Improving access to housing and supportive services for runaway and homeless youth: Reducing vulnerability to human trafficking in New York City

Figures & data

Table 1. List of sets used in mathematical modeling.

Symbol	Definition
$\mathit{Y}$	Set of youth in the system, indexed by y
$\mathit{S}$	Set of RHY organizations in the system, indexed by s
$\mathit{T}$	Time horizon over which services may be scheduled, indexed by t
$\mathit{I}$	Set of service–intensity pairs in system, indexed by i
$\mathit{N}$	Set of demographics attributes in youth and organization profiles (age, gender, child and citizenship status, HT victim or survivor status), indexed by n

Table 3. List of decision variables used in mathematical modeling.

Variable	Type	Definition
$U_{y,s,i}$	Binary	Youth to RHY organization assignment decision variables are equal to 1 if youth $y\in$ $\mathit{Y}$ is assigned to organization $s\in$ $\mathit{S}$ to receive service $i\in$ $\mathit{I},$ 0 otherwise
$X_{y,s,i}^t$	Binary	Time-dependent youth to RHY organization assignment decision variables are equal to 1 if youth $y\in$ $\mathit{Y}$ is assigned to RHY organization $s\in$ $\mathit{S}$ to receive service $i\in$ $\mathit{I}$ at time $t\in$ $\mathit{T}$, 0 otherwise
$E_{s,i}^t$	Continuous	Extra resource decision variables are equal to the amount of extra resources required to fulfill youth’ collective demand at RHY organization $s\in$ $\mathit{S},$ to provide service $i\in$ $\mathit{I}$ at time $t\in$ $\mathit{T}$
$O_{s,i}^t$	Continuous	Overflow shelter decision variables are equal to the number of youth that are directed to the overflow shelter through RHY organization $s\in$ $\mathit{S},$ to receive service $i\in$ $\mathit{I}$ at time $t\in$ $\mathit{T}$

Table 4. Experimental parameters varied; bolded values indicate base expansion model parameters.

Parameter	Symbol	Level
Number of youth	|$\mathit{Y}$|	400, 450, 500, 550, 600
Abandonment percentage of youth	θ	10%, 15%, 20%, 25%, 30%
Duration of service	$d_{y,i}=N(60,15)$	0.8$d_{y,i},$ 0.9$d_{y,i},$ ${\mathit{d}}_{\mathit{y},\mathit{i}},$ 1.1$d_{y,i},$ 1.2$d_{y,i}$
COVID-19 effect	|$\mathit{Y}$|$,c_{s,i,t}$	|$\mathit{Y}$|$=400$ $,c_{s,i,t}^{\prime }=0.5c_{s,i,t}$

Table 5. System overflow metrics comparing number of youth arriving with the base expansion model of 500 youth considering the average across 10 runs.

System Overflow Metrics	Number of Youth
	400	450	500	550	600
Max Overflow Beds	37.0	52.0	87.0	90.0	110.0
Mean Overflow Beds	33.7	46.7	80.4	82.8	102.0
Overflow Cost Change (%)	−65	−29	0	+35	+67
Referral Cost Change (%)	−20.1	−39.3	0	+72.9	+94.5

Figure 1. Illustrative example of sample RHY organizations that provide housing and support services to homeless youth, where X indicates the unavailable resources.

Table 2. List of parameters used in the mathematical modeling.

