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Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1238-1279. doi: 10.3934/mbe.2021067
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Research article

A mathematical model of food intake

  • 1.
    Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
  • 2.
    Center of Excellence in Mathematics, the Commission on Higher Education, Si Ayutthaya Rd., Bangkok 10400, Thailand
  • 3.
    Consiglio Nazionale delle Ricerche, Istituto per la Ricerca e l'Innovazione Biomedica (CNR-IRIB), Palermo, Italy
  • 4.
    Consiglio Nazionale delle Ricerche, Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" (CNR-IASI), Rome, Italy
  • Received: 06 August 2020 Accepted: 30 November 2020 Published: 15 January 2021

  • The metabolic, hormonal and psychological determinants of the feeding behavior in humans are numerous and complex. A plausible model of the initiation, continuation and cessation of meals taking into account the most relevant such determinants would be very useful in simulating food intake over hours to days, thus providing input into existing models of nutrient absorption and metabolism. In the present work, a meal model is proposed, incorporating stomach distension, glycemic variations, ghrelin dynamics, cultural habits and influences on the initiation and continuation of meals, reflecting a combination of hedonic and appetite components. Given a set of parameter values (portraying a single subject), the timing and size of meals are stochastic. The model parameters are calibrated so as to reflect established medical knowledge on data of food intake from the National Health and Nutrition Examination Survey (NHANES) database during years 2015 and 2016.

    Citation: Mantana Chudtong, Andrea De Gaetano. A mathematical model of food intake[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1238-1279. doi: 10.3934/mbe.2021067

    Related Papers:

  • The metabolic, hormonal and psychological determinants of the feeding behavior in humans are numerous and complex. A plausible model of the initiation, continuation and cessation of meals taking into account the most relevant such determinants would be very useful in simulating food intake over hours to days, thus providing input into existing models of nutrient absorption and metabolism. In the present work, a meal model is proposed, incorporating stomach distension, glycemic variations, ghrelin dynamics, cultural habits and influences on the initiation and continuation of meals, reflecting a combination of hedonic and appetite components. Given a set of parameter values (portraying a single subject), the timing and size of meals are stochastic. The model parameters are calibrated so as to reflect established medical knowledge on data of food intake from the National Health and Nutrition Examination Survey (NHANES) database during years 2015 and 2016.


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    [1] C. C. Chow, K. D. Hall, The dynamics of human body weight change, PLOS Comput. Biol., 4 (2008).
    [2] WHO obesity and overweight, 2018. Available from: http://www.who.int/mediacentre/factsheets/fs311/en/.
    [3] H. Gu, Sh. Shao, J. Liu, Zh. Fan, Y. Chen, J. Ni, et.al., Age- and sex-associated impacts of body mass index on stroke type risk: a 27-year prospective cohort study in a low-income population in china, Front. Neurol., 10 (2019).
    [4] K. Cheng, Health oriented lifelong nutrition controls: preventing cardiovascular diseases caused by obesity, SM J. Nutr. Metab., 6 (2020), 1–5.
    [5] K. Mc Namara, H. Alzubaidi, J. K. Jackson, Cardiovascular disease as a leading cause of death: how are pharmacists getting involved?, Integr. Pharm. Res. Pract., 8 (2019), 1–11.
    [6] E. J. Benjamin, P. Muntner, A. Alonso, M. S. Bittencourt, C. W. Callaway, A. P. Carson, et. al., Heart disease and stroke statistics—2019 update: a report from the american heart association, Circulation, 139 (2019), 56–528
    [7] N. Taghizadeh,, H. M. Boezen, J. P. Schouten, C. P. Schröder, E. G. E. de Vries, J. M. Vonk, BMI and lifetime changes in BMI and cancer mortality risk, PLoS ONE, 10 (2015).
    [8] K. Bhaskaran, I. Douglas, H. Forbes, I. dos Santos-Silva, D. A. Leon, and L. Smeeth, Bodymass index and risk of 22 specific cancers: a population-based cohort study of 5.24 million UK adults, Lancet, 384 (2014), 755–765.
    [9] A. S. Barnes, The epidemic of obesity and diabetes, Tex. Heart I. J., 38 (2011), 142–144.
    [10] A. Golay and J. Ybarra, Link between obesity and type 2 diabetes, Best Pract. Res. Cl. En., 19 (2005), 649–663.
    [11] A. S. Al-Goblan, M. A. Al-Alfi, and M. Z. Khan, Mechanism linking diabetes mellitus and obesity, Diabetes Metab. Syndr. Obes., 7 (2014), 587–591.
    [12] N. A. Roper, R. W. Bilous, W. F. Kelly, N. C. Unwin, and V. M. Connolly, Cause-specific mortality in a population with diabetes: south tees diabetes mortality study, Diabetes Care, 25 (2002), 43–48.
    [13] M. Tancredi, A. Rosengren, A.-M. Svensson, M. Kosiborod, A. Pivodic, S. Gudbjörnsdottir, et. al., Excess mortality among persons with type 2 diabetes, New Engl. J. Med., 373 (2015), 1720–1732.
    [14] M. Kalligeros, F. Shehadeh, E. K. Mylona, G. Benitez, C. G. Beckwith, P. A. Chan, et.al., Association of obesity with disease severity among patients with coronavirus disease 2019, Obesity, 28 (2020), 1200–1204.
    [15] R. A. DeFronzo, R. C. Bonadonna, E. Ferrannini, Pathogenesis of NIDDM: a balanced overview, Diabetes Care, 15 (1992), 318–368.
    [16] E. Archer, C. J. Lavie, and J. O. Hill, The contributions of 'diet', 'genes', and physical activity to the etiology of obesity: contrary evidence and consilience, Prog. Cardiovasc. Dis., 61 (2018), 89–102.
    [17] A. D. Baron, G. Brechtel, P. Wallace, S. V. Edelman, Rates and tissue sites of non-insulin- and insulin-mediated glucose uptake in humans, Am. J. Physiol. Endoc. M., 255 (1988), 769–774.
    [18] R. A. DeFronzo, D. Tripathy, Skeletal muscle insulin resistance is the primary defect in type 2 diabetes, Diabetes Care, 32 (2009), S157–S163.
