Structured Modeling and Analysis of Stochastic Epidemics with Immigration and Demographic Effects

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.

Fig 1. Equilibrium distributions of the total population size and the ratio of the number of infectives to the total population size for parameter setting a₁ = 50, β₀ = 30, λ₀ = 5, λ₁ = 5, γ₁ = 5, μ₀ = 4, μ₁ = 15 with immigration rates of susceptibles a₀ = 10, 100, 200, 500 (top to bottom).

Fig 2. Equilibrium distributions of the total population size and the ratio of the number of infectives to the total population size for parameter setting a₀ = 100, β₀ = 30, λ₀ = 5, λ₁ = 5, γ₁ = 5, μ₀ = 4, μ₁ = 15 with immigration rates of infectives a₁ = 10, 50, 100, 200 (top to bottom).

Fig 3. Equilibrium distributions of the total population size and the ratio of the number of infectives to the total population size for parameter setting a₀ = 100, a₁ = 50, a₂ = 0, β₀ = 30, λ₀ = λ₁ = λ₂ = 5, μ₀ = μ₂ = 6, μ₁ = 15, γ₁ = 5 with immunity loss rates γ₂ = 0, 5, 10, 30 (top to bottom).

Fig 4. Equilibrium distributions of the number of susceptibles and the number of removals for parameter setting a₀ = 100, a₁ = 50, a₂ = 0, β₀ = 30, λ₀ = λ₁ = λ₂ = 5, μ₀ = μ₂ = 6, μ₁ = 15, γ₁ = 5 with immunity loss rates γ₂ = 0, 5, 10, 30 (top to bottom).