Recovery and Relapse in Long-Term Care States

Why Recovery and Relapse Need Separate Transitions

A long-term care model should distinguish a person who is receiving care from a person who has recovered and is no longer receiving care. If the person later needs care again, that return is a relapse. By contrast, a person who moves from a moderate-care state to a severe-care state has not recovered and relapsed: care has continued, but its severity has changed.

This distinction affects how transition assumptions are interpreted. A recovery intensity describes movement out of care; a relapse intensity describes a return to care from a recovered state; and an intensity between two care states describes a change in care severity. Combining these transitions can obscure the route by which a person arrives in a state. These distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A recovery transition moves a person from a care state to a recovered state. A relapse transition moves a person from a recovered state back to a care state. A transition between care states, such as moderate care to severe care, is a change in care severity, not a relapse.

Representing Recovery and Relapse

Let \(H\) denote a recovered state, \(C_M\) a moderate-care state, and \(C_S\) a severe-care state. A simple transition graph might allow \(C_M\to H\), \(C_M\to C_S\), \(C_S\to H\), \(H\to C_M\), and \(H\to C_S\). Each arrow represents a possible direct transition; an arrow not included in the graph is not permitted by that model.

For this tutorial, assume transition intensities are constant while the person remains in the current state. Write \(\lambda_{ij}\) for the intensity of a direct transition from state \(i\) to state \(j\). For example, \(\lambda_{H,C_M}\) is the relapse intensity from the recovered state to moderate care. These are transition intensities, not probabilities. In a short interval of length \(h\), the probability of a transition from \(i\) to \(j\) is approximately \(\lambda_{ij}h\), with the probability of two or more transitions being of smaller order than \(h\).

The constant-intensity assumption means that the time until the next transition out of a state has an exponential distribution. It also implies that, conditional on being in the state, the model does not retain the time already spent there. The examples below omit death and other transitions to keep attention on recovery, relapse, and severity changes. A model that includes additional possible exits must include their intensities when calculating the total rate of leaving a state.

Definition: If the possible direct exits from state \(i\) have constant intensities \(\lambda_{ij}\), the total exit intensity is \(\lambda_i=\sum_{j\ne i}\lambda_{ij}\). The time until the first exit from \(i\) has survival probability \(e^{-\lambda_i t}\), and its mean is \(1/\lambda_i\), provided \(\lambda_i>0\).

First-Transition Probabilities

When several transitions can occur from one state, their intensities compete. The total intensity determines how quickly the state is left; the share of that total associated with a particular destination determines the destination of the first transition.

P(\text{first transition is }i\to j)=\frac{\lambda_{ij}}{\lambda_i}, \qquad P(\text{first transition is }i\to j\text{ by time }t) =\frac{\lambda_{ij}}{\lambda_i}\left(1-e^{-\lambda_i t}\right).

These formulas apply to the first transition after entry into state \(i\), under constant intensities and the specified set of possible exits. They are not, in general, the probability of occupying state \(j\) at time \(t\). A person could enter \(j\) and then leave it before \(t\).

Why the Competing-Transition Formula Holds

The probability of no transition from \(i\) during the first \(t\) years is \(e^{-\lambda_i t}\). Therefore, the probability of at least one exit by time \(t\) is \(1-e^{-\lambda_i t}\). Given that an exit occurs, the chance that its destination is \(j\) is proportional to \(\lambda_{ij}\): the share is \(\lambda_{ij}/\lambda_i\). Multiplying the chance of any exit by this destination share gives the stated formula.

Equivalently, the density that the first transition is to \(j\) and occurs at time \(s\) is \(\lambda_{ij}e^{-\lambda_i s}\). Integrating this density from \(0\) to \(t\) gives the same result:

\int_0^t \lambda_{ij}e^{-\lambda_i s}\,ds =\frac{\lambda_{ij}}{\lambda_i}\left(1-e^{-\lambda_i t}\right).

Worked Examples

Worked Example: Recovery Followed by Possible Relapse

A hypothetical contract model has two living states: care \(C\) and recovered \(H\). The constant recovery intensity is \(\lambda_{C,H}=0.40\) per year, and the constant relapse intensity is \(\lambda_{H,C}=0.10\) per year. Find (a) the probability that a person in care experiences a first recovery transition within two years, and (b) the probability that a recovered person relapses within one year. Assume no other exits, no mortality, and constant intensities.

Method. In each starting state there is only one possible exit, so the relevant total exit intensity equals the intensity of that transition.

(a) Starting in care, \(\lambda_C=0.40\). The probability of a first exit from care by two years is \(1-e^{-0.40(2)}=1-e^{-0.80}\). Using \(e^{-0.80}=0.4493289641\), this is \(0.5506710359\), or 0.5507 to four decimal places. Because recovery is the only exit from care in this model, this is the probability of a first recovery within two years.

(b) Starting in \(H\), \(\lambda_H=0.10\). The probability of a first relapse by one year is \(1-e^{-0.10(1)}=1-e^{-0.10}\). Using \(e^{-0.10}=0.904837418036\), the result is \(0.095162581964\), or 0.0952 to four decimal places.

As a check, both probabilities are between zero and one, and the probability over the longer period from care is larger, consistent with its greater cumulative exit intensity. A common mistake is to use the recovery intensity \(0.40\) for the relapse calculation. Relapse starts from \(H\), so its applicable intensity is \(0.10\).

