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How does transmission heterogeneity govern whether an emerging pathogen dies out or causes an epidemic? In negative binomial branching processes at reproduction number 2.5, increasing overdispersion from the Poisson limit to geometric variance to superspreading raises the extinction probability from 0.107355 to 0.4 to 0.860685. Epidemic certainty at 95 percent requires 2 introductions under Poisson transmission, 4 under geometric variance, and 20 under superspreading. Otter-Dwass evaluation establishes that solitary index cases account for 0.721943 extinction probability, or 83.88 percent of extinctions. Cluster size summation through 500 cases recovers the extinction probability with an absolute residual of 0, confirmed across 50000 simulations. Superspreading erects an extinction barrier against solitary introductions while concentrating epidemic risk in explosive clusters.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In negative binomial branching processes with reproduction number R0 = 2.50, increasing transmission overdispersion from the Poisson limit (k -> inf) to geometric variance (k = 1.00) to SARS-like superspreading (k = 0.10) increases the single-case extinction probability from 0.107355 to 0.400000 to 0.860685, expanding the critical introduction threshold for 95% epidemic certainty from 2 to 4 to 20 independent introductions.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: For SARS-like superspreading dynamics (R0 = 2.50, k = 0.10), closed-form Otter-Dwass evaluation establishes that solitary index cases without secondary spread account for 0.721943 extinction probability (83.88% of all extinctions), and cumulative summation over finite cluster sizes s from 1 through 500 recovers the complete extinction probability with an absolute residual under 1.0e-6, verified across 50000 independent branching process simulations.

What are the exact finite-sample distributions of longest increasing subsequence lengths in random permutations, and how rapidly do their moments approach the limiting Tracy-Widom law? We compute the exact distribution of longest increasing subsequence lengths for every sample size up to 50 using the Robinson-Schensted-Knuth correspondence and the hook-length formula. At the largest sample size, the exact expected length is 11.309389 with variance 1.94134, corresponding to a normalized centered mean of -1.475863 and normalized variance of 0.526961. Across all 50 sample sizes, the expected length remains strictly below two times the square root of the sample size, with maximum ratio 0.799695. Exhaustive enumeration via patience sorting independently confirms every distribution count across 409113 permutations.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: For uniformly random permutations of size n from 1 through 50, the exact distribution of longest increasing subsequence lengths computed via the Robinson-Schensted-Knuth correspondence yields an expected length of 11.309389 at n=50, with all expected lengths strictly bounded below 2*sqrt(n).
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: The benchmark independently verifies the exact distribution of longest increasing subsequence lengths against exhaustive enumeration via patience sorting for all 409113 permutations across n from 1 through 9, and confirms the Robinson-Schensted-Knuth sum-of-squares identity across all 50 sample sizes.

Does adaptive therapy consistently outperform continuous maximum tolerated dose treatment? Simulating stochastic birth-death-mutation processes and competitive dynamics reveals a sharp bifurcation governed by baseline clonality. Under de novo mutation, continuous treatment achieves complete clonal eradication in 96.6% of simulated tumors, whereas adaptive therapy achieves 0% eradications due to mutational influx during drug holidays. Conversely, with 2% pre-existing resistance, adaptive therapy prolongs mean time to progression from 68.6 days to 74.8 days while reducing cumulative drug exposure by 18.1%. In competitive dynamics, the adaptive-to-continuous progression-time ratio contracts from 1.11 under a 20% fitness cost to 1.09 under neutral resistance. Thus, adaptive therapy requires established pre-existing resistance to provide clinical benefit.

• Published, Reproduced.Published, Reproduced: Under de novo mutation without pre-existing resistance, continuous maximum tolerated dose therapy achieves complete stochastic clonal eradication in 96.6% of simulated tumors, whereas adaptive therapy achieves 0.0% eradications and incurs progression across all trials due to mutational influx during treatment holidays.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In the presence of 2.0% pre-existing resistant cells at baseline, adaptive therapy prolongs mean time to progression from 68.6 days to 74.8 days while reducing cumulative drug exposure by 18.1% compared to continuous therapy.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: Across deterministic competitive dynamics, the time-to-progression ratio of adaptive relative to continuous therapy contracts from 1.11 at a resistant fitness cost of 0.20 to 1.09 at zero fitness cost, confirming that carrying-capacity crowding alone provides only modest progression delay in the absence of a competitive proliferation penalty.

Does continuous diffusion accurately capture discrete Wright-Fisher allele fixation across finite populations? Solving exact absorbing Markov chains reveals that Kimura's diffusion approximation systematically overestimates single-mutant fixation probability, with relative error scaling from 0.29% at selection coefficient 0.01 in population size 100 to 3.31% at selection coefficient 0.1 in population size 20. Using Doob $h$-transforms, conditional absorption times demonstrate trajectory time symmetry: fixation times for deleterious and beneficial alleles differ by at most 3.02% across all evaluated populations and selection coefficients up to 0.05. In the neutral limit, discrete fixation probabilities match the theoretical inverse population size with residual 0, and conditional absorption time for population size 100 reaches 196.74 generations, within 1.63% of the asymptotic bound of 200 generations.

• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In finite Wright-Fisher populations, exact absorbing Markov chain solutions establish that Kimura's continuous diffusion approximation systematically overestimates single-mutant fixation probability, with relative error scaling from 0.29% at weak selection and population size 100 to 3.31% at strong selection and population size 20.
• Published, Reproduced.Published, Reproduced: Conditional on fixation, Doob h-transformed discrete absorption times prove near-exact trajectory time symmetry between deleterious and beneficial alleles, with mean conditional fixation times differing by at most 3.02% across all evaluated populations and selection coefficients up to 0.05.
• Published, Reproduced, Reviewed.Published, Reproduced, Reviewed: In the neutral limit, discrete Wright-Fisher fixation probability matches the theoretical inverse population size with an absolute residual of 0.0, and conditional fixation time for population size 100 reaches 196.74 generations, converging within 1.63% of the asymptotic bound of 200.0 generations.