Modeling the demand for long-term care services under uncertain information

Abstract

Developing a network of long-term care (LTC) services is currently a health policy priority in many countries, in particular in countries with a health system based on a National Health Service (NHS) structure. Developing such a network requires proper planning and basic information on future demand and utilization of LTC services. Unfortunately, this information is often not available and the development of methods to properly predict demand is therefore essential. The current study proposes a simulation model based on a Markov cycle tree structure to predict annual demand for LTC services so as to inform the planning of these services at the small-area level in the coming years. The simulation model is multiservice, as it allows for predicting the annual number of individuals in need of each type of LTC service (formal and informal home-based, ambulatory and institutional services), the resources/services that are required to satisfy those needs (informal caregivers, domiciliary visits, consultations and beds) and the associated costs. The model developed was validated using past data and key international figures and applied to Portugal at the Lisbon borough level for the 2010–2015 period. Given data imperfections and uncertainties related to predicting future LTC demand, uncertainty was modeled through an integrated approach that combines scenario analysis with probabilistic sensitivity analysis using Monte Carlo simulation. Results show that the model provides information critical for informing the planning and financing of LTC networks.

Appendix A: Assumptions and mathematical formulation of the model

This appendix describes the main assumptions and the complete mathematical formulation of the simulation model.

The following assumptions were used in the proposed simulation model:

• 1. Non-dependent individuals (both non-chronic and chronic without symptoms) do not need to receive any type of LTC service [48];

• 2. Non-dependent and chronic patients with symptoms (with symptoms depending on the chronic disease from which they suffer) need to receive specialized ambulatory care. This type of patients are able to perform their daily activities, and to go to an outpatient unit to see a doctor whenever necessary [67];

• 3. Single people are much more likely to be institutionalized than non-single people, because non-single people can receive care from a companion (ex. spouse) and are much more likely to have children, another important source of informal care [15]. Thus, dependent and single individuals need to receive institutional care, while dependent and non-single individuals need to receive formal and informal home-based care;

• 4. States involving chronic conditions cannot evolve to states with non-chronic conditions, because chronic diseases are long-lasting diseases for which a cure is not expected [50];

• 5. No transitions exist between states characterized by non-single and single individuals. This assumption was used to simplify the model, although the model can be easily changed to include those transitions;

• 6. The level of income is assumed not to vary within the time horizon;

• 7. Chronic patients only die due to their chronic condition.

Additional assumptions were also required for applying the model to Portuguese data:

• 1. Mortality, incidence and dependency rates do not change over time;

• 2. The following chronic diseases were considered: i) oncological diseases, ii) chronic obstructive pulmonary diseases, iii) cerebrovascular diseases and iv) ischemic heart diseases;

• 3. Individuals were defined as dependent if i) are always in bed, ii) are always sitting in a chair, or iii) are limited to their own house;

• 4. Two types of household were used as a proxy for the availability of informal care: single and non-single people;

• 5. Individuals are split into a very low income group (those living below the poverty line) or not a very low income group (the remaining ones).

For the purpose of presenting the mathematical formulation of the model, each of the 20 states of the long-term model was numbered with a state number (see Table 11).

Table 11 Long-term model states

One should note that chronic individuals who have shown some symptoms recently (states 2, 4, 6, 12, 14 and 16) include two main types of individuals: individuals suffering from a chronic disease for the first time (CS _1st ); and chronic individuals who have shown some symptoms during the last year but not for the first time (CS _again ). One needs to distinguish between these two groups of individuals because the Markov states from which they come from are different. E.g., a very low income (VLI) individual with a chronic disease and recent symptoms (CS) at t + 1 (states 2, 4 or 6) might have two different origins: i) he/she might have been non-chronic (NC) at t (belonging to states 7, 8 or 9 at t, which means that he/she is suffering from a chronic disease for the first time at t + 1); or ii) he/she might have already been suffering from a chronic disease at t (belonging to states 1, 2, 3, 4, 5 or 6 at t, which means that he/she is not suffering from a chronic disease for the first time at t + 1).

The sets, parameters and variables used within the model are identified in Tables 12, 13 and 14, respectively.

