Dynamic Vaccine Prioritization via Non-Markovian Final-state Optimization

TL;DR

Memory effects in transmission render long-horizon vaccine optimization challenging. The authors develop an age-stratified non-Markovian epidemic model and a final-state equivalence to a Markovian surrogate, enabling fast real-time prediction of long-term outcomes under vaccination. They introduce Final-state Dynamic Vaccine Prioritization (FS-DVP) with a lookahead window and a residual effective infection rate to balance indirect transmission blocking with direct protection, and quantify the marginal vaccination benefit (MVB) to explain dynamic priority switches. Across simulations and nine-country COVID-19 case studies, FS-DVP outperforms static and short-horizon strategies, with regime shifts toward indirect protection at low $R_0$ and direct protection at high $R_0$, providing actionable guidance for adaptive vaccine deployment.

Abstract

Effective vaccine prioritization is critical for epidemic control, yet real outbreaks exhibit memory effects that inflate state space and make long-term prediction and optimization challenging. As a result, many strategies are tuned to short-term objectives and overlook how vaccinating certain individuals indirectly protects others. We develop a general age-stratified non-Markovian epidemic model that captures memory dynamics and accommodates diverse epidemic models within one framework via state aggregation. Building on this, we map non-Markovian final states to an equivalent Markovian representation, enabling real-time fast direct prediction of long-term epidemic outcomes under vaccination. Leveraging this mapping, we design a dynamic prioritization strategy that continually allocates doses to minimize the predicted long-term final epidemic burden, explicitly balancing indirect transmission blocking with the direct protection of important groups and outperforming static policies and those short-term heuristics that target only immediate direct effects. We further uncover the underlying mechanism that drives shifts in vaccine prioritization as the epidemic progresses and coverage accumulates, underscoring the importance of adaptive allocations. This study renders long-term prediction tractable in systems with memory and provides actionable guidance for optimal vaccine deployment.

Figures (6)

