Optimal timing for staged vaccination campaigns: Insights from scenario tree-based stochastic optimization

Abstract

We develop a mathematical modeling framework to address the challenge in launching an effective staged vaccination campaign during a typical viral infection season to avoid an overwhelmed healthcare system for the entire season. Using the COVID-19 pandemic following its acute phase as a motivating example, our model takes into account the willingness of the public to receive vaccines, as well as the uncertainty of vaccination delivery and administration, to achieve the objective of optimizing the timing and distribution of the vaccination campaign, subject to the constraint that hospitalized cases do not exceed healthcare capacity. The integration of a dynamic transmission model with a scenario tree–based stochastic optimization framework enables the evaluation of future scenarios characterized by uncertainty in vaccination rates, contact mixing, and public adherence to safety measures. Accounting for these future scenarios facilitates the identification of strategies to dynamically adjust the timing and scale of vaccine and contact mixing during distinct phases of the viral season. Our study demonstrates that a well–timed, well–phased vaccination campaign, along with other public health interventions, can prevent overcrowding in hospitalization in a typical viral infection season.

Highlights

• •Developed a mathematical modeling framework for optimizing staged vaccination timing to prevent healthcare system overload.
• •Integrated a dynamic transmission model with scenario tree–based stochastic optimization under uncertainty.
• •Accounted for uncertainty in vaccination rates, contact mixing, and public adherence across seasonal phases.
• •Demonstrated that well–timed, phased vaccination combined with public health interventions prevents hospitalization overcrowding.

1. Introduction

The COVID-19 pandemic has emphasized the importance of the rapid development and deployment of timely and effective vaccination campaigns to reduce disease burden on healthcare systems. The success of such campaigns has been influenced by many factors, including vaccine availability and public willingness to vaccinate (Lun et al., 2023; Marzo et al., 2022), and the capacity of healthcare systems to administer vaccines efficiently during the pandemic (Cuadros et al., 2023).

Vaccine hesitancy remains a significant challenge, identified by the World Health Organization (WHO) as one of the top ten global health threats in 2019, and continues to pose significant challenges in achieving widespread immunization coverage (Nuwarda et al., 2022; Temsah et al., 2021). Vaccine hesitancy is influenced by many factors, including misinformation, mistrust of the healthcare system and governments, and worries about the safety and effectiveness of vaccines (Kricorian et al., 2022; Larson et al., 2018). While much attention has been given to understanding the behavioral and psychological factors that influence vaccine acceptance (Guay et al., 2022; Murphy et al., 2021; Troiano & Nardi, 2021), less focus has been placed on how changes in public willingness over time dynamically impact vaccination strategies. Optimizing vaccination schedules under such uncertainty remains a significant challenge. Most existing optimization models have focused on who should receive vaccines, but do not account for how willingness to vaccinate can change over time (Matrajt et al., 2021). This study examines this connection, offering insights that could improve the effectiveness of vaccination campaigns.

As governments worldwide have faced these challenges and started to prepare for the next pandemic, the need for data-driven approaches to optimize vaccination strategies has become increasingly apparent. Mathematical modeling and optimization techniques provide a framework for evaluating different vaccination scenarios, allowing policymakers to make informed decisions that balance public health goals with practical constraints (Buckner et al., 2021; Moghadas, Fitzpatrick, et al., 2020). Canadian health authorities continue to emphasize the elevated risk of COVID-19 transmission during the fall and winter seasons, particularly in the context of co-circulating respiratory viruses such as influenza and RSV (Public Health Agency of Canada, 2025). Although the acute phase of the pandemic has passed, it remains crucial to effectively mitigate future risks by using the available vaccines. As new vaccines, designed to be effective against mutant strains, become available during the fall and winter seasons that coincide with the seasons of other respiratory infections, strategic vaccination planning remains important.

In this context, the province of Ontario, Canada, presents a valuable case study. With a population exceeding 15 million, Ontario has faced significant challenges in managing COVID-19, particularly in ensuring that its healthcare system can cope with surges in cases while rolling out an effective vaccination program (Cheng et al., 2024). Since the onset of the pandemic in early 2020, Ontario has implemented various public health measures, including lockdowns, social distancing mandates, and mask-wearing protocols, all aimed at curbing the spread of the virus (Abdollahi et al., 2020). However, as vaccination efforts ramped up, the focus has shifted toward optimizing the timing and scale of vaccination to prevent overwhelming the healthcare system, especially the intensive care units (ICUs) (Webster, 2021).