Table 12 Sets

Table 13 Parameters

Table 14 Variables

Since the proposed simulation model includes a short-term decision tree and a long-term model, its mathematical formulation can be split into two main groups of equations: a first group that defines the number of individuals in each one of the 18 branches of the short-term decision tree (Equations 1–6); and a second group for predicting how the number of individuals in each state of the long-term model will evolve over time (Equations 7–12). Each branch and Markov state is divided into different groups of individuals, depending on their age group i ∊ I, gender j ∊ J, borough k ∊ K and type of chronic disease d ∊ D (if any).

The number of individuals in each of the 18 branches of the short-term decision tree corresponds to the initial number of individuals in 18 of the 20 states of the long-term Markov model (the remaining two states correspond to the absorbing states of the long-term model and have no associated branches in the decision tree). This initial number of individuals is denoted by NSt _ijk(t=0)s (individuals from age group i ∊ I, gender j ∊ J and borough k ∊ K that belong to state s ∊ S in the first year of the planning horizon (t = 0)) and is computed according to Equation 1, in which nSt _ijkd(t=0)s and nSt _ijk(t=0)s correspond to the initial number of individuals suffering from a chronic disease d ∊ D (with or without symptoms) and to the initial number of individuals without any chronic disease, respectively (Equations 2 and 3). On the other hand, nSt _ijkd(t=0)s and nSt _ijk(t=0)s depend on the probability of an individual being in states ∊ Sin the first year of the planning horizon (p _ijds and p _ijs , computed based on Equations 4 and 5, respectively).

$$ NS{{t}_{{ijk(t = 0)s}}} = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{d} {nS{{t}_{{ijkd\left( {t = 0} \right)s}}}} } \hfill \\ {nS{{t}_{{ijk(t = 0)s}}}} \hfill \\ \end{array} } \right.\quad \forall i \in I,j \in J,k \in K,s \in S $$

(1)

$$ NS{{t}_{{ijk(t = 0)s}}} = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{d} {nS{{t}_{{ijkd\left( {t = 0} \right)s}}}} } \hfill \\ {nS{{t}_{{ijk(t = 0)s}}}} \hfill \\ \end{array} } \right.\quad \forall i \in I,j \in J,k \in K,s \in S $$

(2)

$$ nS{{t}_{{ijk(t = 0)s}}} = \left\{ {\begin{array}{*{20}{c}} \hfill {nIn{{d}_{{ijk(t = 0)}}} \times {{p}_{{ijs}}}\quad \forall i \in I,j \in J,k \in K,\,s \in \left( {\left\{ {7;8;9} \right\} \cup \left\{ {17;18;19} \right\}} \right)} \\ \hfill {0\,\,\,otherwise} \\ \end{array} } \right. $$

(3)

$$ {{p}_{{ijds}}} = \left\{ {\matrix{ {P{{{(ND|CWS \cap VLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 1} \hfill \\ {P{{{(ND|CS \cap VLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,\,s = 2\,} \hfill \\ {P{{{(S|D \cap CWS \cap VLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,\,s = 3\,} \hfill \\ {P{{{(S|D \cap CS \cap VLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 4} \hfill \\ {P{{{(NS|D \cap CWS \cap VLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 5} \hfill \\ {P{{{(NS|D \cap CS \cap VLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,\,s = 6} \hfill \\ {P{{{(ND|CWS \cap NVLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 11} \hfill \\ {P{{{(ND|CS \cap NVLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 12} \hfill \\ {P{{{(S|D \cap CWS \cap NVLI)}}_{{ijd}}}\,} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 13} \hfill \\ {P{{{(S|D \cap CS \cap NVLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,\,s = 14} \hfill \\ {P{{{(NS|D \cap CWS \cap NVLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 15} \hfill \\ {P{{{(NS|D \cap CS \cap NVLI)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,\,s = 16} \hfill \\ }<!end array> } \right. $$

(4)

$$ {{p}_{{ijs}}} = \left\{ {\matrix{ {P{{{(ND|NC \cap VLI)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J\,,\,s = 7} \hfill \\ {P{{{(S|D \cap NC \cap VLI)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J\,,\,s = 8} \hfill \\ {P{{{(NS|D \cap NC \cap VLI)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J\,,\,s = 9} \hfill \\ {P{{{(ND|NC \cap NVLI)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J\,,\,s = 17} \hfill \\ {P{{{(S|D \cap NC \cap NVLI)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J\,,\,s = 18} \hfill \\ {P{{{(NS|D \cap NC \cap NVLI)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J\,,\,s = 19} \hfill \\ }<!end array> } \right. $$