Worked Example: Relapse Versus Worsening Care Severity

In a hypothetical model, a person in moderate care \(C_M\) can recover to \(H\) with intensity \(0.30\) per year or worsen to severe care \(C_S\) with intensity \(0.20\) per year. From \(H\), relapse can lead to \(C_M\) at intensity \(0.08\) or to \(C_S\) at intensity \(0.02\), both per year. Find the mean time to the next transition from \(C_M\), the probability that its first transition is recovery, and the probability that the first relapse from \(H\) is to severe care within one year. Assume these are the only exits, with constant intensities and no mortality.

Method. Add the intensities of all exits from the starting state, then use their shares of the total for the destination probabilities.

From \(C_M\), the total exit intensity is \(\lambda_{C_M}=0.30+0.20=0.50\). The mean time to its next transition is \(1/0.50=2\) years. The probability that the first transition is recovery is \(0.30/0.50=0.60\). The probability that it is a severity change to \(C_S\) is \(0.20/0.50=0.40\). These two first-transition probabilities sum to one.

From \(H\), the total relapse intensity is \(\lambda_H=0.08+0.02=0.10\). The severe-care destination share is \(0.02/0.10=0.20\). The probability of any first relapse within one year is \(1-e^{-0.10}=0.095162581964\). Thus the probability that the first relapse is specifically to \(C_S\) within one year is \(0.20(0.095162581964)=0.019032516393\), or 0.0190 to four decimal places.

The result is less than the probability of any relapse within one year, as it must be. The common mistake is to call \(C_M\to C_S\) a relapse. It is deterioration while care continues; relapse begins in \(H\).

Worked Example: A Relapse Destination Within a Fixed Period

A hypothetical recovered person can relapse to moderate care at intensity \(0.12\) per year or to severe care at intensity \(0.03\) per year. Find the probability that the first transition out of recovery occurs within two years and is a relapse to moderate care. Also find the probability of any first relapse within two years. Assume these are the only exits, with constant intensities and no mortality.

Method. The total exit intensity from \(H\) is \(\lambda_H=0.12+0.03=0.15\). The share for moderate-care relapse is \(0.12/0.15=0.80\). The probability of any exit within two years is \(1-e^{-0.15(2)}=1-e^{-0.30}\). Since \(e^{-0.30}=0.7408182207\), this is \(0.2591817793\).

Therefore, the probability that the first transition is to moderate care within two years is \(0.80(0.2591817793)=0.2073454234\), or 0.2073 to four decimal places. The probability of any first relapse within two years is \(0.2592\) to four decimal places.

The moderate-care result is below the any-relapse result, and the difference, \(0.2591817793-0.2073454234=0.0518363559\), is the corresponding probability of first relapse to severe care within two years. A common mistake is to use \(0.12\) alone as the total exit intensity; the competing severe-care relapse must be included in the total.

Model Checks and Common Exam Traps

A transition graph and its intensities should describe coherent routes through the model. Before calculating, identify the starting state, list every permitted exit from it, and determine whether the question concerns the first transition or the state occupied at a later time. Keep the following distinctions explicit:

Recovery: movement from a care state to a recovered state.
Relapse: movement from a recovered state to a care state.
Severity change: movement between care states without an intervening recovery.
First-transition probability: the chance that a specified exit is the first transition within a period.
Occupancy probability: the chance of being in a state at a specified time, which can involve multiple transitions.

A first-transition formula does not automatically solve an occupancy question. For example, a person may move from \(H\) to \(C_M\), recover again, and be in \(H\) at the end of the period. The calculation for a first relapse counts that relapse even though the person is not in care at the endpoint. More elaborate occupancy calculations account for possible sequences of transitions and are outside the first-transition calculations developed here.

Key takeaway: Label the state at the start of the transition before assigning its intensity. Care-to-recovered is recovery, recovered-to-care is relapse, and care-to-care is a severity change. For competing exits, use the sum of all relevant intensities as the total exit intensity.

Check Your Understanding

For each problem, state any first-transition interpretation carefully. Assume constant intensities and no mortality unless the question says otherwise.

From a moderate-care state, the only exits are recovery at intensity \(0.24\) per year and deterioration to severe care at intensity \(0.16\) per year. Find the total exit intensity, the mean holding time, and the probability that recovery is the first transition. (Answer: \(0.40\) per year; \(2.5\) years; \(0.6000\).)
From a recovered state, relapse to care is the only exit and has intensity \(0.07\) per year. Find the probability of a first relapse within three years. (Answer: \(1-e^{-0.21}=0.1894\).)
From a recovered state, relapse to moderate care has intensity \(0.045\) per year and relapse to severe care has intensity \(0.015\) per year. Find the probability that the first relapse is to severe care within four years. (Answer: \((0.015/0.060)(1-e^{-0.24})=0.0533\).)
From moderate care, recovery has intensity \(0.20\) per year and deterioration to severe care has intensity \(0.30\) per year. Conditional on deterioration being the first transition, find the probability of a transition from severe care to recovery within the following five years if recovery is the only exit from severe care and its intensity is \(0.10\) per year. Find the probability of this two-stage path, including deterioration being first. (Answer: Conditional probability \(1-e^{-0.50}=0.3935\); path probability \((0.30/0.50)(1-e^{-0.50})=0.2361\).)
From severe care, recovery is the only exit and has intensity \(0.14\) per year. From the recovered state, relapse to moderate care has intensity \(0.05\) per year and relapse to severe care has intensity \(0.02\) per year. Find the probability of recovery within one year, then the probability that the first relapse from the recovered state is to severe care. (Answer: \(1-e^{-0.14}=0.1306\); \(0.02/0.07=0.2857\).)

What You'll Learn