Figures and tables in SNFsim: A Discrete Event Simulator for Decision Support in Skilled Nursing Facilities

Abstract

We introduce SNFsim, an open-source discrete-event simulator for developing and evaluating reinforcement learning (RL) methods for multi-dimensional sequential decision support in Skilled Nursing Facilities (SNFs). SNFs play a vital role in the United States healthcare system, delivering specialized care to individuals with ongoing medical needs. Decision-making within SNFs is often complex due to their fast-paced and stochastic nature. SNFsim provides a modular and extendable simulation of major decision-making processes within SNFs, capturing many of the complexities and uncertainties existing in healthcare environments while still being flexible enough to allow for easy customization. Its potential uses are two-fold: First, as a test bed for the development and comparison of RL algorithms, and second, as the basis of a decision-support system that can be tailored to individual SNFs.

This article contains 13 figures and 20 tables.

Figures

**Figure 1.** Skilled nursing intake flow and staffing process.

**Figure 2.** Code snippet demonstrating an example setup for training an RL agentusing SNFsim.

**Figure 3.** Simplified SNFsim simulation step.

**Figure 4.** Patient information typically available to the SNF at the point of referral.

**Figure 5.** Bivariate relationships between a subset of columns in the generated dataset versus the original dataset.

**Figure 6.** SNF intake flow and joint staffing decision process.

**Figure 7.** Random Forest model performance for patient rehospitalization prediction.

**Figure 8.** Performance comparison of five rollouts between P₁ and P₂ in the snf_v0 environment.

**Figure 9.** Performance comparison of five rollouts between P₁ and P₂ in the snf_v0 environment.

**Figure 10.** PPO agent training performance under varied weight configurations across 200,000 timesteps.

**Figure A1.** PDPM HIPPS code classification build sample.

**Figure A2.** Example of the ICD-10-CM hierarchy for a specific cholera diagnosis.

**Figure A3.** Flow of environment for the snf_v0 Gymnasium environment.

Tables

Table 1. Marginal similarity between original and synthetic data including the metrics mean diff (percentage difference in means), KDE Sim (kernel density overlap), TV dist (total variation distance), JS sim (Jensen-Shannon similarity), and cov (category coverage).

Variable	Mean Diff.	KDE Sim.	TV Dist.	JS Sim.	Cov.
Continuous:
Age	0.02%	88.1%
Length of Stay	13.8%	92.4%
Categorical:
ICD-10-CM			0.194	79.1%	78.7%
Insurance Type			0.150	83.6%	100%
PDPM Code			0.138	83.4%	98.2%
Gender			0.080	87.2%	100%

