By Neil Dubin (auth.)
Stochastic methods frequently pose the trouble that, once a version devi ates from the best different types of assumptions, the differential equations received for the density and the producing features turn into mathematically ambitious. Worse nonetheless, one is especially usually resulted in equations that have no identified resolution and do not yield to plain analytical tools for differential equations. within the version thought of right here, one for tumor progress with an immunological re sponse from the traditional tissue, a nonlinear time period within the transition likelihood for the demise of a tumor mobilephone results in the above-mentioned issues. regardless of the mathematical dangers of this nonlinearity, we will think about a extra subtle version biologically. finally, with the intention to in attaining a extra practical illustration of a classy phenomenon, it can be crucial to check mechanisms which enable the version to deviate from the extra mathematically tractable linear structure. up to now, stochastic versions for tumor development have virtually solely thought of linear transition probabilities.
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Extra info for A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
K Note that it is assumed implicitly that x, the number of tumor cells, takes on nonnegative values only. The assumption that x '" a for our approximation is a difficulty. After all, we would want to be able to calculate the probability of extinction, p(O,t), but here we cannot. Further, lim x+ p(x,t) A-Il ± K which in a general sense suggests a concentration of probability around the deterministic carrying capacity, (A - Il)/K. But p(x,t) is non-integrable and not summable in Cesaro mean (Dubin, 1974 pp.
J_> For J. -11 K' A 1, so that i II (Il+K. j» A diverges. Therefore W, hence S, diverges, which means (19) lim Pr[X(t) t->oo olx(o) 1] L Since an analytic solution for neither the probability generating function nor the density is obtained, it will be useful to recall the above properties once we have derived approximations to the process. The question arises as to whether it might not be preferable to attempt a numerical solution for the state probabilities, Pn(t), as opposed to an approximate analytic solution for these probabilities.
Equating coefficients of e in (11) gives (r=2) t12) (r=3) etc. In general, the i solution for ~l' th ••• equation, '~i d~i for~, contains at least theoretically. ~i+l' otherwise we could obtain a Thus, this method proves abortive, too. That our equations shoulq prove intractable is not surprising since Kendall (1949) found the same in dealing with the logistic process of which our process is a special case. It should be pointed out, however, that the equations for certain nonlinear birth and death processes have been solved.