By Neil Dubin (auth.)
Stochastic tactics frequently pose the trouble that, once a version devi ates from the best different types of assumptions, the differential equations bought for the density and the producing capabilities turn into mathematically ambitious. Worse nonetheless, one is particularly usually resulted 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 development with an immunological re sponse from the traditional tissue, a nonlinear time period within the transition chance for the demise of a tumor mobilephone results in the above-mentioned issues. regardless of the mathematical hazards of this nonlinearity, we can think of a extra subtle version biologically. eventually, for you to in attaining a extra sensible illustration of a sophisticated phenomenon, it will be significant to ascertain mechanisms which permit the version to deviate from the extra mathematically tractable linear layout. up to now, stochastic versions for tumor progress have nearly solely thought of linear transition probabilities.
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Extra resources for A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
The assumptions i) and ii) really tell us that we are finding an asymptotic solution, given that X is expected to be of order N. The values of t for which the approximation is valid are such that $(t) is approximately one. This in turn depends upon the parameters>. 1. See Figures 6, 7, and 8 for plots of the stochastic mean and variance obtained by van Kampen's approach, together with $(t) so as to indicate the region of validity. We see that, by the time $(t) is close to one, both mean and variance have stabilized to their asymptotic values.
Ance c Z c~ 1 c O. ' for the former and i lim = = 6 for the latter. In either case c-l . (A-~) E t t=O c~ c =4 t=O lim i(A-~) . (C;l) = +00 c~ Figures 4 and 5 show plots of c l and Cz for the cumulant method. 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 ....
It should be pointed out, however, that the equations for certain nonlinear birth and death processes have been solved. ) for a process with linear eigen-values. 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.
A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations by Neil Dubin (auth.)