    [19] E. Archer, G. Pavela, S. McDonald, C. J. Lavie, and J. O. Hill, Cell-specific "competition for calories" drives asymmetric nutrient-energy partitioning, obesity, and metabolic diseases in human and non-human animals, Front. Physiol., 9 (2018), 1053.
    [20] R. A. DeFronzo, The triumvirate: $\beta-$cell, muscle, liver: a collusion responsible for NIDDM, Diabetes, 37 (1988), 667–687.
    [21] A. V. Greco, G. Mingrone, A. Giancaterini, M. Manco, M. Morroni, S. Cinti, et. al., Insulin resistance in morbid obesity: reversal with intramyocellular fat depletion, Diabetes, 51 (2002), 144–151.
    [22] O. T. Hardy, M. P. Czech, S. Corvera, What causes the insulin resistance underlying obesity?, Curr. Opin. Endocrinol., 19 (2012), 81–87.
    [23] C. Roberts-Toler, B. T. O'Neill, A. M. Cypess, Diet-induced obesity causes insulin resistance in mouse brown adipose tissue: dio causes bat insulin resistance, Obesity, 23 (2015), 1765–1770.
    [24] R. Firth, P. Bell, H. Marsh, I. Hansen, R. Rizza, Postprandial hyperglycemia in patients with noninsulin-dependent diabetes mellitus, J. Clin. Invest., 77 (1986), 1525–1532.
    [25] Understanding satiety: feeling full after a meal - British nutrition foundation, 2018. Available from: https://www.nutrition.org.uk/healthyliving/fuller/understanding-satiety-feeling-full-after-a-meal.html.
    [26] E. Bilman, E. van Kleef, H. van Trijp, External cues challenging the internal appetite control system—overview and practical implications, Cr. Rev. Food Sci., 57 (2017), 2825–2834.
    [27] EO. G. Edholm, J. G. Fletcher, E. M. Widdowson, R. A. McCance, The energy expenditure and food intake of individual men, Brit. J. Nutr., 9 (1955), 286–300.
    [28] C. B. Saper, T. C. Chou, J. K. Elmquist, The need to feed: homeostatic and hedonic control of eating, Neuron, 36 (2002), 199–211.
    [29] E. Mamontov, Modelling homeorhesis by ordinary differential equations, Math. Comput. Model., 45 (2007), 694–707.
    [30] D. F. Marks, Homeostatic theory of obesity, Health Psychology Open, 2 (2015), 1–30.
    [31] R. D. Palmiter, Is dopamine a physiologically relevant mediator of feeding behavior?, Trends Neurosci., 30 (2007), 375–381.
    [32] D. E. Cummings, Ghrelin and the short- and long-term regulation of appetite and body weight, Physiol. Behav., 89 (2006), 71–84.
    [33] J. Vartiainen, Ghrelin, obesity and type 2 diabetes: genetic, metabolic and epidemiological studies, Ph.D thesis, University of Oulu, 2009.
    [34] I. Nilsson, C. Lindfors, S. O. Fetissov, T. Hökfelt, J. E. Johansen, Aberrant agouti-related protein system in the hypothalamus of the anx/anx mouse is associated with activation of microglia, J. Comp. Neurol., 507 (2008), 1128–1140.
    [35] A. M. Chao, A. M. Jastrebo, M. A. White, C. M. Grilo, R. Sinha, Stress, cortisol, and other appetite-related hormones: prospective prediction of 6-month changes in food cravings and weight, Obesity (Silver Spring, Md.), 25 (2017), 713–720.
    [36] A. Uchida, J. M. Zigman, and M. Perello, Ghrelin and eating behavior: evidence and insights from genetically-modified mouse models, Front. Neurosci., 7 (2013), 121.
    [37] Y. Sun, S. Ahmed, R. G. Smith, Deletion of ghrelin impairs neither growth nor appetite, Mol. Cell. Biol., 23 (2003), 7973–7981.
    [38] K. E. Wortley, K. D. Anderson, K. Garcia, J. D. Murray, L. Malinova, R. Liu, M. Moncrieffe, et.al., Genetic deletion of ghrelin does not decrease food intake but influences metabolic fuel preference, P. Natl. Acad. Sci. USA., 101 (2004), 8227–8232.
    [39] K. E. Wortley, J.-P. del Rincon, J. D. Murray, K. Garcia, K. Iida, M. O. Thorner, et. al., Absence of ghrelin protects against early-onset obesity, J. Clin. Invest., 115 (2005), 3573–3578.
    [40] B. De Smet, I. Depoortere, D. Moechars, Q. Swennen, B. Moreaux, K. Cryns, et. al., Energy homeostasis and gastric emptying in ghrelin knockout mice, J. Pharmacol. Exp. Ther., 316 (2006), 431–439.
    [41] K. Dezaki, H. Sone, M. Koizumi, M. Nakata, M. Kakei, H. Nagai, et. al., Blockade of pancreatic islet-derived ghrelin enhances insulin secretion to prevent high-fat diet-induced glucose intolerance, Diabetes, 55 (2006), 3486–3493.
    [42] P. T. Pfluger, H. Kirchner, S. Günnel, B. Schrott, D. Perez-Tilve, S. Fu, et. al., Simultaneous deletion of ghrelin and its receptor increases motor activity and energy expenditure, Am. J. Physiol. Gastrointest. Liver Physiol., 294 (2008), 610–618.
    [43] T. Sato, M. Kurokawa, Y. Nakashima, T. Ida, T. Takahashi, Y. Fukue, et. al., Ghrelin deficiency does not influence feeding performance., Regul. Peptides, 145 (2008), 7–11.
    [44] Y. Sun, P. Wang, H. Zheng, R. G. Smith, Ghrelin stimulation of growth hormone release and appetite is mediated through the growth hormone secretagogue receptor, P. Natl. Acad. Sci. USA., 101 (2004), 4679–4684.
    [45] A. Abizaid, Z.-W. Liu, Z. B. Andrews, M. Shanabrough, E. Borok, J. D. Elsworth, et. al., Ghrelin modulates the activity and synaptic input organization of midbrain dopamine neurons while promoting appetite, J. Clin. Invest., 116 (2006), 3229–3239.