In this study, we modify the established optimization framework (Xiao et al., 2021) to address the complexities of COVID-19 vaccination planning in Ontario, particularly in the context of a potential future outbreak. We develop a stochastic optimization model to determine the optimal timing and scale of vaccinations over a six-month evaluation horizon, beginning at the start of the vaccination campaign, while considering uncertainty in vaccination uptake and public willingness to vaccinate. By incorporating varying vaccination rates and the ongoing dynamics of disease transmission, the model seeks to identify vaccination pathways that ensure ICU capacity is not exceeded. The findings of this study provide insights into how vaccination strategies can be optimized under uncertainty, offering guidance for policymakers preparing for future outbreaks.

2. Methods

2.1. Transmission dynamics model of COVID-19

We developed a deterministic compartmental model to simulate the transmission dynamics of COVID-19 in Ontario, assuming a constant population size N. The model includes the following compartments: S for susceptible unvaccinated individuals; V for vaccinated individuals; E for exposed individuals; A for asymptomatic individuals; I for symptomatic individuals; D for diagnosed and isolated individuals; H for non-ICU hospitalized individuals; H_ICU for ICU hospitalized individuals; and R for recovered individuals. A schematic diagram of the model structure is provided in Fig. 1.

Fig. 1.

Flowchart of the compartmental model capturing the transmission dynamics of COVID-19.

In the model, susceptible individuals S can be vaccinated at a rate ν (defined in Eq. (3)), moving them to the vaccinated compartment V. When susceptible individuals contract COVID-19, they transition to the exposed compartment E. Vaccinated individuals may still become infected, reflecting imperfect vaccine-induced protection, with reduced infection force λ(1 − η), moving them to the exposed compartment E. In this model, vaccination provides direct protection by reducing the infection force experienced by vaccinated individuals through the factor 1 − η, and indirect protection by lowering overall infection prevalence, which in turn reduces exposure risk in the population. Exposed individuals E can either become symptomatic I at rate σα or asymptomatic A at rate σ(1 − α). Asymptomatic individuals A recover and transition to the recovered compartment R at rate γ_A. A proportion q of symptomatic individuals I will be diagnosed and isolated, entering compartment D at rate δ, while the remaining (1 − q) will recover and move to the recovered compartment R at rate γ_I. Of those diagnosed individuals D, a fraction h will require hospitalization at rate ρ, representing a population-level severity proportion among diagnosed and isolated cases, with f moving to non-ICU beds H and (1 − f) to ICU beds H_ICU. The remaining (1 − h) of the individuals in D will recover and move to R at rate γ_D. Additionally, at rate b_H, some individuals in non-ICU beds H will move to ICU beds H_ICU. The recovery rates for individuals in non-ICU and ICU beds are γ_H and ${\gamma }_{H_{\text{ICU}}}$, respectively.

From the above definitions and variables, we have the following model with non-negative initial conditions:

$$\begin{array}{lll}\frac{dS}{dt}& =& -\lambda S-\nu S,\\ \frac{dV}{dt}& =& \nu S-\lambda (1-\eta )V,\\ \frac{dE}{dt}& =& \lambda (1-\eta )V+\lambda S-\sigma E,\\ \frac{dI}{dt}& =& \sigma \alpha E-(1-q){\gamma }_{I}I-q\delta I,\\ \frac{dA}{dt}& =& \sigma (1-\alpha )E-{\gamma }_{A}A,\\ \frac{dD}{dt}& =& q\delta I-(1-h){\gamma }_{D}D-h\rho D,\\ \frac{dH}{dt}& =& hf\rho D-{\gamma }_{H}H-b_{H}H,\\ \frac{d{H}_{\mathit{ICU}}}{dt}& =& b_{H}H+h(1-f)\rho D-{\gamma }_{H_{\text{ICU}}}{H}_{\mathit{ICU}},\\ \frac{dR}{dt}& =& (1-q){\gamma }_{I}I+{\gamma }_{A}A+(1-h){\gamma }_{D}D+{\gamma }_{H}H+{\gamma }_{H_{\text{ICU}}}{H}_{\mathit{ICU}},\end{array}$$ (1)

where

$$N=S+V+E+I+A+D+H+H_{\mathit{ICU}}+R.$$

The force of infection (λ) is defined as:

$$\lambda =\frac{\beta C(I+aA)}{N}.$$ (2)

We now introduce the average vaccination rate ν over a period T, taking into account the willingness of the population to vaccinate W, a benchmark daily vaccine supply level K, and the number of recipients $N_{r^{(k)}}$ available during each period k, as follows:

$$\nu (W)=\frac{{\sum }_{k=0}^T\frac{\left(1-\frac{1}{T}\right)\cdot K\cdot W}{N_{r^{(k)}}}}{T},$$ (3)

where.

