each patient ρ ∈

From A Chronic Illness From Those Who Do Not, We

refer to the former as chronic patients.

8

Martin Comis et al. Table 3 Attributes of age classes a ∈A.

3.2.3 Appointments

Appointments specify the point in time at which the treatment of a specific patient is scheduled to take

Place. To That End, Appointments B ∈B Are Defined

by the time of the appointment tb ∈T , the attend-

Non-Chronic Patients Can Have At Most One Sched-

uled appointment bact ∈B, called the acute appoint- ment. Acute appointments are intended for the ini- tial treatment of acute illnesses, the follow-up treat- ment of acute illnesses, or both. Chronic patients,

May Have A Regular Appointment Breg ∈B To Treat

their chronic illness in addition to the acute appoint- ment to treat their acute illnesses. While chronic illnesses are only treated during regular appoint- ments, acute illness are treated during any appoint- ment. Thus, all of a patients’ acute illnesses I act are treated during every appointment.

Serve The Purpose Of Defining The Common Charac-

teristics of patients within the respective classes. For patients of age class a ∈A, these characteristics are the deviation from the expected illness duration

∆D

a > 0, the deviation from the expected willingness

Number Of Annual Acute Illnesses Defined Through

the linear function Ia : →R≥0; see Table 3. The deviation from the expected illness duration ∆d

A

is a multiplicative factor, that determines whether

The Expected Willingness To Wait Wfi(Si) ∈T Of An

illnesses changes for patients of age class a ∈A.

The Linear Function Ia : →R≥0 Defines The Ex-

pected number of annual acute illnesses Ia(c) ∈R≥0

Which Can Range From Perfectly Healthy (C = 0) To

extremely delicate (c = 1).

3.2.5 Age Class-Illness Distribution

The age class-illness distribution πact : A ×F act →

Builds The Connection Between The Set Of Age

classes A and the set of acute families of illnesses F act. To that end, πact defines the expected distri- bution of acute illness families for each age class, i.e., among all developed acute illnesses by patients

Of Age Class A ∈A, A Fraction Πact(A, Fi) ∈ Is

expected to belong to illness family fi ∈F act. As a result, πact defines a discrete probability distribu- tion on the set of acute families of illnesses F act for fixed age class a ∈A, i.e., ∑fi∈F act πact(a, fi) = 1.

3.2.6 Patients

Patients are the driving force of the simulation, as their health concerns trigger the events that underly most of the simulation’s processes. All non-chronic patients ρ ∈P are characterized by their geograph-

Pointment Bact ∈B, And Preferences. While The Lo-

cation, health condition, and age class of each pa-

Tient Remain Constant Throughout A Simulation Ex-

periment, a patient’s acute illnesses, acute appoint-

Ment And Preferences Are Variable And Change Over

time. Chronic patients possess all the characteristics of non-chronic patients, but are additionally identi-

Fied By A Constant Chronic Illness Σ ∈I Chro And A

variable regular appointment breg ∈B.

Seeks Treatment With Pcps Outside The Considera-

tion set. Since continuity in the treatment of chronic illnesses is particularly important, chronic patients select a distinguished family physician φfam ∈Gcon

With Whom All Regular Appointments Breg ∈B Are

exclusively arranged. While every patients’ consid-

Modeled Time Period, Patients Reevaluate And Vary

their family physician. Naturally, patients have per-

Availabilities Given By Α: Λ/∼→{0,1}, Where 0

Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

9

encodes unavailability. Finally, patients maintain

For Every Weekly Session [Λ] ∈Λ/∼And Every Con-

sidered physician φ ∈Gcon.

Ratings Are The Means By Which Patients Express

their satisfaction with a physician’s services. When-

Physician Is Determined By The Patient’S Current

ratings. To that end, ratings incorporate patients’

Sense Of Geographic Distance, Matching Of Opening

hours with availabilities, and previous positive and negative experiences. As patients adjust their rat- ings over time, they adjust their choice of PCP. If

A Physician Is Unable To Meet An Appointment Re-

quest, incurs excessive waiting time, or rejects pa- tients due to capacity overruns, patients reduce their rating. Positive experiences such as successful ap-

Pointment Arrangements Or Short Waiting Times In-

crease ratings. In other terms, through their sens- ing of the quality of treatment and the adaptation of their ratings, patients learn about the quality of PCPs throughout the simulation cycle.