(5)

The probabilities of an individual belonging to each state of the long-term model in the first year are computed similarly to Equation 6 (which describes one particular case).

$$ {{p}_{{ijd(s = 1)}}} = P{{\left( {ND|CWS \cap VLI} \right)}_{{ijd}}} = \frac{{P{{{(ND \cap CWS \cap VLI)}}_{{ijd}}}}}{{P{{{(CWS \cap VLI)}}_{{ijd}}}}}\,\,\,\,\forall i \in I,j \in J,d \in D\, $$

(6)

The second group of equations predicts how the number of individuals in each one of the 20 states of the long-term model evolves over time. The number of individuals from age group i ∊ I, gender j ∊ J and borough k ∊ K that belong to state s ∊ S during the time period t ∊ T is denoted by NSt _ijkts and is computed based on Equation 7, in which nSt _ijkdts and nSt _ijkts correspond to the number of individuals suffering from a chronic disease d ∊ D (with or without symptoms) and to the number of individuals without any chronic disease, respectively (Equations 8 and 9). On the other hand, nSt _ijkdts and nSt _ijkts depend on the probability of changing from state m ∊ S to state s ∊ S (p _ijdms and p _ijms , called the state transition probabilities and that are computed based on Equations 10 and 11, respectively).

$$ \begin{array}{*{20}{c}} {NS{{t}_{{ijkts}}} = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{d} {nS{{t}_{{ijkdts}}}} } \\ {nS{{t}_{{ijkts}}}} \\ \end{array} } \right.\forall i \in I,j \in J,k \in K,t \in T,s} \\ { \in S} \\ \end{array} $$

(7)

$$ nS{{t}_{{ijkdts}}} = \left\{ {\matrix{ {\sum\limits_{{m = 1}}^6 {\left[ {nS{{t}_{{(i - 1)jkd(t - 1)m}}}\left( {1 - PDd{{s}_{{(i - 1)jkd(t - 1)}}}} \right){{p}_{{ijdms}}}} \right]} + } \hfill &{} \hfill \\ { + \sum\limits_{{m = 7}}^9 {\left[ {nS{{t}_{{(i - 1)jk(t - 1)m}}}\left( {1 - PDnd{{s}_{{(i - 1)jk(t - 1)}}}} \right){{p}_{{ijdms}}}} \right]} \,} \hfill &{\forall i \in I,j \in J,k \in K\,,d \in D,t \geqslant 1,s \in \left\{ {1;2;3;4;5;6} \right\}} \hfill \\ {\sum\limits_{{m = 11}}^{{16}} {\left[ {nS{{t}_{{(i - 1)jkd(t - 1)m}}}\left( {1 - PDd{{s}_{{(i - 1)jkd(t - 1)}}}} \right){{p}_{{ijdms}}}} \right]} + } \hfill &{} \hfill \\ { + \sum\limits_{{m = 17}}^{{19}} {\left[ {nS{{t}_{{(i - 1)jk(t - 1)m}}}\left( {1 - PDnd{{s}_{{(i - 1)jk(t - 1)}}}} \right){{p}_{{ijdms}}}} \right]} } \hfill &{\forall i \in I,j \in J,k \in K\,,d \in D,t \geqslant 1,s \in \left\{ {11;12;13;14;15;16} \right\}} \hfill \\ {\sum\limits_{{m = 1}}^6 {\left[ {nS{{t}_{{(i - 1)jkd(t - 1)m}}}PDd{{s}_{{(i - 1)jkd(t - 1)}}}{{p}_{{ijdms}}}} \right]} + } \hfill &{} \hfill \\ { + \sum\limits_{{m = 11}}^{{16}} {\left[ {nS{{t}_{{(i - 1)jkd(t - 1)m}}}PDd{{s}_{{(i - 1)jkd(t - 1)}}}{{p}_{{ijdms}}}} \right]} } \hfill &{\forall i \in I,j \in J,k \in K\,,d \in D,t \geqslant 1\,,{ }s = 10} \hfill \\ {0\,\,\,otherwise} \hfill &{} \hfill \\ }<!end array> } \right. $$

