Abstract
Stochastic epidemics with open populations of variable population sizes are considered where due to immigration and demographic effects the epidemic does not eventually die out forever. The underlying stochastic processes are ergodic multi-dimensional continuous-time Markov chains that possess unique equilibrium probability distributions. Modeling these epidemics as level-dependent quasi-birth-and-death processes enables efficient computations of the equilibrium distributions by matrix-analytic methods. Numerical examples for specific parameter sets are provided, which demonstrates that this approach is particularly well-suited for studying the impact of varying rates for immigration, births, deaths, infection, recovery from infection, and loss of immunity.
Highlights
Epidemic processes are important population dynamics describing the outbreak and spread of infectious diseases
We present numerical examples in order to demonstrate that the level-dependent quasi-birth-and-death (LDQBD) modeling formalism in conjunction with matrix-analytic solution methods is well-suited for studying the equilibrium distributions of the introduced stochastic epidemic models
All results presented in the following tables and figures are numerical solutions of the LDQBD models obtained via the matrix-analytic solution method, avoiding the use of costly stochastic simulations
Summary
Epidemic processes are important population dynamics describing the outbreak and spread of infectious diseases. Deterministic models described by ordinary differential equations (ODEs) have the longest tradition and under certain circumstances they provide suitable approximations. Stochastic models, especially Markov chains, are more appropriate. We refer to [1,2,3] for the general background on Markov chains where [1] focuses on biological including epidemic processes, to [4,5,6] for introductory texts on epidemic modeling and related stochastic methods, and to [7,8,9] for extensive surveys of diverse epidemic models. Deterministic and stochastic models are compared in, e.g., [10,11,12]
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