This thesis aims to enhance existing models that infer parameters describing the spread of a virus by analyzing the distribution of empirical cluster sizes of identical genetic sequences. An approach that has gained recent popularity assumes that each individual cluster can be modeled as a Bienaymé-Galton-Watson process, with the distribution of empirical cluster sizes being equal to the law of the final size $\widetilde{Y}_\infty$ of the branching process. By employing the theory of general branching processes counted by characteristics, we demonstrate that the empirical cluster size distribution $C^\alpha$ stochastically dominates $\widetilde{Y}_\infty$ due to the exponential growth of the branching process. Under the assumption that the underlying branching tree follows either a Bienaymé-Galton-Watson process or an age-dependent process, we show that the mean of the empirical cluster size distribution can be used for a (strongly) consistent estimator for the probability of mutation $\nu$. For both branching models, we compute $P(C^\alpha=n)$ for $n=1,2$. We conjecture that $P(C^\alpha=n)$ is independent of the underlying model and that it can be expressed as a function of the mean of the offspring distribution $X$, and the probability mass function of $bin(X, 1-\nu)$. An extension of the model is considered where the probability of mutation is sampled from a distribution $\nu$ for each cluster. We show that under this assumption the empirical mean of the cluster sizes estimates the quantity $\int \nu^{-1}(r) dr$. We also show that the $\nu$ can still be estimated by the empirical mean of the cluster sizes, when the population is divided into a finite number of types with inhomogeneous offspring distributions.

The Curie-Weiss model is a simplification of the Ising model to show the existence of a phase transition for ferromagnetism. In this thesis, we study the behaviour of sums of these dependent variables. We prove in general that under the appropriate assumptions, we can still conclude a version of the Law of Large Numbers. We also find that if there exists a certain m∈R, λ>0 and integer k≥1, we have that (S_n-nm)/n^1/2k converges to exp(-λs^2k/(2k)!) in distribution.
For the Curie-Weiss model this means that for β, which is a constant proportion to inverse temperature, we find that if β∈(0,1) we have S_n/n→δ(s) and S_n/√n→ N(0,σ²) in distribution where σ²=(1-β)^-1-1. At β=1 there occurs a phase transition, we still have that S_n/n→δ(s), but now S_n/n^3/4→\exp(-s⁴/12). When β>1 we can find an m>0 such that S_n/n→½[δ(s-m)+δ(s+m)].
We also study the Curie-Weiss model where we assume that it is under the influence of a magnetic field. We prove that we do not find a phase transition, and we always have S_n/n→δ(s-m) in distribution for some m∈R. Next to this we find that (S_n-nm)/√n always converges to a normal distribution.