• •W is the willingness of the population to receive vaccination, scaled between 0 and 1;
• •K is a benchmark daily vaccination level used in constructing the willingness-based vaccination-rate scenarios;
• •$\left(1-\frac{1}{T}\right)$ represents the proportion of the evaluation period during which an individual is available to receive a vaccine;
• •$N_{r^{(k)}}$ denotes the number of individuals remaining eligible for vaccination after the preceding term in the summation, where $N_{r^{(0)}}=S_0$.

To calculate $N_{r^{(k+1)}}$ at term k of the summation, we define

$$d=\frac{\left(1-\frac{1}{T}\right)\cdot K\cdot W}{N_{r^{(k)}}}.$$

Then,

$$N_{r^{(k+1)}}=N_{r^{(k)}}(1-d).$$

Non-negativity condition: For the function ν(W) to remain non-negative for all k, it is sufficient that

$$N_{r^{(k+1)}}=N_{r^{(k)}}(1-d)>0.$$

2.2. Model parameters and calibration

Table 1 lists the model parameters used in this study. To maintain consistency throughout the analysis, the total baseline population of Ontario was set to N = 15, 608, 369.

Table 1.

Parameter definitions and values used in simulations, with references.

Parameter	Definition	Value	Reference
a	Relative transmission rate of asymptomatic individuals	0.25	He et al. (2020)
β	Transmission coefficient (per-contact probability of infection)	0.076398	Estimated
C	Average contacts per person per day	11.5801	Xiao et al. (2021)
η	Vaccine effectiveness against infection	0.65	Moore et al. (2025)
ν	Vaccination rate	Defined earlier	(3)
σ	Rate from exposed to infectious	1/5	Xiao et al. (2021)
α	Probability of having symptoms among infected individuals	0.7036	Xiao et al. (2021)
γI	Infectious recovery rate	0.1957	Tang et al. (2020)
γA	Asymptomatic recovery rate	0.139	Tang et al. (2020)
δ	Isolation initiation rate	0.1344	Tang et al. (2020)
γH	Hospitalized COVID-19 recovery rate	1/12	Moghadas, Shoukat, et al. (2020)
${\gamma }_{H_{\text{ICU}}}$	ICU COVID-19 recovery rate	1/14	Shoukat et al. (2020)
γD	Recovery rate for isolated individuals	0.2	Tang et al. (2020)
q	Proportion of infectious individuals who enter isolation	0.23	Assumed
f	Proportion of isolated individuals requiring hospitalization in non-ICU units	0.8731	Xiao et al. (2021)
ρ	Hospitalization rate	1/9	Wölfel et al. (2020)
h	Proportion of diagnosed and isolated individuals requiring hospitalization	0.0342	Xiao et al. (2021)
bH	Transfer rate from non-ICU units to ICU units among hospitalized patients	0.0184	Xiao et al. (2021)

The transmission coefficient β was estimated by fitting the model to cumulative reported COVID-19 cases in Ontario from February 26 to April 21, 2020, corresponding to the first wave of the pandemic prior to vaccine availability. For this calibration, we used the Ontario cumulative case time series reported in the supplementary material accompanying Tang et al. (Tang et al., 2020). The remaining baseline parameter values were taken from the literature, as listed in Table 1.

For the contact schedule during this calibration period, we used the Ontario contact pattern reported by Tang et al. (Tang et al., 2020). Using this contact structure, we applied a nonlinear least-squares method to estimate β together with the initial conditions for the exposed, symptomatic infectious, and asymptomatic infectious populations, namely E₀, I₀, and A₀.

