Health Care Management Science

, Volume 13, Issue 2, pp 155–169

A non-homogeneous discrete time Markov model for admission scheduling and resource planning in a cost or capacity constrained healthcare system

Authors

    • School of Computing and Information EngineeringUniversity of Ulster
  • Sally McClean
    • School of Computing and Information EngineeringUniversity of Ulster
  • Brian Meenan
    • School of Electrical and Mechanical EngineeringUniversity of Ulster
  • Peter Millard
    • St George’s, University of London
Article

DOI: 10.1007/s10729-009-9120-0

Cite this article as:
Garg, L., McClean, S., Meenan, B. et al. Health Care Manag Sci (2010) 13: 155. doi:10.1007/s10729-009-9120-0

Abstract

Healthcare resource planners need to develop policies that ensure optimal allocation of scarce healthcare resources. This goal can be achieved by forecasting daily resource requirements for a given admission policy. If resources are limited, admission should be scheduled according to the resource availability. Such resource availability or demand can change with time. We here model patient flow through the care system as a discrete time Markov chain. In order to have a more realistic representation, a non-homogeneous model is developed which incorporates time-dependent covariates, namely a patient’s present age and the present calendar year. The model presented in this paper can be used for admission scheduling, resource requirement forecasting and resource allocation, so as to satisfy the demand or resource constraints or to meet the expansion or contraction plans in a hospital and community based integrated care system. Such a model can be used with both fixed and variable numbers of admissions per day and should prove to be a useful tool for care managers and policy makers who require to make strategic management decisions. We also describe an application of the model to an elderly care system, using a historical dataset from the geriatric department of a London hospital.

Keywords

Resource managementAdmission schedulingNon-homogeneous Markov modelStochastic optimal control

1 Introduction

Admission scheduling [1, 2] and resource planning [3] are fundamental problems which require complex strategies to effectively manage care services ensuring optimum utilization of scarce resources and efficient quality of service delivery. Long waiting lists are considered to be symptomatic of an inefficient care system [46]. To avoid long waiting lists, care professionals and policy makers are required to estimate the resource requirements for a given time in the future and to allocate the budget/ resources to satisfy this demand [7]. Also if the resource availability is limited and cannot meet an ever increasing demand, it is necessary to estimate the number of admissions that can be satisfied each day and schedule accordingly, from the waiting list [8]. This will help the health care managers to optimally exploit capacity and limit the size of the waiting list [7].

Neither admission scheduling nor resource requirements forecasting are new problems and numerous models have been proposed. Some of these [5, 913] are queueing models, others [1419] utilise simulation or population/ratio based approaches which use average lengths of stay to quantify the number of patients requiring care resources, for example [2024]. However, some of these methods, which are mostly stochastic or simulation based, are computationally very complex and require substantial data and logistics [25]. Other simpler methods, on the other hand, do not properly represent case mix [3] or local specificities [25] and thus require more sophisticated models [26]. In addition, as suggested by [3, 27], other approaches are inaccurate or misleading. Also previous models mainly model patient flow in a department or hospital and do not incorporate readmissions and care in the community. Therefore they are not very suitable for admission scheduling and capacity planning in an integrated care system, which includes both hospital and community care [28].

In our previous work [29] we have developed a continuous time non-homogeneous Markov model for admission scheduling and resource forecasting by enumerating patient pathways. The limitation of this model is that it assumes a fixed number of admissions per day, so it cannot be used in many practical scenarios (such as a care system with expansion or contraction plans). A variable number of admissions each day can therefore prove to be a more realistic assumption with correspondingly better solutions that ensure optimal resource utilization. Also we might need to schedule a variable number of admissions each day so as to allocate resources to satisfy the fluctuating demand for care services [12, 30]. In this paper, we present a discrete time Markov model for admission scheduling and capacity planning. Based on our initial homogeneous model, a novel, more realistic non-homogeneous model is developed, incorporating time dependent covariates (in this case the patient’s present age and the present calendar year). This approach can effectively be used for resource requirement forecasting and resource allocation to satisfy the demand or resource constraints or to meet the expansion or contraction plans. We can also use this new model to compare different admission scheduling strategies for a care system. Finally we illustrate use of the model by applying it to a historical dataset of all male patients from a geriatric department of a London hospital admitted during a 16 year period [31]. Throughout the paper we use mathematical notation which is described in Appendix 2 (Table 5).

2 Background

2.1 Patient flow through the care system

An integrated care system is a system in which patient care is provided in both hospital and the community. Patient flow can be characterized by the rate of transition of a patient from one phase to another in this care system. A cohort analysis of patient flows through all phases in a care system can help us to forecast the number of patients in various phases at a given time in the future [32]. In most cases patient flow depends on the patient pathways defined for the care system. We assume that a care system has n hospital phases, such as acute, treatment, rehabilitation, or long stay and m community phases, such as dependent, convalescent, or recovered. These partially observable phases can be real such as various hospital or community care units or conceptual, representing periods in hospital and/or community care depending on the patient phase of care or stage of the illness being treated [33, 34]. Figure 1 is the schematic representation of patient flow through the care system.
https://static-content.springer.com/image/art%3A10.1007%2Fs10729-009-9120-0/MediaObjects/10729_2009_9120_Fig1_HTML.gif
Fig. 1

Patient flow in the care system. Here HPi represents hospital phase i and CPi represents community phase i

A patient pathway for such a system is defined as the way a patient sequentially moves from one hospital phase to the next and similarly one community phase to the next. Patients can be discharged from any hospital phase to the first community phase or the patient can die in any hospital or community phase. Re-admission into the first hospital phase is possible from any community phase. We can therefore represent this care system by an n + m + 1 state discrete time Markov chain with death as an absorbing state. The transition matrix Q for the absorbing Markov chain can then be represented as follows [33, 35]:

Q = {qij} = Transition rate (next transition is to phase j | currently in phase i).
https://static-content.springer.com/image/art%3A10.1007%2Fs10729-009-9120-0/MediaObjects/10729_2009_9120_Equ1_HTML.gif
(1)

Here λi represents the rate of transition from hospital phase i to hospital phase i + 1. νi represents the rate of transition from hospital phase i to the first community phase and μi is the rate of transition from hospital phase i to the absorbing phase death. Similarly αi is the rate of transition from community phase i to community phase i + 1, γi is the rate of transition from community phase i to the first hospital phase and βi is the rate of transition from community phase i to death.