Algorithm 1 Temporal Day in SNFsim (broad visual representation in Figure 3).
1: Input: State $S_{t}=\{P_{t},O_{t},H_{t},C_{t},R_{t}\}$ , Action $a_{t}=\{A_{t},\Delta_{t}\}$ 2: Output: Next state $S_{t+1}=\{P_{t+1},O_{t+1},H_{t+1},C_{t+1},R_{t+1}\}$ 3: where: $P_{t}$ : patients, $O_{t}$ : occupancy, $H_{t}$ : nursing hours, $C_{t}$ : finances, $R_{t}$ : referrals 4: Initialize daily metrics: $\text{Revenue}_{t}\leftarrow 0$ , $\text{Rehospitalizations}_{t}\leftarrow 0$ , $\text{Cost}_{t}\leftarrow 0$ 5: // Update staffing levels based on action 6: for each staff type $s\in\{\text{FullTime},\text{PRN},\text{Agency}\}$ do 7: $\text{Staff}^{s}_{t}\leftarrow\min(\text{Staff}^{s}_{t-1}+\Delta^{s}_{t},\text {MaxStaff}^{s})$ {Enforce staffing constraints} 8: $\text{Cost}_{t}\leftarrow\text{Cost}_{t}+\sum_{i=1}^{3}\text{Staff}^{s}_{t}[i] \cdot\text{Hours}^{s}\cdot\text{Rate}^{s}$ 9: end for 10: // Process referrals 11: $P_{\text{new}}\leftarrow$ Accept referrals based on $A_{t}$ (0 or 1 for each referral in $R_{t}$ ) and bed availability $B_{\max}$ 12: // Calculate nursing hours per patient 13: $\overrightarrow{h}_{t}\leftarrow$ Nursing hours per patient per shift 14: $\alpha_{t}\leftarrow$ Staffing modifier based on ratio of actual to expected care 15: // Update patients and compute rehospitalizations 16: $P_{\text{remove}}\leftarrow\emptyset$ 17: for each patient $p\in P_{t}$ do 18: Update patient day count, accumulate revenue, and record nursing hours received 19: Calculate shift-specific understaffing penalties $\vec{\epsilon}$ 20: $p.r\leftarrow$ Rehospitalization risk based on patient data, staffing, and care quality 21: if $p.r>\text{threshold}$ then 22: Mark patient for rehospitalization and update count 23: end if 24: end for 25: Update patient population: $P_{t+1}\leftarrow(P_{t}\setminus P_{\text{remove}})\cup P_{\text{new}}$ 26: Update occupancy: 27: $O_{t+1}\leftarrow\frac{\|P_{t+1}\|}{B_{\max}}$ {Occupancy as proportion of available beds} 28: Update finances: 29: $C_{t+1}\leftarrow C_{t}+\text{Revenue}_{t}-\text{Cost}_{t}$ 30: Set next-day nursing hours: 31: $H_{t+1}\leftarrow H_{t}$ {Staffing decision at time $t$ determines hours for day $t+1$ } 32: Generate new referrals for next day: $R_{t+1}$ 33: Return $S_{t+1}=\{P_{t+1},O_{t+1},H_{t+1},C_{t+1},R_{t+1}\}$

Algorithm 1 Temporal Day in SNFsim (broad visual representation in Figure 3).

1: Input: State

S_{t}=\{P_{t},O_{t},H_{t},C_{t},R_{t}\}

, Action

a_{t}=\{A_{t},\Delta_{t}\}

2: Output: Next state

S_{t+1}=\{P_{t+1},O_{t+1},H_{t+1},C_{t+1},R_{t+1}\}

3: where:

P_{t}

: patients,

O_{t}

: occupancy,

H_{t}

: nursing hours,

C_{t}

: finances,

R_{t}

: referrals
4: Initialize daily metrics:

\text{Revenue}_{t}\leftarrow 0

\text{Rehospitalizations}_{t}\leftarrow 0

\text{Cost}_{t}\leftarrow 0

5: // Update staffing levels based on action
6: for each staff type

s\in\{\text{FullTime},\text{PRN},\text{Agency}\}

do
7:

\text{Staff}^{s}_{t}\leftarrow\min(\text{Staff}^{s}_{t-1}+\Delta^{s}_{t},\text {MaxStaff}^{s})

{Enforce staffing constraints}
8:

\text{Cost}_{t}\leftarrow\text{Cost}_{t}+\sum_{i=1}^{3}\text{Staff}^{s}_{t}[i] \cdot\text{Hours}^{s}\cdot\text{Rate}^{s}

9: end for
10: // Process referrals
11:

P_{\text{new}}\leftarrow

Accept referrals based on

A_{t}

(0 or 1 for each referral in

R_{t}

) and bed availability

B_{\max}

12: // Calculate nursing hours per patient
13:

\overrightarrow{h}_{t}\leftarrow

Nursing hours per patient per shift
14:

\alpha_{t}\leftarrow

Staffing modifier based on ratio of actual to expected care
15: // Update patients and compute rehospitalizations
16:

P_{\text{remove}}\leftarrow\emptyset

17: for each patient

p\in P_{t}

do
18: Update patient day count, accumulate revenue, and record nursing hours received
19: Calculate shift-specific understaffing penalties

\vec{\epsilon}

20:

p.r\leftarrow

Rehospitalization risk based on patient data, staffing, and care quality
21: if

p.r>\text{threshold}

then
22: Mark patient for rehospitalization and update count
23: end if
24: end for
25: Update patient population:

