For each simplex S [subset or equal to] [S.sup.0] created by the

branching process, this process gives a lower bound LB(S) for the optimal value v(S) of the following problem LMP(S),

Let Z(n) = ([Z.sub.1](n), ..., [Z.sub.p](n)), n = 0, 1, ..., be a p-type Galton-Watson

branching process in a random environment.

Hence, there is inherent stochasticity caused by the finiteness of the number of lineages in applying the method, even if the branching and extinction events exactly follow the

branching process model that leads to equation (5).

The first- and last-birth problems for a multitype age-dependent

branching process. Advances in Appl.

This model can be understood from the discrete time binary Galton-Watson

branching process.

Since, this decision tree technique is not efficacious and suffers from overfitting, a problem which gives distorted results from the training data at the latter stages of the

branching process, the researchers moved on to random-forest technique that instead of calculating the outcome at every branch, calculates the outcome of random branches.

In this section, we also compute the stochastic threshold for disease extinction or invasion by applying the multitype Galton-Watson

branching process. In Section 4, we show the relationship between reproductive number of the deterministic model and the thresholds for disease extinction of the stochastic version; we also illustrate our results using numerical simulations.

The Markov

branching process (MBP) is the discrete-state Markov process ([Z.sub.t] : t [greater than or equal to] 0) on the state-space S = {0,1, ...} whose transition function F(t) = [[f.sub.ij](t)] is standard and satisfies the branching property,

Figure 10 shows the number of partial sets identified during the

branching process normalized by the number of full sets in the entire system, with respect to the sum of the probabilities of the branches that reach a certain value.

The quasistationary limit of this process is the Yaglom limit of its associated

branching process [39, 40] and is exactly geometric.

In our study, we use a

branching process model to estimate the probability distribution of outbreak sizes resulting from the introduction of an Ebola case to a new country where the reproductive number R (i.e., expected number of transmissions per case) would likely be quickly, if not immediately, reduced to <1.

The

branching process improves by using of polyfunctional monomer [19, 21, 22] and the materials known as iniferters such as Thiuram disulfides [23-25] and dithiocarbamates [26].