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Modeling differential equation systems
Modeling differential equation systems












Models try to predict things such as how a disease spreads, or the total number infected, or the duration of an epidemic, and to estimate various epidemiological parameters such as the reproductive number. The models are most often run with ordinary differential equations (which are deterministic), but can also be used with a stochastic (random) framework, which is more realistic but much more complicated to analyze. The Reed-Frost model was also a significant and widely-overlooked ancestor of modern epidemiological modelling approaches. The origin of such models is the early 20th century, with important works being that of Ross in 1916, Ross and Hudson in 1917, Kermack and McKendrick in 1927 and Kendall in 1956. The order of the labels usually shows the flow patterns between the compartments for example SEIS means susceptible, exposed, infectious, then susceptible again. People may progress between compartments. The population is assigned to compartments with labels – for example, S, I, or R, ( Susceptible, Infectious, or Recovered). They are often applied to the mathematical modelling of infectious diseases. Burden and J.Type of mathematical model used for infectious diseasesĬompartmental models are a very general modelling technique. Brown, Methods of Mathematical Modelling: Continous System and Differential Equations, Springer (2015) Higham, Numerical Methods for Ordinary Differential Equations: Initial Value Problems, Springer (2010)

  • Implement and test numerical methods using a scripting language.
  • Be able to derive and analyse fundamental numerical methods.
  • Understand the central concepts of mathematical modelling.
  • Examples of the use of these tools in applications.Īims: By the end of the module the student should be able to:.
  • modeling differential equation systems

    Analysis of discretisation (stability and convergence).Approximation by discretisation (Runge-Kutta and multistep), and the tools needed to analyse there.

    modeling differential equation systems

  • Demonstration of fundamental principles in deriving models ( reaction kinetics and Hamiltonian principle, and fundamental role of dimensional analysis perturbation theory to simplify complex models.
  • Of particular interest and value are their mathematical properties, particularly in respecting properties of the underlying model. Consequently, this module also investigates different methods for approximating the solution to ODEs. Mathematical models, in general, are too complex to solve explicitly, so that approximation methods and computation are essential tools. These principles can also be extended in epidemiology for the modelling of the transmission of diseases. For example, fundamental principles in science like conservation laws and force balances lead to initial value problems. In this module we expose some fundamental aspects of mathematical modelling involving ordinary differential equations. Two types of mathematical models are (i) those arising from the application of physical laws and (ii) those arising from the analysis of data. A fundamental notion is that of Mathematical Modelling in which natural questions are turned into mathematical problems. Leads To: The following modules have this module listed as assumed knowledge or useful background:Ĭontent: Mathematics arises all around us not only is nature but also in social structures.
  • MA269 Asymptotics and Integral Transforms.
  • Useful background: Good working knowledge in linear algebra and analysis
  • Concepts like Taylor expansion and continuity of multivariable functions as discussed in MA259 Multivariable Calculus.
  • Programming in Python as provided e.g.
  • by MA133 Differential Equations or MA113 Differential Equations A

    modeling differential equation systems

    Part of this course work will require some programming using Pythonīasic knowledge on solving differential equations and the structure of solutions for systems of ODEs and DEs as provided e.g. Commitment: 10 x 3 hour lectures + 9 x 1 hour support classesĪssessment: 100% Coursework.














    Modeling differential equation systems