We assume that the duration t of a step is 1 day. Therefore, the discrete time transition probability matrix can be defined as P = {pij} = Probability (next transition is to phase j | currently in phase i), where P can be described as follows [35]:
$$ {\mathbf{P}}{\text{ }} = { \exp }\left( {{\mathbf{Q}}t} \right) = { \exp }\left( {\mathbf{Q}} \right){\text{ }}. $$
(2)

3 Model description

3.1 The homogeneous model

In the previous section we have discussed how we can model a care system as a discrete time absorbing Markov chain, based on our continuous time representation. The vector s0 represents the initial values of number of patients in different phases of the system.
$$ {{\mathbf{s}}_0}{\text{ }} = {\text{ }}\left\{ {{s_{0,{\text{1}}}},{\text{ }}{s_{0,{\text{2}}}},{\text{ }}{s_{0,{\text{3}}}}, \ldots, {\text{ }}{s_{0,n}},{\text{ }}{s_{0,n + {\text{1}}}},{\text{ }}{s_{0,n + {\text{2}}}}, \ldots, {s_{0,n + m}},{\text{ }}{s_{0,n + m + {\text{1}}}}} \right\}{\text{ }}. $$
The mean number of patients in each phase after k days is given by vector sk, where:
$$ {{\mathbf{s}}_k}{\text{ }} = {\text{ }}{{\mathbf{s}}_0}*{{\mathbf{P}}^k}{\text{ }}. $$
We also define a cost vector c = {ci} = daily cost of care in phase i:
$$ {\mathbf{c}} = {\left( {{c_1},{c_2},{c_3},....,{c_{n + m}},0} \right)^\prime } $$
where cn+m+1 = 0 (there is no cost in the death phase).

3.2 The non-homogeneous model

Typically patient flow also depends on time dependent covariates such as the patient’s age and the calendar year. In order to have a more realistic model it is therefore necessary to update parameter values with time [35]. The transition rates λi, μi, νi, αi, βi and γi have been previously assumed to depend log-linearly on the time dependent covariates χ = (χ1χ2 … χn). The log-linear function here takes the form:
$$ { \exp }\left( {\sigma {\text{ }} + {\text{ }}{{\mathbf{b}}^\prime}{\mathbf{\chi }}} \right) $$
(3)
where coefficient parameters σ and vector b are estimated for each of the transition rates using maximum likelihood estimation [33, 35].

In the model presented here we consider two time dependent covariates: the patient’s present age and the present calendar year. In order to have a more realistic model, we update the covariates each day for each patient currently in the system, recalculate the parameter values for each patient separately and recompute the discrete time transition probability matrix P.

The mean number of patients in each phase after k days is then given by:
$$ {{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\prod\limits_{i = 1}^k {{{\rm P}^{(i)}}} $$
(4)
where P(i) represents the value of transition matrix P on day i.
The estimated number of patients in the care system on day k is:
$$ {\eta_k} = \sum\limits_{i = 1}^{n + m} {{s_{k,i}}} = {\text{ }}{{\mathbf{s}}_k}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right){\text{ }} $$
(5)
where h is a column vector of (n + m + 1) elements. Here the first n elements (number of hospital phases) are 1 and the remaining elements are 0, and e is a column vector of (n + m + 1) elements, with first n elements and last element equal to 0 and the remaining m elements (number of community phases) are 1. The column vectors h and e are used to separately estimate the expected number of patients in hospital and community phases.
The expected number of patients in the hospital after k days is:
$$ {\eta_k}^h{\text{ }} = {\text{ }}{{\mathbf{s}}_k}*{\mathbf{h}}{\text{ }}. $$
(6)
The expected number of patients in the community after k days is:
$$ {\eta_k}^c{\text{ }} = {\text{ }}{{\mathbf{s}}_k}*{\mathbf{e}}{\text{ }} $$
(7)
and the expected total daily cost on day k is
$$ {\Omega_k} = {{\mathbf{s}}_k}*{\mathbf{c}}{\text{ }}{\text{.}} $$
(8)

Once the expected numbers of patients in different phases of the system are known, we use these expressions to calculate the optimum rate of admissions to satisfy the given constraints and to estimate the resource requirements based on the given admission schedule.

3.3 Incorporating admissions

For a more realistic model it is necessary to incorporate new admissions into the system. We therefore develop our model to include, in the first instance, a fixed number of admissions each day and subsequently a variable number of admissions (increasing or decreasing) each day are assumed.

3.3.1 Fixed number of admissions

For modelling a fixed number of admissions each day, we first assume that initially there are no patients in the system and on day 0 the first patient is admitted into the first hospital phase. On each subsequent day, one new patient is admitted into the first hospital phase. We will separately estimate the distribution vector for each patient and sum all these to estimate the patients’ distribution on a given day. Here the individual age of each patient is considered. Thus the vector of updated values of patients in different phases after k days is given by:
$$ {{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\sum\nolimits_{j = 1}^k {\left( {\prod\limits_{i = 1}^{k - j + 1} {{\rm P}_j^{(i)}} } \right)} $$
(9)
where s0 = {1,0,0,0,…….,0,…0} representing the fact that on day 0 the first patient is admitted into the first hospital phase and P(i)j is the transition probability matrix for patient j on day i. Here sk represents the number of patients in different phases after k days with 1 admission per day. If the admission rate is a admissions per day, then the expected number of patients in different phases after k days will be a*sk.

3.3.2 Variable number of admissions

For a care system having a variable number of admissions each day we assume that the rate of change in admission rate (growth in service) is r. Here r can be a positive (growth) or negative (contraction) real number. Therefore the expected number of patients in different phases on day k will be:
$$ {{\mathbf{s}}_k} = {\mathbf{x}} + {\mathbf{y}} $$
(10)
where x is the vector of expected number of patients after k days with a fixed number of admissions each day and y is the vector of the change in the expected number of patients in each phase after k days due to the change in admission rate. Therefore
$$ \begin{gathered} {\mathbf{y}} = {{\mathbf{s}}_0}*\sum\nolimits_{j = 1}^k {\left( {\left( {r*j} \right)*\prod\limits_{i = 1}^{k - j + 1} {{\rm P}_j^{(i)}} } \right)} \hfill \\ {\text{ }} = r*{{\mathbf{s}}_0}*\left( {\sum\nolimits_{j = 1}^k {\left( {j*\prod\limits_{i = 1}^{k - j + 1} {{\rm P}_j^{(i)}} } \right)} } \right). \hfill \\ \end{gathered} $$
(11)
Thus the patient distribution on day k will be:
$$ {{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\left\{ {\sum\nolimits_{j = 1}^k {\left( {\prod\limits_{i = 1}^{k - j + 1} {{\rm P}_j^{(i)}} } \right)} + r*\left( {\sum\nolimits_{j = 1}^k {\left( {j*\prod\limits_{i = 1}^{k - j + 1} {{\rm P}_j^{(i)}} } \right)} } \right)} \right\}. $$
(12)
So we can write:
$$ {{\mathbf{s}}_k} = {\mathbf{x}}{\text{ }} + r*{\mathbf{w}}. $$
(13)
where vector w represents the expected change in patient distribution (number of patients in different phases) due to unit change (1 patient per day) in admission rate. Similarly we can have more complex functions for admission rate comprising higher powers of w such as
$$ \begin{array}{*{20}{c}} {{{\mathbf{s}}_k} = = {\text{ }}{\mathbf{x}} + {r_1}*{{\mathbf{w}}_1} + {r_2}*{{\mathbf{w}}_2} + {r_3}*{{\mathbf{w}}_3} + ........ + {r_n}*{{\mathbf{w}}_n}} \hfill \\ {{\text{where }}{{\mathbf{w}}_m} = \sum\nolimits_{j = 1}^k {\left( {{j^m}*\prod\limits_{i = 1}^{k - j + 1} {{\text{P}}_j^{(i)}} } \right)} .} \hfill \\ \end{array} $$
(14)

The results obtained in this sub-section can be used for admission scheduling as discussed in the next sub-section.