Symbol	Definition
ly	Arrival time of youth $y\in$ $\mathit{Y}$ to system
$d_{y,i}$	Duration of service $i\in I$ for youth $y\in$ $\mathit{Y}$
$f_{y,i}$	Number of times service $i\in$ $\mathit{I}$ should be provided to youth $y\in$ $\mathit{Y}$
${\omega }_{y,i}$	Time between appointments while providing service $i\in$ $\mathit{I}$ to youth $y\in$ $\mathit{Y}$
$[a_{y,i},b_{y,i}]$	Earliest and latest possible start times of service $i\in$ $\mathit{I}$ to youth $y\in$ $\mathit{Y}$
ki	Periodicity flexibility of service $i\in$ $\mathit{I}$
αy	Demographics profile of youth $y\in$ $\mathit{Y}$
βs	Demographics profile of RHY organization $s\in$ $\mathit{S}$
ηy	List of requested services by youth $y\in$ $\mathit{Y}$
σs	List of services offered by RHY organization $s\in$ $\mathit{S}$
$c_{s,i,t}$	Capacity of service $i\in \mathit{I}$ in RHY organization $s\in$ $\mathit{S}$ at time $t\in$ T
${\mu }_{s,i}$	Maximum number of in-house resources that RHY organization $s\in$ $\mathit{S}$ can have for service $i\in$ $\mathit{I}$ within their facility
$r_{y,s,i}$	Cost of assigning youth $y\in$ $\mathit{Y}$ to RHY organization $s\in$ $\mathit{S}$ for service $i\in$ $\mathit{I}$
${\gamma }_{s,i}$	Cost of allocating one unit of extra resource to RHY organization $s\in$ $\mathit{S},$ for service $i\in$ $\mathit{I}$
${\lambda }_{s,i}$	Cost of directing youth to overflow shelter from RHY organization $s\in$ $\mathit{S}$ to receive service $i\in$ $\mathit{I}$

Figure 2. Examples of needs profiles for two youth who belong to two different demographics where shading indicates different demographics groups.

Figure 3. Youth are matched with RHY organization 1 and 2 respectively, considering the youth demographics and the accepted demographic at the RHY organizations. The services that are not provided in RHY organizations 1 or 2 are provided through $s_n\in$ ${\mathit{S}}^{\mathit{o}}.$

Table 6. Average system overflow metrics under base expansion model varying $d_{y,i}$ levels.

System Overflow Metrics	Duration of Service
	$0.8d_{y,i}$	$0.9d_{y,i}$	${\mathit{d}}_{\mathit{y},\mathit{i}}$	$1.1d_{y,i}$	$1.2d_{y,i}$
Max Overflow Beds	56.0	62.0	87.0	97.0	117.0
Mean Overflow Beds	48.0	58.7	80.4	96.0	114.0
Overflow Cost Change (%)	−58	−31	0	23	+50
Referral Cost Change (%)	−26	−12	0	+12	+37

Figure 4. Average number of beds received by type, across 10 runs, eight organizations, and 6 months.

Figure 5. Per RHY organization, the average percentage increase in extra in-house resources and number of youth directed to overflow shelter compared with existing in-house capacity over 6 months.

Figure 6. Average number of support services received, by type, over 6 months, across 10 runs.

Figure 7. Average number of extra hours required, per RHY organization and support service, to fulfill demand fom youth, across 10 runs (note that extra in-house resources and referrals are combined).

Figure 8. The effect of total number of youth in the system, on the daily trend in number of youth at the overflow shelter referred from Organization 2 over 6-months, with 90% confidence intervals over 10 runs.

Figure 9. The effect of average length of stay, on the daily trend in number of youth at the overflow shelter referred from Organization 2 over 6 months, with 90% confidence intervals over 10 runs.

Figure 10. The effect of abandonment, on the daily trend in the number of youth at the overflow shelter referred from Organization 2 over 6 months, with 90% confidence intervals over 10 runs.

Table 7. Average system overflow metrics under base expansion model, varying abandonment levels.

System Overflow Metrics	Abandonment %
	30(%)	25(%)	20(%)	15(%)	10(%)
Max Overflow Beds	67.0	75.0	87.0	92.0	97.0
Mean Overflow Beds	64.7	69.7	80.4	88.2	93.0
Overflow Cost Change (%)	−27	−15%	0	14	+29
Referral Cost Change (%)	−18	−8%	0	+10	+16

Figure 11. Average number of beds and different intensity of medical resources received by youth in organizations considering COVID-19, across 10 runs and 6 months.

Data availability

The authors confirm that the aggregated data supporting the findings of this study are available within the article and its supplementary materials. As stated in the Institutional Review Board, due to the nature of this research, study participants (such as RHY organizations) did not agree for their data to be shared publicly; thus detailed, identifiable supporting data is not available.