    [46] I. D. Blum, Z. Patterson, R. Khazall, E. W. Lamont, M. W. Sleeman, T. L. Horvath, et. al., Reduced anticipatory locomotor responses to scheduled meals in ghrelin receptor deficient mice, Neuroscience, 164 (2009), 351–359.
    [47] L. Lin, P. K. Saha, X. Ma, I. O. Henshaw, L. Shao, B. H. J. Chang, E. D. Buras, et. al., Ablation of ghrelin receptor reduces adiposity and improves insulin sensitivity during aging by regulating fat metabolism in white and brown adipose tissues, Aging Cell, 10 (2011), 996–1010.
    [48] X. Ma, L. Lin, G. Qin, X. Lu, M. Fiorotto, V. D. Dixit, et. al., Ablations of ghrelin and ghrelin receptor exhibit differential metabolic phenotypes and thermogenic capacity during aging, PLOS ONE, 6 (2011), 1–10.
    [49] M. D. Klok, S. Jakobsdottir, M. L. Drent, The role of leptin and ghrelin in the regulation of food intake and body weight in humans: a review, Obes. Rev., 8 (2007), 21–34.
    [50] D. P. Figlewicz, S. B. Evans, J. Murphy, M. Hoen, D. G. Baskin, Expression of receptors for insulin and leptin in the ventral tegmental area/substantia nigra (vta/sn) of the rat, Brain Res., 964 (2003), 107–115.
    [51] D. P. Figlewicz, P. Szot, M. Chavez, S. C. Woods, R. C. Veith, Intraventricular insulin increases dopamine transporter mrna in rat vta/substantia nigra, Brain Res., 644 (1994), 331–334.
    [52] What is non-diabetic hypoglycemia?, 2019. Available from: https://www.webmd.com/diabetes/non-diabetic-hypoglycemia.
    [53] Hypoglycemia: signs, risks, causes, and how to raise low blood sugar |everyday health, 2019. Available from: https://www.everydayhealth.com/hypoglycemia/guide/.
    [54] Low blood sugar? 8 warning signs if you have diabetes, 2019. Available from: https://health.clevelandclinic.org/low-blood-sugar-8-warning-signs-diabetes/.
    [55] Understanding hypoglycemia, 2019. Available from: https://www.diabetesselfmanagement.com/managing-diabetes/blood-glucose-management/understanding-hypoglycemia/.
    [56] The effects of low blood sugar on your body, 2019. Available from: https://www.healthline.com/health/low-blood-sugar-effects-on-body.
    [57] Hypoglycemia (low blood sugar): causes and treatment, 2019. Available from: https://www.medicalnewstoday.com/articles/166815.php.
    [58] Non-diabetic hypoglycemia: symptoms, causes, diagnosis, treatment, 2019. Available from: https://www.webmd.com/diabetes/non-diabetic-hypoglycemia#1.
    [59] Polyphagia: the relationship between hunger and diabetes, 2019. Available from: https://www.thediabetescouncil.com/.
    [60] J. L. Jameson, L. J. D. Groot, Endocrinology: adult and pediatric, 7th edition, Elsevier Health Sciences, 2015.
    [61] J. R. Hupp, M. R. Tucker, E. Ellis, Contemporary oral and maxillofacial surgery, 1st edition, Elsevier Health Sciences, 2013.
    [62] D. A. Schatz, M. Haller, M. Atkinson, Type 1 diabetes, an issue of endocrinology and metabolism clinics of north america, 1st edition, Elsevier Health Sciences, 2010.
    [63] L. A. Fleisher, M. F. Roizen, J. Roizen, Essence of anesthesia practice, 4th edition, Elsevier Health Sciences, 2017.
    [64] G. Cheney, Medical management of gastrointestinal disorders, 1st edition, Year Book, 1950.
    [65] M, Manthappa, How to manage your diabetes and lead a normal life, 1st edition, Peacock Books, 2009.
    [66] R. K. Bernstein, Hunger–a common symptom of hypoglycemia, Diabetes Care, 16 (1993), 1049.
    [67] C. Kenny, When hypoglycemia is not obvious: diagnosing and treating under-recognized and undisclosed hypoglycemia, Prim. Care Diabetes, 8 (2014), 3–11.
    [68] J. Morales, D. Schneider, Hypoglycemia, Am. J. Med., 127 (2014), 17–24.
    [69] L. C. Perlmuter, B. P. Flanagan, P. H. Shah, S. P. Singh, Glycemic control and hypoglycemia, Diabetes Care, 31 (2008), 2072–2076.
    [70] B. Schultes, K. M. Oltmanns, W. Kern, H. L. Fehm, J. Born, A. Peters, Modulation of hunger by plasma glucose and metformin, J. Clin. Endocrinol. Metab., 88 (2003), 1133–1141.
    [71] B. Schultes, A. Peters, M. Hallschmid, C. Benedict, V. Merl, K. M. Oltmanns, et. al., Modulation of food intake by glucose in patients with type 2 diabetes, Diabetes Care, 28 (2005), 2884–2889.
    [72] H. A. J. Gielkens, M. Verkijk, W. F. Lam, C. B. H. W. Lamers, A. A. M. Masclee, Effects of hyperglycemia and hyperinsulinemia on satiety in humans, Metabolism, 47 (1998), 321–324.
    [73] B. Schultes, A. K. Panknin, M. Hallschmid, K. Jauch-Chara, B.Wilms, F. de Courbière, et. al., Glycemic increase induced by intravenous glucose infusion fails to affect hunger, appetite, or satiety following breakfast in healthy men, Appetite, 105 (2016), 562–566.
    [74] J. M. McMillin, Blood glucose, 3rd edition, Butterworths, Boston, 1990.
    [75] J. Yokrattanasak, A. De Gaetano, S. Panunzi, P. Satiracoo, W. M. Lawton, Y. Lenbury, A simple, realistic stochastic model of gastric emptying, PLoS ONE, 11 (2016).
    [76] L. K. Phillips, C. K. Rayner, K. L. Jones, M. Horowitz, Measurement of gastric emptying in diabetes, J. Diabetes Complicat., 28 (2014), 894–903.
    [77] S. G. Cao, H. Wu, Z. Z. Cai, Dose-dependent effect of ghrelin on gastric emptying in rats and the related mechanism of action, The Kaohsiung J. Med. Sci., 32 (2016), 113–117.