(8)

$$ nS{{t}_{{ijkts}}} = \left\{ {\matrix{ {\sum\limits_{{m = 7}}^9 {\left[ {nS{{t}_{{(i - 1)jk(t - 1)m}}}\left( {1 - PDnd{{s}_{{(i - 1)jk(t - 1)}}}} \right){{p}_{{ijms}}}} \right]} } \hfill &{\forall i \in I,j \in J,k \in K\,,t \geqslant 1,s \in \left\{ {7;8;9} \right\}} \hfill \\ {\sum\limits_{{m = 17}}^{{19}} {\left[ {nS{{t}_{{(i - 1)jk(t - 1)m}}}\left( {1 - PDnd{{s}_{{(i - 1)jk(t - 1)}}}} \right){{p}_{{ijms}}}} \right]} } \hfill &{\forall i \in I,j \in J,k \in K\,,t \geqslant 1,s \in \left\{ {17;18;19} \right\}} \hfill \\ {\sum\limits_{{m = 7}}^9 {\left[ {nS{{t}_{{(i - 1)jk(t - 1)m}}}PDnd{{s}_{{(i - 1)jk(t - 1)}}}{{p}_{{ijms}}}} \right]} + } \hfill &{} \hfill \\ { + \sum\limits_{{m = 17}}^{{19}} {\left[ {nS{{t}_{{(i - 1)jk(t - 1)m}}}PDnd{{s}_{{(i - 1)jk(t - 1)}}}{{p}_{{ijms}}}} \right]} } \hfill &{\forall i \in I,j \in J,k \in K,t \geqslant 1\,,\,s = 20} \hfill \\ {0\,\,\,otherwise} \hfill &{} \hfill \\ }<!end array> } \right. $$

(9)

$$ {{p}_{{ijdms}}} = \left\{ {\matrix{ {P{{{\left( {ND|CWS \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,m \in \left\{ {1;2;3;4;5;6} \right\},s = 1} \hfill \\ {P{{{\left( {ND|C{{S}_{{again}}} \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {1;2;3;4;5;6} \right\},s = 2} \hfill \\ {P{{{\left( {ND|C{{S}_{{1st}}} \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {7;8;9} \right\},s = 2} \hfill \\ {P{{{\left( {S|D \cap CWS \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {1;2;3;4} \right\},s = 3} \hfill \\ {P{{{\left( {S|D \cap C{{S}_{{again}}} \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {1;2;3;4} \right\},s = 4} \hfill \\ {P{{{\left( {S|D \cap C{{S}_{{1st}}} \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \{ 7;8\}, s = 4} \hfill \\ {P{{{\left( {NS|D \cap CWS \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \{ 1;2;5;6\}, s = 5} \hfill \\ {P{{{\left( {NS|D \cap C{{S}_{{again}}} \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \{ 1;2;5;6\}, s = 6} \hfill \\ {P{{{\left( {NS|D \cap C{{S}_{{1st}}} \cap VLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \{ 7;9\}, s = 6} \hfill \\ 1 \hfill &{\forall i \in I,j \in J,d \in D\,,m \in \left( {\left\{ {1;2;3;4;5;6} \right\} \cup \left\{ {11;12;13;14;15;16} \right\}} \right),s = 10} \hfill \\ {P{{{\left( {ND|CWS \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {11;12;13;14;15;16} \right\},s = 11} \hfill \\ {P{{{\left( {ND|CWS \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {11;12;13;14;15;16} \right\},s = 12} \hfill \\ {P{{{\left( {ND|C{{S}_{{1st}}} \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {17;18;19} \right\},s = 12} \hfill \\ {P{{{\left( {S|D \cap CWS \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D,m \in \left\{ {11;12;13;14} \right\},s = 13} \hfill \\ {P{{{\left( {S|D \cap C{{S}_{{again}}} \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,m \in \left\{ {11;12;13;14} \right\},s = 14} \hfill \\ {P{{{\left( {S|D \cap C{{S}_{{1st}}} \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,m \in \{ 17;18\}, s = 14} \hfill \\ {P{{{\left( {NS|D \cap CWS \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,m \in \{ 11;12;15;16\}, s = 15} \hfill \\ {P{{{\left( {NS|D \cap C{{S}_{{again}}} \cap NVLI} \right)}}_{{ijd}}}} \hfill &{\forall i \in I,j \in J,d \in D\,,m \in \{ 11;12;15;16\}, s = 16} \hfill \\ {P{{{\left( {NS|D \cap C{{S}_{{1st}}} \cap NVLI} \right)}}_{{ijd}}}\,} \hfill &{\,\forall i \in I,j \in J,d \in D\,,m \in \{ 17;19\}, s = 16} \hfill \\ }<!end array> } \right. $$

