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

Area of Science:

• Epidemiology
• Mathematical Biology
• Stochastic Processes

Background:

• Stochastic epidemic models often assume closed populations, limiting their applicability to real-world scenarios with continuous population dynamics.
• Open populations with immigration and demographic changes (births, deaths) present challenges for traditional epidemic modeling.
• Ensuring disease persistence requires understanding complex population dynamics and their impact on epidemic spread.

Purpose of the Study:

• To develop a mathematical framework for analyzing stochastic epidemics in open populations with variable sizes.
• To investigate the long-term behavior and equilibrium distributions of such epidemics.
• To provide an efficient computational method for assessing the influence of various demographic and disease-related parameters.

Main Methods:

• Modeling epidemics as level-dependent quasi-birth-and-death processes.
• Utilizing matrix-analytic methods for efficient computation of unique equilibrium probability distributions.
• Analyzing multi-dimensional continuous-time Markov chains representing population dynamics.

Main Results:

• Demonstrated that stochastic epidemics in open populations can reach unique, stable equilibrium distributions.
• Showcased the efficiency of matrix-analytic methods for computing these distributions.
• Numerical examples confirmed the model's suitability for analyzing parameter impacts.

Conclusions:

• The proposed modeling approach, using quasi-birth-and-death processes and matrix-analytic methods, is effective for studying persistent stochastic epidemics in open populations.
• This framework allows for a comprehensive analysis of how immigration, birth rates, death rates, infection, recovery, and immunity loss affect epidemic dynamics.
• The findings provide valuable insights for public health strategies and disease management in dynamic populations.