Modeling Admission, Discharge, and Readmission

Why hospital care can require a cycle

A patient can be admitted, discharged, and later admitted again. A diagram that ends the patient’s pathway at the first discharge cannot represent that repeated course. A multi-state diagram instead describes the patient’s current state and the transitions allowed by the study’s definitions. When discharge makes readmission possible, the diagram needs a path back from an outside-hospital state to a hospital state.

The diagram is only as clear as its state definitions. In particular, “outside hospital” alone does not distinguish a patient who has not yet had a study-period admission from one who has been discharged and could be readmitted. The model below separates those histories. It treats death as an absorbing state: once entered, it cannot be left.

Definition: Let \(X(t)\) denote a patient’s state at follow-up time \(t\). Use four states: \(O_0\), alive and outside hospital with no admission or discharge episode during follow-up; \(H\), alive and in hospital; \(O_R\), alive and outside hospital after a discharge during follow-up; and \(D\), dead. An admission from \(O_0\) is the first study-period admission; an admission from \(O_R\) is a readmission. The state \(D\) is absorbing.

These are history-based definitions for a specified follow-up period. They make \(O_0\) and \(O_R\) distinct even though both describe someone outside hospital now. Under this setup, a patient who enters follow-up alive and outside hospital starts in \(O_0\). A patient already hospitalized at entry starts in \(H\), not \(O_0\); if discharged during follow-up, that patient moves to \(O_R\). Thus, a baseline admission does not create conflicting state assignments.

The states and their transitions

The allowed transitions follow directly from the definitions. A first admission takes the patient from \(O_0\) to \(H\); discharge takes the patient from \(H\) to \(O_R\); and readmission takes the patient from \(O_R\) back to \(H\). Death can occur from any living state. There is no transition from \(D\) back to a living state.

O_0 \longrightarrow H \longrightarrow O_R \longrightarrow H \longrightarrow O_R \longrightarrow \cdots

O_0 \longrightarrow D,\qquad H \longrightarrow D,\qquad O_R \longrightarrow D

The repeated \(O_R \longrightarrow H \longrightarrow O_R\) segment is the cycle: each return to \(H\) represents another admission, and the following move to \(O_R\) represents discharge. The ellipsis means the pattern can repeat as often as the observed follow-up permits; it does not assert that any patient must be readmitted. Death can interrupt the sequence from any living state.

This diagram treats a hospital stay as one continuous inpatient episode. A transfer between facilities counts as part of the same episode if the study protocol defines it that way; otherwise the protocol must say how transfers are classified. The diagram also assumes that the relevant admission, discharge, and death events can be observed and assigned consistently. Those are design and measurement requirements, not consequences of drawing the arrows.

Worked Example: Tracing a Patient’s Repeat Admission

Hypothetical timeline. A patient is alive and outside hospital at follow-up day 0. The patient is admitted on day 12, discharged on day 16, readmitted on day 41, discharged on day 45, and dies on day 60. Assign the state path and calculate the time spent in each state.

From day 0 until the first admission, the patient is in \(O_0\): \(12-0=12\) days. The first admission changes the state to \(H\), occupied from day 12 to day 16: \(16-12=4\) days. Discharge changes the state to \(O_R\), occupied from day 16 to day 41: \(41-16=25\) days.

The day-41 admission is a readmission, so the patient returns to \(H\) for \(45-41=4\) days. After the day-45 discharge, the patient is again in \(O_R\), for \(60-45=15\) days, until death. The full path is \(O_0 \to H \to O_R \to H \to O_R \to D\). The living-state durations sum to \(12+4+25+4+15=60\) days, matching the follow-up time to death.

The arithmetic checks the timeline’s partition into consecutive intervals: there is no overlap and no unassigned time. The death at day 60 is a state entry, not additional time alive in \(O_R\).

Constructing and checking the diagram

A reliable construction starts with the study question and protocol rather than with a familiar diagram template. Specify the follow-up origin, decide what counts as an admission episode and discharge, and decide whether patients hospitalized at entry are included. Then define each state in terms of observable status and relevant follow-up history. Only after that should the arrows be added.

Fix the cohort and time origin.
For example, follow patients from a defined study-entry date. State whether entry requires being outside hospital or permits patients already admitted.

Write state definitions that partition the cohort.
At each time, a patient must have one and only one state. Here, \(O_0\), \(H\), and \(O_R\) distinguish living patients by current location and study-period discharge history; \(D\) records death.

Add arrows for protocol-defined events.
Use \(O_0 \to H\) for a first study-period admission, \(H \to O_R\) for discharge, and \(O_R \to H\) for readmission. Add death arrows from each living state.

Trace plausible paths against the definitions.
Check a first admission, a discharge, a repeat admission, death in each living state, and any permitted baseline condition. Revise the states if a path cannot be assigned unambiguously.

The logic behind the cycle is simple but worth making explicit. By definition, an admission from \(O_R\) occurs after a discharge during follow-up, so it is a readmission and enters \(H\). If that hospital episode ends in a discharge during follow-up, the patient is outside hospital after a follow-up discharge and therefore returns to \(O_R\). Applying the same definitions again yields \(O_R \to H \to O_R\), which can repeat. This is why one discharge does not make \(O_R\) an endpoint.

Worked Example: Counting Transitions in Hypothetical Paths

Consider four hypothetical patients followed from day 0, all alive and outside hospital at entry. Their observed paths are: Patient A, \(O_0 \to H \to O_R \to H \to O_R \to D\); Patient B, \(O_0 \to H \to D\); Patient C, \(O_0 \to D\); Patient D, \(O_0 \to H \to O_R \to D\). Count the observed transitions by type.