3.4 Admission scheduling

Admission scheduling can be defined as designing an admission policy to ensure optimum utilization of the future available resources satisfying given constraints or meeting a given target. Possible constraints can be availability based on budgets, beds or other resources (such as nursing staff, physicians, or specialists) at a given time in the future or the total daily, monthly or yearly expenditure. We will consider two scenarios of interest: first a new care system and second a pre-existing care system.

3.4.1 A new care system

In a new care system the first patient is admitted into hospital phase 1 on day 0 (at t = 0). Therefore for a new care system:
$$ {{\mathbf{s}}_0}{\text{ }} = \left\{ {{\text{1}},0,0,0, \ldots \ldots .,0, \ldots 0} \right\}. $$

We assume that the constraint is to satisfy the available number of beds, so there will be only B(tgiven) beds available at time tgiven.

Therefore the expected number of admissions allowed each day can be calculated as:
$$ {A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{B\left( {{t_{\text{given}}}} \right)}}{{{\eta_{{t_{\text{given}}}}}}}{\text{ per day}} $$
(15)
where, \( {\eta_{{t_{\text{given}}}}} \)is the expected number of patients in the care system at time tgiven and is defined in (5).
If the constraint is the total number of beds available in a hospital but the beds availability in the community is not constrained, then the expected number of admissions is:
$$ {A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{B\left( {{t_{\text{given}}}} \right)}}{{\eta_{{t_{\text{given}}}}^h}}{\text{ per day}} $$
(16)
where \( \eta_{{t_{\text{given}}}}^h \)is the number of patients in hospital at time tgiven and is defined in (6).
If the total daily cost of care is a constraint then the expected number of admissions is:
$$ {A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{C\left( {{t_{\text{given}}}} \right)}}{{{\Omega_{{t_{\text{given}}}}}}}{\text{ per day}} $$
(17)
where C(tgiven) is the available daily budget (cost) at time tgiven. \( {\Omega_{{t_{\text{given}}}}} \)is the expected total daily cost at time tgiven and is defined in (8).
Here it is important to note that we are required to define the rate of change in admission rate (r) so as to calculate the number of admissions allowed each day. For a system with a fixed number of admissions each day, r = 0. We can also calculate the value of r if the resource constraints are available for two different dates (t1 and t2) in the future. For example if the total daily cost constraint is given for two dates in the future, with C(t1) the cost at time t1 and C(t2) the cost at time t2, from (8) and (13):
$$ {\Omega_{{t_1}}}{\text{ = }}\left( {{{\mathbf{x}}_{\text{1}}}{\text{ + r*}}{{\mathbf{w}}_{\text{1}}}} \right){\text{*}}{\mathbf{c}} $$
(18)
and similarly
$$ {\Omega_{{t_2}}}{\text{ = }}\left( {{{\mathbf{x}}_{\text{2}}}{\text{ + r*}}{{\mathbf{w}}_{\text{2}}}} \right){\text{*}}{\mathbf{c}}{\text{ }}. $$
(19)
As the initial rates of admission should be the same, using (17) we get:
$$ \frac{{C\left( {{t_1}} \right)}}{{{\Omega_{{t_1}}}}} = \frac{{C\left( {{t_2}} \right)}}{{{\Omega_{{t_2}}}}} $$
and this gives:
$$ r = \left( {\frac{{\left( {C\left( {{t_1}} \right)*\left( {{{\mathbf{x}}_2}*{\mathbf{c}}} \right)} \right) - \left( {C\left( {{t_2}} \right)*\left( {{{\mathbf{x}}_1}*{\mathbf{c}}} \right)} \right)}}{{\left( {C\left( {{t_2}} \right)*\left( {{{\mathbf{w}}_1}*{\mathbf{c}}} \right)} \right) - \left( {C\left( {{t_1}} \right)*\left( {{{\mathbf{w}}_2}*{\mathbf{c}}} \right)} \right)}}} \right). $$
(20)

From (20) it is clear that the value of r will be negative for a hospital with a contraction plan.

If we want to keep the total cost constant for a given duration in the future (from t1 to t2):
$$ C\left( {{t_{\text{1}}}} \right){\text{ }} = {\text{ }}C\left( {{t_{\text{2}}}} \right){\text{ }} = {\text{ }}C(t), $$
so
$$ r = \left( {\frac{{\left( {{{\mathbf{x}}_2}*{\mathbf{c}}} \right) - \left( {{{\mathbf{x}}_1}*{\mathbf{c}}} \right)}}{{\left( {{{\mathbf{w}}_1}*{\mathbf{c}}} \right) - \left( {{{\mathbf{w}}_2}*{\mathbf{c}}} \right)}}} \right){\text{ }}{\text{.}} $$
(21)
Similarly, if the bed availability is given for two dates in the future, B(t1) at time t1 and B(t2) at time t2, from (5) and (13) we obtain:
$$ {\eta_{{t_{\text{1}}}}} = \left( {{{\mathbf{x}}_1} + r*{{\mathbf{w}}_1}} \right)*\left( {{\mathbf{h}} + {\mathbf{e}}} \right){\text{, }} $$
(22)
$$ {\eta_{{t_{\text{2}}}}} = \left( {{{\mathbf{x}}_2} + r*{{\mathbf{w}}_2}} \right)*\left( {{\mathbf{h}} + {\mathbf{e}}} \right) $$
(23)
and the initial rates of admission should be the same. Using (15) we obtain:
$$ \frac{{B\left( {{t_1}} \right)}}{{{\eta_{{t_1}}}}} = \frac{{B\left( {{t_2}} \right)}}{{{\eta_{{t_2}}}}}, $$
therefore:
$$ r = \left( {\frac{{\left( {B\left( {{t_1}} \right)*\left( {{{\mathbf{x}}_2}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right)} \right) - \left( {B\left( {{t_2}} \right)*\left( {{{\mathbf{x}}_1}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right)} \right)}}{{\left( {B\left( {{t_2}} \right)*\left( {{{\mathbf{w}}_1}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right)} \right) - \left( {B\left( {{t_1}} \right)*\left( {{{\mathbf{w}}_2}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right)} \right)}}} \right). $$
(24)
If the bed availability remains constant for a given duration in future (from t1 to t2), then:
$$ B\left( {{t_{\text{1}}}} \right){\text{ }} = {\text{ }}B\left( {{t_{\text{2}}}} \right) = {\text{ }}B(t). $$
Hence:
$$ r = \left( {\frac{{\left( {{{\mathbf{x}}_2}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right) - \left( {{{\mathbf{x}}_1}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right)}}{{\left( {{{\mathbf{w}}_1}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right) - \left( {{{\mathbf{w}}_2}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)} \right)}}} \right){\text{ }}{\text{.}} $$
(25)

