Solution Of Pde Models

While some aspects of PDE models may be amenable to an analytic approach, or, in very straightforward cases, solution, in general it is necessary to resort to numerical approaches to solution. Although good computer modeling tools are available for the solution of ODE models, the picture is different for PDE models for which the usual approach is for computer programs to be written ad hoc in a high-level programming language such as C, C++ or Fortran; although often tedious, this approach is usually relatively straightforward. Suitable algorithms for numerical solutions can be found in reference works such as Numerical Recipes series (see Recommended Resources). Essentially the procedure is for a set of arrays, or one multidimensional array to be used to represent the proportion of the population in each compartment of the model with consecutive locations in the arrays representing consecutive age steps. As before, at each time step the appropriate transitions occur between the different epidemiological compartments; but also at each time step, starting from the oldest age step, the contents of adjacent age steps are moved to the next (i.e., older) age step (the contents of oldest age step being discarded on the assumption that for this age step death is the next step). As a rule of thumb the time step used should, for convenience, be of the same duration as the age step and, to avoid numerical errors, preferably be about an order of magnitude less than the mean duration corresponding to the fastest transition rate in the equation system; that is, if the time from infection to the development of symptoms is 10 days and is the shortest period represented in the model, the time and age steps should be around 1 day.

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