The fitting procedure yielded β = 0.076398, E₀ = 130.93, I₀ = 21.02, and A₀ = 5.00. Fig. 2 shows the agreement between the fitted model trajectory and the observed cumulative case data. This early-wave contact schedule was used only to estimate the baseline value of β. In the subsequent scenario-based stochastic optimization analysis, phase-specific contact rates and transmission levels were specified separately to represent prospective seasonal conditions. Because the main goal of this study is to illustrate a stochastic optimization framework for vaccination timing, transmission was calibrated to reported case data, while hospitalization and ICU trajectories were treated as model-generated outcomes used to evaluate healthcare burden under stressed scenarios.

Fig. 2.

Calibration of the transmission coefficient β using cumulative reported COVID-19 cases in Ontario during the pre-vaccination first wave (February 26 to April 21, 2020), used solely to establish a baseline estimate of β for the subsequent scenario-based analysis. The fitted model trajectory is shown alongside observed cumulative case data.

2.3. Willingness to vaccinate

We will use model (1) to investigate the effect of willingness on vaccination uptake in the population based on the function (3). We will focus on different phases where the level of willingness to vaccinate is different from the studies of (Nah et al., 2020; Xiao et al., 2021) where the focus was on social distancing escalation or deescalation pathways. We consider different phases of low, moderate, and high willingness. Incorporating phases based on vaccination willingness is crucial at the beginning of the assumed seasonal vaccination campaign for COVID-19, informed by data on dose availability during the fall (Public Health Agency of Canada, 2025), which coincides with the influenza season. We use these three assumed phases to model willingness to vaccinate under the assumption that vaccination uptake may increase as perceived infection risk rises during periods of elevated respiratory virus activity (Baumgaertner et al., 2020). Accordingly, initial willingness is assumed to be relatively low at the start of the campaign, with increases expected as risk perception changes over time. Individual willingness is a key factor in increasing vaccination rates (Basta et al., 2022); however, willingness alone does not fully explain variations in daily vaccination numbers, suggesting that whether or not an individual receives a vaccine is more complex and influenced by a variety of factors (Jiang & Feng, 2022). Given this complex nature of vaccination rates, we employ a stochastic approach to account for uncertainties. To model uncertainty in vaccination uptake, we first specified phase-specific willingness values or ranges and converted these into corresponding average vaccination rates using Eq. (3), which links willingness W, the benchmark daily vaccine supply K, and the remaining eligible population over the evaluation horizon. For Phase 0, a fixed willingness value was used to determine the initial vaccination rate ν₀. For Phases 1–3, willingness ranges were mapped to corresponding vaccination-rate ranges, and symmetric triangular distributions were defined on these ranges. Two representative realizations were then selected from each of Phases 1–3, resulting in 2 × 2 × 2 = 8 scenarios. Phase 0 was treated as a common initial decision across all scenarios, while the subsequent phase durations were allowed to adapt according to the realized scenario path.

2.4. Scenario tree-based stochastic optimization model

In this section, we analyze the impact of uncertainty on vaccination rates in Ontario by considering four distinct phases of the COVID-19 vaccination campaign during the fall and winter seasons. As discussed earlier, a primary factor contributing to this uncertainty is the willingness of the population to receive the COVID-19 vaccine. To account for this uncertainty, we assume various percentage ranges representing the population's willingness to vaccinate for each phase. Using the average vaccine rate function defined by Eq. (3), and considering the different ranges of willingness, we can determine the ranges for the average vaccination rate. To further address the uncertainty in vaccination rates across all phases, we applied a symmetric triangular distribution on these ranges. While the phase structure captures systematic changes in population willingness over the vaccination season, scenarios are introduced to represent uncertainty in the realized vaccination rate within each phase, arising from variability in individual uptake and vaccine availability. Accordingly, the scenario tree consists of eight terminal paths formed by the phase-specific combinations of two representative realizations in each of Phases 1–3, while the Phase 0 vaccination rate remains common across all scenarios.

