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Optimization problems in epidemiology, biomechanics & medicine. (English) Zbl 1342.49062
Summary: Mathematical simulations are of increasing relevance for applications in engineering and the life sciences. Disciplines like epidemiology, biomechanics, medical image processing, just to name a few, are subject of academic research since decades. With modern numerical methods and the advent of increasing computing power not just simulations of biological systems are within reach, but also questions of optimizing the systems to aim at a certain goal can be addressed. In this paper, we will discuss some examples from epidemiology as well as biomechanical models for muscles. All these models are based on a set of differential equations. Defining a suitable cost functional to measure the distance to the goal of our optimization, mathematical tools from constrained optimization can be applied to solve optimization problems and to derive suitable numerical algorithms.

49N90 Applications of optimal control and differential games
74L15 Biomechanical solid mechanics
92B05 General biology and biomathematics
92D30 Epidemiology
74P10 Optimization of other properties in solid mechanics
Full Text: DOI
[1] Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen, 311 pp. Vieweg + Teubner, Wiesbaden (2009)
[2] Aldila, D; Götz, T; Soewono, E, An optimal control problem arising from a dengue disease transmission model, Math. Biosci., 242, 9-16, (2013) · Zbl 1316.92079
[3] Wijaya, K.P., Götz, T., Soewono, E.: An optimal control model of mosquito reduction management in a dengue endemic region. Int. J. Biomath. 7(5), 1450,056 (2014). doi:10.1142/S1793524514500569 · Zbl 1125.92007
[4] Rockenfeller, R., Goetz, T.: Optimal control of isometric muscle dynamics. J. Math. Fundam. Sci. 47(1), 12-30 (2015). doi:10.5614/j.math.fund.sci.2015.47.1.2
[5] Hinze, M., Pinnau, R., Ulbrich, S.: Optimization with PDE Cconstraints, Mathematical Modelling: Theory and Application, vol. 23. Springer-Verlag, New York (2009) · Zbl 1167.49001
[6] Bailey, N.: The mathematical theory of infectious diseases and its applications, 2nd Edition. Charles Griffin & Company Ltd (1975) · Zbl 0334.92024
[7] Dietz, K.: Transmission and control of arbovirus diseases. In: Ludwig, D., Cooke, K.L. (eds.) Epidemiology, pp. 104-121. SIAM, Philadelphia (1975) · Zbl 0322.92023
[8] Wijaya, KP; Götz, T; Soewono, E, Advances in mosquito dynamics modeling, Arxiv, 1503, 02573, (2015) · Zbl 1351.49053
[9] Bauer, S; Gruber, K; Kilian, F, 3d-computermodell der menschlichen lendenwirbelsäule—entwicklung und anwendungsmöglichkeiten in der medizin, (2010), Rostock
[10] Grujicic, M., Pandurangan, B., Xie, X., Gramopadhye, A., Wagner, D., Ozen, M.: Musculoskeletal computational analysis of the influence of car-seat design/adjustments on long-distance driving fatigue. Int. J. Ind. Ergon. 40(3), 345-355 (2010). doi:10.1016/j.ergon.2010.01.002http://www.sciencedirect.com/science/article/pii/S016981411000003X · Zbl 1316.92079
[11] Seyfarth, A., Grimmer, S., Häufle, D.F.B., Kalveram, K.T.: Can robots help to understand human locomotion? Automatisierungstechnik 60(11), 653-661 (2012). URL http://dblp.uni-trier.de/db/journals/at/at60.htmlSeyfarthGHK12
[12] Keppler, V.: Biomechanische modellbildung zur simulation zweier mensch-maschinen-schnittstellen. Ph.D. thesis, Eberhard-Karls-Universität zu Tübingen, Fakultät für Mathematik und Physik (2003) · Zbl 1351.49053
[13] Hill, AV, The heat of shortening and the dynamic constants of muscle, Proc. R. Soc. Lond. B, 126, 136-195, (1938)
[14] Günther, M; Schmitt, S; Wank, V, High-frequency oscillations as a consequence of neglected serial damping in Hill-type muscle models, Biol. Cybern., 1, 63-79, (2007) · Zbl 1125.92007
[15] Haeufle, D; Günther, M; Bayer, A; Schmitt, S, Hill-type muscle model with serial damping and eccentric force-velocity relation, J. Biomech., 47, 1531-1536, (2014)
[16] Hatze, H, A myocybernetic control model of skeletal muscle, Biol. Cybern., 25, 103-119, (1977) · Zbl 0346.92011
[17] Wank, V, Muscle growth and fiber type composition in hind limb muscles during postnatal development in pigs, Cells Tissues Organs, 182, 171-181, (2006)
[18] Hawkins, D; Hull, ML, Muscle forces as affected by fatigue: mathematical model and experimental verification, J. Biomech., 26, 1117-1128, (1993)
[19] Zajac, FE, Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control, Crit. Rev. Biomed. Eng., 17, 359-411, (1989)
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