Patient A contributes one \(O_0 \to H\), two \(H \to O_R\), one \(O_R \to H\), and one \(O_R \to D\). Patient B contributes one \(O_0 \to H\) and one \(H \to D\). Patient C contributes one \(O_0 \to D\). Patient D contributes one \(O_0 \to H\), one \(H \to O_R\), and one \(O_R \to D\).

Adding these contributions gives 3 transitions \(O_0 \to H\), 3 transitions \(H \to O_R\), 1 transition \(O_R \to H\), 1 transition \(H \to D\), 1 transition \(O_0 \to D\), and 2 transitions \(O_R \to D\). The total is \(3+3+1+1+1+2=11\), which equals the total number of arrows traversed across the four paths: \(5+2+1+3=11\). This check confirms that both admissions in Patient A’s path are represented rather than truncating the record at the first discharge.

Entry conditions, assumptions, and interpretation

If a study includes people already hospitalized at entry, their initial state is \(H\). A discharge during follow-up moves them to \(O_R\), because the definition of \(O_R\) is based on a discharge during follow-up, not on whether the hospital episode began after entry. This convention is coherent even though that discharge is not from an admission that began during follow-up. The protocol should say whether the resulting admission after discharge is called a readmission for the study’s purposes.

An alternative is to restrict the cohort to patients outside hospital at entry. Then everyone begins in \(O_0\), and a baseline inpatient is excluded. Either design can be used, but mixing the two without defining initial states creates ambiguity. In particular, defining \(O_0\) as “no admission since entry” while also assigning a baseline inpatient to \(O_0\) after discharge would conflict with a separate rule that discharge leads to \(O_R\).

A diagram specifies which state changes are represented; it does not by itself establish that the process is Markov. The Markov assumption concerns whether the present state contains the relevant information for the future distribution of the process. If readmission depends on the number of previous admissions or on time since the most recent discharge, the four-state representation may not capture that dependence. This possibility can motivate different state definitions or time-history specifications, but the arrows alone cannot rule it out.

Worked Example: Classifying a Patient Hospitalized at Entry

Hypothetical setup. A study includes a patient already in hospital at day 0. The patient is discharged on day 3, readmitted on day 20, discharged again on day 23, and remains alive outside hospital through day 30. Assign states and check that there is no overlap at discharge.

At day 0, the patient is in \(H\), because the patient is alive and hospitalized. From day 0 to day 3, the patient remains in \(H\), for \(3-0=3\) days. The day-3 discharge moves the patient to \(O_R\), because a discharge occurred during follow-up, even though the hospital stay began before follow-up. The patient remains in \(O_R\) until the day-20 readmission, for \(20-3=17\) days.

The day-20 readmission moves the patient to \(H\) for \(23-20=3\) days. The day-23 discharge returns the patient to \(O_R\), which is occupied through day 30 for \(30-23=7\) days. The state path is \(H \to O_R \to H \to O_R\), with total living time \(3+17+3+7=30\) days. The patient is never assigned to \(O_0\), so the pre-entry admission creates no conflict with the state definitions.

Common pitfalls and alternatives

Ending the diagram at discharge. Discharge is not necessarily an endpoint. If readmission is part of the question, include the arrow from \(O_R\) back to \(H\).
Using one undifferentiated outside-hospital state. That state hides the distinction between no study-period discharge and being outside hospital after a discharge. Separate states when that distinction matters to the question.
Leaving baseline status undefined. Decide whether baseline inpatients are excluded or start in \(H\). If included, a discharge during follow-up leads to \(O_R\) under the definitions used here.
Treating the diagram as proof of a Markov model. A list of states and arrows describes the proposed structure; it does not show that the present state is a sufficient summary of history.

A first-admission analysis is simpler when the question concerns only time to the first admission, but it discards later admission and discharge paths. A recurrent-event summary can count multiple admissions, but by itself may not describe how much time patients spend hospitalized or outside hospital between admissions. The multi-state representation makes those changing states explicit. It is a representation of the process, not a particular estimation method or a guarantee that all relevant clinical history has been captured.

Key takeaway: Define the outside-hospital states by follow-up history, make baseline assignment explicit, and include the return arrow from the discharged state to hospital when readmission is possible. The resulting cycle represents repeated admission and discharge without making discharge an absorbing endpoint.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A patient enters follow-up alive and outside hospital, is admitted on day 5, discharged on day 9, and readmitted on day 18. Give the state path through day 18 and state how many days were spent in each earlier state.
Under the definitions in this tutorial, where does a patient hospitalized at study entry begin? What state follows a discharge on day 2?
Why is a transition from \(O_R\) to \(H\) needed when repeat admissions are part of the study question?
A patient is discharged and later dies outside hospital without readmission. Give the path from the first admission through death.
Does drawing the cycle \(O_R \to H \to O_R\) establish the Markov assumption? Explain briefly.

Solutions. (1) The path is \(O_0 \to H \to O_R \to H\). The patient spends \(5-0=5\) days in \(O_0\) and \(9-5=4\) days in \(H\) before the readmission; the readmission occurs at day 18. (2) The patient starts in \(H\); discharge during follow-up moves the patient to \(O_R\). (3) That arrow represents a new admission after a follow-up discharge; without it, the diagram cannot represent readmission. (4) The path is \(O_0 \to H \to O_R \to D\), assuming the patient entered follow-up outside hospital and the first admission occurred during follow-up. (5) No. The diagram specifies state labels and possible transitions, but does not show that the current state alone contains all history relevant to the future.