Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

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Im Modellierungsseminar diskutieren wir Fragestellungen der Physik, der Biologie, der Geologie, der Wirtschaft ... und lernen dadurch die Mathematik von einer völlig neuen Seite kennen: Mathematik ist in vielen Bereichen unseres Lebens enthalten!

Im diesem Modellierungsseminar diskutieren und lösen die TeilnehmerInnen in Kleingruppen mit Hilfe der Mathematik Probleme und Phänomene aus der Medizin, z.B. Auswirkungen und Effektivität von Impfstrategien. Die im Seminar erhaltenen Modelle werden anhand von frei zugänglichen Daten kalibriert. Sie erlauben einen Ausblick auf die (mögliche) zukünftige Entwicklungen und auch den Einfluss von Parametern auf die Problemgrössen.

Dieses Seminar steht in gleicher Weise MathematikerInnen, WirtschaftsmathematikerInnen, LehramtskandidatInnen und Medizinern offen.

Schwerpunkt dieses interdisziplinären Seminars werden nichtlineare gewöhnliche bzw. partielle Differentialgleichungen (Modellierung, Analysis und Numerik) sein. Von medizinischer Seite wird es von Prof. Dr. med. Helmut Brunner, Facharzt für Mikrobiologie und Infektionsepidemiologie, Universität Düsseldorf, betreut.

Im Rahmen des Seminars werden wir das Mathematica Software Paket EpiModel der Polytechnischen Universität Valencia kennenlernen.

Literatur:

  Einführung
• R.M. Anderson, R.M. May, Infectious diseases of humans: dynamics and control, Oxford University Press, 1991.
• N.J.T. Bailey, The mathematical theory of infectious diseases and its application, London, Griffin, 1975.
• E. Beretta, V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Comput. Math. Appl. 12A (1986) 677-694.
• F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001.
• V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, Springer, 1993.
• C. Castillo-Chavez , S. Blower, P. Van Den Driessche (eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, IMA Volumes in Mathematics and Its Applications, Springer, 2002.
• H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000), 599-653.
• W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics-I, Proc. Roy. Soc. 115A (1927), 700-721. Reprinted in Bull. Math. Biol. 53 (1991), 33-55.
  Impfstrategien
• M.E. Alexander, C. Bowman, A.B. Gumel, S.M. Moghadas, B.M. Sahai, R. Summers, A vaccination model for transmission dynamics of influenza, SIAM J. Applied Dynamical Systems 3 (2004), 503-524.
• M.E. Alexander, C. Bowman, Z. Feng, M. Gardam, S.M. Moghadas, G. Röst, J. Wu, P. Yan, Emergence of drug resistance: implications for antiviral control of pandemic influenza, Proc. R. Soc. B 274 (2007), 1675-1684.
• A.J. Arenas, J.A. Moraño, J.C. Cortés, Non-standard numerical method for a mathematical model of RSV epidemiological transmission, Comput. & Math. Appl. 56 (2008), 670-678.
• A.J. Arenas, G. González-Parra, B.M. Chen-Charpentier A nonstandard numerical scheme of predictor-corrector type for epidemic models, Comput. & Math. Appl. 59 (2010), 3740-3749.
• S. Gao , Y. Liu, J.J. Nieto, H. Andrade, Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission, Math. Comput. Simul., in press 2011.
• M. Iannelli, F. Milner, A. Pugliese, Analytical and numerical results for the age structured SIS epidemic model with mixed inter-intra-cohort transmission, SIAM J. Math. Anal. 23 (1992), 662-688.
• L. Jódar, R.J. Villanueva, A.J. Arenas, G.C. González, Nonstandard numerical methods for a mathematical model for influenza disease, Mathematics and Computers in Simulation 79 (2008), 622-633.
• C. Kribs-Zaleta, J. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Mathematical Biosciences 164 (2000), 183-201.
• S. Liu, Y. Pei, C. Li, L. Chen, Three kinds of TVS in a SIR epidemic model with saturated infectious force and vertical transmission, Applied Mathematical Modelling 33 (2009), 1923-1932.
• S.M. Moghadas, A.B. Gumel, A mathematical study of a model for childhood diseases with non-permanent immunity, Journal of Computational and Applied Mathematics 157 (2003), 347-363.
  Modelle zur Dosisfindung von Antibiotika (Thema von Helmut Brunner und Axel Dalhoff)