Sidered Pcps Gcon. Appointment Requests Are One

of the ways in which patients and PCPs interact. Patients attempt up to two appointment requests in

Assign To The Considered Primary Care Physicians

φ ∈Gcon. If both requested PCPs fail to offer a fea- sible appointment within the patient’s willingness to wait, patients resort to their second way of in-

Teracting With Physicians: They Forgo An Appoint-

ment and visit a PCP as a walk-in patient. The se- lection of the PCP for the walk-in visit is based on

(Φ,[Λ]) Of The

targeted session λ ∈Λ.

Upon Arrival, Patients May Be Rejected By Physi-

cians due to, e.g., capacity overloads. Following a rejection, patients update their rating of the reject- ing PCP and attempt a new visit as walk-in patient at the then-highest-rated PCP. Rejected patients are

Unsuccessfully Continue To Seek Treatment. In Our

model, this emergency state does not enforce a par-

Ticular Pcp Behavior. Instead, Pcps May Include

the emergency state in their decision making. Until an illness i ∈I act subsides, patients continu- ously try to arrange follow-up appointments to the initial treatment with the attending physician in the Table 4 Attributes of (chronic) patients ρ ∈P.

Variable

follow-up interval νi ∈T . Analogously, chronic pa- tients continuously try to arrange regular appoint- ments with their family physician φfam ∈Gcon in the

Illness Σ ∈I Chro. Only If The Arrangement Of A

follow-up or regular appointment fails and the after- care of the patients is endangered, do patients seek follow-up treatment as walk-in patients. As a result, a chronic patient’s chronic illness ς ∈I chro can be treated by a physician other than the family physi-

Cian Φfam ∈Gcon, But Only Through A Walk-In Visit

triggered by the unavailability of a regular appoint- ment. In SiM-Care, patients do not directly interact with other patients. However, an indirect form interac-

Tion Emerges As Patients Compete With Each Other

for timely treatment by their preferred PCP. The attributes shared by all patients as well as the

Attributes Specific To Chronic Patients Are Summa-

rized in Table 4.

Primary Care Physicians Operate Practices Featur-

ing an uncapacitated waiting room to offer medical services to patients in need. The model character-

Izes Physicians Φ ∈G By Their Geographic Location

ℓ∈L, opening hours, as well as an individual set of strategies to schedule appointments, manage patient admission, and organize treatments.

Sim-Care Assumes That All Physicians Operate In

clinical sessions. Opening hours for these sessions

The Session Of The Week Via O: Λ/∼→H × H

where H denotes the set of decimal times defined in

10

Martin Comis et al. Section 3.1. Opening hours specify for each session λ ∈Λ the time window o([λ]) := [o([λ]),o([λ])] dur- ing which a physician generally admits patients for

Is Defined As O(Λ) := (Δ,O([Λ]) ∈T , The Session’S

end as o(λ) := (δ,o([λ]) ∈T . To encode that a PCP

Is Closed For A Weekly Session [Λ] ∈Λ/∼, We Set

o([λ]) = /0. Physicians utilize the first hour after the

End Of Each Session As Time Buffer To Compensate

for possible delays and walk-in patients. Buffers are

Considered Anticipated Working Time So That Only

service time that extends beyond the buffer consti- tutes overtime. Figure 3 provides a schematic visu- alization of a PCP’s working day.

Primary care physicians implement a set of strate-

Gies To Schedule Appointments, Decide On Patient

admissions, and organize the treatment of patients. These strategies govern the physicians’ interactions with patients and incorporate all of their sensing, predicting, adapting, and learning.

Appointment Slots And How The Resulting Slots Are

assigned to requesting patients. The feasible set of

Via The Interface Shown In Figure 4. That Is, Ev-

ery appointment scheduling strategy S ∈Sapp has to provide the functionality to answer appointment re- quests with an appointment suggestion (that can be

Empty). Thereby, Every Appointment Request Speci-

fies the requesting patient, earliest possible appoint- ment time, willingness to wait, whether the request is for a regular appointment, and whether patient’s availabilities have to be respected. Furthermore, ev- ery appointment scheduling strategy S ∈Sapp has to provide the functionality to schedule previously of- fered appointments as well as the functionality to cancel previously scheduled appointments. Finally,

Pointments Within A Session That Are Scheduled To

take place after specified point in time.

