A mathematical model of HPA axis dynamics and impacts of alcohol consumption

Parameter	Value	Description	Source(s)
${\bar{c}}_{\infty}$	0.2	Minimal stored baseline CRH	[28-30]
b	0.6	Stored CRH decay rate as a function of cortisol	[28-30]
$t_{c}$	69.3	CRH biosynthesis timescale	[28-30]
$q_{0}$	28.0	Maximum release rate of CRH in basal state	[28-30]
$I_{0}$	1.0	Basal level of the external stimuli	[28-30]
k	2.83	Relates stored CRH to CRH release rate	[28-30]
$g_{c, m a x}$	42.0	Maximum auto/paracrine effect of CRH in the pituitary	[28-30]
n	5	Hill coefficient describing the self-up-regulation of CRH	[28-30]
$q_{1}^{- 1}$	25.0	Circulating CRH conc. at half-maximum self-up-regulation	[28-30]
$q_{2}$	1.8	Ratio of CRH and cortisol decay rates	[28-30]
$p_{2}^{- 1}$	0.067	or-complex conc. for half-maximum negative feedback	[28-30]
$p_{3}$	7.2	Ratio of ACTH and cortisol decay rates	[28-30]
$p_{4}$	0.05	(or-complex conc.) $^{2}$ at half-maximum positive feedback on r production	[28-30]
$p_{5}$	0.11	Basal GR production rate by pituitary	[28-30]
$p_{6}$	2.9	Ratio of GR and cortisol decay rates	[28-30]
$ω$	0.045	Frequency of 24hr circadian rhythm	[28-30]
$z_{B}$	0.6	Fluid ounces of alcohol for 12oz. beer	[25]
$z_{W}$	0.75	Fluid ounces of alcohol for 6.25oz. wine	[25]
$z_{L}$	0.8	Fluid ounces of alcohol for 2oz. liquor	[25]
$β$	0.00015	The alcohol elimination rate in kg/L/hr	[24,31]
$σ$	0.6	Volume of distribution (a constant relating the distribution of water in the body in L/kg)	[24]
$δ$	0.8	The density of ethanol (0.8 oz. per fluid ounce)	[24]
$W_{f}$	2732.8	Average weight of a female in oz.	[32]
$W_{m}$	3196.8	Average weight of a male in oz.	[32]
A	30	Number of days in the period of concern
$B A C_{m a x}$	0.0008	Legal limit of BAC in the United States in kg/L	[33]
$λ$	(varies) with $λ \in [0, 7]$	Number of days per week (7 days) where alcohol was consumed of any quantity and type.

College	Everyday	Weekdays	Weekend	Sporadic
0001111	1111111	1111100	0000011	$α α α α α α α$

T The dotted curves represent the solutions to u(c) in the full model, Eq.(21). The solid curve represents the solutions to u(c) in the simplified model, Eq.(23). Parameters from Table 2.

The grey region represents bistability in the $(y_{1}^{*}, y_{2}^{*})$ plane for $K = \frac{1}{3} \times 10^{4}$ . The three solid blue lines are defined as $y_{2}^{*} = 3 y_{1}^{*}$ , $y_{2}^{*} = y_{1}^{*}$ , and $y_{2}^{*} = \frac{1}{3} y_{1}^{*}$ . The red dotted and dashed curves are defined by Eq.(35) and Eq.(36). Parameters used are from Table 2, and the red dot indicates the intersection of Eq.(35) and Eq.(36) when h = 1 for reference.

Dotted red - Fast c nullcline, full model; Solid red - Fast y nullcline, simplified model; Dashed blue - Slow *c_s* nullcline, full model; Solid blue - Slow x nullcline, simplified model.

$h_{2} (t, 33, μ)$ : These plots represent the variation in the rate h₂, using g₂, when fixing an age to x = 33 and applying different drinking schedules and alcohol types/concentrations (See Appendix 7). Recall t = 0 corresponds to 8:00 AM. (g₂ and x = 33 were chosen for simulation purposes. The plots utilizing g₁ can be found in the GitHub within Appendix 7).