(10)

$$ {{p}_{{ijms}}} = \left\{ {\matrix{ {P{{{\left( {ND|NC \cap VLI} \right)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J,m \in \left\{ {7;8;9} \right\},s = 7} \hfill \\ {P{{{\left( {S|D \cap NC \cap VLI} \right)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J,m \in \{ 7;8\}, s = 8} \hfill \\ {P{{{\left( {NS|D \cap NC \cap VLI} \right)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J,m \in \{ 7;9\}, s = 9} \hfill \\ {P{{{\left( {ND|NC \cap NVLI} \right)}}_{{ij}}}\,} \hfill &{\forall i \in I,j \in J,m \in \left\{ {17;18;19} \right\},s = 17} \hfill \\ {P{{{\left( {S|D \cap NC \cap NVLI} \right)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J,m \in \{ 17;18\}, s = 18} \hfill \\ {P{{{\left( {NS|D \cap NC \cap NVLI} \right)}}_{{ij}}}} \hfill &{\forall i \in I,j \in J,m \in \{ 17;19\}, s = 19} \hfill \\ 1 \hfill &{\forall i \in I,j \in J,m \in \left( {[7;9] \cup [17;19]} \right),s = 20} \hfill \\ }<!end array> } \right. $$

(11)

Transition probabilities are computed similarly to Equation 12 (which describes one particular case).

$$ {{p}_{{ijdm(s = 1)}}} = P{{\left( {ND|CWS \cap VLI} \right)}_{{ijd}}} = \frac{{P{{{(ND \cap CWS \cap VLI)}}_{{ijd}}}}}{{P{{{(CWS \cap VLI)}}_{{ijd}}}}}\,\,\,\,\forall i \in I,j \in J,d \in D\,,m \in \left\{ {1;2;3;4;5;6} \right\} $$

(12)

Based on the number of individuals in each state and per time period (nSt _ijkdts and nSt _ijkts ), it is possible to compute:

• 1. The total number of individuals from borough k ∊ K who are in need of formal home-based care during time period t ∊ T (NFHC _kt , Equation 13), in which nFHC _kdt and nFHC _kt correspond to the number of individuals suffering from a chronic disease d ∊ D (with or without symptoms) and without any chronic disease, respectively, who are in need of this service (Equations 14 and 15);

• 2. The total number of individuals from borough k ∊ K who are in need of informal home-based care during time periodt ∊ T(NIHC _kt , Equation 16), in which nIHC _kdt and nIHC _kt correspond to the number of individuals suffering from a chronic disease d ∊ D (with or without symptoms) and without any chronic disease, respectively, who are in need of this service (Equations 17 and 18);

• 3. The total number of individuals from borough k ∊ K who are in need of ambulatory care during time period t ∊ T (NAC _kt , Equation 19), in which nAC _kdt corresponds to the number of individuals suffering from a chronic disease d ∊ D (with or without symptoms) who are in need of this service (Equation 20);

• 4. The total number of individuals from borough k ∊ K who are in need of institutional care during time period t ∊ T (NIC _kt , Equation 21), in which nIC _kdt and nIC _kt correspond to the number of individuals suffering from a chronic disease d ∊ D (with or without symptoms) and without any chronic disease, respectively, who are in need of this service (Equations 22 and 23).

$$ NFH{{C}_{{kt}}} = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{d} {nFH{{C}_{{kdt}}}} } \hfill \\ {nFH{{C}_{{kt}}}} \hfill \\ \end{array} } \right.\quad \forall k \in K\,,t \in T $$

