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### Appendix C. Technical Appendix: Modeling DVC Operations

**Note:** In the following discussion, the term "DVC patient flow rate" refers to the planning concept of an average rate of patient processing over the duration of the mass prophylaxis response, not as an actual measure of the unpredictable rate at which patients may show up at DVCs in the aftermath of a bioterrorist attack. The equations and spreadsheet models presented here use this average patient flow rate for 2 reasons. First, there is no good data to guide prediction of patient surge arrivals at DVCs, so any model that tried to estimate surge arrivals would be inherently prone to error. Second, it is likely that, with appropriate use of law enforcement and public information campaigns, planners could maintain constant patient flow rates at their DVCs by controlling entry.

The goal of mass prophylaxis planning is to ensure that dispensing of necessary antibiotics, vaccines, or other medical supplies to target populations occurs within a designated time frame. In certain cases, the time frame for response will be fixed (e.g., in a widespread smallpox attack wherein vaccination of all potential contacts should take place within 4 days of exposure). However, most other factors in the response scenario will either be variable (e.g., population affected) or under planners' control (e.g., number of DVC sites, number of staff, and station process times).

#### A. Modeling Approach

The spreadsheet programs included with this Planning Guide allow planners to model DVC activities based on 2 assumptions. First, all actions in the DVC are considered deterministic, rather than stochastic, processes. While this eliminates naturalistic variability from elements like patient interarrival time and station processing times, it greatly enhances the simplicity and understandability of model estimates. Second, these spreadsheet programs give results for DVCs at what is called "steady-state operation." The definition of steady-state in this setting is that queues occurring at any station in any given DVC in the system do not experience a net increase in length over the course of the prophylaxis campaign. Another way of saying this is that the rate of arrivals equals the rate of departures from the DVC as a whole and from every station in the DVC, as shown in the figure:

**Average Entering (PT/Min) =
Average
Exiting (PT/Min)**

The first step in modeling DVC activities is to determine this average patient flow.

**1. Determining Campaign, DVC, and Station Flow**

Under the deterministic steady-state assumption, individuals arrive at each DVC station at a constant rate throughout operation of the prophylaxis campaign. The flow at these stations can be calculated from features of the campaign as a whole (i.e., overall processing rate across a community), the number of DVCs, and the DVC patient flow plan. For ease in calculations and to avoid errors, these flows should all be in the same unit of time (e.g., per minute).

**Average Campaign Flow**
Average campaign flow represents the total number of individuals processed per unit of time across the entire affected community. It is a function of the total population in the target community (e.g., town population) and the length of the prophylaxis campaign. Algebraically,

i) R_{Campaign} = Pop ÷ T

Where: R_{Campaign} = Average campaign flow (or rate)

Pop = Total size of population (or number of patients)

T = Length of Time for campaign

This calculation will give a campaign flow rate of patients per unit time of campaign. *T* can be days, hours, or minutes. To set *T* at minutes, first determine how many hours per day the campaign will be operating (e.g., the DVCs will be open 24 hours per day). The equation for *T* in terms of minutes becomes:

ii) T = D × H × M
Where: D = Length of campaign in days

H = Hours of operation per day

M = Minutes of operation per hour

Combining i) and ii) gives a calculation of campaign flow in terms of Patients per Minute, as follows:

iii) R_{Campaign} = Pop ÷ (D × H × M)

For example, a campaign targeting 10,000 people over 5 days, operating at 8 hours per day will have an average flow of 10,000÷(5×8×60) = 4.17 pts/min.

Assuming R_{Campaign} is fixed and constant, it becomes the variable to which all staffing calculations ultimately become tethered. Consequently, changes in either staff per DVC, number of DVCs, or station process times, for example, will necessarily cause changes in each other such that the campaign flow remains constant.

**Average DVC Flow**
The average DVC flow (denoted as R_{DVC}) is a measure of the total patients per unit of time each DVC in a campaign can process. Three methods of determining the average DVC flow include a) User-defined; b) Briefing-defined; c) DVC number-defined.