To evaluate phased vaccination strategies under uncertainty, we partition the campaign timeline into four intervals with start times t₀, t₁, t₂, and t₃, each marking the onset of a vaccination phase. According to the Public Health Agency of Canada, increases in COVID-19 test positivity may precede the rollout of seasonal vaccination campaigns, and updated COVID-19 vaccine doses are typically made available in the fall (Public Health Agency of Canada, 2025). Motivated by this guidance, we assume that the initial start time for vaccine availability, t₀, corresponds to September 1 (day–month format). For the vaccination rate, we denote ν₀, ν₁, ν₂, and ν₃ through the four phases. To represent uncertainty in vaccination uptake within each phase, we consider a finite set of scenarios indexed by j = 1, 2, …, 8, where each scenario corresponds to a particular realization of the vaccination rate across phases. For the initial phase, we employ a fixed percentage of willingness as a baseline to determine the initial vaccination rate. The evaluation period T concludes on the 1st of March. A vaccination strategy is defined by the decision variables (ϵ₀, ϵ₁, ϵ₂), where ϵ_i = t_i+1 − t_i for i = 0, 1, 2 denotes the duration of Phases 0, 1, and 2, respectively. The duration of Phase 3, ϵ₃, is determined implicitly by ϵ₀ + ϵ₁ + ϵ₂ + ϵ₃ = T.

To explicitly manage ICU bed capacity, the ICU hospitalization component of the COVID-19 transmission model was incorporated as a constraint within the optimization framework. Our objective is to ensure that the number of hospitalized patients in ICU units at any time t, denoted H_ICU(t), does not exceed the threshold $\underset{\_}{H_{\mathit{ICU}}}$, representing the available ICU bed capacity.

$$H_{\mathit{ICU}}(t)\le \underset{\_}{H_{\mathit{ICU}}},\text{for}{t}_0<t<t_0+T,$$

where t₀ + T is the endpoint of the evaluation period, H_ICU(t) represents the number of hospitalized individuals at time t, and $\underset{\_}{H_{\mathit{ICU}}}$ denotes the hospital capacity threshold. Based on publicly reported ICU occupancy levels during periods of elevated COVID-19 activity in Ontario, we set the ICU capacity threshold to $\underset{\_}{H_{\mathit{ICU}}}=566$ (Government of Ontario, 2022), corresponding to the reported ICU occupancy level during peak activity. Therefore, we used this number as the threshold for the capacity of the COVID-19 ICU beds. For the decision variables (ϵ₀, ϵ₁, ϵ₂), we assume a lower bound such that ${ϵ}_0^j,{ϵ}_1^j,{ϵ}_2^j\ge 14$ for each scenario j = 1, 2, 3, …, 8. This accounts for the minimum time needed to assess the willingness of individuals to vaccinate, which involves conducting surveys and is estimated to take at least 14 days. Furthermore, we impose an inequality constraint: ${ϵ}_0^j+{ϵ}_1^j+{ϵ}_2^j\le T$ for each j = 1, 2, 3, …, 8. Consistent with standard scenario-based stochastic programming formulations, Phase 0 represents a here-and-now decision that is common across scenarios, whereas subsequent phase durations may adapt to scenario-dependent vaccination rates.

To ensure the realism and applicability of the stochastic model to real-world decision-making, decisions must be made in the face of uncertainty and adjusted over time as that uncertainty is resolved. We applied non-anticipativity constraints on our decision variables. Thus, ϵ₀ (the duration of Phase 0) represents a here-and-now decision made before any uncertainty is revealed, based on the information available at the current time. In contrast, ${ϵ}_1^j$ and ${ϵ}_2^j$ (the durations of Phases 1 and 2 for scenario j) are recourse decisions made after some uncertainties have been resolved. ${ϵ}_1^j$ is determined after ${\nu }_1^j$ is observed and is adapted accordingly. Similarly, ${ϵ}_2^j$ is decided after ${\nu }_2^j$ is observed and also adapted.

Objective Function: Our goal is to minimize the vaccination effort associated with overly optimistic assumptions about vaccination rates, while determining the optimal initiation times for the vaccination pathway during each of the four phases: 0, 1, 2, and 3. In addition, our goal is to ensure that the number of hospitalizations remains below a predefined threshold throughout these phases. To achieve this, we employed a modified version of the scenario-based stochastic programming model in (Nah et al., 2020), which is expressed as follows:

$$\sum _j{p}^j\left(\sum _{i=0}^3{\nu }_i^j{ϵ}_i^j+u\cdot g\left({ϵ}_0^j,{ϵ}_1^j,{ϵ}_2^j\right)\right)$$

where

$$g\left({ϵ}_0^j,{ϵ}_1^j,{ϵ}_2^j\right)=-\log \left(\underset{\_}{H_{\mathit{ICU}}}-{\max }_{t_0<t<t_0+T}{H}_{\mathit{ICU}}(t)\right),$$

Given that u is a penalty parameter associated with the penalty function g and is set to 10⁻⁶, and p^j represents the likelihood of scenarios with j = 1, 2, …, 8, the objective function balances vaccination effort across phases with a penalty for exceeding ICU capacity.