    [78] L. X. Yu, G. L. Amidon, A compartmental absorption and transit model for estimating oral drug absorption, Int. J. Pharm., 186 (1999), 119–125.
    [79] K. Ogungbenro, L. Aarons, A semi-mechanistic gastric emptying pharmacokinetic model for (13)C-octanoic acid: an evaluation using simulation, Eur. J. Pharm. Sci., 45 (2012), 302–310.
    [80] J. D. Berke, S. E. Hyman, Addiction, dopamine, and the molecular mechanisms of memory, Neuron, 25 (2000), 515–532.
    [81] R. A. Wise, Addictive drugs and brain stimulation reward, Annu. Rev. Neurosci., 19 (1996), 319–340.
    [82] P. M. Milner, Brain-stimulation reward: a review, Can. J. Psychol., 45 (1991), 1–36.
    [83] J. M. Liebman, Discriminating between reward and performance: a critical review of intracranial self-stimulation methodology, Neurosci. Biobehav. R., 7 (1983), 45–72.
    [84] R. A. Wise, Brain reward circuitry: insights from unsensed incentives, Neuron, 36 (2002), 229–240.
    [85] A. E. Kelley, V. P. Bakshi, S. N. Haber, T. L. Steininger, M. J.Will, M. Zhang, Opioid modulation of taste hedonics within the ventral striatum, Physiol. Behav., 76 (2002), 365–377.
    [86] R. Coccurello, M. Maccarrone, Hedonic eating and the "delicious circle": from lipid-derived mediators to brain dopamine and back, Front. Neurosci-Switz., 12 (2018).
    [87] S. L. Teegarden, T. L. Bale, Decreases in dietary preference produce increased emotionality and risk for dietary relapse, Biol. Psychiat., 61 (2007), 1021–1029.
    [88] B. G. Hoebel, N. M. Avena, M. E. Bocarsly, P. Rada, Natural addiction: a behavioral and circuit model based on sugar addiction in rats, J. Addict. Med., 3 (2009), 33–41.
    [89] J.W. Dalley, B. J. Everitt, T.W. Robbins, Impulsivity, compulsivity, and top-down cognitive control, Neuron, 69 (2011), 680–694.
    [90] E. N. Pothos, V. Davila, D. Sulzer, Presynaptic recording of quanta from midbrain dopamine neurons and modulation of the quantal size, J. Neurosci., 18 (1998), 4106–4118.
    [91] S. Fulton, P. Pissios, R. P. Manchon, L. Stiles, L. Frank, E. N. Pothos, et. al., Leptin regulation of the mesoaccumbens dopamine pathway, Neuron, 51 (2006), 811–822.
    [92] K. Toshinai, Y. Date, N. Murakami, M. Shimada, M. S. Mondal, T. Shimbara, J. L. Guan, et. al., Ghrelin-induced food intake is mediated via the orexin pathway, Endocrinology, 144 (2003), 1506–1512.
    [93] H. Y. Chen, M. E. Trumbauer, A. S. Chen, D. T. Weingarth, J. R. Adams, E. G. Frazier, et. al., Orexigenic action of peripheral ghrelin is mediated by neuropeptide y and agouti-related protein, Endocrinology, 145 (2004), 2607–2612.
    [94] S. Luquet, C. T. Phillips, R. D. Palmiter, NPY/AgRP neurons are not essential for feeding responses to glucoprivation, Peptides, 28 (2007), 214–225.
    [95] K. Bugarith, T. T. Dinh, A. J. Li, R. C. Speth, S. Ritter, Basomedial hypothalamic injections of neuropeptide Y conjugated to saporin selectively disrupt hypothalamic controls of food intake, Endocrinology, 146 (2005), 1179–1191.
    [96] Y. Date, T. Shimbara, S. Koda, K. Toshinai, T. Ida, N. Murakami, et. al., Peripheral ghrelin transmits orexigenic signals through the noradrenergic pathway from the hindbrain to the hypothalamus, Cell Metab., 4 (2006), 323–331.
    [97] M. L. Westwater, P. C. Fletcher, H. Ziauddeen, Sugar addiction: the state of the science, Eur. J. Nutr., 55 (2016), 55–69.
    [98] P. C. Fletcher, P. J. Kenny, Food addiction: a valid concept?, Neuropsychopharmacol., 43 (2018), 2506–2513.
    [99] T. L. Davidson, S. Jones, M. Roy, R. J. Stevenson, The cognitive control of eating and body weight: it's more than what you "think", Front. Psychol., 10 (2019).
    [100] A. De Gaetano, T. A. Hardy, A novel fast-slow model of diabetes progression: insights into mechanisms of response to the interventions in the diabetes prevention program, PLOS ONE, 14 (2019), 1–39.
    [101] J. Ha, L. S. Satin, A. S. Sherman, A mathematical model of the pathogenesis, prevention, and reversal of type 2 diabetes, Endocrinology, 157 (2016), 624–635.
    [102] A. Borri, S. Panunzi, A. De Gaetano, A glycemia-structured population model, J. Math. Biol., 73 (2016), 39–62.
    [103] P. Palumbo, S. Ditlevsen, A. Bertuzzi, A. De Gaetano, Mathematical modeling of the glucose-insulin system: a review, Math. Biosci., 244 (2013), 69–81.
    [104] I. Ajmera, M. Swat, C. Laibe, N. L. Nov'ere, V. Chelliah, The impact of mathematical modeling on the understanding of diabetes and related complications, CPT: Pharmacometrics Syst. Pharmacol., 2 (2013).
    [105] T. Hardy, E. Abu-Raddad, N. Porksen, A. De Gaetano, Evaluation of a mathematical model of diabetes progression against observations in the diabetes prevention program, Am. J. Physiol. Endoc. M., 303 (2012), 200–212.
    [106] J. Ribbing, B. Hamrén, M. K. Svensson, M. O. Karlsson, A model for glucose, insulin, and beta-cell dynamics in subjects with insulin resistance and patients with type 2 diabetes, J. Clin. Pharmacol., 50 (2010), 861–872.
    [107] A. De Gaetano, T. Hardy, B. Beck, E. Abu-Raddad, P. Palumbo, J. Bue-Valleskey, et. al., Mathematical models of diabetes progression, Am. J. Physiol. Endoc. M., 295 (2008), 1462–1479.