*User-defined DVC Flow*
The average number of patients per unit of time processed can be based on past experiences or live exercises (denoted as R_{DVC-UD}). However, as explained in more detail below in Section 3: Staffing Calculations, the number of staff per station and per DVC is directly proportional to this flow. Consequently, a higher flow (i.e., larger number of patients processed per unit time) demands a larger number of working staff. Spatial constraints (i.e., number of staff a given DVC can accommodate) may not allow for this number and thus the DVC flow may need to be decreased.

*Briefing-defined*
On-site briefings to ensure patient education and consent may be required by Federal, State, or local regulations (e.g., as currently required for all Investigational New Drug (IND) protocols). Because of both their duration (i.e., briefings likely will have the longest process time of all DVC stations) and their scope (i.e., all patients will have to be briefed), briefings will determine the patient flow for each DVC. Regardless of their placement within a DVC flow plan, the briefing will impact other stations both upstream and downstream. Upstream stations should be capable of achieving the briefing flow in order to fill the briefing space to capacity (and thus prevent wasted space and materials). At the same time, upstream stations should not operate faster than the briefing flow as this will produce queues of increasing length outside of the briefing area. Downstream stations should also be capable of achieving the briefing flow to prevent queues of increasing length.

Consequently, planners creating DVCs with on-site briefings should determine their DVC flow by equating it to the briefing flow (denoted as R_{DVC-BD}). The briefing flow is a function of 2 characteristics: the number of patients simultaneously briefed (the product of number of briefing rooms and room capacity) and the length of the briefing. Algebraically,

iv) R_{DVC} = R_{DVC-BD} = (N_{Rooms} × N_{Patients per room}) ÷ T_{Briefing}
Where: N_{Rooms} = Number of briefing rooms

N_{Patients per room} = Capacity of each room

T_{Briefing} = Length of each briefing (in minutes for R_{DVC-BD} to be equal to patients per minute)

*DVC Number-defined DVC Flow*
In certain cases, planners may decide on a maximum number of DVCs in their campaign prior to calculating patient flow. The average DVC flow can be calculated as follows:

v) R_{DVC} = R_{DVC-ND} = R_{Campaign} ÷ N_{DVC}

Where: R_{DVC-ND} = Average DVC patient flow using the number-defined method

N_{DVC} = Maximum number of DVCs within the campaign.

Combining equation iii) with v) will allow calculation of DVC flow in patients per minute as follows:

vi) R_{DVC} = R_{DVC-ND} = Pop ÷ (D × H × M × N_{DVC})

**Average Station-Specific Flow**
The station-specific flow is a function of two variables: the average DVC flow and the proportion of that flow that arrives at the station of interest. This proportion is determined by features of DVC patient flow plan. The DVC flow plan determines the paths that patients may travel. The proportion of patients who take a given path is determined by calculating the percentage taking that path at each branch point along the way (percentages which must be assigned by planners). These station-specific probabilities are then multiplied by the overall patient flow for the DVC (R_{DVC}) as calculated above. Algebraically, for any station *i*, within a DVC pathway containing a total of *I* sequentially numbered stations, the corresponding station-specific flow (R_{Si}) can be calculated as follows:

I

vii) R_{Si} = R_{DVC} x Π (P*i*)

*i* = 1

Where: *i* = Sequential number of station located within flow path of DVC

I = Total number of stations within flow path containing this station

P*i* = Proportion of patients entering into station *i*

The following example demonstrates this method. This diagram represents a simple DVC layout. Circles represent individual stations within the DVC and the station of interest is highlighted.

[D] Select for Text Description.

To calculate the station-specific average flow of Station 4, first identify the pathway a patient would follow from entrance into the DVC to reach the station (represented by the thick lines) and multiply the corresponding estimated probabilities. Finally, multiply this result by the average DVC flow. By example:

Assume: R_{DVC} = 10 pts/min

*p1* = 1.0 User-defined value

*p2* = .80 User-defined value

*p3* = .8 User-defined value

*p4* = .5 User-defined value

Then: R_{S4} = (*p4* × *p3* × *p2* × *p1*) × R_{DVC} or

= (.5 × .8 × .8 × 1.0) × 10 pts/min

= 3.2 pts/min

Certain stations may have multiple pathways of entrance. In such case, the product of the chain of probabilities of each associated pathway should be added and this total then multiplied by R_{DVC}.

