Complex intervention modelling should capture the dynamics of adaptation

Background Complexity has been linked to health interventions in two ways: first as a property of the intervention, and secondly as a property of the system into which the intervention is implemented. The former recognizes that interventions may consist of multiple components that act both independently and interdependently, making it difficult to identify the components or combinations of components (and their contexts) that are important mechanisms of change. The latter recognizes that interventions are implemented in complex adaptive systems comprised of intelligent agents who modify their behaviour (including any actions required to implement the intervention) in an effort to improve outcomes relative to their own perspective and objectives. Although an intervention may be intended to take a particular form, its implementation and impact within the system may deviate from its original intentions as a result of adaptation. Complexity highlights the challenge in developing interventions as effective health solutions. The UK Medical Research Council provides guidelines on the development and evaluation of complex interventions. While mathematical modelling is included in the guidelines, there is potential for mathematical modeling to play a greater role. Discussion The dynamic non-linear nature of complex adaptive systems makes mathematical modelling crucial. However, the tendency is for models of interventions to limit focus on the ecology of the system - the ‘real-time’ operation of the system and impacts of the intervention. These models are deficient by not modelling the way the system reacts to the intervention via agent adaptation. Complex intervention modelling needs to capture the consequences of adaptation through the inclusion of an evolutionary dynamic to describe the long-term emergent outcomes that result as agents respond to the ecological changes introduced by intervention in an effort to produce better outcomes for themselves. Mathematical approaches such as those found in economics in evolutionary game theory and mechanism design can inform the design and evaluation of health interventions. As an illustration, the introduction of a central screening clinic is modeled as an example of a health services delivery intervention. Summary Complexity necessitates a greater role for mathematical models, especially those that capture the dynamics of human actions and interactions. Electronic supplementary material The online version of this article (doi:10.1186/s12874-016-0149-8) contains supplementary material, which is available to authorized users.

Additional file 1: . Model details -referral to next available specialist via centralized screening clinics.

Part A. The ecological dynamics
We develop a simple high-level model describing the operation of health system with two options for referring patients from primary to specialty care. The model is presented in Figure 1 (main article) and notation is summarized in Table 1.
Patients that can be treated by the specialist are added to the specialists practice for treatment, → . The proportion of patients that receive specialty care is given by and for patients who were screened at the central intake clinic and patients referred directly to specialist respectively. That is, and can be thought of the positive predictive values of the central screening clinic, and specialist screening processes respectively. Treatment is provided at rate → . Thus, the movement of patients from consult into treatment is given by: Finally, we note that specialists have limited appointments that must be allocated amongst patients needing first consults, and those needing treatment. Appointments for new consults are freed as treated patients leave the specialist's care (at rate ), and are booked at a rate proportional to the consult rate, The dynamics describing the specialist's appointment availability are: Since, ̇= −̇, it follows that the number of appointments available is constant, = + .

Analysis of the ecological dynamics
The ecological dynamics of the system are straightforward. If capacity exceeds demand throughout the health service network, ( (1 − ) +̂) < for all specialists, then the system will reach an equilibrium state given by:

Performance analysis
The ecological dynamics are now analyzed to describe system performance. Here, we measure performance through wait times. The expected wait times from referral to specialist consult for a patient referred via the central intake clinic, , and for a patient referred directly to a specialist, are, respectively: = 1 + ∑̂ 1 * , and = 1 + 1 * .
Thus, the expected wait time for patients seeking specialty care is: Here, ∑ is minimized when referrals are equally distributed amongst specialists, = 1− , which is assumed. Referral via the central intake screening clinic creates additional wait time improvements if the wait time resulting from the central intake screening process, 1/ is less than the wait time resulting from the screening via a direct referral, 1/ . However, wait time improvements will be dependent on how specialists allocate available consult appointments between referrals received from the central intake clinic and direct referrals as captured in the first consult rate, * . The optimized state of the system -with the intervention in place-occurs when = 1, = 0 and = 1 for all S. That is, the referrals from primary care to specialty care are optimized when all specialists are referred patients via the central intake clinic; the expected wait time is

