By Neil Dubin (auth.)
Stochastic methods usually pose the trouble that, once a version devi ates from the easiest types of assumptions, the differential equations got for the density and the producing services turn into mathematically ambitious. Worse nonetheless, one is particularly usually 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 progress with an immunological re sponse from the conventional tissue, a nonlinear time period within the transition chance for the dying of a tumor phone results in the above-mentioned problems. regardless of the mathematical negative aspects of this nonlinearity, we can examine a extra subtle version biologically. finally, in an effort to in achieving a extra real looking illustration of a classy phenomenon, it will be important to check mechanisms which enable the version to deviate from the extra mathematically tractable linear layout. to this point, 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
E. 9.. 00 L at n=l N 2 n! 1) N (70) (~2 + + ~ + ~)lq(y,t). N2 N We now equate, in (70), terms in NO: (71) (2~ - l)~y(yq) + a H[(H)1) (A-)1) + ~l~ ay2 As per van Kampen (1961), we can transform this into an equation for which the solution is known. (72) seT) -log Let ~(l and note that ds/dT (73) an = -a ay -=. as 2~ ~) - log 4, - 1. Under this transformation, (71) becomes (qy) + Applying the additional transformations y = ze -s s ,q = e Q(z,s), we obtain (74) e 2s~ az 2 The solution to this equation is given by Chandrasekhar (1943): (75) where I 1 Q(z,s) = - - - exp [- (47f1)!
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. 27-29), which is highly undesirable since we require, for p(x,t) to be a probability distribution, that ! p(x,t)dx l. ). 2) Using log p(x,t). Bailey (1964, pp. (x,t) log p(x,t) yields better results than attempting a solution with method A for p(x,t) itself.
The deterministic formulation of our model can be written (44) dX We wish to consider small fluctuations from the equilibrium point X = x o = A-~ K This state is quasi-stationary because there is always the possibility of extinction, assuming that there will be stochastic fluctuations around that point (Pielou, 1969, p. 27). (45) Considering stochastic fluctuations we can write the equation dX where dz is the stochastic displacement from the deterministic rate of change. We assume (46) Pr[dz +lIX] AXdt, Pr[dz -lIX] (~X Pr[dz OIX] = 1 - (AX + + KX 2 )dt, ~X + KX2)dt, as dt + O.
A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations by Neil Dubin (auth.)