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

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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.

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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.

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Correspondence to Lalit Garg.

Appendices

Appendix 1: Equations for sensitivity analysis

1.1 The mean number of patients in each phase after k days

1.1.1 The change in the transition rates

From Eq. 4, the mean number of patients in each phase after k days is given by:

$$ {{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\prod\limits_{i = 1}^k {{{\rm P}^{(i)}}} $$

where P (i) represents the value of transition matrix P on day i which is defined in Eq. 2 as follows

$$ {{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\prod\limits_{i = 1}^k {\left( {\exp \left( {{{\mathbf{Q}}^{(i)}}} \right)} \right)} $$

Therefore,

$$ \begin{array}{*{20}{c}} { \Rightarrow {{\mathbf{s}}_k} + {\Delta_{\mathbf{Q}}}{{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\prod\limits_{i = 1}^k {\left( {\exp \left( {{{\mathbf{Q}}^{(i)}} + \Delta {{\mathbf{Q}}^{(i)}}} \right)} \right)} = \left. {{{\mathbf{s}}_0}*\prod\limits_{i = 1}^k {(\exp \left( {{{\mathbf{Q}}^{(i)}}} \right)*\exp \left( {\Delta {{\mathbf{Q}}^{(i)}}} \right)} } \right)} \hfill \\ { \Rightarrow {{\mathbf{s}}_k} + {\Delta_{\mathbf{Q}}}{{\mathbf{s}}_k} = {{\mathbf{s}}_0}*\prod\limits_{i = 1}^k {\left( {\exp \left( {{{\mathbf{Q}}^{(i)}}} \right)} \right)} *\prod\limits_{i = 1}^k {\left( {\exp \left( {\Delta {{\mathbf{Q}}^{(i)}}} \right)} \right)} } \hfill \\ { \Rightarrow {{\mathbf{s}}_k} + {\Delta_{\mathbf{Q}}}{{\mathbf{s}}_k} = {{\mathbf{s}}_k}*\prod\limits_{i = 1}^k {\left( {\exp \left( {\Delta {{\mathbf{Q}}^{(i)}}} \right)} \right)} .} \hfill \\ \end{array} $$
(35)

Here ∆Q represents the change in the transition matrix Q defined by Eq. 1. There might be equal changes in all transition rates which represents a global shift in transition rates, or there might be just change in one transition rate such as a change in the value of μ l . If the latter is the case then,

$$ \exp \left( {\Delta {\mathbf{Q}}} \right) = \exp \left( {\Delta {\mu_l}} \right) $$

Therefore

$$ {{\mathbf{s}}_k} + {\Delta_{\mathbf{Q}}}{{\mathbf{s}}_k} = {{\mathbf{s}}_k}*\prod\limits_{i = 1}^k {\left( {\exp \left( {\Delta {\mu_l}^{(i)}} \right)} \right)} . $$
(36)

1.1.2 The change in the initial number of admissions per day

If the initial number of admissions per day is a with a variable rate of admissions per day, then from Eq. 30, the expected number of patients in different phases after a given number of days,

$$ \begin{array}{*{20}{c}} {{{\mathbf{s}}_k} = a*\left( {{\mathbf{x}}{\text{ }} + r*{\mathbf{w}}} \right)} \hfill \\ {{{\mathbf{s}}_k} + {\Delta_a}{{\mathbf{s}}_k} = \left( {a + \Delta a} \right)*\left( {{\mathbf{x}}{\text{ }} + r*{\mathbf{w}}} \right)} \hfill \\ {{{\mathbf{s}}_k} + {\Delta_a}{{\mathbf{s}}_k} = {{\mathbf{s}}_k} + \Delta a*\left( {x + r*{\mathbf{w}}} \right)} \hfill \\ {\frac{{{\Delta_a}{{\mathbf{s}}_k}}}{{{{\mathbf{s}}_k}}} = \frac{{\Delta a}}{a}} \hfill \\ \end{array} $$
(37)

1.1.3 The change in the rate of change in the admission rate

Again from Eq. 30, for calculating the sensitivity of r the rate of change in the admission rate.

$$ \begin{array}{*{20}{c}} {{{\mathbf{s}}_k} = a*\left( {{\mathbf{x}}{\text{ }} + r*{\mathbf{w}}} \right)} \hfill \\ {{{\mathbf{s}}_k} + {\Delta_r}{{\mathbf{s}}_k} = a*\left( {{\mathbf{x}}{\text{ }} + \left( {r + \Delta r} \right)*{\mathbf{w}}} \right)} \hfill \\ {{{\mathbf{s}}_k} + {\Delta_r}{{\mathbf{s}}_k} = a*\left( {{\mathbf{x}}{\text{ }} + \left( {r + \Delta r} \right)*{\mathbf{w}}} \right) = a*\left( {{\mathbf{x}}{\text{ }} + r*{\mathbf{w}}} \right) + a*\Delta r*{\mathbf{w}}} \hfill \\ {{{\mathbf{s}}_k} + {\Delta_r}{{\mathbf{s}}_k} = {{\mathbf{s}}_k} + a*\Delta r*{\mathbf{w}}} \hfill \\ {\frac{{{\Delta_r}{{\mathbf{s}}_k}}}{{{{\mathbf{s}}_k}}} = \frac{{\Delta r*{\mathbf{w}}}}{{\left( {{\mathbf{x}}{\text{ }} + r*{\mathbf{w}}} \right)}}} \hfill \\ \end{array} $$
(38)

