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The design and implementation of new configurations of mental health services to meet local needs is a challenging problem. In the UK, services for common mental health disorders such as anxiety and depression are an example of a system running near or at capacity, in that it is extremely rare for the queue size for any given mode of treatment to fall to zero. In this paper we describe a mathematical model that can be applied in such circumstances. The model provides a simple way of estimating the mean and variance of the number of patients that would be treated within a given period of time given a particular configuration of services as defined by the number of appointments allocated to different modes of treatment and the referral patterns to and between different modes of treatment. The model has been used by service planners to explore the impact of different options on throughput, clinical outcomes, queue sizes, and waiting times. We also discuss the potential for using the model in conjunction with optimisation techniques to inform service design and its applicability to other contexts.

Health treatment activities where arriving patients might have to wait for treatment and where duration of treatment follows a certain probability distribution have often been modelled using queueing theory. A classic example is the study of accident and emergency departments in acute hospitals [

The provision of mental health care for depression and anxiety in the primary care system is one such complex system. A configuration for mental health care delivery called “stepped care” is advocated for patients with common mental health problems [

A diagram highlighting the difference between a traditional care model and the stepped care model. GP: General Practitioner.

The introduction of stepped care within an existing mental health care framework is challenging. Planning the delivery of stepped care requires decisions concerning the treatments to be offered, the number and type of staff, the protocol for how patients transfer between treatments, and the balance of provision between low and high intensity treatments. The other key feature of mental health care systems other than their complexity is that they are often operating at capacity. This is more feasible than in acute hospital environments since patients in the queue effectively wait “at home” using little resource.

In this paper we describe a mathematical model we have developed to help planners design a stepped care mental health system [

The approach used is complementary to traditional queueing theory and is most suited to systems where traffic intensities are greater than or equal to one or to a system where the starting states have large queues. It complements recent work on queueing systems where some of the servers are always busy [

The unit of capacity we consider in this analysis is a time slot in a diary (e.g., a therapy session) and we assume that patients are treated in discrete sessions. A given time slot in a diary is assumed to be devoted to one, and only one, distinct treatment type. We further assume that a patient takes at least one session to be treated, and that at the start of the modeled period there is no patient currently undergoing treatment (i.e., at

For

For

For

For

We begin by considering

We can then define

Thus we have derived an expression for the probability that at some time

We now extend the concept of a single treatment slot to a network of treatment slots that can be thought of as representing a given system.

Consider a treatment slot of type

Let

Define the random variable

Let

Define the random variable

Define the random variable

Define the random variable

Define

Define

For a constant number

If a positive integer-valued distribution

Additionally the expectation and variance of

Proof of these results can be found in Grimmett and Stirzaker [

Consider a treatment in a general network as shown in Figure

Flows in and out of treatment slots of type

As shown in Figure

For each person leaving a particular treatment slot of type

From (

Equation (

In circumstances where the network is always full, the output from units of capacity of type

We are now in a position to consider the expectation and variance of the change in the queue size

For a patient waiting to receive treatment for a mental health problem, waiting time in a queue is more likely to be of concern than the actual number of people waiting. Let

The linear nature of the equations above in terms of

Maximise:

Total number of therapy sessions per week:

Total number of sessions for a treatment of type

Specify a maximum increase in waiting time of

This mathematical model has been implemented as part of a project examining the implementation of stepped care systems [

In this system there are three types of appointments available: an initial screening appointment, a low-intensity therapy appointment, and a high-intensity therapy appointment. People are either referred by their GP (General Practitioner) into the system in which case they begin with an assessment appointment or they can self-refer directly to low-intensity therapy treatment.

The proportion of patients moving between different types of appointment is shown in Table

Flows of patients between different types of appointment and two endpoints of the stepped care system.