3.4.2 A pre-existing care system

For a pre-existing care system there are already some patients in various hospital and community phases of the care system at time t = 0. Therefore, one or more of the elements of s0 is non-zero. For such a care system first we should find the expected number of patients in each phase at time t = tgiven days without any new admissions. Then we find the distribution of the new patients. Therefore, the distribution of the patients without new admissions (from (4)) is:
$$ {{\mathbf{\tilde s}}_{{t_{\text{given}}}}} = {{\mathbf{\tilde s}}_0}*\prod\limits_{i = 1}^{{{\text{t}}_{\text{given}}}} {{{\rm P}^{(i)}}} {\text{, }} $$
(26)
$$ {\tilde \eta_{{t_{\text{given}}}}} = {{\mathbf{\tilde s}}_{{t_{\text{given}}}}}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right) = {{\mathbf{\tilde s}}_0}*\prod\limits_{i = 1}^{{t_{\text{given}}}} {{{\rm P}^{(i)}}} *\left( {{\mathbf{h}} + {\mathbf{e}}} \right) $$
(27)
and the distribution of newly admitted patients (from (5)) is:
$$ {\eta_{{t_{\text{given}}}}} = {{\mathbf{s}}_{{t_{\text{given}}}}}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right) $$
where \( {{\mathbf{s}}_{{t_{\text{given}}}}} \)can be calculated from (12) with
$$ {{\mathbf{s}}_0}{\text{ }} = \left\{ {{\text{1}},0,0,0, \ldots \ldots .,0, \ldots 0} \right\}. $$
Thus the expected number of admissions allowed is:
$$ {A_{\text{req}}} = \frac{{B\left( {{t_{\text{given}}}} \right) - {{\tilde \eta }_{{t_{\text{given}}}}}}}{{{\eta_{{t_{\text{given}}}}}}}{\text{ per day}}{\text{.}} $$
(28)
Similarly if the total daily cost of care is a constraint, the expected number of admissions is:
$$ {A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{C\left( {{t_{\text{given}}}} \right) - {{\tilde \Omega }_{{t_{\text{given}}}}}}}{{{\Omega_{{t_{\text{given}}}}}}}{\text{ per day}} $$
(29)
where \( {\tilde \Omega_{{t_{\text{given}}}}} = {{\mathbf{\tilde s}}_{{t_{\text{given}}}}}*{\mathbf{c}}. \)

We can use (28) and (29) to estimate the expected admission rate to satisfy the given bed availability constraints and the expected cost of care constraints respectively.

If a care system has variable number of admissions each day the initial rate of admission a is given then from Eq. 13, and the rate of change in admission rate is
$$ r = \frac{{\left( {{{\mathbf{s}}_{{t_{given}}}} - {\mathbf{x}}} \right)}}{{\mathbf{w}}}. $$
From (28)
$$ {{\mathbf{s}}_{{t_{given}}}} = \frac{{B\left( {{t_{\text{given}}}} \right) - {{\tilde \eta }_{{t_{\text{given}}}}}}}{{a*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)}}. $$
Therefore the rate of change in admission rate is
$$ r = \frac{{\left( {\frac{{B\left( {{t_{\text{given}}}} \right) - {{\tilde \eta }_{{t_{\text{given}}}}}}}{{a*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)}} - {\mathbf{x}}} \right)}}{{\mathbf{w}}} = \frac{{B\left( {{t_{\text{given}}}} \right) - {{\tilde \eta }_{{t_{\text{given}}}}} - a*{\mathbf{x}}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)}}{{a*{\mathbf{w}}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right)}}. $$
(30)
Similarly from (29) the rate of change in admission rate is
$$ r = \frac{{\left( {\frac{{C\left( {{t_{\text{given}}}} \right) - {{\tilde \Omega }_{{t_{\text{given}}}}}}}{{a*{\mathbf{c}}}} - {\mathbf{x}}} \right)}}{{\mathbf{w}}} = \frac{{C\left( {{t_{\text{given}}}} \right) - {{\tilde \Omega }_{{t_{\text{given}}}}} - a*{\mathbf{x}}*{\mathbf{c}}}}{{a*{\mathbf{w}}*{\mathbf{c}}}}. $$
(31)

4 An illustration

We illustrate the use of our model using the example of an 8-phase integrated care system proposed by Faddy and McClean [33] comprising of four hospital phases (acute, treatment, rehabilitative and long-stay), three community phases (dependent, convalescent and recovered) and one absorbing phase, death. Faddy and McClean [35] estimated parameters for the Coxian distribution fitting this 8-phase model to the dataset of all male patients admitted between 1969 and 1984 at the geriatric department of a London hospital [31]. In this paper [35] the authors used penalized maximum likelihood estimation with a penalty against multi-model distributions (penalty against distributions where the transition matrix Q had disparate eigenvalues) thus avoiding over-fitting which might underestimate the effects of time dependent covariates. They used the Nelder Mead algorithm [36] routine of MATLAB software [37] to perform the minimization. Figure 2 is the schematic representation of this care system.
https://static-content.springer.com/image/art%3A10.1007%2Fs10729-009-9120-0/MediaObjects/10729_2009_9120_Fig2_HTML.gif
Fig. 2

Patient flow in the eight phase care system

4.1 Daily resource requirement

We can estimate the daily resource requirements in a care unit or a care system if we know the expected number of patients in each phase in the care unit at a given time in future. To calculate the expected cost we have used the indicative cost for the 8-phase care system estimated by McClean & Millard [38] using relative weightings for each phase based on the then current data for geriatric patients in the UK. These relative weightings are 15, 10, 8 and 6 for hospital phases acute, treatment, rehabilitative and long-stay respectively and 5, 4 and 0.5 for community phases dependent, convalescent and recovered respectively [38]. Also McClean & Millard [38] estimated average daily cost of the care in acute phase as £150. Here we will consider 2 cases, one with a fixed number of admissions each day and the other with a variable number of admissions each day.

