Adaptive and non-adaptive models of depression: A comparison using register data on antidepressant medication during divorce

Divorce is associated with an increased probability of a depressive episode, but the causation of events remains unclear. Adaptive models of depression propose that depression is a social strategy in part, whereas non-adaptive models tend to propose a diathesis-stress mechanism. We compare an adaptive evolutionary model of depression to three alternative non-adaptive models with respect to their ability to explain the temporal pattern of depression around the time of divorce. Register-based data (304,112 individuals drawn from a random sample of 11% of Finnish people) on antidepressant purchases is used as a proxy for depression. This proxy affords an unprecedented temporal resolution (a 3-monthly prevalence estimates over 10 years) without any bias from non-compliance, and it can be linked with underlying episodes via a statistical model. The evolutionary-adaptation model (all time periods with risk of divorce are depressogenic) was the best quantitative description of the data. The non-adaptive stress-relief model (period before divorce is depressogenic and period afterwards is not) provided the second best quantitative description of the data. The peak-stress model (periods before and after divorce can be depressogenic) fit the data less well, and the stress-induction model (period following divorce is depressogenic and the preceding period is not) did not fit the data at all. The evolutionary model was the most detailed mechanistic description of the divorce-depression link among the models, and the best fit in terms of predicted curvature; thus, it offers most rigorous hypotheses for further study. The stress-relief model also fit very well and was the best model in a sensitivity analysis, encouraging development of more mechanistic models for that hypothesis.

In the main manuscript we used a relatively simply version of the evolutionary adaptive model, which is in line with the general notation that "models should be as simple as possible but no simpler". Although there are compelling reasons to avoid exact modern-day parametrization of the selective environments for behavioural states whose point of origin is unclear (see main text), such parametrization nevertheless serves as a good way to study robustness of the results. At least we know that the modern-day parameters are something that can happen in reality. This supplementary text present a more complex version of evolutionary state dependent model that uses a modern-day parametrization for the studied population. The simpler model in the main manuscript is nested within (a restricted version) of the more complex model; for brevity, we also discuss general aspects of both models in this text. We call the larger model a demographical parameterization of the optimal-behaviour model.

Demographically parameterized model
The demographical parameterization was based on the data released by Statistics Finland [1][2][3], and on previous studies using their data [4,5]. Below Figure summarizes these data as they were used. Mortality rate was well described by fitted constant, linear, and quadratic terms (cubic term no longer improved the fit: p > 0.5), and the model prediction was used rather than the somewhat idiosyncratic raw estimates (see panel A for the data and model predictions). Similarly, a convenient parametrization was chosen for fertility and marriage rates so as to approximate the published graphs under visual inspection (panels B-C). 1 Figure S1D shows that the fit of the demographically 1 Specifically, the fertility-rate function for men was given by the density function of the log-normal distribution (location 0 and scale ½) between values 0.3 and 2.0 spread to the interval of ages 20 to 40 by an affine transformation, and re-scaled so that the maximum annual fertility approximately corresponded to the observed rate (150 births per 1000 persons). Female fertility was otherwise the same, but with the scale parameter 0.4 and a 3-year translation of the x-axis, to reflect the earlier reproductive period compared to men. Female marriage rate was also log-normal on the interval 0.4 to 1.6 with scale 0.4 (males 'delayed' by two years). In addition to noise reduction, the parametric representation was useful for the below-described sensitivity analysis ( Figure S2E-F) with respect to age-patterns, where we extrapolated mortality, fertility and marrying rates to older age groups. determination than for the simple model may be due to decreasing fertility at the old age. Overall, however, we took the good fit of both the simple and the demographically parameterized model to indicate robustness of the main results under different conditions. We also performed further sensitivity analyses on the demographically parameterized model, as described below. Before the results, the definition of the demographically parameterized model is given.
Supplementary Figure 1. Demographic data and the fit of the state-dependent optimality model in the implied 'realistic' environment. The panels show age-dependent (A) mortality rate, (B) rate of offspring production, and (C) marrying rate, whereas panel (D) shows the optimality model fit in these age-dependent environments, separately for male and female data.