The scenario-based model is interpreted within a rolling-horizon framework. Initially, we solve the model and implement the optimal Phase 0 duration ϵ₀ immediately. Near the end of Phase 0, the model can be re-solved using updated information from the completed phase, such as the current system state and observed vaccination uptake. This yields an updated Phase 0 duration to be implemented at that time. Although ${ϵ}_1^j$ and ${ϵ}_2^j$ are never implemented, their inclusion ensures that the implemented ϵ₀ does not lead to future infeasibility under any scenario considered in the model.

3. Results

In this section, we present our findings based on our stochastic optimization modeling. Our goal is to evaluate the impact of different vaccination strategies in the province of Ontario during the fall and winter seasons under outbreak conditions, using pandemic-era parameters. The objective is to determine the optimal initial timing for the vaccination pathway for the four phases while ensuring that hospitalizations in ICU beds remain manageable. We consider two epidemiological cases corresponding to high and low initial willingness to vaccinate.

To address the goal of optimal vaccination timing, we consider the administration of a seasonal vaccine dose beginning in early September. The concept of a fall vaccination campaign is supported by national guidance and established public health practice in Canada, where updated COVID-19 vaccine formulations are typically made available ahead of the fall–winter respiratory virus season (Public Health Agency of Canada, 2025). Accordingly, we assume that a seasonal dose is administered in the fall to mitigate the anticipated increase in transmission risk.

3.1. COVID-19 outbreak with initially high willingness to vaccinate

In this subsection, we examine the effects of an initially high willingness among individuals to receive an updated COVID-19 vaccine dose on the first of September. We consider a scenario with a gradual decrease in the contact rate (C) without enforcing a lockdown. Additionally, we assume that recommendations related to personal protective equipment (PPE) and other protective behaviors are initially followed to some extent at the start of the evaluation period (T). However, over time, adherence to PPE usage diminishes, either due to reduced public compliance or the absence of enforced PPE guidelines. This is reflected in a slightly increased transmission rate (β). We also examined a second high-willingness case in which the transmission rate and contact rate were fixed across all phases to assess how the optimized vaccination timing responds under sustained transmission pressure.

Using the stochastic optimization model defined in Section 2.4, we analyzed an initially high-willingness setting to determine a phased vaccination strategy under uncertainty. The scenario construction for this setting is summarized in Table 2, which reports the benchmark daily vaccination level, phase-specific willingness assumptions, the resulting vaccination rates, and the optimized phase durations across the eight scenarios. The eight scenarios arise from the phase-specific combinations of two representative realizations in each of Phases 1–3. In this case, we assumed an initially high willingness to vaccinate, with a gradual decrease in contact rate after Phase 0 and a modest increase in transmission intensity across later phases to reflect reduced adherence to protective behaviors over time. In this case, the contact rate decreases from C₀ = 11.5801 in Phase 0 to C_r = 8.4644 in the remaining evaluation period, while the transmission rate increases from β₀ = 0.0702 to β₁ = 0.0723 and then to β_r = 0.076398. Additional parameter details for this case are provided in Appendix A, Table A.1.

Table 2.

Consolidated summary of the benchmark daily vaccination level K, willingness assumptions W, scenario-specific vaccination rates ν, optimized phase durations ϵ (days), and initiation dates for the high-willingness setting with phase-varying transmission. Phase 0 quantities are common to all scenarios.

Shared input	Value
Benchmark daily vaccination level K	100, 000 doses/day (planning value)
W0 (Phase 0 willingness, fixed)	75%
W1 range (Phase 1)	70%–75%
W2 range (Phase 2)	75%–80%
W3 range (Phase 3)	80%–85%
ν0 (Phase 0 vaccination rate)	0.0109

Quantity (phase-specific)	s1	s2	s3	s4	s5	s6	s7	s8
ν1	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101
ν2	0.0122	0.0122	0.0126	0.0126	0.0122	0.0122	0.0126	0.0126
ν3	0.0165	0.0165	0.0165	0.0165	0.0176	0.0176	0.0176	0.0176
ϵ0 (days)	16.50	16.50	16.50	16.50	16.50	16.50	16.50	16.50
ϵ1 (days)	16.99	16.52	16.99	16.52	16.99	16.52	16.99	16.52
ϵ2 (days)	16.43	16.14	16.02	15.81	16.43	16.14	16.02	15.81
ϵ3 (days)	131.08	131.84	131.49	132.17	131.08	131.84	131.49	132.17
t0	01 Sep
t1	17 Sep
t2	04 Oct
t3	20 Oct	20 Oct	20 Oct	19 Oct	20 Oct	20 Oct	20 Oct	19 Oct