P_{t+1}\leftarrow(P_{t}\setminus P_{\text{remove}})\cup P_{\text{new}}

26: Update occupancy:
27:

O_{t+1}\leftarrow\frac{|P_{t+1}|}{B_{\max}}

{Occupancy as proportion of available beds}
28: Update finances:
29:

C_{t+1}\leftarrow C_{t}+\text{Revenue}_{t}-\text{Cost}_{t}

30: Set next-day nursing hours:
31:

H_{t+1}\leftarrow H_{t}

{Staffing decision at time

t

determines hours for day

t+1

}
32: Generate new referrals for next day:

R_{t+1}

33: Return

S_{t+1}=\{P_{t+1},O_{t+1},H_{t+1},C_{t+1},R_{t+1}\}

Table 2. Multivariate Logistic Regression Results

	Model A		Model B		Model C		Model D
Feature	Coef.	P	Coef.	P	Coef.	P	Coef.	P
LoS			-3.173	***	-3.180	***	-3.193	***
meanHours					-0.216	***	-0.215	***
reimbursement	+0.188	***			-0.344	*	-0.329	*
age	-0.028				-0.005		-0.005
NPG			-0.230	***	-0.274	***	-0.293	***
SLP			+0.077		+0.100		+0.083
PT_OT					+0.033		+0.035
NTA					_0.448	**	_0.428	**
days_since_start							+0.052
gender							+0.017
Additional categorical features in Model D (not included):
ICD-10-CM Chapter (***), facility, insurance type
Number of features	2		3		8		13
AUC-ROC	0.550		0.867		0.873		0.867
Max VIF	14.7		2.1		33.3		36.6

Model D contains full feature set used in Random Forest (RF AUC=0.921 vs. Logistic Regression AUC=0.867). ***p < 0.001, **p < 0.01, *p < 0.05. N=7,948; 588 (7.4%) readmissions.

Table 3. Required SNF Configuration Parameters

Parameter	Description	Default Value
total_beds	Number of beds in the facility	100
occupancy_bounds	Target min/max occupancy range	75%/90%
nursing_hours_target	Target min/max nursing hours per patient	2.5/3.5
full_time_cna	Total available full-time nursing assistants	20
prn_cna	Total available pro re nata (as-needed) staff	10
agency_cna	Total available agency-contracted staff	15
referral_rate	Mean number of daily referrals	5
min_cna_hours	Mandated nursing hours per patient day	2.8
fac_state	State where the SNF is located	New York
readmission_threshold	Threshold for readmission prediction	0.7

Table 4. Summary of training performance across different PPO weight configurations.

Metric	Balanced	Financial Focus	Care Focus
Weights	[1.0, 1.0, 1.0, 1.0]	[1.6, 1.6, 0.4, 0.4]	[0.4, 0.4, 1.6, 1.6]
Average Reward	1.41	0.40	2.54
First Quarter Avg	1.07	0.13	1.88
Last Quarter Avg	1.53	0.53	2.94
Improvement (%)	42.74%	312.67%	56.00%
Trend Slope	0.5982	0.5307	1.3929

Table 5. Outcomes by policy configuration over 10 different 365-day episodes. Weight vectors indicate a priori selection of importance for [reimbursement, nursing cost, rehospitalization, occupancy] objectives.

Metric	Balanced	Financial Focus	Care Focus
Weights	[1.0, 1.0, 1.0, 1.0]	[1.6, 1.6, 0.4, 0.4]	[0.4, 0.4, 1.6, 1.6]
Occupancy Rate	55.62% ± 0.24%	53.89% ± 0.20%	72.97% ± 0.39%
Rehospitalization Rate	37.3% ± 4.1%	42.8% ± 3.6%	13.6% ± 3.4%
Referral Acceptance Rate	80.0%	100.0%	60.1%
Daily Revenue	$31, 618 ± $131	$32, 199 ± $136	38, 467 ± $141
Daily Cost	$6, 938 ± $1.81	$6, 922 ± $1.75	$9, 933 ± $1.44
Daily Profit	$24, 680	$25,277	$28,534

Table A1. PDPM payment groups to code value.