Order Of Treatment Among Patients From The Wait-

ing room. Physicians sense their patients’ waiting times as input for their strategy. To account for the

Cians Adjust Their Consultation Speed And Thereby

service times. The feasible set of treatment strate- gies Stmt is defined via the interface shown in Fig- Table 5 Attributes of PCPs φ ∈G.

S ∈S Tmt

ure 4. That is, every treatment strategy S ∈Stmt has to keep track of admitted patients, count the number

Of Waiting Patients With And Without Appointment,

and define how the treatment strategy is affected by the beginning of a session. Moreover, every treat- ment strategy S ∈Stmt has to determine the next pa- tient to be treated (that might not exist) as well as the PCP’s current consultation speed which is thor- oughly discussed in Section 3.7.4.

Waiting Room. In Sim-Care, Pcps Are Required To

treat all admitted patients. Thus, physicians under- estimating their workload due to faulty predictions

Overestimate Their Workload Reject Too Many Pa-

tients and fail to fully utilize their available time.

At The End Of Every Session’S Buffer, Physicians

learn by reevaluating their predictions and adapt- ing their admission policy. The feasible set of ad- mission strategies Sadm is defined via the interface shown in Figure 4. That is, every admission strategy S ∈Sadm has to be able to decide whether an arriv- ing patient is admitted or not given the PCP’s treat-

Over, Every Admission Strategy S ∈Sadm Has To De-

fine the adaptive traits that are performed at the end

Of Every Session’S Buffer And Depend On The Pcp’S

treatment strategy. Physicians do not directly interact with other physi-

Emerges As Pcps Compete For The Patients’ Favor

while striving for optimal utilization. The attributes of PCP’s are summarized in Table 5.

Within Sim-Care, The Progression Of Time Is Mod-

eled via the discrete event paradigm. That is time Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

Overtime

Fig. 3 Schematic representation of a PCP’s morning (λ0) and afternoon (λ1) session visualizing service-, idle- and overtime.

Iadmissionstrategy

boolean acceptPatient(IAppointmentSchedulingStrategy as,

Itreatmentstrategy Ts, Arrivalevent Ae)

void adaptStrategy(Session session, ITreatmentStrategy ts) Fig. 4 Interfaces implemented by strategies. is a continuum which is traversed between discrete

Events At Which The System State Is Updated. The

model stores events of the form (t,e) in a sequential

Queue Q Where T ∈T Is The Point In Time An Event

of type e ∈E occurs.

Events In Q Happen Chronologically, I.E., Q =

{(t1,e1),...,(tn,en)} with ti ≤ti+1 for 1 ≤i ≤n−1.

As Soon As An Event (Ti,Ei) ∈Q Occurs, The Simu-

lation advances from time ti−1 to time ti, compare

Point In Time T ∈T , I.E., When The First Element

(ti,ei) ∈Q with ti ≥T occurs.

Any Event (T,E) ∈Q Can Generate New Events Or

delete existing ones. To be introduced or affected

By Event (T,E) ∈Q , Events (T′,E′) ∈Q Must Hap-

pen after time t, i.e., we require t′ > t, so that time progresses in a consistent fashion. By construction, event queue Q never runs empty.

Every simulation run follows the structure depicted in Figure 6, chronologically processing the events in Q until time T ∈T is reached. In this, the spe-

Cific Process Depends On The Event Type E ∈E. We

now describe the different event types. Arrival events are indicated by earv(φ,ρ). As illus- trated in Figure 7(a), they mark the event of patient ρ arriving at physician φ’s practice, either for an ap- pointment or as a walk-in. The physician’s decision to admit or reject arriving patients depends on φ’s admission strategy. Every admitted patient is guar-

Anteed To Receive Treatment And Enters The Physi-

cian’s waiting room. If the physician is currently idle, this triggers the physician’s treatment strategy and treatment commences.