4.1.1 A fixed number of admissions

For a new care system [9] with one admission each day (fixed), the expected number of patients in different phases (after a given number of days) is presented in Table 1. Here the average age of patients at the time of admission is 80 years and the initial year of the care system is 1976. From Table 1, we can also calculate the expected number of patients in any phase or expected cost with any (fixed) number of admissions per day. If the number of admissions per day are a then we need to multiply the corresponding entry in Table 1 by a. For example, if in a new hospital, each day on average 100 new patients are admitted, then after 1,000 days, the expected numbers of patients in the hospital and community are estimated as 5051 and 39981 respectively and the expected total daily cost of care is estimated as £1,165,560. Similarly after 2,000 days, the expected numbers of patients in the hospital and community are estimated as 6420 and 57785 respectively and the expected total daily cost of care is estimated as £1,253,450.
Table 1

Estimated number of patients and cost (new system with 1 admission/ day)

Days

Phase1

Phase2

Phase3

Phase4

Phase5

Phase6

Phase7

Phase8

Total Daily Cost(In £)

100

11.07

8.95

10.16

0.27

11.16

31.65

2.85

23.89

5,223.0

200

11.97

9.67

14.29

0.88

12.50

70.13

16.28

64.27

7,471.3

300

12.60

10.16

15.94

1.50

12.95

95.47

38.94

112.44

8,931.8

400

13.04

10.50

16.94

2.06

13.10

110.38

68.06

165.92

9,895.4

500

13.36

10.74

17.70

2.55

13.09

118.00

101.21

223.35

10,527.1

600

13.60

10.93

18.32

2.97

12.99

120.78

136.51

283.90

10,940.3

700

13.81

11.08

18.87

3.34

12.84

120.51

172.51

347.04

11,214.3

800

13.99

11.22

19.39

3.66

12.66

118.41

208.18

412.47

11,403.4

900

14.18

11.36

19.91

3.94

12.49

115.29

242.78

480.04

11,543.0

1,000

14.38

11.51

20.43

4.19

12.32

111.66

275.83

549.68

11,655.6

1,100

14.59

11.66

20.98

4.42

12.15

107.82

307.00

621.37

11,754.1

1,200

14.82

11.83

21.55

4.62

12.00

103.93

336.12

695.12

11,846.0

1,300

15.07

12.01

22.16

4.80

11.86

100.08

363.08

770.93

11,934.9

1,400

15.33

12.21

22.80

4.99

11.74

96.32

387.82

848.80

12,022.4

1,500

15.61

12.41

23.47

5.16

11.62

92.64

410.34

928.74

12,109.1

1,600

15.91

12.63

24.18

5.32

11.51

89.07

430.66

1,010.71

12,195.2

1,700

16.22

12.86

24.92

5.49

11.41

85.59

448.82

1,094.68

12,280.8

1,800

16.55

13.10

25.70

5.66

11.31

82.21

464.88

1,180.59

12,365.8

1,900

16.89

13.35

26.51

5.82

11.22

78.92

478.91

1,268.38

12,450.3

2,000

17.24

13.61

27.36

5.99

11.14

75.72

490.99

1,357.96

12,534.5

4.1.2 A variable number of admissions

Fiems et al. [13] describe how we can calculate the expected number of patients in different phases, after a given number of days, with a variable rate of admissions per day. Again from (13) we here obtain:
$$ {{\mathbf{s}}_k} = {\mathbf{x}}{\text{ }} + r*{\mathbf{w}} $$
where the vector x represents the expected number of patients in different phases, after a given number of days, for a new care system with one admission per day and r is the rate of change in the admission rate. The vector w is the expected change in the patients’ distribution due to unit (1 patient per day) change in admission rate. However, it is not necessary to start the admission schedule with one admission per day as we can have multiple admissions per day. Therefore, we can generalize (13) to:
$$ {{\mathbf{s}}_k} = a*\left( {{\mathbf{x}}{\text{ }} + r*{\mathbf{w}}} \right) $$
(32)
where a represents the initial number of admissions per day.
Table 2 presents the value of the vector w for a number of time points in the future. As a and r are scalars, we can calculate the estimated number of patients in each phase as a linear combination of x and w and similarly we can calculate the cost. For example, with initial rate of admission 100 patients per day and this rate increasing by 1 patient per day (r = 0.01), the expected total daily cost of care after 500 days is estimated as £2,593,937 (100*(10527.1 + 0.01*1541227)) and after 1,000 days it is estimated as £3,898,469 (100*(11655.6 + 0.01*2732909)).
Table 2

Estimated values of change multipliers for different phases (new system with 1 admission/ day)

Days

Phase1

Phase2

Phase3

phase4

phase5

Phase6

phase7

Phase8

Total daily cost (in £)

100

167.3

210.4

516.6

18.4

417.1

1,975.9

210.2

1,434.1

189,509

200

310.4

321.8

1,084.7

110.1

636.1

7,582.6

2,262.6

7,591.6

518,548

300

484.2

455.4

1,463.7

263.2

817.0

13,819.4

7,770.0

19,777.1

883,538

400

665.4

594.2

1,777.9

455.7

978.6

19,163.7

17,421.6

38,742.9

1,231,388

500

845.8

731.4

2,069.0

671.6

1,123.0

23,077.0

31,232.7

64,999.4

1,541,227

600

1,026.9

868.2

2,351.0

899.2

1,254.4

25,572.0

48,793.5

98,934.8

1,812,459

700

1,215.0

1,009.1

2,634.8

1,130.1

1,379.7

26,921.5

69,466.6

140,893.1

2,054,930

800

1,417.2

1,159.9

2,932.3

1,359.2

1,505.8

27,477.1

92,528.9

191,219.5

2,281,731

900

1,640.1

1,325.7

3,255.0

1,583.3

1,638.4

27,564.7

117,264.4

250,278.3

2,504,815

1,000

1,888.2

1,510.2

3,612.6

1,801.3

1,780.8

27,436.4

143,015.9

318,454.7

2,732,909

1,100

2,164.2

1,715.4

4,011.9

2,013.0

1,934.6

27,259.1

169,206.2

396,145.7

2,971,016

1,200

2,468.7

1,941.9

4,457.2

2,219.2

2,099.1

27,124.0

195,340.7

483,749.1

3,220,849

1,300

2,800.9

2,189.4

4,950.4

2,421.2

2,272.9

27,066.0

220,998.7

581,650.5

3,481,661

1,400

3,159.0

2,456.4

5,491.5

2,620.1

2,453.7

27,083.2

245,822.4

690,213.8

3,751,133

1,500

3,540.3

2,741.0

6,079.0

2,817.4

2,638.7

27,153.9

269,506.6

809,773.2

4,026,123

1,600

3,941.8

3,041.0

6,710.4

3,014.2

2,825.2

27,248.3

291,791.6

940,627.5

4,303,206

1,700

4,360.3

3,353.9

7,382.8

3,211.8

3,010.6

27,336.0

312,458.2

1,083,036.5

4,579,022

1,800

4,792.2

3,677.1

8,092.5

3,411.0

3,192.4

27,390.4

331,326.2

1,237,218.1

4,850,474

1,900

5,234.3

4,008.0

8,835.7

3,612.7

3,368.4

27,389.5

348,253.7

1,403,347.7

5,114,825

2,000

5,682.8

4,343.9

9,608.4

3,817.4

3,536.7

27,317.5

363,136.8

1,581,556.5

5,369,748

4.2 Resource constraints

4.2.1 A new care system

If resource availability at a given time in the future is a constraint it is necessary to schedule admissions in advance such that at a given time in the future the number of patients requiring care resources should be equal to the amount of resources available. In Table 3, we have provided a few examples of resource constraints: the total daily cost limit and the number of beds available in the hospital (summing all beds available in all hospital phases acute, treatment, rehabilitative and long-stay) at a given time in the future. We have also calculated the expected number of admissions per day (assuming a fixed number of admissions each day) allowed to meet these constraints.
Table 3