Plots for consuming ten drinks following the everyday schedule for beer, wine, and liquor. The time of drinking is 9:00 PM, 13 hours after peak cortisol levels (8:00 AM). The top subplot for each plot shows the linear decay of the BAC level for each occurrence of drinking. In the bottom plots, the solid purple line represents the individual's circadian rhythm while the black dotted line represents the circadian rhythm if no alcohol was consumed. The vertical red dotted lines ( $\tilde{t} = 24, 48, \dots, 168$ hours) represent a 24 hour block of time (8:00AM - 8:00AM (next day)).

Table 3

Maximum values of $\bar{φ} (\tilde{t})$ which occur when $\tilde{t} = t_{i} = 0$ in Eq.(5) with $N_{j} = 10$ for each d_j and changing z and W as needed. In the everyday schedule, the value of $\bar{φ} (\tilde{t})$ is uniform across each day since N_j is fixed. This is not the case in Figure 6, however. See Table 4.

	Beer	Wine	Liquor
Male	$m a x (\bar{φ} (\tilde{t}))$ = 3.182 hours	$m a x (\bar{φ} (\tilde{t}))$ = 3.910 hours	$m a x (\bar{φ} (\tilde{t}))$ = 4.171 hours
Female	$m a x (\bar{φ} (\tilde{t}))$ = 3.659 hours	$m a x (\bar{φ} (\tilde{t}))$ = 4.574 hours	$m a x (\bar{φ} (\tilde{t}))$ = 4.879 hours

Plots for consuming a random number of drinks each day $(N_{j} = randint {1, 10})$ , following the everyday schedule for beer, wine, and liquor and varying W as needed. The value of *N_j* for each day can be found in Appendix 7.

Table 4

Ranges of $\bar{φ} (\tilde{t})$ by setting $\tilde{t} = t_{i} = 0$ in Eq.(5) with $N_{j} = randint {1, 10}$ (See Appendix 7) for each d_j and changing z and W as needed. In the everyday schedule, the value of $\bar{φ} (\tilde{t})$ varies daily, since N_j is not fixed. See Figure 6.

	Beer	Wine	Liquor
Male	0.3128 ≤ $\bar{φ} (\tilde{t})$ ≤ 2.5025 (hours)	0.3910 ≤ $\bar{φ} (\tilde{t})$ ≤ 3.5191 (hours)	0.4171 ≤ $\bar{φ} (\tilde{t})$ ≤ 3.7538 (hours)
Female	0.3659 ≤ $\bar{φ} (\tilde{t})$ ≤ 2.9274 (hours)	0.9148 ≤ $\bar{φ} (\tilde{t})$ ≤ 4.1167 (hours)	0.4879 ≤ $\bar{φ} (\tilde{t})$ ≤ 3.4153 (hours)

Table 5

Maximum values of $\bar{φ} (\tilde{t})$ which occur when $\tilde{t} = t_{i} = 0$ in Eq.(5) with $N_{j} = 10$ for each d_j and changing z and W as needed. In the everyday schedule, the value of $\bar{φ} (\tilde{t})$ is uniform across each day. This is not the case in Fig. 8, however. See Table 6.

	Beer	Wine	Liquor
Male	$m a x (\bar{φ} (\tilde{t}))$ = 3.182 hours	$m a x (\bar{φ} (\tilde{t}))$ = 3.910 hours	$m a x (\bar{φ} (\tilde{t}))$ = 4.171 hours
Female	$m a x (\bar{φ} (\tilde{t}))$ = 3.659 hours	$m a x (\bar{φ} (\tilde{t}))$ = 4.574 hours	$m a x (\bar{φ} (\tilde{t}))$ = 4.879 hours

Plots for consuming a random number of drinks each day, following the everyday schedule for beer, wine, and liquor. The value of *N_j* for each day can be found in Appendix 7.

Table 6

	Beer	Wine	Liquor
Male	0.3128 ≤ $\bar{φ} (\tilde{t})$ ≤ 2.8153 (hours)	0.3910 ≤ $\bar{φ} (\tilde{t})$ ≤ 3.5191 (hours)	0.8342 ≤ $\bar{φ} (\tilde{t})$ ≤ 2.9196 (hours)
Female	0.3659 ≤ $\bar{φ} (\tilde{t})$ ≤ 3.2933 (hours)	1.3722 ≤ $\bar{φ} (\tilde{t})$ ≤ 4.1167 (hours)	0.9758 ≤ $\bar{φ} (\tilde{t})$ ≤ 2.9274 (hours)