Follow-up events are indicated by efol(φ,ρ,i). Some families of illnesses fi ∈F cannot be treated via a single visit. Instead, the related illnesses i ∈I re-

Quire Follow-Up Treatments In Intervals Defined By

the parameter νi̸ = /0. Ensuring continuous follow-

Up Treatments, Patients Always Try To Arrange A

follow-up appointment immediately after the treat- ment of illnesses requiring follow-up consultation. To account for the fact that no feasible follow-up

Appointment Might Be Available, Sim-Care Gener-

ates a follow-up event efol(φ,ρ,i) at time ttreat + νi every time illness i ∈I with νi̸ = /0 suffered by pa-

Tient Ρ ∈P Is Treated By Physician Φ ∈G At Time

ttreat ∈T . Follow-up events serve as the patient’s reminder to actively re-pursue follow-up consulta- tion for illness i after the duration of the follow-up interval. Triggered by a follow-up event efol(φ,ρ,i), patient ρ reattempts to arrange a follow-up appoint-

Unable To Provide A Suitable Appointment, Ρ Seeks

follow-up consultation as a walk-in patient. Every follow-up treatment of an illness i ∈I invalidates all associated existing follow-up events, as the follow- up interval is reset. Therefore, SiM-Care deletes all

Ated With Illnesses I ∈I That Were Treated During

a visit before the new follow-up events are gener- ated. As a result, follow-up events only trigger if an illness has not been treated for the duration of its follow-up interval νi ∈T .

12

Martin Comis et al.

E4

Q = {(t1,e1),(t2,e2),(t3,e3),(t4,e4),(t5,e5),(t6,e6),(t7,e7)}

T4 −T3

Fig. 5 Progression of time induced by the processing of event queue Q via the discrete event paradigm.

T < T

Fig. 6 Structure of simulation run with time horizon T.

Begins, The Sampled Service Time Determines The

time of the subsequent release event erel(φ,ρ). All treated illnesses i ∈I act without duration (di = /0) are cured through a one-time treatment and thus re- moved from I act. Subsequently, all existing follow-

In The Previously Described Manner. The Success-

ful treatment revokes existing emergency flags, i.e., we set ε = 0. If the patient’s chronic illness ς ∈I was treated, the next recurrent regular appointment

Breg ∈B With Physician Φfam Is Requested At Time

ttreat + νς. Then, patients request an acute appoint-

Up Treatment Of The Persisting Acute Illness I∗=

argmini∈I act:νi̸=/0 νi with smallest follow-up interval.

The Requested Appointments Ensure The Follow-Up

treatment of all illnesses suffered by patient ρ and

Will Preempt The Previously Generated Follow-Up

events. Finally, physicians implement their treat-

Ment Strategy To Select The Next Patient From The

waiting room if the latter is non-empty. Otherwise, physicians remains idle until the next arrival event triggers the treatment strategy. As a result of this behavior, physicians are never intentionally idle.

Illness events are indicated by eill(ρ). As illustrated in Figure 7(c), they describe that patient ρ starts to suffer from a new acute illness. This means that the

Model Generates A New Acute Illness I ∈I Act With

stochastic qualities that depend on the patient’s age and health condition and adds it to their set of ill- nesses I. To treat the emerged illness, patients re-

Quest An Appointment From Their Preferred Physi-

cians or, in case this does not succeed, directly visit the preferred physician as a walk-in. As a result, each illness event generates a corresponding arrival event earv(φ,ρ) and adds it to the queue Q . Finally, each illness event generates a future illness event eill(ρ) for patient ρ and adds it to the queue Q to

Mark The Next Point In Time Patient Ρ Develops An

acute illness.

Illness I ∈I Act. Whenever The Model Generates A

new acute illness i ∈I act with di̸ = /0, it also gen- erates a corresponding recovery event erec(ρ,i) at time till + di, where till ∈T is the point in time illness i is developed. Illnesses without duration (di = /0) do not require a recovery event as they are immediately cured through their initial treatment. A recovery event removes illness i from I and deletes any associated follow-up event efol(φ,ρ,i) ∈Q . If patient ρ does not suffer from acute illnesses fol- lowing the removal of illness i, i.e., I act = /0, the

Uled Acute Appointments. Such Cancellations Oc-

Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

(C)

Fig. 7 Processing of (a) arrival events earv(φ,ρ), (b) release events erel(φ,ρ), and (c) illness events eill(ρ); ρ ∈P and φ ∈G. cur with the patient’s age-class-specific probability

Pa ∈ And Consequently Delete The Associated

arrival event earv(φ,ρ). As a result, some patients

Debriefing. Should Patient Ρ Be Currently Seeking

walk-in treatment due to persisting chronic illness ς ∈I, this effort is continued. Otherwise, current walk-in attempts are canceled and the associated ar- rival event earv(φ,ρ) is deleted.

