Section: Model of Stochastic Processes and Network Models

Notations

\(u,v\) for locations; discrete, usually subscript
\(s,t\) for times; continuous, usually function inputs/variables
\(T, \omega, \Omega\) for characteristic times
\(N\) as a size variable
\(x, y, z\) are state variables; often functions of location and time

General Sketch

Here, \(u\) and \(v\) are locations (potentially identical locations), \(s\) and \(t\) are times. We take that \(t \geq s\). \(\emptyset\) represents patients that are not observed to return to the hospital system after their discharge from location \(u\). We assume the associated hazard for this process (\(\zeta_u(t)\)) to be known for all locations \(u\) at all times \(t\): the modelling of this process is the same for all network models, and will be neglected in the following.

Further, we define the function \(w_{uv}(s, t)\) to be the number of individuals that are observed to discahrge from \(u\) at time \(s\) and readmit at time \(t\). We allow a maximum of one of the pairs \(u,v\) and \(s,t\) to be reflexive, i.e. \(u=v\) or \(s=t\), but not both.

We will also assume that we will have quantities \(N_u\) that represent the “size” of a location \(u\). We assume that all locations \(u\) will have \(N_u\) individuals in them at all times, and that the observed mvoements \(w_{uv}(s, t)\) can be thought of as being scaled by \(N_u\).

The governing equations can be roughly expressed as a function of the hazards of each process.

\[\begin{aligned} \lambda(u(s) \to \emptyset) &= \zeta_u(s)\\ \lambda(u(s) \to v(t)) &= d_{uv}(s, t)\\ \lambda(u(s) \to z_{uv}(t)) &= \eta{uv}(s, t)\\ \lambda(z_{uv}(t) \to v(t)) &= \rho_{uv}(s, t) \end{aligned}\]

Note that

\[ \lambda(U \to V) = \lambda \]

represents that in some time step \(\Delta t\), the number of individuals that move from state \(U\) to state \(V\) is distributed as

\[ \mathrm{Poisson}(\lambda x_U \Delta t) \].

Naïve Static Model

For both the naïve and improved static network models, we have that the observations of \(w\) are taken over a time period of \(T_\Sigma\).

In the naïve static model , we have that:

\[d_{uv}(s,t) = d_{uv} = \frac{\sum_{s,t} w_{uv}(s,t)}{T_\Sigma N_u}\] \[\eta_{uv}(s,t) = \eta_{uv} = 0\] \[\rho_{uv}(s,t) = \rho_{uv} = 0\]

That is, all movements are instantaneous, and the rate of movement is the mean rate of movement over the entire observation period.

Improved Static Model

For the improved static model, we choose some threshold value \(\omega\) that delineates direct and indirect transfers.

We will have:

\[d_{uv}(s,t) = d_{uv} = \frac{\sum_{s,t: (t-s) < \omega} w_{uv}(s,t)}{T_\Sigma N_u}\] \[\eta_{uv}(s,t) = \eta_{uv} = \frac{\sum_{s,t: (t-s) \geq \omega} w_{uv}(s,t)}{T_\Sigma N_u}\] \[\rho_{uv}(s,t) = \rho_{uv} = \left[ \frac{\sum_{s,t: (t-s) \geq \omega} (t-s) w_{uv}(s,t)}{\sum_{s,t: (t-s) \geq \omega} w_{uv}(s,t)} \right]^{-1}\]

Here, we model the rate of indirect transfers \(\eta\) similarly to the rate of direct transfers \(d\), by counting the average rate of observed movements. For the rate of readmission, \(\rho\), we use the inverse of the mean readmission duration/delay.

Snapshot Model

For the snapshot model, we choose a snapshot duration \(\omega\). This defines the threshold duration of an indirect transfer.

This introduces an new intermediary state variable \(z'\) that contains the individuals that would enter \(z\) within the duration of a given snapshot \([t, t+\omega]\), so that they do not immediately readmit to their next hospital. This alters the indirect transfer processes to

\[\begin{aligned} \lambda(u(s) \to {z'}_{uv}(t)) &= \eta_{uv}(s, t)\\ \lambda({z'}_{uv}(t) \to z_{uv}(t)) &= \delta(t \mathrm{ mod } \omega)\\ \end{aligned}\]

where \(\delta(\cdot)\) is the Dirac delta function.

Alongside this, we also compute from the data, the number of patients “at home” at a given time, in order to inform the denominator for the process from \(z'\) to \(z\). We denote this quantity \(H_{uv}(t)\)

We will have that:

\[d_{uv}(s,t) = d_{uv}(s) = \frac{\sum_{t: (t-s) < \omega} w_{uv}(s,t)}{\omega N_u}\] \[\eta_{uv}(s,t) = \eta_{uv}(s) = \frac{\sum_{t: (t-s) \geq \omega} {w_{uv}(s, t)}}{\omega N_u}\] \[\rho_{uv}(s,t) = \rho_{uv}(t) = \frac{\sum_{s: (t-s) \geq \omega} {w_{uv}(s, t)}}{\omega H_{uv}}\]

Temporal Model

For the temporal network, we make a choice of the time discretisation \(\omega\). That is, events that occur within a time window \([\tau, \tau+\omega)\) are collapsed and aggregated to occur at time \(\tau\). Over the observation (and simulation) period, we will have a series of time window boundaries \(\{\tau_0, \tau_1, \dots\}\). We denote the smallest value in that set larger than a given time \(t\) to be \(\ceil{t}_\omega\)

For the readmission process, we assume that individuals that would readmit within a given time window \([\tau, \tau+\omega)\) do so uniformly.

This yields:

\[d_{uv}(s, t) = \frac{\sum_{t: (t-s) < \omega} w_{uv}(s, t)}{\omega N_u}\] \[\eta_{uv}(s, t) = \frac{\sum_{t: (t-s) \geq \omega} w_{uv}(s, t)}{\omega N_u}\] \[\rho_{uv}(s,t) = \frac{1}{\ceil{t}_\omega - t}\]

The readmission process thus has a uniform hazard.