Part B. The evolutionary dynamics
We now describe the evolutionary dynamics that create change in the system's ecological dynamic.
Specifically, we model how specialists allocate available consult appointments between referrals received from the central intake clinic and direct referrals using a utility driven feedback mechanism. Specifically, we assume that the specialist seeks to maximize throughput so as to treat as many patients per unit time as is possible, where throughput is given by the utility function: To examine how a specialist's utility is affected by allocation strategy x we can examine the marginal return, which describes the gain in utility awarded when through a small change in the allocation strategy.
Here, we must recognize that a game is established, as specialist utility is not only determined by the specialist's allocation strategy, but also by the strategy of the other specialists. To calculate the marginal return, we consider a focal specialist, and ask how that specialist's payoff is altered by a marginal increase (above the average, * ) in the allocation of consult appointments to patients choosing next available specialist, = * + . The altered payoff to our focal specialist (as a first order approximation) is: In words, a small increase in the allocation of consults to patients choosing next available specialist, results in the gain of throughput acquired through central intake at the expense of throughput lost from direct referrals.
If we assume a small amount of variation among the x of our specialist group, normally distributed about a mean * then the evolutionary dynamic describing the adaptation process can be written as * = ( ) | = * .
Which simply states that the change in the mean of * is determined as the product of the variation in x, and the marginal return produced when using strategy x, in a system where the strategy * is typically used. For this to be true, the change in x must be purely driven by the adaptation process. Note the above is a simple form of a replicator dynamic; aptly named because decision strategies that produce higher payoffs are "replicated" and increase in frequency, whereas decision strategies with lower payoffs decrease in frequency.

Analysis of the evolutionary dynamic.
The direction of change observed in * is determined by the sign of the marginal return: The challenge of implementation is revealed by examining / | = * ≈0 . If referrals to the central intake clinic are low (i.e. < * as * → 0), then specialists will prefer to allocate an increased proportion of consults to direct referrals. In such cases, the introduction of the central screening clinic will be difficult if not impossible to implement.
Implementation of the central intake clinic requires that increased participation of specialists be beneficial to them. For this to be achieved the proportion of referrals sent to the central screening clinic at start up, coupled with the benefit offered to the specialist by the screening process, must be sufficiently high, > * . This may be achievable if referrals to the central intake clinic are at least equal to the initial specialist allocations, ≥ * as * → 0.
Assuming that the central intake clinic offers some benefit to specialists, then the system will eventually settle to an evolutionary equilibrium given by: * = .
This simply states that the proportion of available consults allocated to referrals received from the central intake clinic is given by the quotient of throughput acquired through central intake clinic and throughput acquired from direct referrals. The existence of an evolutionary equilibrium means that the intervention can be sustained long term (from the point of view of our evolutionary dynamic), However, it does not guarantee that the intervention is effective in its sustained state. The expected equilibrium specialist participation rate as a function of the of the proportion of referrals that are sent to the central intake clinic.
If the proportion of a specialist's throughput that is acquired through central intake is sufficiently high, then full participation, * = 1, can be maintained, allowing optimal system performance to be achieved.
However, if specialists are able to maintain a sufficient level of throughput from direct referrals relative to that generated from the central intake clinic, then * may be low at equilibrium. In such cases, the wait time benefits generated by the central intake clinic may be small and may not be sufficient to justify the continued maintenance of the central intake clinic.

Summary
Our model of the ecological dynamics of the system demonstrated that the introduction of referral to next available specialist through an optional centralized screening clinic can produce shorter wait times between referral and specialist consult. However, it also revealed that the effectiveness and sustainability of the intervention is linked to the adaptive behaviour of specialists. The inclusion of an evolutionary dynamic to model the adaptive behaviour of specialists allows us to assess whether the intervention remains effective over the long term. In doing so, we observe that there is a risk that the effectiveness of the intervention is not sustainable. This occurs when the objectives of the specialists are at odds with what is best for the system. In minimizing wait times, the system prefers to distribute workload evenly amongst the specialists, but an even share of referrals might be less than what each specialist desires. In such cases, if a specialist can acquire a bigger share through direct referrals then the specialist will do exactly that. However, if the system generates enough referrals to the central intake clinic, then specialists will prefer to increase their allocation of consults to the central intake clinic for scheduling.