From | Assessment | Low intensity | High intensity | Completed treatment | Dropped out of treatment |
---|---|---|---|---|---|

Assessment | 0% | 40% | 20% | 10% | 30% |

Low intensity | 0% | 0% | 20% | 50% | 30% |

High intensity | 0% | 0% | 0% | 80% | 20% |

The health system managers have 100 weekly appointments available to cover all types of appointment. For the purposes of this example, we assume that the number of weekly booked sessions a patient uses for each type of treatment is exponential with means of 1, 3, and 6 for assessments, low-intensity and high-intensity sessions respectively. The arrivals and capacity allocation for the current system are given in Table

Arrivals to the system and allocation of resources within the system.

Appointment type | Average number of new, external arrivals every week | Weekly appointment slots allocated |
---|---|---|

Assessment | 20 | 30 |

Low intensity | 10 | 40 |

High intensity | 0 | 30 |

Although the waiting times for screening and high-intensity treatments are acceptable, the increase in waiting time for a low-intensity appointment over 6 months is unacceptably long (12 weeks) (see columns 2 and 3 of Table

There can be a maximum of 100 total allocated appointments.

The maximum increase in average waiting time for an assessment is 4 weeks.

The maximum increase in average waiting time for either low or high intensity treatment is 8 weeks.

There must be at least 20 assessment sessions, 30 low-intensity, and 20 high-intensity sessions every week.

Example use of optimisation to allocate available treatment slots to treatment types.

Appointment type/End point | Current weekly appointment slots allocated | Average increase in waiting time (weeks) | Suggested weekly appointment slots allocated | Average increase in waiting time (weeks) |
---|---|---|---|---|

Assessment | 30 | 1.3 | 26 | 4 |

Low intensity | 40 | 11.7 | 45 | 6 |

High intensity | 30 | 7 | 29 | 7 |

— | — | |||

— | — |

We ran this optimisation problem using Microsoft Excel Solver (version 2003). We note that Microsoft Solver is a standard add-in to Microsoft Excel and there exist several resources on its use within Excel (e.g., [

The suggested appointment schedule has resulted in a more even distribution of the expected waits for each type of treatment and increased the expected total number of people completing treatment over the 6-month time frame.

A clear limitation is the assumption that the system is always busy. However, application of the model to any given system would still provide the maximum possible throughput of the system over a given period of time. In the context of a mental health system, other limitations apply. Firstly, all patients are considered to be homogeneous and no allowance is made for patients with different characteristics (for instance presenting problem) having different duration of stay distributions or different pathways through the system. Secondly, in this analysis, time is considered to be defined by the number of treatment slots and thus application to a system where sessions are not regularly spaced in time is more complicated. Finally, “holding” or “blocking back” behaviour is not accounted for in the model, in that both the duration of treatment and the destination of patients from each treatment are assumed independent of the state of the system.

This mathematical model was developed in response to a specific problem within the configuration of mental health care services [

We have described a simple way of analysing throughput and flows for a networked system in the situation where a system is always busy or where this is a reasonable approximation. We note that this is not a steady-state model and instead considers changes in mean output, queue sizes and waiting times over a relatively short (6 months) time period. We have also shown how optimisation techniques might be applied to the subsequent design of a network in the context of a mental health system.

This approach could be useful in other health systems where “servers” (whether beds, clinicians or other resources are always busy) but a key assumption that needs to be met is that there will always be a queue. In practice this assumption is less likely to be valid for systems where queues involve people waiting in a physical allocated space (for instance, in an emergency department) than where people can “virtually” wait at home. Nonetheless, considering such busy systems over a short amount of time using these sorts of models can provide insight into the allocation of resources and management of arrivals to complement standard steady state queuing theory.

Remember that the random variable

For each person leaving a particular treatment slot of type

From (

and thus

It is a standard result that

This project was funded by the National Institute for Health Research Service Delivery and Organisation programme (Project no. 08/1504/109). The views expressed in this publication/presentation are those of the authors and not necessarily those of the NHS, the NIHR, or the Department of Health. The NIHR SDO programme is funded by the Department of Health.