**2. Determining the Number of DVCs**

The total number of DVCs must be sufficient to process the total population within the given time frame or the campaign will not be a success. Consequently, the most direct method of calculating the total number of DVCs is to divide the average campaign flow by the average DVC flow, as follows:

viii) N_{DVC} = R_{CAMPAIGN} ÷ R_{DVC}

The total number of DVCs within a campaign is inversely proportional to the average flow of each DVC. Decreasing the average DVC flow will increase the number of necessary DVCs. Decreasing the number of DVCs (e.g., because of resource limitations) will increase the necessary average DVC flow to process a population within the given time frame of the campaign. Fixing the number of DVCs or the DVC flow rate (e.g., by mandating that all DVCs must operate at 100 patients per minute) will force a change in the overall campaign flow and therefore in the overall time needed to complete the prophylaxis campaign.

**3. Staffing Calculations**

The number of staff required for a prophylaxis campaign can be calculated for each station within a DVC, for the DVC as a whole, and for the campaign in total. The number of staff is a function of patient flow, average process time, and the ratio of staff to patient. Under the deterministic representation of a steady-state (where queues, if existent, are constant in length), staff can be calculated using the following general formula:

ix) S = R × T × I
Where: S = Staff

R = Entering patient flow

T = Process time

I = Ratio of staff to patients

Calculating staff then becomes a matter of plugging in the appropriate *R* as explained in Section 1, ensuring the unit of time measure for *T* and *R* are consistent, and determining the ratio of staff to patients for the activity.

**Station-specific staffing**
Two factors determine the optimal number of staff at a DVC station: patient flow (the average number of patients arriving at a station per unit time) and the station-specific processing time (the time needed to process the average patient at that station). When a DVC is running at steady-state operation, staff activities and patient arrivals are balanced so that no *new* bottlenecks or queues form. (Note: a system that is functioning at steady-state can have queues, but they do not get any longer during the steady-state operation.) A simple formula shows how these 2 factors determine the optimal number of staff for each station under steady-state operation:

x) S_{Station} = R_{Station} × T_{Station} × I_{Station}
Where S_{Station} = Staff at station

R_{Station} = Patient flow arriving at station (patients per minute)

T_{Station} = Processing time for station

I_{Station} = Staff-to-Patient ratio at station (e.g., I=1 if one staff member is required for the entire duration of processing of each patient)

**Total DVC staffing**
The total number of staff needed to run a DVC is the sum of the number of staff needed at each station:

xi) S_{DVC} = Σ S_{Station}.

#### B. Definitions of DVC Efficiency

Two measures reflect the efficiency of DVC design: bottlenecks and staff utilization. If more patients arrive than can be processed by DVC staff, a bottleneck will occur at one or more of the stations inside the DVC. A bottleneck at a single station can decrease efficiency of the entire DVC by reducing processing rates at other stations in 1 of 2 ways: long lines at one station may interfere with operations at other stations (e.g., by blocking access), and staff may be shifted to the affected station, thereby compromising efficiency of other areas. To solve bottlenecks, DVC managers may need to increase the total number of DVC staff or decrease processing times (e.g., by shortening forms or protocols).

If the DVC plan overestimates either the need for staff at a given station or the need for entire DVCs to achieve community-wide prophylaxis, waste in the form of staff underutilization or excess "down-time" will occur. As noted, in a large-scale mass prophylaxis operation staff will be one of the resources in shortest supply. In that case, inefficient use of staff at one station or DVC can be expected to decrease the efficiency of some other aspect of the prophylaxis campaign. In plain language, if staff at a DVC or station find themselves idle during a large-scale event, the DVC plan that assigned them to that station needs reevaluation.

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*Current as of August 2004*

AHRQ Publication No. 04-0044