1.2 The expected total daily cost on day k

1.2.1 The change in cost matrix

From Eq. 8, the expected total daily cost on day k is given as follows

$$ \begin{array}{*{20}{c}} {{\Omega_k} = {{\mathbf{s}}_k}*{\mathbf{c}}{\text{ }}} \hfill \\ { \Rightarrow {\Omega_k} + {\Delta_{\mathbf{c}}}{\Omega_k} = {{\mathbf{s}}_k}*\left( {{\mathbf{c}} + \Delta {\mathbf{c}}} \right){\text{ }}} \hfill \\ { \Rightarrow {\Omega_k} + {\Delta_{\mathbf{c}}}{\Omega_k}{\text{ = }}{\Omega_k} + {{\mathbf{s}}_k}*\Delta {\mathbf{c}}} \hfill \\ { \Rightarrow \frac{{{\Delta_{\mathbf{c}}}{\Omega_k}}}{{{\Omega_k}}}{\text{ = }}\frac{{\Delta {\mathbf{c}}}}{{\mathbf{c}}}.} \hfill \\ \end{array} $$
(39)

Here ∆c represents the change in cost matrix c defined in Section 3.1. Section 4.1 defines c in the form of relative weightings. In such a case ∆c represents the change in one or more weightings.

1.3 The expected number of admissions allowed each day

1.3.1 The change in the transition rates

From Eq. 15 the expected number of admissions allowed each day can be calculated as:

$$ {A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{B\left( {{t_{\text{given}}}} \right)}}{{{\eta_{{t_{\text{given}}}}}}}{\text{ per day}} $$

where \( {\eta_{{t_{\text{given}}}}} \)is defined in the Eq. 5

$$ {\eta_{{t_{\text{given}}}}} = \sum\limits_{i = 1}^{n + m} {{s_{{t_{\text{given}}},i}}} = {\text{ }}{{\mathbf{s}}_{{t_{\text{given}}}}}*\left( {{\mathbf{h}} + {\mathbf{e}}} \right){\text{ }} $$

Therefore

$$ \begin{array}{*{20}{c}} {{A_{\text{req}}}{\text{ + }}{\Delta_{\mathbf{Q}}}{A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{B\left( {{t_{\text{given}}}} \right)}}{{{\eta_{{t_{\text{given}}}}}*\prod\limits_{i = 1}^{{t_{\text{given}}}} {\left( {\exp \left( {\Delta {{\mathbf{Q}}^{(i)}}} \right)} \right)} }}{\text{ per day}}} \hfill \\ { \Rightarrow {A_{\text{req}}}{\text{ + }}{\Delta_{\mathbf{Q}}}{A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{{A_{\text{req}}}}}{{\prod\limits_{i = 1}^{{t_{\text{given}}}} {\left( {\exp \left( {\Delta {{\mathbf{Q}}^{(i)}}} \right)} \right)} }}{\text{ per day}}} \hfill \\ \end{array} $$
(40)

If \( \Delta {\mathbf{Q}} \) represents the change in only one transition rate such as μ l then

$$ {A_{\text{req}}}{\text{ + }}{\Delta_{\mathbf{Q}}}{A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{{A_{\text{req}}}}}{{\prod\limits_{i = 1}^{{t_{\text{given}}}} {\left( {\exp \left( {\Delta {\mu_l}^{(i)}} \right)} \right)} }}{\text{ per day}} $$
(41)

1.3.2 The change in cost matrix

If the total daily cost of care is a constraint then the expected number of admissions is calculated in Eq. 17:

$$ {A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{C\left( {{t_{\text{given}}}} \right)}}{{{\Omega_{{t_{\text{given}}}}}}}{\text{ per day}} $$

\( {\Omega_{{t_{\text{given}}}}} \)is the expected total daily cost at time t given and is defined in (8). Therefore

$$ \begin{array}{*{20}{c}} {{A_{\text{req}}}{\text{ + }}{\Delta_{\mathbf{c}}}{A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{C\left( {{t_{\text{given}}}} \right)}}{{{\Omega_{{t_{\text{given}}}}} + {{\mathbf{s}}_{{t_{\text{given}}}}}*\Delta {\mathbf{c}}}}{\text{ per day}}} \hfill \\ {{A_{\text{req}}}{\text{ + }}{\Delta_{\mathbf{c}}}{A_{\text{req}}}{\text{ }} = {\text{ }}\frac{{{A_{\text{req}}}}}{{1 + \left( {\Delta {\mathbf{c}}/{\mathbf{c}}} \right)}}{\text{ per day}}} \hfill \\ {\frac{{{\Delta_{\mathbf{c}}}{A_{\text{req}}}}}{{{A_{\text{req}}}}} = \frac{{ - \Delta {\mathbf{c}}/{\mathbf{c}}}}{{1 + \left( {\Delta {\mathbf{c}}/{\mathbf{c}}} \right)}}{\text{ per day }} = \frac{{ - 1}}{{1 + \left( {{\mathbf{c}}/\Delta {\mathbf{c}}} \right)}}{\text{ per day }}} \hfill \\ \end{array} $$
(42)

If ∆c < <c , then

$$ \frac{{{\Delta_{\mathbf{c}}}{A_{\text{req}}}}}{{{A_{\text{req}}}}} \approx \frac{{ - \Delta {\mathbf{c}}}}{{\mathbf{c}}}{\text{ per day}} $$
(43)

Appendix 2

Table 5 Terms/ parameters used in the paper

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Garg, L., McClean, S., Meenan, B. et al. A non-homogeneous discrete time Markov model for admission scheduling and resource planning in a cost or capacity constrained healthcare system. Health Care Manag Sci 13, 155–169 (2010). https://doi.org/10.1007/s10729-009-9120-0

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