Estimated number of admissions per day according to budget or beds availability

Day

Estimating the number of admissions per day if the constraint is the daily cost

Estimating the number of admissions per day if the constraint is the beds availability

Daily cost limit (in £)

Number of beds available in the hospital

Average admissions per day

Number of beds available in the hospital

Daily cost (in £)

Average admissions per day

500

800,000

3,370

76

6,000

628,690

135

1,000

400,000

1,734

34

8,000

822,597

158

1,500

500,000

2,340

41

4,500

458,330

79

2,000

750,000

3,841

60

1,700

171,991

26

With a variable number of admissions per day we can have a number of different expected admission schedules to satisfy the same constraints depending on the expected initial rate of admissions (a) and expected rate of change in admission rate (r). For such a care system, usually these decisions are made concerning the resource availability at some time points in the future. We can use (15)–(17) to calculate the expected initial rate of admissions to satisfy the resource constraint for the first given time point in the future. By using (20), (21), (24) and (25) we can then calculate the expected rate of change in admission rate (r) to satisfy constraints relating to two given time points. In Table 4 we have provided some examples of resource constraints for total daily cost constrained at two time points in the future. We have calculated the expected value of r to satisfy these constraints.
Table 4

Estimated rate of change in admission rate (r) and estimated number of admissions per day (a) to satisfy given budget constraints at two time points in the future

t1 in days

C(t1) in £

t2 in days

C(t2) in £

Areq = a (Estimated initial admission rate)

r (estimated rate of change in admission rate)

500

800,000

1,000

800,000

88.23

−0.00095

500

800,000

1,000

120,240

30.83

0.01

1,000

400,000

1,500

500,000

17.64

0.04

1,000

500,000

1,200

400,000

100

−0.0024354

1,500

500,000

2,000

610,000

15.73

0.049

1,600

200,000

2,000

200,000

18.47

−0.0032

4.2.2 A pre-existing care system

For a pre-existing care system, first we allocate the care resources to the expected remaining patients from the patients already in the system at time t = 0 (or today/ present time). Then we allocate the remaining resources to the expected new patients in the same manner as in the case of a new care system discussed above. Using (4), we can compute the expected number of remaining patients in different phases at a given time in the future (tgiven) from patients already in the system at time t = 0. Daily care cost for such patients can be computed using (8). Again from (4) we obtain:
$$ {{\mathbf{s}}_{{t_{given}}}} = {{\mathbf{s}}_0}*\prod\limits_{i = 1}^{{t_{given}}} {{{\rm P}^{(i)}}} = {{\mathbf{s}}_0}*{{\mathbf{H}}_{{t_{given}}}} $$
(33)
where
$$ {{\mathbf{H}}_{{t_{given}}}} = \prod\limits_{i = 1}^{{t_{given}}} {{{\rm P}^{(i)}}} $$
(34)
and \( {{\mathbf{H}}_{{t_{given}}}} = \left\{ {{h_{ij}}} \right\} \)is the matrix of the expected number of patients in phase j at time tgiven if there is one patient in phase i at time t = 0. A typical value of \( {{\mathbf{H}}_{{t_{given}}}} \)for tgiven = 1,000 days is:
$$ {{\mathbf{H}}_{{t_{given}} = 1000}} = \left\{ {\begin{array}{*{20}{c}} {{\text{0}}{\text{.0022}}}{{\text{0}}{\text{.0017}}}{{\text{0}}{\text{.0029}}}{{\text{0}}{\text{.0020}}}{{\text{0}}{\text{.0017}}}{{\text{0}}{\text{.0182}}}{{\text{0}}{\text{.2253}}}{{\text{0}}{\text{.7460}}} \\ {{\text{0}}{\text{.0024}}}{{\text{0}}{\text{.0018}}}{{\text{0}}{\text{.0032}}}{{\text{0}}{\text{.0024}}}{{\text{0}}{\text{.0019}}}{{\text{0}}{\text{.0197}}}{{\text{0}}{\text{.2437}}}{{\text{0}}{\text{.7251}}} \\ {{\text{0}}{\text{.0018}}}{{\text{0}}{\text{.0013}}}{{\text{0}}{\text{.0023}}}{{\text{0}}{\text{.0050}}}{{\text{0}}{\text{.0014}}}{{\text{0}}{\text{.0149}}}{{\text{0}}{\text{.1789}}}{{\text{0}}{\text{.7943}}} \\ {{\text{0}}{\text{.0006}}}{{\text{0}}{\text{.0005}}}{{\text{0}}{\text{.0008}}}{{\text{0}}{\text{.1107}}}{{\text{0}}{\text{.0009}}}{{\text{0}}{\text{.0095}}}{{\text{0}}{\text{.0430}}}{{\text{0}}{\text{.8341}}} \\ {{\text{0}}{\text{.0030}}}{{\text{0}}{\text{.0023}}}{{\text{0}}{\text{.0040}}}{{\text{0}}{\text{.0014}}}{{\text{0}}{\text{.0023}}}{{\text{0}}{\text{.0245}}}{{\text{0}}{\text{.3079}}}{{\text{0}}{\text{.6547}}} \\ {{\text{0}}{\text{.0032}}}{{\text{0}}{\text{.0024}}}{{\text{0}}{\text{.0043}}}{{\text{0}}{\text{.0013}}}{{\text{0}}{\text{.0025}}}{{\text{0}}{\text{.0262}}}{{\text{0}}{\text{.3319}}}{{\text{0}}{\text{.6282}}} \\ {{\text{0}}{\text{.0044}}}{{\text{0}}{\text{.0033}}}{{\text{0}}{\text{.0058}}}{{\text{0}}{\text{.0012}}}{{\text{0}}{\text{.0034}}}{{\text{0}}{\text{.0324}}}{{\text{0}}{\text{.4664}}}{{\text{0}}{\text{.4831}}} \\ {\text{0}}{\text{0}}{\text{0}}{\text{0}}{\text{0}}{\text{0}}{\text{0}}{{\text{1}}{\text{.0000}}} \\ \end{array} } \right\} $$
and a typical value of \( {{\mathbf{H}}_{{t_{given}}}} \) for tgiven = 2,000 days is:
$$ {{\mathbf{H}}_{{t_{given}} = 2000}} = \left\{ {\begin{array}{*{20}{c}} {{\text{0}}{\text{.0013}}}{{\text{0}}{\text{.0009}}}{{\text{0}}{\text{.0018}}}{{\text{0}}{\text{.0006}}}{{\text{0}}{\text{.0007}}}{{\text{0}}{\text{.0050}}}{{\text{0}}{\text{.0829}}}{{\text{0}}{\text{.9069}}} \\ {{\text{0}}{\text{.0014}}}{{\text{0}}{\text{.0010}}}{{\text{0}}{\text{.0019}}}{{\text{0}}{\text{.0006}}}{{\text{0}}{\text{.0008}}}{{\text{0}}{\text{.0054}}}{{\text{0}}{\text{.0897}}}{{\text{0}}{\text{.8992}}} \\ {{\text{0}}{\text{.0010}}}{{\text{0}}{\text{.0008}}}{{\text{0}}{\text{.0014}}}{{\text{0}}{\text{.0008}}}{{\text{0}}{\text{.0006}}}{{\text{0}}{\text{.0040}}}{{\text{0}}{\text{.0662}}}{{\text{0}}{\text{.9253}}} \\ {{\text{0}}{\text{.0004}}}{{\text{0}}{\text{.0003}}}{{\text{0}}{\text{.0005}}}{{\text{0}}{\text{.0123}}}{{\text{0}}{\text{.0003}}}{{\text{0}}{\text{.0018}}}{{\text{0}}{\text{.0242}}}{{\text{0}}{\text{.9603}}} \\ {{\text{0}}{\text{.0017}}}{{\text{0}}{\text{.0013}}}{{\text{0}}{\text{.0024}}}{{\text{0}}{\text{.0006}}}{{\text{0}}{\text{.0010}}}{{\text{0}}{\text{.0068}}}{{\text{0}}{\text{.1131}}}{{\text{0}}{\text{.8731}}} \\ {{\text{0}}{\text{.0018}}}{{\text{0}}{\text{.0014}}}{{\text{0}}{\text{.0026}}}{{\text{0}}{\text{.0006}}}{{\text{0}}{\text{.0010}}}{{\text{0}}{\text{.0073}}}{{\text{0}}{\text{.1219}}}{{\text{0}}{\text{.8633}}} \\ {{\text{0}}{\text{.0026}}}{{\text{0}}{\text{.0019}}}{{\text{0}}{\text{.0037}}}{{\text{0}}{\text{.0008}}}{{\text{0}}{\text{.0014}}}{{\text{0}}{\text{.0102}}}{{\text{0}}{\text{.1699}}}{{\text{0}}{\text{.8096}}} \\ {\text{0}}{\text{0}}{\text{0}}{\text{0}}{\text{0}}{\text{0}}{\text{0}}{{\text{1}}{\text{.0000}}} \\ \end{array} } \right\} $$