Open- And Close Events Are Indicated By Eopn(Φ)

and eclo(φ), respectively. They mark the beginning and ending (including buffer) of a session λ ∈Λ op-

Strategies Become Aware Of A Session’S Beginning,

e.g., to allow for strategies that do not treat early- arriving patients before o(λ), and that overtime is

Incurred For All Treatments Performed Beyond The

anticipated buffer time of λ.

Frequency Of Events. This Applies To Aspects Of

illnesses as well as to patient arrivals, appoint-

Quence, Every Simulation Experiment Includes Mul-

tiple stochastic repetitions of the modeled time pe-

Riod, Termed Simulation Runs. When Examining Sim-

ulation output, we account for the resulting variabil- ity through confidence intervals.

In The Following, We Highlight The Aspects Of The

model that are probabilistic rather than determinis- tic and discuss how the distributions underlying the random values are parameterized.

Frequency Of Acute Illnesses. The Occurrence Of

acute illnesses in SiM-Care is modeled via a Pois- sion process. Patients develop acute illnesses at a frequency that depends on their age and health con-

Health Condition C ∈, The Expected Number Of

acute illnesses per year is given by the parameter Ia(c). The intensity (or rate) of the Poission proc- cess is thus Ia(c)/364 per day. Moreover, the dura- tion between two consecutive illness events eill(ρ)

For Patient Ρ Can Be Sampled From An Exponential

distribution with rate Ia(c)/364; see [20, chapter 2]. Type of Acute Illnesses. Whenever an illness event eill(ρ) occurs and patient ρ ∈P falls ill, the model

C ∈. The Model Assumes A Probabilistic Link

between illness family fi ∈F act and the patient’s age class a that is expressed via the age class-illness distribution πact; see Section 3.2.5. To that end, any emerging acute illness of patient ρ is randomly as- signed to an illness family according to the discrete probability distribution f 7→πact(a, f) for f ∈F act.

Care, Its Seriousness Si ∈ Depends On A Tri-

angular distribution defined on the closed interval

14

Martin Comis et al.

A (Fi,Si) = 2

Fig. 8 Weibull distributions of ωi ∈T for different values of patient’s age adjusted expected willingness to wait Eω a (fi,si).

i. Thus, patients with a bad health condition tend to develop more serious illnesses.

The Duration Di ∈T Of Illness I Depends On A Log-

normal distribution. Given i’s family of illnesses

A(Fi,Si) := ∆D

a · D fi(si).

Therefore, Sim-Care Samples The Illness’ Duration

di from a log-normal distribution with sdlog σ = 0.3

And Meanlog Μ = Log(Ed

a(fi,si))−σ2/2. Patient ρ’s willingness to wait for the initial treat-

Ment Of Illness I As Specified By Ωi ∈T Depends

on a Weibull distribution. Given i’s family of ill-

Oping Patient’S Age Class A ∈A, The Age-Adjusted

expected willingness to wait of illness i is defined

Et. Al , We Sample Ωi From A Weibull Distribu-

tion with shape parameter p = 2 and derive the scale

Γ Denotes The Gamma Function. Figure 8 Visualizes

the resulting density functions for various choices of the age-adjusted expected willingness to wait. Patient Punctuality. Patients do not always arrive on time for their scheduled appointments b ∈B. In- stead, SiM-Care allows for patient arrivals to vary

And Standard Deviation Of Σ = 6 Minutes Such That

roughly 20% of all patients are expected to arrive late for their appointments which is consistent with the observations reported in .

Beta(1.93,2.94)

Fig. 9 Histogram and beta distributed maximum-likelihood fit for empirical walk-in arrival rates from .

Walk-In Arrivals. Walk-In Patients Have No Pre-

specified time at which they are expected to arrive. Instead, SiM-Care defines for every walk-in patient

Arrival Time B ∈T Which Are Both Situational And

thoroughly discussed in Section 3.7.3. The walk-in patients’ actual arrival within the given feasible ar- rival interval [a,b] depends on a beta distribution.

