The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect

Area of Science:

• Epidemiology
• Mathematical Biology
• Public Health

Background:

• Critical vaccination coverages aim to reduce the effective reproduction ratio to one.
• Heterogeneous populations present complex challenges for optimal vaccine distribution.
• Existing strategies may not be efficient in minimizing vaccine requirements.

Purpose of the Study:

• To define and solve the optimization problem for the most efficient critical vaccination coverage.
• To establish the equivalence between minimizing vaccine use and maximizing the herd effect.
• To develop a generalizable method for optimal vaccine allocation in diverse populations.

Main Methods:

• Formulated an optimization problem to minimize vaccine doses for an effective reproduction ratio of one.
• Utilized Perron-Frobenius theory for an efficient general solution.
• Derived specific algorithms for separable mixing and two-population interactions.

Main Results:

• The optimization problem is equivalent to maximizing the proportion of uninfected individuals (herd effect).
• An efficient algorithm prioritizes subpopulations based on their size-to-reproduction ratio for separable mixing.
• An explicit analytical solution was found for two interacting populations.

Conclusions:

• Optimal vaccine allocation significantly reduces the number of vaccines needed to achieve critical coverage.
• The proposed method offers a more efficient approach to pandemic preparedness and disease control.
• Perron-Frobenius theory provides a powerful framework for solving complex vaccination optimization problems.