Die Pharmakologie der Antibiotika läßt sich in die Pharmakokinetik (PK) und die Pharmakodynamik (PD) unterteilen. Eine antibakterielle Therapie sollte im Idealfall zur klinischen und zur bakteriologischen Heilung führen. Werden nicht alle pathogenen Organismen durch die Therapie beseitigt, kann es während oder am Ende einer Therapie zur Vermehrung von Subpopulationen der Erreger mit höheren MIC-Werten (MIC Minimal Inhibitory Concentration) und potenzieller Resistenzentwicklung der Erreger gegen das Antibiotikum kommen. Daher wird versucht, mit Hilfe des PK/PD-Modells, das die in vivo ermittelten Parameter der Pharmakokinetik (meist als Blutspiegel) mit dem in vitro bestimmten MIC-Wert verbindet, geeignete Dosierungen für eine bessere Wirkung der Therapie zu finden.

• A PK/PD Approach to Antibiotic Therapy
• P.G. Ambrose, S.M. Bhavnani, C.M. Rubino et al., Pharmacokinetics-pharmacodynamics of antimicrobial therapy: It's not just for mice anymore, Clinical Infectious Diseases 44 (2007), 79-86.
• A. Dalhoff, P.G. Ambrose, J.W. Mouton, A long journey from minimum inhibitory concentration testing to clinically predictive breakpoints: deterministic and probabilistic approaches in deriving breakpoints, Infection 37 (2009), 296-305.
• J.W. Mouton, Relationship between pharmacodynamic indices and killing patterns in vitro, Future Microbiol. 6 (2011), 613-619.
• V.H. Tam, M. Nikolaou, A novel approach to pharmacodynamic assessment of antimicrobial agents: New insights to dosing regimen design, PLoS Comput Biol. 7 (2011), e1001043.
  Meningitis (Hirnhautentzündung)
• C.L. Trotter, N.J. Gay, W.J. Edmunds, The natural history of meningococcal carriage and disease, Epidemiol. Infect. 134 (2006), 556-566.
• C.L. Trotter, N.J. Gay, W.J. Edmunds, Dynamic Models of Meningococcal Carriage, Disease, and the Impact of Serogroup C Conjugate Vaccination, Amer. J. Epidemiol. 162 (2005), 89-100.
  Humane Papillomviren (HPV)
• M.C. Boily, B. Masse, Mathematical models of disease transmission: a precious tool for the study of sexually transmitted diseases, Can. J. Public Health 88 (1997), 255-65.
• E.J. Dasbach, E.H. Elbasha, R.P. Insinga, Mathematical Models for Predicting the Epidemiologic and Economic Impact of Vaccination against Human Papillomavirus Infection and Disease, Epidemiologic Reviews 28 (2006), 88-100.
• E.H. Elbasha, E.J. Dasbach, R.P. Insinga, Model for Assessing Human Papillomavirus Vaccination Strategies, Emerg. Infect. Dis. 13 (2007) 28-41.
• J.P. Hughes, G.P. Garnett, L. Koutsky, The theoretical population level impact of a prophylactic human papilloma virus vaccine, Epidemiology 13 (2002), 631-639.
• E.R. Myers, D.C. McCrory, K. Nanda et al., Mathematical model for the natural history of human papillomavirus infection and cervical carcinogenesis, Am. J. Epidemiol. 151 (2000), 1158-1171.
• E.R. Myers, Mathematical models as research tools for HPV disease, Papillomavirus Rep. 13 (2002), 141-144.
  Rotavirus
• E. Shim, Z. Feng, M. Martcheva, C. Castillo-Chavez, An age-structured epidemic model of rotavirus with vaccination, J. Math. Biol. 53 (2006), 719-746.
  SARS
• A.B. Gumel, S. Ruan, T. Day, et al., Modelling strategies for controlling SARS outbreaks, Proc.Royal Soc. B: Bio. Sci. 271 (2004), 2223-2232.
• R.G. McLeod, J.F. Brewster, A.B. Gumel, D.A. Slonowsky, Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs, Math. Biosci. Engrg 3 (2006), 527-544.
• S. Ruan, W. Wang, S.A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Engrg 3 (2006), 205-218.
• X. Yan, Y. Zou, Optimal and sub-optimal quarantine and isolation control in SARS epidemics, Math. Comput. Modelling 47 (2008), 235-245.
• Y. Zhou, Z. Ma, F. Brauer, A discrete epidemic model for SARS transmission and control in China, Math. Comput. Modelling 40 (2004), 1491-1506.
  Herpes
• A.B. Gumel, Numerical modeling of the transmission dynamics of drug-sensitive and drug-resistant HSV-2, Commun. Nonlin. Sci. Numer. Simul. 6 (2001), 23-27.
  Modellierung des programmierten Zelltods (Apoptosis)
• M. Fussenegger, J.E. Bailey, J. Varner, A mathematical model of caspase function in apoptosis, Natur. Biotechnol. 18 (2000), 768-774.
  Modellierung des Immunsystems
• S.M. Andrew, C.T.H. Baker, G.A. Bocharov, Rival approaches to mathematical modelling in immunology, J. Comput. Appl. Math. 205 (2007), 669-686.
• I.M. Rouzine, K. Murali-Krishna, R. Ahmed, Generals die in friendly fire, or modeling immune response to HIV, J. Comput. Appl. Math. 184 (2005), 258-274.
  Transportmodelle für Sauerstoff