Specifically, we fit a beta distribution using maxi- mum likelihood estimation to the empirical arrival

Rates Reported By Shan Et Al. . As A Result, We

sample the arrival times of walk-in patients from the interval [a,b] of feasible arrival times according to a beta distribution with shape parameters p = 1.93 and q = 2.94; cf. Figure 9.

Service Time. SiM-Care treats the service time per patient, i.e., the duration of treatments, as a random

Practice. As Suggested In Literature , We Di-

vide the sample into patients with and without ap-

Hood Fit. Histograms Of Our Empirical Samples And

the resulting distributions for walk-ins and patients with appointment are depicted in Figures 10 and 11. Based on the fitted distributions, we sample the ser- vice times of patients with appointment from a log-

Normal Distribution With Meanlog Μ = 1.82 And Sd-

log σ = 0.692 and the service times for walk-in pa- tients from a log-normal distribution with a mean- log µ = 1.254 and sdlog σ = 0.723. As our collected data set does not incorporate transition times, we prolong all sampled service times by one minute.

Appointment Cancellations. Patients That Recover

from all their current acute illnesses, i.e., I act = /0,

Cancel Their Existing Acute Appointment Bact ∈B

Patients, Primary Care, and Policy: Simulation Modeling for Health Care Decision Support

6.174

Fig. 10 Histogram and log-normal maximum-likelihood fit for empirical service times of patients with appointment.

3.505

Fig. 11 Histogram and log-normal maximum-likelihood fit for empirical service times of walk-in patients.

Compare Section 3.2.4. As Long As Patients Suffer

from acute illnesses, they only cancel their acute ap- pointment if they require earlier treatment due to a newly emerged acute illness. All patients that have not canceled their appointment will arrive for it. As chronic illnesses are static within the model, regular appointments are never canceled.

Sim-Care Tracks Key Performance Indicators From

the point of view of patients, primary care physi- cians, and policy makers. Thereby, it aims to illus- trate the trade-offs between the stakeholders’ objec- tives. As these indicators emerge from agent inter- actions based on patients’ evolving preferences and physicians’ evolving strategies, they are difficult to predict in general.

From The Patients’ Point Of View, Key Performance

indicators include access time, travel distance, and

Waiting Time. Access Time Measures The Time A Pa-

tient has to wait for an appointment, i.e, given the

Time Of The Arranged Appointment Tb ∈T It Is De-

fined as ac-time := tb −t. The travel distance mea-

Sures The One-Way Distance Patient Ρ ∈P Has To

travel when visiting physician φ ∈G, i.e., tr-dist := dist(ℓρ,ℓφ), where dist(ℓ1,ℓ2) denotes the driving

Distance Between Locations ℓ1 ∈L And ℓ2 ∈L In

kilometers. The patient’s waiting time measures the time spent on-site before the actual treatment com- mences. For walk-in patients, we define the waiting time for given walk-in arrival tarr ∈T and treatment commencement ttreat ∈T as wait-time :=ttreat−tarr.

Ing Time For Given Time Of The Appointment Tb ∈

T , patient’s arrival at the practice tarr ∈T , and

Treatment Commencement Ttreat ∈T As Wait-Time :=

max{ttreat −max{tb,tarr}, 0}. To evaluate patient’s indicators, SiM-Care keeps track of the total access

Time Of Arranging Acute And Regular Appointments,

the total number of arranged acute and regular ap-

Pointments, The Total Number Of Attended Appoint-

ments, the total number of walk-in patients, the total distance traveled by patients to access physicians, and the total waiting time for both patients with ap- pointment and walk-ins.

From The Physicians’ Point Of View, Key Perfor-

mance indicators include the utilization, overtime,

Number Of Treatments, And Number Of Rejected Pa-

tients with and without appointment. A physician’s utilization describes the percentage of the available working time spent treating patients, i.e., for a ses-

24). Note That

our definition of utilization clearly underestimates a physician’s actual utilization as we do not account for additional tasks such as reporting, accounting,

In Sim-Care. Overtime Describes The Physician’S

working time beyond the anticipated buffer, i.e., if the last patient in session λ ∈Λ is released at time

1

24, 0}; see Figure 3. To evaluate the physician’s in- dicators, SiM-Care collects on physician level the total service time spent treating patients, the total number of performed treatments, the total overtime, and the total number of rejected patients with and

Can Be Derived From The Opening Hours O And The

modeled time horizon T.

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