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feedback mechanism; distributed time delay; discrete time delay; asymptotic stability; catatonic schizophrenia; hormone therapy
In the present paper, a mathematical model, originally proposed by Danziger and Elmergreen and describing the thyroid-pituitary homeostatic mechanism, is modified and analyzed for its physiological and clinical significance. The influence of different system parameters on the stability behavior of the system is discussed. The transportation delays of different hormones in the bloodstream, both in the discrete and distributed forms, are considered. Delayed models are analyzed regarding the stability and bifurcation behavior. Clinical treatment of periodic catatonic schizophrenia is discussed in presence of transportation delays. Numerical simulations are presented to support analytic results.
[1] J.  Cronin: The Danziger-Elmergreen theory of periodic catatonic schizophrenia. Bull. Math. Biol. 35 (1973), 689–707. DOI 10.1007/BF02458370 | MR 0339842 | Zbl 0253.92002
[2] J.  Cronin: Mathematical aspects of periodic catatonic schizophrenia. Bull. Math. Biol. 39 (1977), 187–199. MR 0484501 | Zbl 0366.92007
[3] L.  Danziger, G. L.  Elmergreen: Mathematical theory of periodic relapsing catatonia. Bull. Math. Biophys. 16 (1954), 15–21. DOI 10.1007/BF02481809
[4] L. Danziger, G. L.  Elmergreen: The thyroid-pituitary homeostatic mechanism. Bull. Math. Biophys. 18 (1956), 1–13. DOI 10.1007/BF02477840
[5] L.  Danziger, G. L.  Elmergreen: Mathematical models of endocrine systems. Bull. Math. Biophys. 19 (1957), 9–18. DOI 10.1007/BF02668288 | MR 0086728
[6] L.  Danziger, G. L.  Elmergreen: Mechanism of periodic catatonia. Confinia Neurol. 18 (1958), 159–166. DOI 10.1159/000105050
[7] J.  Durrell: Biological rhythms and psychiatry: psychoendocrine mechanisms. Cycles Biologiques et Psychiatrie, J.  de Ajuriaguerra (ed.), Masson, Paris, 1968.
[8] B. N. Ghosh, D. M.  Woodbury, and G. Sayers: Quantitative effects of thyrotropic hormone on iodine  131 accumulation in thyroid and plasma proteins of hypophysectomised rats. Endocrin. 48 (1951), 631–642. DOI 10.1210/endo-48-5-631
[9] R.  Gjessing, A.  Bernhardsen, and H. Froshaug: Investigation of amino acids in periodic catatonic patients. J. Ment. Sci. 104 (1958), 188–200. DOI 10.1192/bjp.104.434.188
[10] B. D.  Hassard, N. D.  Kazarinoff, and Y. H. Wan: Theory and Applications of Hopf bifurcation. Cambridge University Press, Cambridge, 1981. MR 0603442
[11] Y. Kuang: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston, 1993. MR 1218880 | Zbl 0777.34002
[12] N. MacDonald: Biological Delay System: Linear Stability Theory. Cambridge University Press, Cambridge, 1989. MR 0996637
[13] M.  Maeda, O.  Borud, and L. R. Gjessing: Investigation of cholesterol and fatty acid in periodic catatonia. Br. J.  Psychiat. 115 (1969), 81–93. DOI 10.1192/bjp.115.518.81
[14] B.  Mukhopadhyay, R.  Bhattacharyya: A delayed mathematical model for testosterone secretion with feedback control mechanism. Int. J.  Math. Math. Sci. 34 (2004), 105–115. DOI 10.1155/S0161171204307271 | MR 2038801
[15] C. P.  Richter: Biological Clocks in Medicine and Psychiatry. C. C.  Thomas, Springfield, 1965.
[16] S.  Takahashi, L. R.  Gjessing: Fluorometric determination of catecholamines in human urine combined with thin-layer chromatography. Clinical Chim. Acta 36 (1972), 369. DOI 10.1016/0009-8981(72)90011-3
[17] W .P. Vanderlaan, M. A.  Greer: Some effects of the hypophysis on iodine metabolism by the thyroid gland of the rat. Endocrinology 47 (1950), 36–47. DOI 10.1210/endo-47-1-36
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