By Neil Dubin (auth.)
Stochastic techniques usually pose the trouble that, once a version devi ates from the best varieties of assumptions, the differential equations acquired for the density and the producing capabilities develop into mathematically bold. Worse nonetheless, one is especially frequently resulted in equations that have no identified answer and do not yield to straightforward analytical equipment for differential equations. within the version thought of the following, one for tumor development with an immunological re sponse from the conventional tissue, a nonlinear time period within the transition chance for the loss of life of a tumor phone results in the above-mentioned problems. regardless of the mathematical dangers of this nonlinearity, we will examine a extra refined version biologically. finally, with a view to in attaining a extra real looking illustration of a sophisticated phenomenon, it can be crucial to check mechanisms which enable the version to deviate from the extra mathematically tractable linear layout. so far, stochastic types for tumor development have nearly solely thought of linear transition probabilities.
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Extra info for A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
Efforts to apply the methods used in the above references were not successful for the present model. Lack of boundedness is a problem, as well as the fact that both births and deaths are possible. Despite the fact that an exact solution for the density or generating functions is not obtained, certain properties of process can be derived. We can write the differential equation for the deterministic formulation of the model as dx = AX - (~+KX)X, X dt (13) 1 at t where x is the number of tumor cells.
This is analogous to what we found earlier for the moments. Whittle (1957) and Bailey (1964) have both suggested neglecting cumulants of order higher than two. Whittle feels that the approximation is natural; after all, the assumption that cumulants higher than order one are zero leads to a deterministic formulation for the model. Allowing second order cumulants can then be considered a refinement of that first step. Clearly the resulting stochastic process will be Gaussian, the normal distribution being the unique distribution with only first and second cumulants non-zero.
These are all drawbacks, and in total we have no insight into the change of the nature of the stochastic process that has been effected. encouraging. We have lim c l (t) t-(43) limc 2 (t) = t-lim M(a, t) t-- _00 _00 , , O. Analysis of the result is not 40 The first of these follows because a 4 , the coefficient of its leading term as t is negative. The second and third derive from similar reasoning for d 6 • + 00, Since our original stochastic process is non-negative, an asymptotic mean at negative infinity does not recommend the cumulant method.