Generality of endemic prevalence formulae

Area of Science:

• Epidemiology
• Mathematical Biology
• Infectious Disease Modeling

Background:

• The susceptible proportion at endemic equilibrium (s*) is often approximated as 1/R0 in simple disease models.
• Realistic epidemiological models require relaxing assumptions of exponential lifetime and infectious periods.

Purpose of the Study:

• To investigate the validity of the s(*)=1/R0 relationship under more complex and realistic modeling assumptions.
• To explore the impact of non-exponential distributions, general recruitment, multiple infection stages, and population heterogeneity on disease dynamics.

Main Methods:

• Developed a generalized mathematical framework for infectious disease dynamics.
• Analyzed homogeneous and multi-group (heterogeneous) population models.
• Introduced a modified reproduction number, R0(e), to account for non-exponential processes.

Main Results:

• In homogeneous populations, s(*) = 1/R0(e), where R0(e) approximates R0 under specific conditions.
• Infection stage proportions correlate with expected time spent in each stage.
• The s(*)=1/R0 formula is robust for many human infections but fails in certain scenarios.
• For heterogeneous populations, the s(*)=1/R0 formula generally does not hold, except under specific symmetry conditions.

Conclusions:

• The simple s(*)=1/R0 relationship is an approximation that requires careful consideration of model assumptions.
• A modified reproduction number, R0(e), provides a more accurate measure in generalized models.
• Population structure significantly impacts the validity of basic epidemiological formulas, necessitating advanced modeling for accurate predictions.