    [108] C. C. Mason, R. L. Hanson, W. C. Knowler, Progression to type 2 diabetes characterized by moderate then rapid glucose increases, Diabetes, 56 (2007), 2054–2061.
    [109] W. de Winter, J. DeJongh, T. Post, B. Ploeger, R. Urquhart, I. Moules, et. al., A mechanism-based disease progression model for comparison of long-term effects of pioglitazone, metformin and gliclazide on disease processes underlying type 2 diabetes mellitus, J. Pharmacokinet. Phar., 33 (2006), 313–343.
    [110] A. Bagust, M. Evans, S. Beale, P. D. Home, A. S. Perry, M. Stewart, A model of long-term metabolic progression of type 2 diabetes mellitus for evaluating treatment strategies, PharmacoEconomics, 24 Suppl 1 (2006), 5–19.
    [111] B. Topp, K. Promislow, G. deVries, R. M. Miura, D. T. Finegood, A model of beta-cell mass, insulin, and glucose kinetics: pathways to diabetes, J. Theor. Biol., 206 (2000), 605–619.
    [112] T. Okura, R. Nakamura, Y. Fujioka, S. Kawamoto-Kitao, Y. Ito, K. Matsumoto, et. al., Body mass index $\geq23$ is a risk factor for insulin resistance and diabetes in japanese people: a brief report, PLOS ONE, 13 (2018).
    [113] Y. H. Cheng, Y. C. Tsao, I. S. Tzeng, H. H. Chuang, W. C. Li, T. H. Tung, et. al., Body mass index and waist circumference are better predictors of insulin resistance than total body fat percentage in middle-aged and elderly Taiwanese, Medicine, 96 (2017).
    [114] J. A. Hawley, Exercise as a therapeutic intervention for the prevention and treatment of insulin resistance, Diabetes Metab. Res. Rev., 20 (2004), 383–393.
    [115] R. N. Bergman, Y. Z. Ider, C. R. Bowden, C. Cobelli, Quantitative estimation of insulin sensitivity., Am. J. Physiol. Endoc. M., 236 (1979).
    [116] G. Toffolo, R. N. Bergman, D. T. Finegood, C. R. Bowden, C. Cobelli, Quantitative estimation of beta cell sensitivity to glucose in the intact organism: a minimal model of insulin kinetics in the dog, Diabetes, 29 (1980), 979–990.
    [117] C. Dalla Man, R. A. Rizza, C. Cobelli, Meal simulation model of the glucose-insulin system, IEEE trans. Biomed. Eng., 54 (2007), 1740–1749.
    [118] W. Liu, F. Tang, Modeling a simplified regulatory system of blood glucose at molecular levels, J. Theor. Biol., 252 (2008), 608–620.
    [119] W. Liu, C. Hsin, F. Tang, A molecular mathematical model of glucose mobilization and uptake, Math. Biosci., 221 (2009), 121–129.
    [120] Z. Wu, C. K. Chui, G. S. Hong, S. Chang, Physiological analysis on oscillatory behavior of glucose–insulin regulation by model with delays, J. Theor. Biol., 280 (2011), 1–9.
    [121] M. Lombarte, M. Lupo, G. Campetelli, M. Basualdo, A. Rigalli, Mathematical model of glucose–insulin homeostasis in healthy rats, Math. Biosci., 245 (2013), 269–277.
    [122] A. C. Pratt, J. A. D. Wattis, A. M. Salter, Mathematical modelling of hepatic lipid metabolism, Math. Biosci., 262 (2015), 167–181.
    [123] J. Girard, The incretins: from the concept to their use in the treatment of type 2 diabetes. part a: incretins: concept and physiological functions, Diabetes Metab., 34 (2008), 550–559.
    [124] J. J. Holst, C. F. Deacon, T. Vilsbøll, T. Krarup, S. Madsbad, Glucagon-like peptide-1, glucose homeostasis and diabetes, Trends Mol. Med., 14 (2008), 161–168.
    [125] J. J. Holst, T. Vilsbøll, C. F. Deacon, The incretin system and its role in type 2 diabetes mellitus, Mol. Cell. Endocrinol., 297 (2009), 127–136.
    [126] K. Kazakos, Incretin effect: GLP-1, GIP, DPP4, Diabetes Res. Clin. Pr., 93 (2011), 32–36.
    [127] J. J. Holst, C. F. Deacon, Is there a place for incretin therapies in obesity and prediabetes?, Trends Endocrin. Met., 24 (2013), 145–152.
    [128] S. Masroor, M. G. J. van Dongen, R. Alvarez-Jimenez, K. Burggraaf, L. A. Peletier, M. A. Peletier, Mathematical modeling of the glucagon challenge test, J. Pharmacokinet. Phar., 46 (2019), 553–564.
    [129] S. J. Russell, F. H. El-Khatib, M. Sinha, K. L. Magyar, K. McKeon, L. G. Goergen, et. al., Outpatient glycemic control with a bionic pancreas in type 1 diabetes, New Engl. J. Med., 371 (2014), 313–325.
    [130] G. Zhao, D. Wirth, I. Schmitz, M. Meyer-Hermann, A mathematical model of the impact of insulin secretion dynamics on selective hepatic insulin resistance, Nat. Commun., 8 (2017), 1–10.
    [131] A. De Gaetano, O. Arino, Mathematical modelling of the intravenous glucose tolerance test, J. Math. Biol., 40 (2000), 136–168.
    [132] Y. Lenbury, S. Ruktamatakul, S. Amornsamarnkul, Modeling insulin kinetics: responses to a single oral glucose administration or ambulatory-fed conditions, Biosystems., 59 (2001), 15–25.
    [133] A. Mukhopadhyay, A. De Gaetano, O. Arino, Modeling the intra-venous glucose tolerance test: a global study for a single-distributed-delay model, Discrete Contin. Dyn. S., 4 (2004), 407.
    [134] U. Picchini, A. De Gaetano, S. Panunzi, S. Ditlevsen, G. Mingrone, A mathematical model of the euglycemic hyperinsulinemic clamp, Theor. Biol. Med. Model., 2 (2005), 44.