Definition of the demographically parameterized model
The demographically parameterized model is defined by the below figure, which extends that of the main manuscript; furthermore, the below table provides the explicit representation in terms of transition matrices of probability theory.
Supplementary Figure 2. Possible state transitions in the evolutionary state-dependent model. Note that it is also possible to stay in the same state for longer than one time step, but for clarity the selfloops are not shown (but see below Table). A 'strategy' π(x,t) defines whether to be in a 'depressed' mode u 1 or not (mode u 0 ) given the state x and time t. The star superscript refers specifically to the optimal strategy that maximizes the reproductive value. The choice of mode dictates the transitionprobability structure for the next time step. The effect of the 'depressed' mode is to decrease the probability of a divorce-like transition from the relationship-at-risk state to the unpartnered state by the value s, to increase the (now possibly time-and state-dependent) probability of dying am t (or bm t ) by z, and to remove the (possibly time-dependent) probability of marrying ρ t (removal of ρ t had no consequences here, but for some, it seems a logical outcome of 'depression'). For the demographically parameterized model, coefficients a and b reflected the fact the mortality rates are higher for unpartnered individuals (a > b), but for the simpler model we assumed a = b = 1. We assumed that as many relationships at risk end up in divorce as in reconciliation on average (d 2 ); the function f(d 1 , d 2 ) captured the overall divorce rate, as illustrated below. Table 1. The full stochastic state-transition matrices for the two behavioural modes, 'normal' (u 0 ) and 'depressed' (u 1 ), and the states 'seeking partner' (1), 'married' (2), 'relationship at risk' (3), and 'dead' (4)

State-transition matrix …
… when in mode u 0 … when in mode u 1 Going from state 2 (partnered, no relationship risk) to state 1 (unpartnered) takes two state transitions in our model, and the probability of making those within four 3-month periods (i.e. within a year) is given by Thus, there are two parameters (p 23 and p 31 , mapped respectively to d 1

Sensitivity analyses
We explored values of d 1 in the range from 0.015 to 0.15, which implied a (declining) range from 0.359 to 0.031 for d 2 under the condition of a constant divorce rate (note that the transition structure of above Table limits  Nevertheless, there is a region for d 1 where the model fits particularly well (R 2 > 0.9); that is, for both sexes, the region 0.03 < d 1 < 0.08 yields a good fit, as long as s is sufficiently high (i.e., s > 0.004). This is the parameter regime where our model can explain the observed data reasonably well, and observing values outside of that range would constitute evidence against the model and its underlying hypotheses.
New qualitative insights may also arise from a systematic investigation of temporal changes in optimal behaviour, such as age-dependent patterns in reactions to divorce. We first studied how age and expected future reproductive value affected the choice of behavioural mode u 1 (depressed) in the simple model (dotted line in Supplementary Figure 3D). Considering a simulated population in which most people are initially (at age 20) unpartnered, they increasingly enter into marriage and then in some cases transition to the relationship-at-risk state in which u 1 is beneficial. This leads to a rapid initial rise in the general prevalence of u 1 , followed by a long, relatively stable prevalence and a subsequent peak and drop in the last age periods. The peak towards the end reflects the fact that after the period the modelled reproductive resource 'ends'; keeping the resource is thus worth everything just before the final period and nothing in the exact final period. While conceptually relevant, these start-and end-game dynamics did not affect the divorce estimates in our simple constant-fertility model, because we excluded the first 20 and last 20 periods.
The simple model dealt more with the potential for reproduction (fecundity) than actual number of offspring (completed fertility), whereas our demographically parameterized model was set using data on completed fertility. In the latter model, the 'end effect' is much more pronounced due to a smoother, more rapid decline in the amount of completed fertility remaining (Supplementary Figure 3D). But there may be more at play than just end-period dynamics, because almost half the simulated population becomes depressed at the end. We thus roughly extrapolated the fertility, mortality, and partner-finding rates, simulating age periods 30-50, 40-60, and 50-70 years, in addition to the original 20-40 years. What happens is that completed male fertility gets very low and female fertility practically ends at the oldest age periods. Because the only currency that guides behaviour in our optimality model is the production of offspring, the use of the