• Figure 1: Model Construction. a Age-stratified Contact. The population is divided into subgroups according to their age. Contacts occur both within age groups and between different age groups. b General Non-Markovian Epidemic Dynamics. Each individual is in one of four states: susceptible ($\textbf{S}$), infected ($\textbf{I}$), removed ($\textbf{R}$; recovery $\textbf{W}$ or death $\textbf{D}$), or protected ($\textbf{P}$). Transitions occur via: (i) infection of susceptibles through contact with infectious individuals ($\textbf{S}\rightarrow\textbf{I}$); (ii) removal of infected individuals by recovery or death ($\textbf{I}\rightarrow\textbf{R}$), with death governed by age-stratified infection fatality rates (IFRs); and (iii) vaccination-induced protection ($\textbf{S}\rightarrow\textbf{P}$), occurring $\delta$ days after vaccination with probability $\eta$ (vaccine efficacy). Time-varying infectiousness and removal are characterized by the hazard functions $\omega_{\mathrm{inf}}(\tau)$ and $\omega_{\mathrm{rem}}(\tau)$, respectively. c Optimal Vaccine Allocation Strategy (Goal). Based on the model, this study aims to develop an optimal, model-informed strategy for dynamically allocating vaccines among different age groups over time under limited supply, with the objective of minimizing the overall epidemic burden. d State Aggregation (Generality of Non-Markovian Models, Illustrative Example). As an illustrative example of the generality of non-Markovian models, a detailed model with exposed ($\textbf{I}_{\mathrm{exp}}$), asymptomatic ($\textbf{I}_{\mathrm{asym}}$), presymptomatic ($\textbf{I}_{\mathrm{presym}}$), and symptomatic ($\textbf{I}_{\mathrm{sym}}$) substates is aggregated to a single infected class $\textbf{I}$. Simulations (blue: detailed; red $\times$: general) show show perfect agreement in the time evolution of cumulative infections, symptomatic cases and deaths without vaccination (upper panels) and with vaccination (lower panels). In the vaccination scenario, an idealized one-time campaign vaccinates 30% of the total population; the dash-dotted and dotted lines indicate the vaccination time and the onset of vaccine-induced protection, respectively.
• Figure 2: Dynamic vaccine prioritization via final-state Optimization. a Final-state equivalence. (i): At an arbitrary time $t$, for an individual infected at $t'$, only hazards beyond the infection age $a = t-t'$ influence subsequent transmission, thereby defining residual effective infection rate $\lambda_{\dagger}(a)$. Replacing hazard functions with $\omega_{\mathrm{inf}}(\tau)\equiv \mu\lambda_{\dagger}(a)$ and $\omega_{\mathrm{rem}}(\tau)\equiv \mu$ ($\mu>0$) yields a Markovian surrogate with the same final state. (ii): In $100$ Monte Carlo simulations (red curves), switching to the Markovian surrogate at an intermediate time $t$ yields trajectories (blue curves) that converge to identical final cumulative infections, confirming final-state equivalence. b Final-state prediction without vaccination. (i): By solving the Markovian surrogate, the equivalence enables final-state prediction at any time without vaccination. (ii): Predictions made at distinct times (blue $\times$) match the final value (horizontal dashed line) of the cumulative-infection curve (red curve), including at the age-group level (bars). c Final-state prediction under vaccination. (i): Vaccination reduces susceptibles, but protection is realized after a fixed $\delta$-day delay, complicating analysis. (ii): A lookahead sliding window precomputes $\delta$-ahead states, making protection realized at $t+\delta$ available for prediction already at vaccination time $t$. (iii): With this window, final state is predictable at any time under vaccination. (iv): Predictions at and after vaccination (blue $+$ before, red $\star$ at, orange $\times$ after vaccination) align with the final value (horizontal dashed line) of the cumulative infection curve (red curve) under vaccination, including at the age-group level (bars); the blue curve illustrates the decline of susceptible population over time and the abrupt reduction due to the delayed protection (dotted vertical line) following the vaccination (dash-dotted vertical line). d Vaccine Allocation Optimization. (i): Age-stratified final-state infections allow computation of total cumulative infections, deaths, and YLL, defining a cost function maping candidate vaccine allocation to final epidemic burden. (ii): The allocation that minimizes the predicted final epidemic burden is solved using Sequential Least Squares Programming (SLSQP) optimization algorithm. (iii): The optimal strategy outperforms five empirical reference strategies. e Dynamic Vaccine Prioritization. (i): At each vaccination time, we determine and administer the vaccine allocation that minimizes the predicted final epidemic burden at that time, and then repeat the process at the next decision point. (ii): After three rounds, the red curve shows cumulative infections; yellow, blue, and red arrows mark final‑state predictions at the 1st, 2nd, and 3rd rounds, each using only information available up to that time and equaling the realized final size after one (yellow dash‑dotted), two (blue dashed), and three (red solid) rounds.
• Figure 3: a Epidemic trajectories (cumulative infections, deaths, YLL) under non-Markovian final-state dynamic vaccine prioritization (FS-DVP, red solid curve) versus static prioritization strategies targeting specific age groups, for $R_0 = 2.5$, $\theta = 0.35\%$, and a 60-day vaccination campaign. The gray shaded region indicates the vaccination period. b Final-state cumulative infections, deaths, and YLL as a function of $R_0$, comparing FS-DVP to five empirical static strategies; insets show the difference from no-vaccination baseline. Vertical dotted line in middle panel indicates where the performance ranking of static strategies changes substantially for reducing deaths. c Time evolution of cumulative infections for static versus dynamic allocation when $R_0$ is constant (upper: $R_0 = 1.5$) or changes mid-outbreak (lower: $R_0$ shifts from 1.5 to 2.3 at day 150). The gray shaded region indicates the vaccination period. d Final-state outcomes for each control objective (cumulative infections, deaths, YLL) under both constant and varied $R_0$ scenarios. FS-DVP (red dashed) consistently achieves the lowest epidemic burden, while the optimal static strategy varies with scenario and objective.
• Figure 4: a Comparison of final epidemic burdens, i.e., cumulative infections, deaths, and years of life lost (YLL), under a non‐Markovian model across a range of $R_0$ values (from $1.1$ to $2.9$). The primary comparison is between two dynamic strategies: Final‐State Dynamic Vaccine Prioritization (FS‐DVP, red solid) and Transient‐State Dynamic Vaccine Prioritization (TS‐DVP, blue solid). Their performance is benchmarked against static strategies and a no‐vaccination baseline. Vertical dashed lines indicate $R_0=1.4$ and $2.5$, which are the values used for analysis in (b)--(c). b The direct and indirect effects detected by FS-DVP and the resulting optimal, age-stratified dynamic vaccine allocations. Panels are arranged by objective (columns: minimizing cumulative infections, deaths, YLL) and transmission level (rows: $R_0 = 1.4$, $2.5$). In each panel, the top two heatmaps show the direct and indirect effects detected by FS-DVP (entries normalized by the maximum value of the total-effects matrix, i.e., the combined direct and indirect effects, with embedded marginal bars indicating column sums), and the temporal heatmap below shows the age-stratified, dynamic allocations produced by FS-DVP. c The direct effects detected by TS-DVP and the resulting optimal, age-stratified dynamic vaccine allocations. Panels are arranged by objective (columns: minimizing cumulative infections, deaths, YLL). Because TS-DVP lacks access to indirect effects, each panel displays two heatmaps of direct effects detected by TS-DVP at $R_0 = 1.4$ and $R_0 = 2.5$ (each matrix max-normalized to $1$ with embedded column-sum bars), and the two heatmap below show the age-stratified, dynamic vaccine allocations produced by TS-DVP at $R_0 = 1.4$ and $2.5$.
• Figure 5: a--b Comparison of the time evolution curves for the cumulative infected fraction under different strategies, with the basic reproduction number $R_0$ set to $1.5$ (a) and $2.5$ (b). The daily vaccine supply $\theta$ is $0.14\%$ of the total population, and the gray area indicates the vaccination period of 200 days (a) and 80 days (b). c--d The upper panels display our dynamic optimal vaccine allocations over time, with three dashed rectangles marking the first three vaccine shift time points. The dotted area in d highlights the disorder during the vaccination period. The lower panels show the marginal vaccination benefit (MVB) $\xi^{\mathrm{c}}_l$ for each age group $l$ over time, and arrows indicating where the highest MVB curves, together with shifts in vaccine allocation, merge when $R_0$ equals $1.5$ with smaller value (c) or intersect when $R_0$ equals $2.5$ with larger value (d). Inset in the lower panel of (d) zooms in on the MVB curves when only vaccinating the age group 0–9 between the second and third shift time points, demonstrating the highest MVB value for 0–9 during that period. e--f Show the final‐state cumulative infected fraction of each age group relative to the total population under different strategies with $R_0$ set to $1.5$ (e) and $2.5$ (f). g--h illustrate the sum of the time‐varying optimal vaccine allocation for each age group.