The stochastic general epidemic model revisited and a generalization

Area of Science:

• Epidemiology
• Mathematical Biology
• Statistical Modeling

Background:

• The classical general epidemic model, established by Bartlett (1949), has theoretical origins dating back to Ross (1911).
• Explicit state probabilities for this model were found by Kryscio (1975), but were computationally limited for larger populations.
• Previous mathematical frameworks struggled with scalability due to population size constraints.

Purpose of the Study:

• To develop a more practical and scalable mathematical framework for epidemic modeling.
• To derive explicit solutions for epidemic state probabilities irrespective of population size.
• To extend epidemic modeling to handle generalized processes and time-dependent rates.

Main Methods:

• Shifted focus from (susceptibles, infectives) to (infectives, removals) state variables.
• Derived new formulae for state probabilities in a generalized epidemic process.
• Applied derived solutions to the classical Bartlett epidemic model.
• Considered extensions for time-dependent transition rates.

Main Results:

• Obtained significantly simpler and more manageable formulae for epidemic state probabilities.
• The new formulae are not restricted by population size, offering broader applicability.
• Demonstrated the utility of the (infectives, removals) approach for generalized epidemic processes.
• Successfully applied the method to the classical model and considered time-varying rates.

Conclusions:

• The (infectives, removals) state representation provides a more computationally tractable approach to epidemic modeling.
• This method overcomes the population size limitations of previous models.
• The generalized framework and extensions offer enhanced flexibility for analyzing various epidemic scenarios.