    [135] U. Picchini, S. Ditlevsen, A. De Gaetano, Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations, J. Math. Biol., 53 (2006), 771–796.
    [136] S. Panunzi, P. Palumbo, A. De Gaetano, A discrete single delay model for the intra-venous glucose tolerance test, Theor. Biol. Med. Model., 4 (2007), 35.
    [137] D. V. Giang, Y. Lenbury, A. De Gaetano, P. Palumbo, Delay model of glucose–insulin systems: global stability and oscillated solutions conditional on delays, J. Math. Anal. Appl., 343 (2008), 996–1006.
    [138] J. Li, M. Wang, A. De Gaetano, P. Palumbo, S. Panunzi, The range of time delay and the global stability of the equilibrium for an IVGTT model, Math. Biosci., 235 (2012), 128–137.
    [139] P. Palumbo, P. Pepe, S. Panunzi, A. De Gaetano, Time-delay model-based control of the glucose-insulin system, by means of a state observer., Eur. J. Control, 6 (2012), 591–606.
    [140] P. Toghaw, A. Matone, Y. Lenbury, A. De Gaetano, Bariatric surgery and T2DM improvement mechanisms: a mathematical model, Theor Biol Med Model, 9 (2012).
    [141] A. De Gaetano, S. Panunzi, A. Matone, A. Samson, J. Vrbikova, B. Bendlova, et. al., Routine OGTT: a robust model including incretin effect for precise identification of insulin sensitivity and secretion in a single individual, PLOS ONE, 8 (2013).
    [142] P. Palumbo, G. Pizzichelli, S. Panunzi, P. Pepe, A. De Gaetano, Model-based control of plasma glycemia: tests on populations of virtual patients, Math Biosci, 257 (2014), 2–10.
    [143] K. Juagwon, Y. Lenbury, A. De Gaetano, P. Palumbo, Application of modified watanabe's approach for reconstruction of insulin secretion rate during OGTT under non-constant fraction of hepatic insulin extraction, Int. J. Math. Comp. Simul., 7 (2013), 304–313.
    [144] A. De Gaetano, S. Panunzi, D. Eliopoulos, T. Hardy, G. Mingrone, Mathematical modeling of renal tubular glucose absorption after glucose load, PLOS ONE, 9 (2014).
    [145] S. Sakulrang, E. J. Moore, S. Sungnul, A. De Gaetano, A fractional differential equation model for continuous glucose monitoring data, Adv. Differ. Equ-NY., (2017).
    [146] P. Palumbo, A. De Gaetano, An islet population model of the endocrine pancreas, J. Math. Biol., 61 (2010), 171–205.
    [147] A. De Gaetano, C. Gaz, P. Palumbo, S. Panunzi, A unifying organ model of pancreatic insulin secretion, PLOS ONE, 10 (2015).
    [148] A. De Gaetano, C. Gaz, S. Panunzi, Consistency of compact and extended models of glucose-insulin homeostasis: the role of variable pancreatic reserve, PLOS ONE, 14 (2019).
    [149] T. Hardy, E. Abu-Raddad, N. Porksen, A. De Gaetano, Evaluation of a mathematical model of diabetes progression against observations in the diabetes prevention program, Am. J. Physiol. Endocrinol. Metab., 303 (2012), 200–212.
    [150] A. Hinsberger, B. K. Sandhu, Digestion and absorption, Current Paediatrics, 14 (2004), 605–611.
    [151] A. D. Jackson, J. McLaughlin, Digestion and absorption, Surgery, 27 (2009), 231–236.
    [152] I. Campbell, Digestion and absorption, Anaesth. Intens. Care Med., 13 (2012), 62–63.
    [153] P. R. Kiela, F. K. Ghishan, Physiology of intestinal absorption and secretion, Best Practice & Research Clinical Gastroenterology, 30 (2016), 145–159.
    [154] A. B. Strathe, A. Danfær, A. Chwalibog, A dynamic model of digestion and absorption in pigs, Anim. Feed Sci. Tech., 143 (2008), 328–371.
    [155] S. Salinari, A. Bertuzzi, G. Mingrone, Intestinal transit of a glucose bolus and incretin kinetics: a mathematical model with application to the oral glucose tolerance test, Am. J. Physiol. Endoc. M., 300 (2011), 955–965.
    [156] M. Taghipoor, G. Barles, C. Georgelin, J. R. Licois, P. Lescoat, Digestion modeling in the small intestine: impact of dietary fiber, Math. Biosci., 258 (2014), 101–112.
    [157] E. D. Lehmann, T. Deutsch, , A physiological model of glucose-insulin interaction in type 1 diabetes mellitus, J. Biomed. Eng., 14 (1992), 235–242.
    [158] A. Roy, R. S. Parker, Dynamic modeling of exercise effects on plasma glucose and insulin levels, J. Diabetes Sci. Tech., 1 (2007), 338–347.
    [159] R. Hovorka, V. Canonico, L. J. Chassin, U. Haueter, M. Massi-Benedetti, M. O. Federici, et. al., Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes, Physiol. Meas., 25 (2004), 905–920.
    [160] G. M. Barnwell, F. S. Stafford, Mathematical model for decision-making neural circuits controlling food intake, B. Psychonomic Soc., 5 (1975), 473–476.
    [161] R. C. Boston, P. J. Moate, K. C. Allison, J. D. Lundgren, A. J. Stunkard, Modeling circadian rhythms of food intake by means of parametric deconvolution: results from studies of the night eating syndrome, Am. J. Clin. Nutr., 87 (2008), 1672–1677.
    [162] F. Cameron, G. Niemeyer, B. A. Buckingham, Probabilistic evolving meal detection and estimation of meal total glucose appearance, J. Diabetes Sci. Tech., 3 (2009), 1022–1030.
    [163] N. P. Balakrishnan, L. Samavedham, G. P. Rangaiah, Personalized mechanistic models for exercise, meal and insulin interventions in children and adolescents with type 1 diabetes, J. Theor. Biol., 357 (2014), 62–73.
    [164] M. Jacquier, F. Crauste, C. O. Soulage, H. A. Soula, A predictive model of the dynamics of body weight and food intake in rats submitted to caloric restrictions, PLOS ONE, 9 (2014).