• J.P. Whiteley, D.J. Gavaghana, C.E.W. Hahn, Oxygen transport to muscle tissue where regions of low oxygen tension exist, Math. Comput. Modelling 42 (2005), 1113-1122.
  Hepatitis A
• M. Ajelli, L. Fumanelli, P. Manfredi, S. Merler. Spatiotemporal dynamics of viral hepatitis A in Italy, Theoret. Popul. Biol. 79 (2011), 1-11.
• M. Ajelli, M. Iannelli, P. Manfredi, M.C.D. Atti, Basic mathematical models for the temporal dynamics of HAV in medium-endemicity Italian areas, Vaccine 26 (2008), 1697-1707.
  Hepatitis B
• L. Du, D. Huang, Q.Xie, A mathematical model for acute hepatitis B virus infection, 3rd International Conference on Biomedical Engineering and Informatics (BMEI), 2010, 1109-1113.
• J.R. Williams, D.J. Nokes, G.F. Medley, R.M. Anderson, The transmission dynamics of hepatitis B in the UK: A mathematical model for evaluating costs and effectiveness of immunization programmes, Epidemiol. Infect. 116 (1996), 71-89.
  Dengue-Fieber
• L. Cai, S. Guo, X. Li, M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos, Solitons & Fractals 42 (2009), 2297-2304.
• M. Derouich, A. Boutayeb, E.H. Twizell, A model of dengue fever, Biomed. Eng. Online 2 (2003) 4.
• M. Derouich, A. Boutayeb, Dengue fever: Mathematical modelling and computer simulation, Appl. Math. Comput. 177 (2006), 528-544.
• L. Esteva, C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci. 150 (1998), 131-151.
• L. Esteva, C. Vargas, A model for dengue disease with variable human population, J. Math. Biol. 38 (1999), 220-240.
• Z. Feng, V. Vealsco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol. 35 (1997) 523-544.
• A. Tran, M. Raffy, On the dynamics of dengue epidemics from large-scale information, Theoret. Popul. Biol. 69 (2006), 3-12.
  Aids / HIV
• S. Busenberg, C. Castillo-Chavez, A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. & Biol. 8 (1991), 1-29.
• S. Kovács, Dynamics of an HIV/AIDS model - The effect of time delay, Appl. Math. Comput. 188 (2007), 1597-1609.
• A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review 41 (1999), 3-44.
• S. Gupta, M. Anderson, R.M. May, Mathematical models and the design of public health policy: HIV and anti-viral therapy, SIAM Review 35 (1993), 1-16.
• Th. Hoffmann, H. Brunner, Model for simulation of HIV/AIDS and cost-effectiveness of preventing non-tuberculous mycobacterial (MAC-disease), Eur. J. Health Econom. 5 (2004), 129-135.
• M. Nasri, M. Dehghan, M.J. Douraki, Study of a system of non-linear difference equations arising in a deterministic model for HIV infection, Appl. Math. Comput. 171 (2005), 1306-1330.
• A.R. McLean, S.M. Blower, Imperfect vaccines and herd immunity, Royal Soc. London Ser. B 253 (1993), 9-13.
• A.B. Gumel, Xuewu Zhang, P.N. Shivakumar, M.L. Garba, B.M. Sahai, A new mathematical model for assessing therapeutic strategies of HIV infection. J. Theoret. Med. 4 (2002), 147-155.
• A.B. Gumel, A competitive numerical method for a chemotherapy model of two HIV subtypes, Appl. Math. Comput. 131 (2002), 327-335.
• A.B. Gumel, R.E. Mickens, B. Corbett, A nonstandard finite-difference scheme for a model of HIV transmission and control, J. Comput. Meth. Sci. Engrg. 3 (2003), 91-98.
• A.B. Gumel, S.M. Moghadas, R.E. Mickens, Effect of a preventive vaccine on the dynamics of HIV transmission, Commun. Nonlin. Sci. Numer. Simul. 9 (2004), 649-659.
• M.Y. Ongun, The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4⁺T cells, Math. Comput. Modelling 53 (2011), 597-603.
  Tuberkulose
• C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1 (2004), 361-404.
• Feng, Z., Huang, W., Castillo-Chavez, On the role of variable latent periods in mathematical models for tuberculosis, J. Dyn. Differential Equations 13 (2001), 425-452.
• D. Gammack, S. Ganguli, S. Marion, J. Segovia-Juarez, D.E. Kirschner, Understanding the immune response in tuberculosis using different mathematical models and biological scales, SIAM J. Multiscale Model. Simul. 3 (2005), 312-345.
• D. Okuonghae, S.E. Omosigho, Analysis of a mathematical model for tuberculosis: What could be done to increase case detection, J. Theoret. Biol. 269 (2011), 31-45.
• B. Song, C. Castillo-Chavez, J.P. Aparicio, Tuberculosis models with fast and slow dynamics: the role of close and casual contacts, Math. Biosci. 180 (2002), 187-205.
  Diabetes
• A.M. Albisser, J. Tiran, A Mathematical Modeling Study of Insulin Dynamics with Closed- and Open-Loop Control, Proc. Symp. Circuits Syst. (1980), 489-492.
• G.W. Swan, An optimal control model of diabetes mellitus, Bull. Math. Biol. 44 (1982), 793-808.
  (Schweine-)Grippe

