By Neil Dubin (auth.)
Stochastic approaches usually pose the trouble that, once a version devi ates from the best sorts of assumptions, the differential equations got for the density and the producing features turn into mathematically bold. Worse nonetheless, one is particularly usually ended in equations that have no recognized resolution and do not yield to straightforward analytical equipment for differential equations. within the version thought of the following, one for tumor progress with an immunological re sponse from the conventional tissue, a nonlinear time period within the transition chance for the dying of a tumor cellphone results in the above-mentioned problems. regardless of the mathematical negative aspects of this nonlinearity, we can examine a extra subtle version biologically. finally, for you to in achieving a extra lifelike illustration of a classy phenomenon, it is crucial to ascertain mechanisms which enable the version to deviate from the extra mathematically tractable linear layout. up to now, stochastic versions for tumor progress have nearly completely thought of linear transition probabilities.
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Extra info for A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
If we could integrate (28), we would have two answers, but because of the complicated nature of "d" such integration proved beyond the capabilities of the author and standard tables. As a result several further approximations were tried, but proved abortive (Dubin, 1974 pp. 32-36). 3) Laplace transform approach. The motivation for this method is the awkwardness of incorporating initial conditions in an approximate solution. By using the Laplace transform with respect to time, initial conditions are included in the basic difference equation.
All the factors above, taken with the heavy algebra required to obtain the approximation, lead to the conclusion that some other approximation technique needs to be found. 5. 4. 5. 4. 5. 4. 0E-04 110 VALUES FIGURE Sa 44 STOCHASTIC VARIANCE FOR CUMULANT METHOD o~------------------------------------~ -20000 -40000 -60000 x gJ .... 5. 4. OE-07 200 VALUES FIGURE 5b 200 45 6. Stochastic Linearization Here we consider small fluctuations about an equilibrium point. The object is to obtain the mean and variance of those stochastic fluctuations, rather than a stochastic process which approximates our exact model.
4. 5. 4. 5. 4. 0E-04 110 VALUES FIGURE Sa 44 STOCHASTIC VARIANCE FOR CUMULANT METHOD o~------------------------------------~ -20000 -40000 -60000 x gJ .... 5. 4. OE-07 200 VALUES FIGURE 5b 200 45 6. Stochastic Linearization Here we consider small fluctuations about an equilibrium point. The object is to obtain the mean and variance of those stochastic fluctuations, rather than a stochastic process which approximates our exact model. A discussion of stochastic linearization is considered here because it bears a relation to van Kampen's approximation method, which follows it.
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