PT/OT	SLP	NURS	NPG	Code Value
TA	SA	ES3	NA	A
TB	SB	ES2	NB	B
TC	SC	ES1	NC	C
TD	SD	HDE2	ND	D
TE	SE	HDE1	NE	E
TF	SF	HBC2	NF	F
TG	SG	CBC2		G
TH	SH	CA2		H
TI	SI	CBC1		I
TJ	SJ	CA1		J
TK	SK	BAB2		K
TL	SL	BAB1		L
TM		HBC1		M
TN		LDE2		N
TO		LDE1		O
TP		LBC2		P
		LBC1		Q
		CDE2		R
		CDE1		S
		PDE2		T
		PDE1		U
		PBC2		V
		PA2		W
		PBC1		X
		PA1		Y

Table A2. PT and OT case mix groups and PT and OT CMIs based on clinical category and PT and OT function score.

Clinical Category	PT & OT Function Score	PT & OT Case Mix Group	PT CMI	OT CMI
Major Joint Replacement or Spinal Surgery	0-5	TA	1.53	1.49
Major Joint Replacement or Spinal Surgery	6-9	TB	1.69	1.63
Major Joint Replacement or Spinal Surgery	10-23	TC	1.88	1.68
Major Joint Replacement or Spinal Surgery	24	TD	1.92	1.53
Other Orthopedic	0-5	TE	1.42	1.41
Other Orthopedic	6-9	TF	1.61	1.59
Other Orthopedic	10-23	TG	1.67	1.64
Other Orthopedic	24	TH	1.16	1.15
Medical Management	0-5	TI	1.13	1.17
Medical Management	6-9	TJ	1.42	1.44
Medical Management	10-23	TK	1.52	1.54
Medical Management	24	TL	1.09	1.11
Non-Orthopedic Surgery and Acute Neurologic	0-5	TM	1.27	1.30
Non-Orthopedic Surgery and Acute Neurologic	6-9	TN	1.48	1.49
Non-Orthopedic Surgery and Acute Neurologic	10-23	TO	1.55	1.55
Non-Orthopedic Surgery and Acute Neurologic	24	TP	1.08	1.09

Table A3. PDPM assessment type to code value.

Assessment Type	Code Value
Initial Patient Assessment	0
PPS 5-Day Assessment	1

Table A4. PT and OT case mix groups and PT and OT CMIs based on clinical category and PT and OT function score.

Clinical Category	PT & OT Function Score	PT & OT Case Mix Group	PT CMI	OT CMI
Major Joint Replacement or Spinal Surgery	0-5	TA	1.53	1.49
Major Joint Replacement or Spinal Surgery	6-9	TB	1.69	1.63
Major Joint Replacement or Spinal Surgery	10-23	TC	1.88	1.68
Major Joint Replacement or Spinal Surgery	24	TD	1.92	1.53
Other Orthopedic	0-5	TE	1.42	1.41
Other Orthopedic	6-9	TF	1.61	1.59
Other Orthopedic	10-23	TG	1.67	1.64
Other Orthopedic	24	TH	1.16	1.15
Medical Management	0-5	TI	1.13	1.17
Medical Management	6-9	TJ	1.42	1.44
Medical Management	10-23	TK	1.52	1.54
Medical Management	24	TL	1.09	1.11
Non-Orthopedic Surgery and Acute Neurologic	0-5	TM	1.27	1.30
Non-Orthopedic Surgery and Acute Neurologic	6-9	TN	1.48	1.49
Non-Orthopedic Surgery and Acute Neurologic	10-23	TO	1.55	1.55
Non-Orthopedic Surgery and Acute Neurologic	24	TP	1.08	1.09

Table A5. SLP case mix groups and SLP CMIs based on whether patient has a mechanically altered diet or swallowing disorder and the presence of acute neurological conditions, SLP-related comorbidity, or cognitive impairment.

