Mathematical Analysis of Spread Models: From the viewpoints of Deterministic and random cases

Abstract

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.