Wir werden zeigen, wie man generell ansteckende Krankheiten mathematisch modelliert, um eine Prognose zu berechnen. Diese sog. SIR-Modelle (Susceptible-Infectious-Recovered) sind Kammer-Modelle, die zu Systemen von nichtlinearen Differentialgleichungen führen. Diese werden numerisch mit nichtstandard Methoden gelöst, die speziell die Positivität der Lösungen garantieren können.

• H. Jansen, E.H. Twizell, An unconditionally convergent discretization of the SEIR model, Math. Comput. Simul. 58 (2002), 147-158.
• G. Li, Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, Solitons & Fractals 25 (2005), 1177-1184.
• W.-M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiology models, J. Math. Biol. 23 (1986), 187-204.
• W.-M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987), 359-380.
• L. Jódar, R.J. Villanueva, A.J. Arenas, G.C. González, Nonstandard numerical methods for a mathematical model for influenza disease, Math. Comput. Simul. 79 (2008), 622-633.
• R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme, J. Diff. Eqs. Appl. 16 (2010), 209-216.
  Malaria

Bei der Modellierung der Ausbreitung von Malaria spielen sozial-ökonomische Ursachen eine große Rolle.

• L.F. Berkman, I. Kawachi, A historical framework for social epidemiology, in: L.F. Berkman, I. Kawachi (eds.), Social epidemiology, Oxford University Press, Oxford (2000).
• M.H. Craig, R.W. Snow, D. le Sueur, A climate-based distribution model of malaria transmission in Africa, Parasitol Today 15 (1999), 105-111.
• R. Ngom, A. Siegmund, Urban malaria in Africa: an environmental and socio-economic modelling approach for Yaounde, Cameroon, Nat. Hazards (2009).
  Toxoplasmose
• A.J. Arenas, G.C. González-Parra, R.-J. Villanueva, Modeling toxoplasmosis spread in cat population under vaccination, Theoret. Popul. Biol. 77 (2010), 227-237.
• D.F. Aranda, R.-J. Villanueva, A.J. Arenas, G.C. González-Parra, Mathematical modeling of Toxoplasmosis disease in varying size populations, Comput. Math. Appl. 56 (2008), 690-696.
• G.C. González-Parra, A.J. Arenas, D.F. Aranda, R.-J. Villanueva, L. Jódar, Dynamics of a model of Toxoplasmosis disease in human and cat populations, Comput. Math. Appl. 57 (2009), 1692-1700.
• D. Trejos, I. Duarte, A mathematical model of dissemination of toxoplasma Gondii by cats, Brazilian Symposium on Mathematical and Computational Biology 27 (2006), 7-13.
  West-Nil-Virus
• G. Cruz-Pacheco, L. Esteva, J. Montaño-Hirose, C. Vargas, Modelling the dynamics of west nile virus, Bull. Math. Biol. 67 (2005), 1157-1172.
  Agenten-basierte Modellierung in der Medizin (allgemeine Artikel)
• G. An, Dynamic knowledge representation using agent-based modeling: ontology instantiation and verification of conceptual models, Methods Mol. Biol. 500 (2009), 445-468.
• S. Coakley, R. Smallwood, M. Holcombe, From molecules to insect communities - how formal agent based computational modelling is uncovering new biological facts, Scientiae Mathematicae Japonicae 64 (2006), 185-198.
