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We consider a simple branching random walk in which each individual performs an asymmetric random walk on the real line and record the positions of all individuals in each generation. In this paper, we show that the set of occupied positions is eventually an interval almost surely on the event of non-extinction, extending the results of Grill and Johnson. [ABSTRACT FROM AUTHOR]

In a multitype ( d types) supercritical positively regular Galton–Watson branching process, let { X n , X n − 1 , … , X 0 } denote the types of a randomly chosen (i.e., uniform distribution) individual from the n th generation and this individual’s n ancestors. It is shown here that this sequence converges in distribution to a Markov chain { Y 0 , Y 1 , … } with transition probability matrix ( p i j ) 1 ≤ i , j ≤ d and having the stationary distribution. We also consider the critical case conditioned on non-extinction. [ABSTRACT FROM AUTHOR]

• Two spread models are proposed from the deterministic and random aspects based on the substitution dynamic system and the theory of branching processes. • The long-term behaviors of the spread rates are found in both deterministic and random models. • The transition of two spread patterns during the pandemic period can be described. • Our methodology provides a way to compare two spread patterns which can be used as a reference for decision making. This paper models the spread of the pandemic with mathematical analysis to provide predictions for different classes of individuals. We consider the spread by using a branching process and a substitution dynamical system as random and deterministic models, respectively, to approximate the pandemic outbreak. Both approaches are based on the assumption of Markov processing. The deterministic model provides an explicit estimate for the proportion of individuals of a certain type in the particular generation given any initial condition, where a generation means a unit of observation time. The proportion relates to the matrix derived from the Markov setting. In addition, the methodology reveals the efficiency of epidemic control policies, such as vaccine injections or quarantine, by the relative spread rate that is used for the prediction of the number of individuals of a certain type. On the other hand, the stochastic approximation has more of an empirical impact than the deterministic one does. Our investigation explicitly exhibits the spread rate of a certain type with respect to an initial condition of any type. After estimating the average spread rate, the effect of adopting a particular policy can be evaluated. The novelty of this elucidation lies in connecting these two models and introducing the idea of the transition spread model between two topological spread models to capture the change of the spread patterns, which is a real-world phenomenon during the epidemic periods due to changes in the environment or changes in disease control policies. Roughly speaking, the deterministic model is a special case of the stochastic model under some particular probability. Most importantly, with the help of the stochastic model, we establish the transition processing of two deterministic models, which is called a transition model. In other words, any stochastic model is "bounded" by two deterministic models. Moreover, a computable way has been established to predict the long-term spread rate due to the Markov properties of the models and matrix representations for the spread patterns. [ABSTRACT FROM AUTHOR]