11. TECHNICAL APPENDIX

NOTE: In the following discussion, the term "clinic 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 clinics in the aftermath of a bioterrorist attack. The equations and spreadsheet models presented here use this average patient flow rate for two reasons. First, there is no good data to guide prediction of patient surge arrivals at dispensing sites, 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 clinics 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 dispensing clinic sites, number of staff, and station process times).

A. Modeling Approach

These spreadsheet programs allow planners to model clinic activities based on two assumptions. First, all actions in the clinic 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 clinics 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 clinic 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 clinic as a whole and from every station in the clinic as shown in the figure:

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

1. Determining Campaign, per-Clinic and per-Station Flow

Under the deterministic steady-state assumption, individuals arrive at each clinic 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 clinics, and the clinic 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).

a. 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: RCampaign = 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 clinics 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 clinic, number of clinics, or station process times, for example, will necessarily cause changes in each other such that the campaign flow remains constant.

b. Average Clinic Flow

The average clinic flow (denoted as R_DVC for "Dispensing/Vaccination Clinic) is a measure of the total patients per unit of time each clinic in a campaign can process. Three methods of determining the average clinic flow include: a) User-defined; b) Briefing-defined; c) clinic number-defined.

1) User-defined Clinic 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, the number of staff per station and per clinic 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 clinic can accommodate) may not allow for this number and thus the clinic flow may need to be decreased.

2) 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 clinic stations) and their scope (i.e., all patients will have to be briefed), briefings will determine the patient flow for each clinic. Regardless of their placement within a clinic flow plan, the briefing will affect 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 clinics with on-site briefings should determine their clinic flow by equating it to the briefing flow (denoted as R_DVC-BD). The briefing flow is a function of two characteristics: the number of patients simultaneously briefed (the product of number or 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)

3) Clinic Number-defined patient Flow

In certain cases, planners may decide on a maximum number of clinics in their campaign prior to calculating patient flow. The average clinic flow can be calculated as follows:

v) R_DVC = R_DVC-ND = R_Campaign ÷ NDVC

Where: R_DVC-ND = Average clinic patient flow using the number-defined method
NDVC = Maximum number of clinics within the campaign.

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

vi) R_DVC = R_DVC-ND = Pop ÷ (D × H × M × N_DVC)

c. Average Station-Specific Flow

The station-specific flow is a function of 2 variables: the average clinic flow and the proportion of that flow that arrives at the station of interest. This proportion is determined by features of clinic patient flow plan. The clinic 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 clinic (R_DVC) as calculated above. Algebraically, for any station i, within a clinic pathway containing a total of j sequentially numbered stations, the corresponding station-specific flow (R_Si) can be calculated as follows:
                 j
vii) R_Si = ∑(Π) × R_DVC
                i=1
Where: i = Sequential number of station located within flow path of clinic
j = Total number of stations within flow path containing this station
Pi = Proportion of patients entering into station i

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

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

Assume: R_DVC = 10 pts/min
p1 = 1.0
p2 = 0.8
p3 = 0.8
p4 = 0.5

Then: R_S4 = (p4 × p3 × p2 × p1) × R_DVC or
= (0.5 × 0.8 × 0.8 × 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 Clinics

The total number of clinics 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 clinics is to divide the average campaign flow by the average clinic patient flow, as follows:

viii) N_DVC = R_CAMPAIGN ÷ R_DVC

The total number of clinics within a campaign is inversely proportional to the average flow of each clinic. Decreasing the average clinic flow will increase the number of necessary clinics. Decreasing the number of clinics (e.g., because of resource limitations) will increase the necessary average clinic flow to process a population within the given time frame of the campaign. Fixing the number of clinics patient flow rate (e.g., by mandating that all clinics 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 clinic, for the clinic 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.

a. Station-specific staffing

Two factors determine the optimal number of staff at a clinic 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 clinic 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.) The following formula shows how these two 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)

b. Total Clinic staffing

The total number of staff needed to run a dispensing clinic is the sum of the number of staff needed at each station:

xi) S_DVC = ∑ S_Station

B. Definitions of Clinic Efficiency

Two measures reflect the efficiency of clinic design: bottlenecks and staff utilization. If more patients arrive than can be processed by clinic staff, a bottleneck will occur at one or more of the stations. A bottleneck at a single station can decrease efficiency of the entire clinic by reducing processing rates at other stations in one of two 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, clinic managers may need to increase the total number of staff or decrease processing times (e.g., by shortening forms or protocols).

If the dispensing clinic plan overestimates either the need for staff at a given station or the need for whole clinics 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 clinic can be expected to decrease the efficiency of some other aspect of the prophylaxis campaign. In plain language, if staff at a dispensing clinic or station find themselves idle during a large-scale event, the clinic plan that assigned them to that station needs re-evaluation.