Optimal Vaccination Strategies for an Heterogeneous SIS Model

TL;DR

The paper develops a rigorous bi-objective framework for optimal vaccination in heterogeneous SIS populations, framing the problem as minimizing the pair $(C(\eta), L(\eta))$ with $L$ chosen as either the effective reproduction number $R_e$ or the endemic infection level $\mathfrak{I}$ and using $\eta(x)$ to denote the non-vaccinated fraction by feature. It establishes the existence and continuous parametrization of the Pareto frontier for both loss choices, analyzes the anti-Pareto frontier, and derives structural properties (compactness, connectedness, convexity under regularity) of the feasible outcome set. The SIS model is treated in a general infinite-dimensional setting with a weak-* topology to ensure compactness of strategy sets, and key results connect Pareto-optimal solutions to fixed-cost and fixed-loss single-objective problems via value functions $C_\star$ and $L_\star$. An illustrative multipartite graph example demonstrates the practical form of Pareto optima, while companion papers supply proofs of continuity, model couplings, and several extensions. Overall, the work provides a foundational framework for designing targeted vaccination policies with provable trade-off characterizations between vaccine usage and epidemiological impact.

Abstract

We study in a general mathematical framework the optimal allocation of vaccine in an heterogeneous population. We cast the problem of optimal vaccination as a bi-objective minimization problem min(C($η$),L($η$)), where C and L stand respectively for the cost and the loss incurred when following the vaccination strategy $η$, where the function $η$(x) represents the proportion of non-vaccinated among individuals of feature x.To measure the loss, we consider either the effective reproduction number, a classical quantity appearing in many models in epidemiology, or the overall proportion of infected individuals after vaccination in the maximal equilibrium, also called the endemic state. We only make few assumptions on the cost C($η$), which cover in particular the uniform cost, that is, the total number of vaccinated people.The analysis of the bi-objective problem is carried in a general framework, and we check that it is well posed for the SIS model and has Pareto optima, which can be interpreted as the ``best'' vaccination strategies. We provide properties of the corresponding Pareto frontier given by the outcomes (C($η$), L($η$)) of Pareto optimal strategies.

Figures (3)

• Figure 1: Example of optimization with $\mathrm{L} = R_e$ for a multipartite graph with no intra-population transmission.
• Figure 2: An example of the possible aspects of the feasible region $\mathbf{F}$ (in light blue), the value functions $\mathrm{L}_\star$, $\mathrm{L}^\star$, $C_\star$, $C^\star$, and the Pareto and anti-Pareto frontier (in red) under Assumption \ref{['hyp:loss+cost']}.
• Figure 3: Typical shape of the Pareto frontier $\mathcal{F}$ in solid red line, the anti-Pareto frontier $\mathcal{F}^\mathrm{Anti}$ in red dashed line and the feasible region $\mathbf{F}$ in light blue for the SIS model with a monatomic kernel.

Theorems & Definitions (48)

• Theorem 1.1: Properties of the Pareto frontier
• Remark 1.2: Optimal critical strategies do not depend on the loss
• Remark 2.1: Other properties of the various sets of vaccination strategies
• Definition 2.2
• Remark 2.3: Affine cost functions
• Proposition 2.4: Optimal solutions for fixed cost or fixed loss
• proof
• Proposition 2.5: Elementary properties of $\mathrm{L}_\star,\mathrm{L}^\star,C_\star,C^\star$
• proof
• Proposition 2.6: Single-objective and bi-objective problems