Constrained minimization problems for the reproduction number in meta-population models.

Area of Science:

• Epidemiology
• Mathematical Biology
• Infectious Disease Modeling

Background:

• Heterogeneous mixing in SIR models can significantly increase the basic reproduction number (R0) compared to homogeneous mixing.
• Previous studies used meta-population models and the gradient of the effective reproduction number (Re) to find optimal vaccination strategies.
• Optimal solutions derived from gradients may not always be feasible due to parameter constraints.

Purpose of the Study:

• To derive the analytic conditions for feasible optimal vaccination strategies in SIR models with heterogeneous mixing.
• To obtain explicit expressions for optimal solutions in multi-population models.
• To establish bounds for optimal solutions under general mixing functions.

Main Methods:

• Analysis of SIR models with heterogeneous mixing functions.
• Gradient-based optimization of the effective reproduction number (Re).
• Derivation of analytic conditions for solution feasibility and explicit solution expressions for N sub-populations.

Main Results:

• Conditions for the feasibility of optimal vaccination strategies are analytically derived.
• Explicit optimal solutions are obtained for N=2 sub-populations.
• Bounds for optimal solutions are established for general mixing functions, including proportionate and preferential mixing.

Conclusions:

• The study provides crucial insights into the feasibility of optimal vaccination strategies in heterogeneous populations.
• General mixing schemes result in bounds for Re and R0 influenced by proportionate and isolated mixing.
• Findings are vital for developing robust and practical public health interventions against infectious diseases.