We can estimate the expected resource requirements for such a care system by using (33) and (34). For example if there are no new admissions to a care system with initially only 500 patients in the acute phase (and no patient in any other phase) then the expected number of patients left in care system after 2,000 days will be 0.65, 0.45, 0.9, 0.3, 0.35, 0.25 and 41.45 patients respectively in the acute phase, treatment phase, rehabilitative phase, long-stay phase, dependent phase, convalescent phase and recovered phase.

We can use (28) and (29) to estimate the expected admission rate to satisfy the given bed availability constraints and the expected cost of care constraints respectively. For example if a care system with initially 100 patients in each phase has an expansion plan to double the expected total daily cost of care after 1,000 days, then
$$ {{\mathbf{\tilde s}}_{\text{0}}}{\text{ = }}\left\{ {\begin{array}{*{20}{c}} {\text{100}} & {\text{100}} & {\text{100}} & {\begin{array}{*{20}{c}} {\text{100}} & {\text{100}} & {\text{100}} & {\begin{array}{*{20}{c}} {\text{100}} & {\text{0}} \\ \end{array} } \\ \end{array} } \\ \end{array} } \right\} $$
and
$$ {\tilde \Omega_{\text{0}}}{\text{ }} = \pounds{\text{48500}}{\text{.}} $$
The constraint is
$$ {\text{C}}\left( {{t_{given}}{\text{ = 1000}}} \right){\text{ = }}2*{\widetilde\Omega_{\text{0}}} = \pounds97000. $$
From (33) and (34) we obtain\( {\widetilde{\mathbf{s}}_{{t_{given}}{\text{ = 1000}}}}{\text{ = }}{\widetilde{\mathbf{s}}_{\text{0}}}*{{\mathbf{H}}_{{t_{given}} = 1000}} = \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {1.76} & {1.33} \\ \end{array} } & {2.33} & {12.4} & {1.41} \\ \end{array} } & {14.54} & {179.71} & {486.55} \\ \end{array} } \right\}\,{\text{and}}\,{\widetilde\Omega_{{t_{given}} = 1000}} = \pounds{\text{2878}}{\text{.05 }}{\text{.}} \)From (29), the mean rate of admissions allowed (with a fixed number of admissions each day) will be
$$ {A_{\text{req}}} = \frac{{C\left( {{t_{\text{given}}} = 1000} \right) - {{\tilde \Omega }_{{t_{\text{given}}} = 1000}}}}{{{\Omega_{{t_{\text{given}}} = 1000}}}}{\text{ per day = }}\frac{{97000 - 2878.05}}{{{\Omega_{{t_{\text{given}}} = 1000}}}}. $$

From Table 1, \( {\Omega_{{t_{\text{given}}}}}_{ = 1000} = \pounds11655.60 \)

Therefore, Areq = 8.0753 patients/day.

For a care system with variable number of admissions each day, if the initial rate of admission (a) is 10 patients/day, then using (31) the rate of change in admission rate (r) can be estimated as follows:
$$ r = \frac{{C\left( {{t_{\text{given}}} = 1000} \right) - {{\tilde \Omega }_{{t_{{\text{given = 1000}}}}}} - a*\left( {{\mathbf{x}}*{\mathbf{c}}} \right)}}{{a*\left( {{\mathbf{w}}*{\mathbf{c}}} \right)}}. $$
Putting the values of \( \left( {{\mathbf{x}}*{\mathbf{c}}} \right) \) and \( \left( {{\mathbf{w}}*{\mathbf{c}}} \right) \) from the Tables 1 and 2 respectively we get
$$ r = \frac{{97000 - 2878.05 - \left( {10*{\text{11655}}{\text{.6}}} \right)}}{{10*{\text{2732909}}}} = {\text{ - 0}}{\text{.000821}}. $$
If the same care system has a contraction plan to half the expected total daily cost of care after 1,000 days, then the constraint is
$$ {\text{C}}\left( {{t_{given}}{\text{ = 1000}}} \right){\text{ = }}\frac{{{{\widetilde\Omega }_{\text{0}}}}}{2} = \pounds24250. $$
From (29), the mean rate of admissions allowed (with a fixed number of admissions each day) will be
$$ {A_{\text{req}}} = \frac{{C\left( {{t_{\text{given}}} = 1000} \right) - {{\widetilde\Omega }_{{t_{\text{given}}} = 1000}}}}{{{\Omega_{{t_{\text{given}}} = 1000}}}}{\text{ per day = }}\frac{{24250 - 2878.05}}{{{\Omega_{{t_{\text{given}}} = 1000}}}}. $$

From Table 1, \( \left( {{\mathbf{x}}*{\mathbf{c}}} \right)\,{\Omega_{{t_{\text{given}}} = 1000}} = \pounds11655.60 \)

Therefore, Areq = 1.8336 patients/day.