Condition*	Mechanically Altered Dietor Swallowing Disorder	SLP Case Mix Group	SLP CMI
None	Neither	SA	0.68
None	Either	SB	1.82
None	Both	SC	2.66
Any One	Neither	SD	1.46
Any One	Either	SE	2.33
Any One	Both	SF	2.97
Any Two	Neither	SG	2.04
Any Two	Either	SH	2.85
Any Two	Both	SI	3.51
All Three	Neither	SJ	2.98
All Three	Either	SK	3.69
All Three	Both	SL	4.19

Presence of Acute Neurological Condition, SLP-Related Comorbidity, or Cognitive Impairment.

Table A6. Nursing payment group (CMG) and corresponding CMIs based on RUG-IV Nursing RUG, extensive services status, clinical conditions, depression status, and restorative nursing services.

RUG-IV Nursing RUG	Extensive Services	Clinical Conditions	Depression	RNS	Function Score	CMG	CMI
ES3	Trach & Ventilator				0-14	ES3	4.04
ES2	Trach or Ventilator				0-14	ES2	3.06
ES1	Infection Isolation				0-14	ES1	2.91
HE2/HD2		SMC	Yes		0-5	HDE2	2.39
HE1/HD1		SMC	No		0-5	HDE1	1.99
HC2/HB2		SMC	Yes		6-14	HBC2	2.23
HC1/HB1		SMC	No		6-14	HBC1	1.85
LE2/LD2		RMC	Yes		0-5	LDE2	2.07
LE1/LD1		RMC	No		0-5	LDE1	1.72
LC2/LB2		RMC	Yes		6-14	LBC2	1.71
LC1/LB1		RMC	No		6-14	LBC1	1.43
CE2/CD2		CRC	Yes		0-5	CDE2	1.86
CE1/CD1		CRC	No		905	CDE1	1.62
CC2/CB2		CRC	Yes		6-14	CBC2	1.54
CA2		CRC	Yes		15-16	CA2	1.08
CC1/CB1		CRC	No		6-14	CBC1	1.34
CA1		CRC	No		15-16	CA1	0.94
BB2/BA2		BCS		2+	11-16	BAB2	1.04
BB1/BA1		BCS		0-1	11-16	BAB1	0.99
PE2/PD2		ADL		2+	0-5	PDE2	1.57
PE1/PD1		ADL		0-1	0-5	PDE1	1.47
PC2/PB2		ADL		2+	6-14	PBC2	1.21
PA2		ADL		2+	15-16	PA2	0.7
PC1/PB1		ADL		0-1	6-14	PBC1	1.13
PA1		ADL		0-1	15-16	PA1	0.66

Table A7. Non-Therapy Ancillaries (NTA) CMIs based on NTA case mix groups and NTA score ranges.

NTA Score Range	NTA Case Mix Group	CMI
12+	NA	3.25
9-11	NB	2.53
6-8	NC	1.85
3-5	ND	1.34
1-2	NE	0.96
0	NF	0.72

Table A8. Urban rate components.

Rate Component	PT	OT	SLP	Nursing	NTA	Non-Case-Mix (NCM)
Per Diem Amount	$62.84	$58.49	$23.46	$109.55	$82.64	$98.10

Table A9. Rural rate components.

Rate Component	PT	OT	SLP	Nursing	NTA	Non-Case-Mix (NCM)
Per Diem Amount	$71.63	$65.79	$29.56	$104.66	$78.96	$99.91

Table A10. Day in stay adjustment factor.

Day in Stay	Adjustment Factor
1-20	1.00
21-27	0.98
28-34	0.96
35-41	0.94
42-48	0.92
49-55	0.90
56-62	0.88
63-69	0.86
70-76	0.84
77-83	0.82
84-90	0.80
91-97	0.78
98-150	0.76

Table A11. NTA component adjustment factor.

Day in Stay	Adjustment Factor
1-3	3.00
4-150	1.00

Table A12. Estimated Hourly Wages for Full-time, PRN, and Agency CNAs by State based on (Nursa, 2024).