(13)

$$ nFH{{C}_{{kdt}}} = \sum\limits_i {\sum\limits_j {\left( {nS{{t}_{{ijkdt(s = 5)}}} + nS{{t}_{{ijkdt(s = 6)}}} + nS{{t}_{{ijkdt(s = 15)}}} + nS{{t}_{{ijkdt(s = 16)}}}} \right)} } \,\,\,\,\,\forall k \in K\,,d \in D,t \in T $$

(14)

$$ nFH{{C}_{{kt}}} = \sum\limits_i {\sum\limits_j {\left( {nS{{t}_{{ijkt(s = 9)}}} + nS{{t}_{{ijkt(s = 19)}}}} \right)} } \,\,\,\,\,\forall k \in K\,,t \in T $$

(15)

$$ NIH{{C}_{{kt}}} = NFH{{C}_{{kt}}} $$

(16)

$$ nIH{{C}_{{kdt}}} = nFH{{C}_{{kdt}}} $$

(17)

$$ nIH{{C}_{{kt}}} = nFH{{C}_{{kt}}} $$

(18)

$$ NA{{C}_{{kt}}} = \sum\limits_d {nA{{C}_{{kdt}}}\,\,\,\,} \forall k \in K\,,t \in T $$

(19)

$$ nA{{C}_{{kdt}}} = \sum\limits_i {\sum\limits_j {\left( {nS{{t}_{{ijkdt(s = 2)}}} + nS{{t}_{{ijkdt(s = 12)}}}} \right)\,\,\,\,\forall k \in K\,,d \in D,t \in T} } $$

(20)

$$ NI{{C}_{{kt}}} = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{d} {nI{{C}_{{kdt}}}} } \hfill \\ {nI{{C}_{{kt}}}} \hfill \\ \end{array} } \right.\quad \forall k \in K\,,t \in T $$

(21)

$$ nI{{C}_{{kdt}}} = \sum\limits_i {\sum\limits_j {\left( {nS{{t}_{{ijkdt(s = 3)}}} + nS{{t}_{{ijkdt(s = 4)}}} + nS{{t}_{{ijkdt(s = 13)}}} + nS{{t}_{{ijkdt(s = 14)}}}} \right)\,\,\,\,\,\forall k \in K\,,d \in D,t \in T} } $$

(22)

$$ nI{{C}_{{kt}}} = \sum\limits_i {\sum\limits_j {\left( {nS{{t}_{{ijkt(s = 8)}}} + nS{{t}_{{ijkt(s = 18)}}}} \right)\,\,\,\,\forall k \in K\,,t \in T} } $$

(23)

The volume of resources/services required to meet the needs predicted for each borough k ∊ K and time period t ∊ T can thus be computed using as a basis the total number of individuals in need of each type of service – the number of informal caregivers is denoted by NInfC _kt , and is computed based on Equation 24; the number of domiciliary visits to be provided by doctors and by nurses are denoted by NDVd _kt and NDVn _kt , respectively, and are computed based on Equations 25 and 26; the number of consultations is denoted by NC _kt , and is computed based on Equation 27; and the number of beds is denoted by NB _kt , and is computed based on Equation 28.

$$ NInf{{C}_{{kt}}} = NIH{{C}_{{kt}}} \times nInfCp $$

(24)

$$ NDV{{d}_{{kt}}} = NFH{{C}_{{kt}}} \times nVdmp $$

(25)

$$ NDV{{n}_{{kt}}} = NFH{{C}_{{kt}}} \times nVnmp $$

(26)

$$ N{{C}_{{kt}}} = NA{{C}_{{kt}}} \times nCmp $$

(27)

$$ N{{B}_{{kt}}} = NI{{C}_{{kt}}} \times LOS $$

(28)

Finally, the total cost associated with meeting predicted demand in borough k ∊ K during time period t ∊ T is computed using Equation 29.

$$ TotalCos{{t}_{{kt}}} = NFH{{C}_{{kt}}} \times UCdp + N{{C}_{{kt}}} \times UCc + N{{B}_{{kt}}} \times UCb $$

(29)