• C.A. Hunt, G.E. Ropella, T.N. Lam, J. Tang, S.H. Kim, J.A. Engelberg, S. Sheikh-Bahaei, At the biological modeling and simulation frontier, Pharm. Res. 26 (2009), 2369-2400.
• N. Moreira, In Pixels and In Health: Computer modeling pushes the threshold of medical research, Science News 169 (2006), 40-44.
• J.N. Tegnér, A. Compte, C. Auffray, G. An, G. Cedersund, G. Clermont, B. Gutkin, Z.N. Oltvai, K.E. Stephan, R. Thomas, P. Villoslada, Computational disease modeling - fact or fiction?, BMC Syst. Biol. 3 (2009), 56.
• B.C. Thorne, A.M. Bailey, S.M. Peirce, Combining experiments with multi-cell agent-absed modeling to study biological tissue patterning, Briefings in Bioinformatics. (2007)
• Y. Vodovotz, G. Constantine, J. Rubin, M. Csete, E.O. Voit, G. An, Mechanistic simulations of inflammation: Current state and future prospects, Math. Biosci. 217 (2009), 1-10.
  Agenten-basierte Modellierung von ansteckenden Krankheiten
• A.H. Auchincloss, A.V. Diez Roux, A new tool for epidemiology: the usefulness of dynamic-agent models in understanding place effects on health, Am. J. Epidemiol. 168 (2008), 1-8.
• C. Carpenter, L. Sattenspiel, The design and use of an agent-based model to simulate the 1918 influenza epidemic at Norway House, Manitoba, Am. J. Hum. Biol. (2008)
• H. Devillers, J.R. Lobry, F. Menu, An agent-based model for predicting the prevalence of Trypanosoma cruzi I and II in their host and vector populations, J. Theor. Biol. 255 (2008), 307-315.
• Z. Guo, P.M. Sloot, J.C. Tay, A hybrid agent-based approach for modeling microbiological systems, J. Theor. Biol.255 (2008), 163-75.
• N. Hupert, W. Xiong, A. Mushlin, The virtue of virtuality: the promise of agent-based epidemic modeling, Transl. Res. 151 (2008), 273-274.
• B.Y. Lee, V.L. Bedford, M.S. Roberts, K.M. Carley, Virtual epidemic in a virtual city: simulating the spread of influenza in a US metropolitan area, Transl. Res. 151 (2008), 275-287.
• B. Roche, J.F.Guégan, F. Bousquet, Multi-agent systems in epidemiology: a first step for computational biology in the study of vector-borne disease transmission, BMC Bioinformatics. 9 (2008), 435.



Scheinkriterium:

  Präsentation

• Erläuterung des Anwendungsproblems
• Mathematische Modellierung (evtl. mehrstufig)
• Lösung / Analyse / Numerik
• Diskussion der Resultate und ihrer Relevanz für die Anwendung
• Für die Präsentation wird Prosper bzw. HA-Prosper oder PPower4 oder Beamer empfohlen
  ( Verschiedene Präsentationsprogramme)
• Evtl. numerische Simulationen (möglichst in Matlab, Scilab oder Octave)
  Schriftliche Ausarbeitung (ca. 10 Seiten, inkl. Beispiele)
  Regelmässige Teilnahme am Seminar

Vorkenntnisse: Basiswissen mathematischer Grundvorlesungen wird vorausgesetzt.

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University of Wuppertal Faculty of Mathematics and Natural Sciences Department of Mathematics Applied Mathematics & Numerical Analysis Group

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