Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T09:11:53.199Z Has data issue: false hasContentIssue false
  • English
  • Français

Patient-dependent effects in disease control: a mathematical model

Published online by Cambridge University Press:  17 February 2009

Jean M. Tchuenche
Jean M. Tchuenche
Affiliation:
Mathematics Department, University of Dar es Salaam, P.O. Box 35062, Dar es Salaam, Tanzania; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The state of a patient is an important concept in biomedical sciences. While analytical methods for predicting and exploring treatment strategies of disease dynamics have proven to have useful applications in public health policy and planning, the state of a patient has attracted less attention, at least mathematically. As a result, models constructed in relation to treatment strategies may not be very informative. We derive a patient-dependent parameter from an age-physiology dependent population model, and show that a single treatment strategy is not always optimal. Also, we derive a function which increases with the patient dependence parameter and describes the effort expended to be in good health.

Keywords

population dynamicsagecost functionalHamiltonianpatientdependent parametereffort
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 4 , April 2007 , pp. 583 - 596
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Afenya, A., “Cancer treatment strategies and mathematical modeling”, in Mathematical models in medical and health sciences (eds Horn, M. A., Simonett, G. and Webb, G.), (Vanderbilt University Press, Nashville TN, 1998) 1–15.Google Scholar
[2]Drasdo, D. and Loeffler, M., “Individual-based models to growth and folding in one-layered tissues: Intestinal crypts and early development”, Nonlinear Anal. 47 (2001) 245–256.CrossRefGoogle Scholar
[3]Friedman, A., Variational principles and free-boundary problems, 2nd ed. (Robert E. Krieger Publishing Co., Inc., Malabar Florida, 1988).Google Scholar
[4]GÖhde, W., Cassens, U., Lehman, L. G., Gohde, W., Berkes, P., Westerberg, C. and Greve, B., “Individual patient-dependent influence of erythrocyte lysing procedures on flow-cytometric analysis of leukocyte subpopulations”, Transfus. Med. Hemother 30 (2003) 165–170.CrossRefGoogle Scholar
[5]Goldestein, J. A., Semigroups of linear operators and applications (Oxford University Press, New York, Clarendon Press, Oxford, 1985).Google Scholar
[6]Hale, J. K., Asymptotic behaviour of dissipative systems, Mathematical Surveys Monographs (AMS, Providence, Rhode Island, 1998).Google Scholar
[7]Inaba, H., “Threshold stability results for an age-structured epidemic model”, J.Math. Biol. 28 (1990) 411–434.CrossRefGoogle ScholarPubMed
[8]Joshi, H. R., Lenhart, S., Li, M. Y. and Wang, L., “Optimal control methods applied to disease models”, Contemp. Math. 410 (2006) 187–207.CrossRefGoogle Scholar
[9]Kot, M., Elements of mathematical ecology (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
[10]Lemos, J. M., Magalhāes, H., Mendonca, T. and Rocha, P., “Observer dynamics in switching control: Application to neuromuscular blockade”, 2005, www.partec.com/partec/lysing.pdf, Retrieved February, 28, 2006.Google Scholar
[11]Oster, G. and Takahasi, Y., “Models for age-specific interactions in a periodic environment”, Ecol. Monogr. 44 (1974) 483–501.CrossRefGoogle Scholar
[12]Da Prato, G. and Iannelli, M., “Boundary control problem for age-dependent equations”, in Evolution equations, control theory and biomathematics (eds Clement, P. and Lumers, G.), Volume 155 of Lecture Notes in Pure and Applied Mathematics, (Marcel Dekker, Inc., New York, 1994) 91–100.Google Scholar
[13]Prüb, J., “Equilibrium solutions of age-specific population dynamics of several species”, J. Math. Biol. 11 (1981) 65–83.Google Scholar
[14]Rosen, R., “On models and modeling”, Appl. Math. Comput. 56 (1993) 359–372.Google Scholar
[15]Sinko, J. W. and Streifer, W., “A new model for age-size structure of a population”, Ecology 48 (1967) 910–918.CrossRefGoogle Scholar
[16]Sneddon, I. N., The use of integral transform (McGraw-Hill, New York, 1972).Google Scholar
[17]Sowunmi, C. O. A., “Time discrete 2-sex population model with gestation period”, in Mathematical modelling of population dynamics (ed. Rudnicki, R.), (Banach Center Publications, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2004) 259–266.Google Scholar
[18]Tchuenche, J. M., “Mathematical population dynamics of sickle-cell anaemia, a genetically transmitted disease”, Ph. D. Thesis, Fac. Sci., University of Ibadan, 2002.Google Scholar
[19]Tchuenche, J. M., “Theoretical population dynamics model of a genetically transmitted disease: Sickle-cell anaemia”, Bull. Math. Biol. 69 (2006) 699–730.CrossRefGoogle ScholarPubMed
[20]Thieme, H., “Semiflows generated by Lipschitz perturbations of non-densely defined operators”, Diff. Integral Eqns. 3 (1990) 1035–1066.Google Scholar
[21]Yong, J., “Optimal control for semi-linear evolution equations with point-wise state constraints”, in Proceedings of the Conference on Nonlinear Evolution and Infinite-Dimensional Dynamical Systems, Jul. 12–16, 1995 (ed. Ta-Tsien, ), (World Scientific, Singapore, New Jersey, London, Hongkong, 1997).Google Scholar