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A Stochastic Model for Immunological Feedback in by Neil Dubin (auth.)

By Neil Dubin (auth.)

Stochastic tactics frequently pose the trouble that, once a version devi­ ates from the easiest forms of assumptions, the differential equations got for the density and the producing features turn into mathematically bold. Worse nonetheless, one is particularly frequently resulted in equations that have no recognized 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 traditional tissue, a nonlinear time period within the transition chance for the loss of life of a tumor telephone ends up in the above-mentioned issues. regardless of the mathematical negative aspects of this nonlinearity, we can examine a extra refined version biologically. finally, on the way to in achieving a extra sensible illustration of a classy phenomenon, it will be significant to check mechanisms which permit 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

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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. At the equilibrium point 0, (47) Var(dz) (+1)2 AXdt + (-1)2(~X + KX2)dtlx=xo (AX O + ~XO + KX 0 2 )dt. 46 We want to find the mean and variance of X about the equilibrium value, xo under the operation of dz, the stochastic fluctuation. e. X - m, where m (48) E(X).

For the probability generating function, P{Z,t) L n=O we have (4) ap{Z,t) at If we could solve either the difference equation for the density or the partial differential equation for the probability generating function, the process would be completely described. The standard approach to equations such as (4) is separation of variables. P{z,t) = m{z)n{t). This gives the separated equations (5) n' (t) n{t) = constant, [A{Z2_Z) + (\l+K) (l-z) ]m' (z) + K{z-z2)m"{z) m{z) constant. The first is easily solved to yield (6) n{t) where c l is a constant to be determined from the initial conditions.

50) But also [(A-~)m - Km 2 - KVar(X)]dt (51) implies, taken with (50) (52) (A - ~)m which can be written as (53) Clearly then m < x o ' and assuming that m ~ xo we obtain the estimate (54) To find Var(X) , assume that X departs only slightly from x O' and let X with u small. We have (A-~) K ' 47 and hence we write the stochastic equation (45) as (56) Kx 0 2 [-u(1+u)]dt + dz, since Xo = (A - ~)/K. du = -KxOu(l + u)dt + since, for u small, 1 + u (58) Xo gives ' 1. From (57) we can obtain u (1 - KxOdt) We note that E(dz) E[(dz)2] ~ ~ dz 2 (u + du)2 Xo Dividing by 2 2u + x(dz) 2 + --(1 - KxOdt)dz 2 Xo 0 0 implies Var(dz) and further that because we are in a state of stochastic equilibrium, E(u) Var(u).

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