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

Matthew Gergley; Vinodh Chellamuthu

doi:10.2478/ijmce-2026-0006

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

International Journal of Mathematics and Computer in Engineering

Volume 4 (2026): Issue 1 (June 2026)

By: Matthew Gergley and Vinodh Chellamuthu

Open Access

|May 2026

Figures & Tables

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 (y1*,y2*)(y_1^*,y_2^*) plane for κ=13×104\kappa = {1 \over 3} \times {10^4}. The three solid blue lines are defined as y2*=3y1*,y2*=y1*y_2^* = 3y_1^*,y_2^* = y_1^*, and y2*=13y1*y_2^* = {1 \over 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 cs nullcline, full model; Solid blue - Slow x nullcline, simplified model.

h2(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, g3 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 (t˜=24,48,⋯,168 hours)(\tilde t = 24,48, \cdots,168{\rm{ hours}}) represent a 24 hour block of time (8:00AM - 8:00AM (next day)).

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

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

Maximum values of φ¯(t˜)\overline{\varphi}(\tilde{t}) which occur when t˜=ti=0\tilde{t} = t_i = 0 in Eq_(5) with Nj=10N_j = 10 for each dj and changing z and W as needed_ In the everyday schedule, the value of φ¯(t˜)\overline{\varphi}(\tilde{t}) is uniform across each day since Nj is fixed_ This is not the case in Figure 6, however_ See Table 4_

	Beer	Wine	Liquor
Male	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 3.182 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 3.910 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 4.171 hours
Female	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 3.659 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 4.574 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 4.879 hours

Dimensional parameter values_

Parameter	Value	Description	Source(s)
c¯∞\overline{c}_{\infty}	0.2	Minimal stored baseline CRH	[28-30]
b	0.6	Stored CRH decay rate as a function of cortisol	[28-30]
tct_c	69.3	CRH biosynthesis timescale	[28-30]
q0q_0	28.0	Maximum release rate of CRH in basal state	[28-30]
I0I_0	1.0	Basal level of the external stimuli	[28-30]
k	2.83	Relates stored CRH to CRH release rate	[28-30]
gc,maxg_{c,\max}	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]
q1−1q_1^{-1}	25.0	Circulating CRH conc. at half-maximum self-up-regulation	[28-30]
q2q_2	1.8	Ratio of CRH and cortisol decay rates	[28-30]
p2−1p_2^{-1}	0.067	or-complex conc. for half-maximum negative feedback	[28-30]
p3p_3	7.2	Ratio of ACTH and cortisol decay rates	[28-30]
p4p_4	0.05	(or-complex conc.)2^2 at half-maximum positive feedback on r production	[28-30]
p5p_5	0.11	Basal GR production rate by pituitary	[28-30]
p6p_6	2.9	Ratio of GR and cortisol decay rates	[28-30]
ω\omega	0.045	Frequency of 24hr circadian rhythm	[28-30]
zBz_B	0.6	Fluid ounces of alcohol for 12oz. beer	[25]
zWz_W	0.75	Fluid ounces of alcohol for 6.25oz. wine	[25]
zLz_L	0.8	Fluid ounces of alcohol for 2oz. liquor	[25]
β\beta	0.00015	The alcohol elimination rate in kg/L/hr	[24,31]
σ\sigma	0.6	Volume of distribution (a constant relating the distribution of water in the body in L/kg)	[24]
δ\delta	0.8	The density of ethanol (0.8 oz. per fluid ounce)	[24]
WfW_f	2732.8	Average weight of a female in oz.	[32]
WmW_m	3196.8	Average weight of a male in oz.	[32]
A	30	Number of days in the period of concern
BACmaxBAC_{max}	0.0008	Legal limit of BAC in the United States in kg/L	[33]
λ\lambda	(varies) with λ∈[0,7]\lambda \in [0,7]	Number of days per week (7 days) where alcohol was consumed of any quantity and type.

Ranges of φ¯(t˜)\overline{\varphi}(\tilde{t}) by setting t˜=ti=0\tilde{t} = t_i = 0 in Eq_(5) with Nj=randint{1,10}N_j = \text{randint}\{1,10\} (See Appendix 7) for each dj and changing z and W as needed_ In the everyday schedule, the value of φ¯(t˜)\overline{\varphi}(\tilde{t}) varies daily_ See Figure 8_

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

Maximum values of φ¯(t˜)\overline{\varphi}(\tilde{t}) which occur when t˜=ti=0\tilde{t} = t_i = 0 in Eq_(5) with Nj=10N_j = 10 for each dj and changing z and W as needed_ In the everyday schedule, the value of φ¯(t˜)\overline{\varphi}(\tilde{t}) is uniform across each day_ This is not the case in Fig_ 8, however_ See Table 6_

	Beer	Wine	Liquor
Male	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 3.182 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 3.910 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 4.171 hours
Female	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 3.659 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 4.574 hours	max(φ¯(t˜))\overline{\varphi}(\tilde{t}) = 4.879 hours

j_ijmce-2026-0006_tab_003

College	Everyday	Weekdays	Weekend	Sporadic
0001111	1111111	1111100	0000011	ααααααα\alpha\alpha\alpha\alpha\alpha\alpha\alpha

Correspondence between t values and times of day_

t (hours past 8:00 AM)	Time of Day
0	8:00 AM
π/2	2:00 PM
π	8:00 PM
3π/2	2:00 AM (next day)
2π	8:00 AM (next day)

Ranges of φ¯(t˜)\overline{\varphi}(\tilde{t}) by setting t˜=ti=0\tilde{t} = t_i = 0 in Eq_(5) with Nj=randint{1,10}N_j = \text{randint}\{1,10\} (See Appendix 7) for each dj and changing z and W as needed_ In the everyday schedule, the value of φ¯(t˜)\overline{\varphi}(\tilde{t}) varies daily, since Nj is not fixed_ See Figure 6_

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