For a care system with variable number of admissions each day, if the initial (or present) rate of admission (a) is 10 patients/day, then using (31) the rate of change in admission rate (r) can be estimated as follows:
$$ r = \frac{{C\left( {{t_{\text{given}}} = 1000} \right) - {{\tilde \Omega }_{{t_{{\text{given = 1000}}}}}} - a*\left( {{\mathbf{x}}*{\mathbf{c}}} \right)}}{{a*\left( {{\mathbf{w}}*{\mathbf{c}}} \right)}}. $$
Putting the values of and \( \left( {{\mathbf{w}}*{\mathbf{c}}} \right) \) from Tables 1 and 2 respectively we get
$$ r = \frac{{24250 - 2878.05 - \left( {10*{\text{11655}}{\text{.6}}} \right)}}{{10*{\text{2732909}}}} = {\text{ - 0}}{\text{.003483}}. $$

If the bed availability is a constraint, then we can use (28) to estimate the mean rate of admissions allowed for a care system with a fixed number of admissions each day and (30) to estimate the rate of change in admission rate (r) for a care system with variable number of admissions each day.

5 Discussion

In our model, we assume that there is always a waiting list of patients who can be admitted to the first phase of the care system whenever there is a bed available. This assumption is realistic in many practical situations. However, as the transition probabilities are estimated separately for each patient in the care system, the model can be modified for resource requirements planning in other scenarios where patients arrive arbitrarily and admissions can be modeled as a Poisson stream. Another limitation of our model is that it is based largely on expected values. The model can therefore be used to predict the expected number of patients in each phase and expected daily cost of care. However, these expected numbers represent long term averages, but do not reflect variability. We are currently working on enhancing our model to estimate the variability of these averages and to examine its effects. Such variability and its effects can also be estimated using a stochastic simulation models (such as a discrete event simulation).

An obvious application of our model is to facilitate decision making of health and social care planners, managers and policy makers for resource planning and budget allocation, especially for the elderly care. However, this model can be customized for other care systems as well. Our model equips care planners, managers and policy makers with a tool for resource planning, budget allocation and strategic decision making for care management and service improvement. In a community based setting, they can use our model with demographic data and regional health statistics data, to allocate budget for health and social care for the population of their region. Bed availability, optimum utilization of the allocated budget, minimization of waiting time and the size of patient queues can thus be assured Individual resource requirements for hospital care and community care can also be estimated. Another application of the tool is in health insurance where an insurance company might use the model to forecast resource requirements and then decide the optimum premium for the patient’s health insurance policy.

Our model can also be used for better informed budget allocation and resource planning decisions evaluating the combined cost measure of the dead-alive trade off, quality of life and economic costs [39]. This can be achieved by using our model coupled with a partially observable Markov decision process based on, for example, the patient management model proposed by [39] to select the care pathway which optimizes the combined cost for the integrated care system. This will also facilitate cost effectiveness analysis (CEA) [4042] and cost-utility analysis (CUA) [40, 43] of different admission scheduling policies and resource management strategies. For example we might decide if it is cost-effective to discharge (from a hospital phase) a patient to a care home or nursing home or his/her own home, with a care package such as is described in [44]. We can estimate the resource availability and schedule the admissions accordingly. Quality of life can be measured in terms of quality adjusted life years gained (QALY) [45], disability adjusted life years (DALYs) [45], life years gained (LYs gained) [45] or Sen’s capability approach [46, 47]. It can also be measured for individual patients in the care system facilitating combined cost effectiveness analysis for the whole the system as in the model the transition probabilities are estimated separately for each patient.

To understand the effect of change in the various input parameters we carried out the sensitivity analysis of various input parameters (see Appendix 1 for the mathematical details). The results of this analysis can be summarized as follows:
  • The mean number of patients in each phase on a given day in the future increases exponentially with the change in a transition rate (a parameter of transition matrix Q) (see Eqs. 35 and 36, Appendix 1).

  • The mean number of patients in each phase on a given day in the future increases linearly with the change in the initial (or fixed) admission rate (see Eq. 37, Appendix 1).

  • The mean number of patients in each phase on a given day in the future increases linearly with the change in the rate of change in the admission rate (see Eq. 38, Appendix 1).

  • Changes in the expected total daily cost on a given day in the future increases linearly with changes in daily cost of care in different phases (and similarly with the change in the relative weighting for a phase for indicative cost) (see Eq. 39, Appendix 1).

  • If the beds availability on a given day in the future is a constraint, then the expected number of admissions allowed each day decreases exponentially with the change in a transition rate (see Eqs. 40 and 41, Appendix 1).

  • If the daily cost of care on a given day in the future is a constraint, then the expected number of admissions allowed each day decreases approximately linearly with changes in daily cost of care in different phases (and similarly with the change in the relative weighting for a phase for indicative cost) (see Eqs. 42 and 43, Appendix 1).

Our model is very efficient in terms of algorithm complexity. Implementation of the algorithm discussed in the last section for resource requirements forecasting and/or admission scheduling would require matrix multiplications for each new admission and for each day throughout the duration we wish to forecast the resource requirement. Also the number of admissions is equal to the time horizon of the resource requirement. Therefore the time complexity of the algorithm is O(k2) where k is the number of days ahead for which the resource requirement requires to be forecasted. Also, we require to store the previous value of matrix sk for each iteration. Therefore the space complexity of the algorithm is O(n).

6 Conclusion

We have demonstrated how a discrete time non-homogeneous Markov models can be effectively used in more sophisticated admission scheduling and resource requirement forecasting and allocation. We allow both fixed and variable rate of admissions to satisfy the demand or resource constraints. This can be a very useful tool for care managers and policy makers so as to facilitate strategic decision making for care management and service improvement. We are currently working to develop our approach as a decision (what-if analysis) tool.

Acknowledgements

The authors acknowledge support for this work from the Engineering and Physical Sciences Research Council funded RIGHT and MATCH projects (Grant References EP/E019900/1 and GR/S29874/01). Any views or opinions presented herein are those of the authors and do not necessarily represent those of RIGHT or MATCH, their associates or their sponsors.

Copyright information

© Springer Science+Business Media, LLC 2009