    [165] A. L. Murillo, M. Safan, C. Castillo-Chavez, E. D. C. Phillips, D. Wadhera, Modeling eating behaviors: the role of environment and positive food association learning via a ratatouille effect, Math. Biosci. Eng., 13 (2016), 841–855.
    [166] NHANES 2015-2016 dietary data, 2019. Available from: https://wwwn.cdc.gov/Nchs/Nhanes/Search/DataPage.aspx?Component=Dietary&CycleBeginYear=201.
    [167] D. E. Cummings, J. Q. Purnell, R. S. Frayo, K. Schmidova, B. E. Wisse, D. S. Weigle, A preprandial rise in plasma ghrelin levels suggests a role in meal initiation in humans, Diabetes, 50 (2001), 1714–1719.
    [168] P. Toghaw, A. Matone, Y. Lenbury, A. De Gaetano, Bariatric surgery and T2DM improvement mechanisms: a mathematical model, Theor. Biol. Med. Model., 9 (2012).
    [169] L. Beaugerie, B. Flourié, P. Marteau, P. Pellier, C. Franchisseur, J. C. Rambaud, Digestion and absorption in the human intestine of three sugar alcohols, Gastroenterology, 99 (1990), 717–723.
    [170] Y. Tsuchida, S. Hata, Y. Sone, Effects of a late supper on digestion and the absorption of dietary carbohydrates in the following morning, J. Physiol. Anthropol., 32 (2013).
    [171] R. M. Atkinson, B. J. Parsons, D. H. Smyth, The intestinal absorption of glucose, J. Physiol., 135 (1957), 581–589.
    [172] Big mac®: calories and nutrition |mcdonald's, 2019. Available from: https://www.mcdonalds.com/us/en-us/product/big-mac.htm.
    [173] The nutritional content of beer, 2018. Available from: http://www.dummies.com/food-drink/drinks/beer/ the-nutritional-content-of-beer/.
    [174] T. M. S. Wolever, Carbohydrate and the regulation of blood glucose and metabolism, Nutr. Rev., 61 (2003), 40–48.
    [175] P. J. Randle, P. B. Garland, C. N. Hales, E. A. Newsholme, The glucose fatty-acid cycle. its role in insulin sensitivity and the metabolic disturbances of diabetes mellitus, Lancet, 1 (1963), 785–789.
    [176] S. C.Walpole, D. Prieto-Merino, P. Edwards, J. Cleland, G. Stevens, I. Roberts, The weight of nations: an estimation of adult human biomass, BMC Public Health, 12 (2012), 439.
    [177] M. E. Clegg, M. Pratt, O. Markey, A. Shafat, C. J. K. Henry, Addition of different fats to a carbohydrate food: impact on gastric emptying, glycaemic and satiety responses and comparison with in vitro digestion, Food Res. Int., 48 (2012), 91–97.
    [178] J. G. Moore, P. E. Christian, J. A. Brown, C. Brophy, F. Datz, A. Taylor, et. al., Influence of meal weight and caloric content on gastric emptying of meals in man, Digest. Dis. Sci., 29 (1984), 513–519.
    [179] L. Achour, S. Méance, A. Briend, Comparison of gastric emptying of a solid and a liquid nutritional rehabilitation food, Eur. J. Clin. Nutr., 55 (2001), 769–772.
    [180] I. Locatelli, A. Mrhar, M. Bogataj, Gastric emptying of pellets under fasting conditions: a mathematical model, Pharm. Res., 26 (2009), 1607–1617.
    [181] Fasting blood sugar levels, 2017. Available from: https://www.diabetes.co.uk.
    [182] R Core Team, R: a language and environment for statistical computing, 1st edition, Vienna, Austria, 2019.
    [183] M. Macht, G. Simon, Emotions and eating in everyday life, Appetite, 35 (2000), 65–71.
    [184] L. Canetti, E. Bachar, E. M. Berry, Food and emotion, Behav. Process., 60 (2002), 157–164.
    [185] S. Peciña, K. S. Smith, Hedonic and motivational roles of opioids in food reward: implications for overeating disorders, Pharmacol. Biochem. Behav., 97 (2010), 34–46.
    [186] S. Fulton, Appetite and reward, Front. Neuroendocrin., 31 (2010), 85–103.
    [187] A. H. Sam, R. C. Troke, T. M. Tan, G. A. Bewick, The role of the gut/brain axis in modulating food intake, Neuropharmacology, 63 (2012), 46–56.
    [188] E. L. Gibson, Emotional influences on food choice: sensory, physiological and psychological pathways, Physiol. Behav., 89 (2006), 53–61.
    [189] D. A. Zellner, S. Loaiza, Z. Gonzalez, J. Pita, J. Morales, D. Pecora, et. al., Food selection changes under stress, Physiol. Behav., 87 (2006), 789–793.
    [190] P. M. A. Desmet, H. N. J. Schifferstein, Sources of positive and negative emotions in food experience, Appetite, 50 (2008), 290–301.
    [191] M. Macht, How emotions affect eating: a five-way model, Appetite, 50 (2008), 1–11.
    [192] L. E. Stoeckel, R. E. Weller, E. W. Cook, D. B. Twieg, R. C. Knowlton, J. E. Cox, Widespread reward-system activation in obese women in response to pictures of high-calorie foods, NeuroImage, 41 (2008), 636–647.
    [193] E. Näslund, P. M. Hellström, Appetite signaling: from gut peptides and enteric nerves to brain, Physiol. Behav., 92 (2007), 256–262.
    [194] L. Brondel, M. Romer, V. Van Wymelbeke, N. Pineau, T. Jiang, C. Hanus, et. al., Variety enhances food intake in humans: role of sensory-specific satiety, Physiol. Behav., 97 (2009), 44–51.
    [195] R. C. Havermans, N. Siep, A. Jansen, Sensory-specific satiety is impervious to the tasting of other foods with its assessment, Appetite, 55 (2010), 196–200.
    [196] G. Finlayson, A. Arlotti, M. Dalton, N. King, J. E. Blundell, Implicit wanting and explicit liking are markers for trait binge eating. a susceptible phenotype for overeating, Appetite, 57 (2011), 722–728.
    [197] R. J. Stevenson, M. Mahmut, K. Rooney, Individual differences in the interoceptive states of hunger, fullness and thirst, Appetite, 95 (2015), 44–57.