State	Full-time CNA ($/hr)	PRN CNA ($/hr)	Agency CNA ($/hr)
Alabama	15.04	18.05	20.30
Alaska	22.63	27.16	30.55
Arizona	19.69	23.63	26.58
Arkansas	15.41	18.49	20.80
California	22.63	27.16	30.55
Colorado	20.95	25.14	28.28
Connecticut	20.70	24.84	27.95
Delaware	18.57	22.28	25.07
Florida	17.67	21.20	23.85
Georgia	16.77	20.12	22.64
Hawaii	21.63	25.96	29.20
Idaho	17.92	21.50	24.19
Illinois	19.87	23.84	26.82
Indiana	18.10	21.72	24.44
Iowa	18.45	22.14	24.91
Kansas	17.32	20.78	23.38
Kentucky	17.30	20.76	23.36
Louisiana	14.63	17.56	19.75
Maine	20.65	24.78	27.88
Maryland	19.60	23.52	26.46
Massachusetts	21.22	25.46	28.65
Michigan	18.71	22.45	25.26
Minnesota	19.40	23.28	26.19
Mississippi	12.35	14.82	16.67
Missouri	15.07	18.08	20.34
Montana	16.83	20.20	22.72
Nebraska	17.00	20.40	22.95
Nevada	19.89	23.87	26.85
New Hampshire	20.36	24.43	27.49
New Jersey	19.02	22.82	25.68
New Mexico	16.20	19.44	21.87
New York	19.56	23.47	26.41
North Carolina	16.24	19.49	21.92
North Dakota	18.33	22.00	24.75
Ohio	16.05	19.26	21.67
Oklahoma	13.39	16.07	18.08
Oregon	18.67	22.40	25.20
Pennsylvania	17.23	20.68	23.26
Rhode Island	19.39	23.27	26.18
South Carolina	15.47	18.56	20.88
South Dakota	14.90	17.88	20.12
Tennessee	15.56	18.67	21.01
Texas	16.94	20.33	22.87
Utah	16.12	19.34	21.76
Vermont	18.60	22.32	25.11
Virginia	17.10	20.52	23.09
Washington	20.80	24.96	28.08
West Virginia	14.22	17.06	19.20
Wisconsin	17.58	21.10	23.73
Wyoming	17.38	20.86	23.46

Table A13. ICD-10-CM Chapters and corresponding categories.

Chapter	Title	Categories
1	Certain Infectious and Parasitic Diseases	A00-B99
2	Neoplasms	C00-D49
3	Diseases of the Blood and Blood-Forming Organs	D50-D89
4	Endocrine, Nutritional, and Metabolic Diseases	E00-E89
5	Mental, Behavioral, and Neurodevelopmental Disorders	F01-F99
6	Diseases of the Nervous System	G00-G99
7	Diseases of the Eye and Adnexa	H00-H59
8	Diseases of the Ear and Mastoid Process	H60-H95
9	Diseases of the Circulatory System	I00-I99
10	Diseases of the Respiratory System	J00-J99
11	Diseases of the Digestive System	K00-K95
12	Diseases of the Skin and Subcutaneous Tissue	L00-L99
13	Diseases of the Musculoskeletal System and Connective Tissue	M00-M99
14	Diseases of the Genitourinary System	N00-N99
15	Pregnancy, Childbirth, and the Puerperium	O00-O9A
16	Certain Conditions Originating in the Perinatal Period	P00-P96
17	Congenital Malformations, Deformations, and Chromosomal Abnormalities	Q00-Q99
18	Symptoms, Signs, and Abnormal Clinical and Laboratory Findings	R00-R99
19	Injury, Poisoning, and Certain Other Consequences of External Causes	S00-T88
20	External Causes of Morbidity	V00-Y99
21	Factors Influencing Health Status and Contact with Health Services	Z00-Z99
22	Codes for Special Purposes	U00-U85, U89

Table A14. PPO Hyperparameters

Hyperparameter	Value
Learning rate (α)	3 × 10^-4
Batch size	64
Steps per update	2048
Epochs per batch	10
Clipping parameter (ε)	0.2
Max gradient norm	0.5
Discount factor (γ)	0.99
GAE λ	0.95
Entropy coefficient	0.01
Total timesteps	200,000
Policy architecture	MultiInputPolicy