    [198] H. Wang, J. Li, Y. kuang, Mathematical modeling and qualitative analysis of insulin therapies, Math. Biosci., 210 (2007), 17–33.
    [199] D. M. Thomas, A. Ciesla, J. A. Levine, J. G. Stevens, C. K. Martin, A mathematical model of weight change with adaptation, Math. Biosci. Eng., 6 (2009), 873–887.
    [200] C. L. Chen, H. W. Tsai, Modeling the physiological glucose–insulin system on normal and diabetic subjects, Comput. Meth. Prog. Bio., 97 (2010), 130–140.
    [201] C. C. Y. Noguchi, E. Furutani, S. Sumi, Enhanced mathematical model of postprandial glycemic excursion in diabetics using rapid-acting insulin, 2012 Proceedings of SICE Annual Conference (SICE), Akita, (2012), 566–571.
    [202] H. Zheng, H. R. Berthoud, Eating for pleasure or calories, Curr. Opin. Pharmacol., 7 (2007), 607–612.
    [203] O. B. Chaudhri, C. J. Small, S. R. Bloom, The gastrointestinal tract and the regulation of appetite, Drug Discov. Today, 2 (2005), 289–294.
    [204] B. M. McGowan, S. R. Bloom, Gut hormones regulating appetite and metabolism, Drug Discov. Today, 4 (2007), 147–151.
    [205] B. Meister, Neurotransmitters in key neurons of the hypothalamus that regulate feeding behavior and body weight, Physiol. Behav., 92 (2007), 263–271.
    [206] S. Higgs, J. Thomas, Social influences on eating, Curr. Opin. Behav. Sci., 9 (2016), 1–6.
    [207] S. Griffioen-Roose, G. Finlayson, M. Mars, J. E. Blundell, C. de Graaf, Measuring food reward and the transfer effect of sensory specific satiety, Appetite, 55 (2010), 648–655.
    [208] K. C. Berridge, T. E. Robinson, J. W. Aldridge, Dissecting components of reward: 'liking', 'wanting', and learning, Curr. Opin. Pharmacol., 9 (2009), 65–73.
    [209] K. C. Berridge, 'liking' and 'wanting' food rewards: brain substrates and roles in eating disorders, Physiol. Behav., 97 (2009), 537–550.
    [210] K. C. Berridge, C. Y. Ho, J. M. Richard, A. G. DiFeliceantonio, The tempted brain eats: pleasure and desire circuits in obesity and eating disorders, Brain Res., 1350 (2010), 43–64.
    [211] R. C. Havermans, "You say it's liking, i say it's wanting …". on the difficulty of disentangling food reward in man, Appetite, 57 (2011), 286–294.
    [212] R. C. Havermans, How to tell where 'liking' ends and 'wanting' begins, Appetite, 58 (2012), 252–255.
    [213] G. Finlayson, M. Dalton, Current progress in the assessment of 'liking' vs. 'wanting' food in human appetite. comment on "you say it's liking, i say it's wanting...". on the difficulty of disentangling food reward in man, Appetite, 58 (2012), 373–378; 252–255.
    [214] P. W. J. Maljaars, H. P. F. Peters, D. J. Mela, A. a. M. Masclee, Ileal brake: a sensible food target for appetite control. a review, Physiol. Behav., 95 (2008), 271–281.
    [215] H. S. Shin, J. R. Ingram, A. T. McGill, S. D. Poppitt, Lipids, CHOs, proteins: can all macronutrients put a 'brake' on eating?, Physiol. Behav., 120 (2013), 114–123.
    [216] A. M. Wren, L. J. Seal, M. A. Cohen, A. E. Brynes, G. S. Frost, K. G. Murphy, et. al., Ghrelin enhances appetite and increases food intake in humans, J. Clin. Endocrinol. Metab., 86 (2001), 5992.
    [217] K. A. Levin, Study design III: cross-sectional studies, Evid. Based Dent., 7 (2006), 24–25.
    [218] J. Tack, K. J. Lee, Pathophysiology and treatment of functional dyspepsia, J. Clin. Gastroenterol., 39 (2005), 211–216.
    [219] S. A. Murray, M. Kendall, K. Boyd, A. Sheikh, Illness trajectories and palliative care, BMJ, 330 (2005), 1007–1011.
    [220] M. Binn, C. Albert, A. Gougeon, H. Maerki, B. Coulie, M. Lemoyne, et. al., Ghrelin gastrokinetic action in patients with neurogenic gastroparesis, Peptides, 27 (2006), 1603–1606.
    [221] A. Abizaid, T. L. Horvath, Brain circuits regulating energy homeostasis, Regul. Peptides, 149 (2008), 3–10.
    [222] M. Traebert, T. Riediger, S. Whitebread, E. Scharrer, H. A. Schmid, Ghrelin acts on leptin-responsive neurones in the rat arcuate nucleus, J. Neuroendocrinol., 14 (2002), 580–586.
    [223] Y. C. L. Tung, A. K. Hewson, S. L. Dickson, Actions of leptin on growth hormone secretagogue-responsive neurones in the rat hypothalamic arcuate nucleus recorded in vitro, J. Neuroendocrinol., 13 (2001), 209–215.
    [224] N. Sáinz, J. Barrenetxe, M. J. Moreno-Aliaga, J. A. Martínez, Leptin resistance and diet-induced obesity: central and peripheral actions of leptin, Metabolism, 64 (2015), 35–46.
    [225] NHANES - participants - why I was selected, 2019. Available from: https://www.cdc.gov/nchs/nhanes/participant/participant-selected.htm.
    [226] E. Archer, G. A. Hand, S. N. Blair, Validity of U.S. nutritional surveillance: national health and nutrition examination survey caloric energy intake data, 1971–2010, PLoS ONE, 8 (2013).
    [227] E. Archer, G. Pavela, C. J. Lavie, The inadmissibility of what we eat in America and NHANES dietary data in nutrition and obesity research and the scientific formulation of national dietary guidelines, Mayo Clin. Proc., 90 (2015), 911–926.
    [228] E. Archer, C. J. Lavie, J. O. Hill, The failure to measure dietary intake engendered a fictional discourse on diet-disease relations, Front. Nutr., 5 (2018), 105.
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