The optimized vaccination pathway for the initially high-willingness case is summarized in Table 2. Across all eight scenarios, the optimal duration of Phase 0 is approximately 16.5 days. Phase 1 varies between about 16.5 and 17.0 days, while Phase 2 ranges from about 15.8 to 16.4 days. Phase 3 is the longest phase, lasting approximately 131–132 days, with an optimal start date of 19–20 October. The corresponding phase-wise vaccination totals are reported in Table 3. Phase 0 accounts for approximately 2.35 million vaccinations (15.08% of the Ontario population), while total vaccinations across the full evaluation horizon range from 9.66 to 10.06 million across scenarios. The epidemiological dynamics under these optimized pathways are shown in Fig. 3. The detailed scenario tree for this case is provided in Fig. A.1 in Appendix A.

Table 3.

Total vaccinations administered during each phase across eight scenarios. Values are reported as total vaccinations (in millions), followed by the percentage of the population in parentheses.

Scenario	Phase 0	Phase 1	Phase 2	Phase 3	Total
s1	2.35 (15.08%)	2.05 (13.14%)	2.06 (13.18%)	3.20 (20.49%)	9.66 (61.89%)
s2	2.35 (15.08%)	2.11 (13.54%)	1.94 (12.43%)	3.29 (21.06%)	9.69 (62.11%)
s3	2.35 (15.08%)	2.05 (13.14%)	2.13 (13.64%)	3.19 (20.44%)	9.72 (62.30%)
s4	2.35 (15.08%)	2.11 (13.54%)	2.01 (12.87%)	3.28 (21.01%)	9.75 (62.50%)
s5	2.35 (15.08%)	2.05 (13.14%)	2.06 (13.17%)	3.50 (22.43%)	9.96 (63.82%)
s6	2.35 (15.08%)	2.11 (13.54%)	1.94 (12.43%)	3.59 (23.00%)	10.00 (64.05%)
s7	2.35 (15.08%)	2.05 (13.14%)	2.13 (13.64%)	3.49 (22.37%)	10.03 (64.23%)
s8	2.35 (15.08%)	2.11 (13.54%)	2.01 (12.87%)	3.58 (22.97%)	10.06 (64.46%)

Fig. 3.

Epidemiological dynamics under the optimized vaccination pathways for the initially high-willingness setting with phase-varying transmission. Panels show ICU hospitalizations, cumulative ICU hospitalizations, non-ICU hospitalizations, infections, and vaccine rate over calendar months, together with the no-vaccination reference trajectory.

As shown in Fig. 3, in the absence of vaccination, assuming the late-phase transmission level used in this case, β = 0.076398, together with C = 8.4644, ICU hospitalizations peak in late November and substantially exceed the capacity threshold of 566 beds. In contrast, under the optimized vaccination scenarios, peak ICU occupancy is substantially reduced and remains below the healthcare capacity threshold. Additionally, we observe fewer infections and a reduced need for non-ICU hospitalization beds.

To evaluate how the optimized vaccination timing responds to sustained transmission pressure, we also considered a second high-willingness case in which the transmission rate and contact rate were fixed across all phases, with β = 0.076398 and C = 8.4644. The optimized pathway for this case is summarized in Table 4. Under this assumption, Phase 0 lasts approximately 21.4 days across all scenarios, Phase 1 ranges from 22.18 to 25.15 days, Phase 2 ranges from 17.29 to 20.04 days, and Phase 3 lasts approximately 114.79–119.31 days, beginning between 01 and 06 November. Compared with the phase-varying transmission case, vaccination effort is redistributed toward the earlier phases, with Phase 0 accounting for about 3.06 million vaccinations and total vaccinations ranging from 9.68 to 10.22 million across scenarios, as reported in Table 5. Under the same fixed-transmission assumptions, the no-vaccination reference trajectory shown in Fig. 4 exceeds the ICU capacity threshold, whereas the optimized vaccination scenarios remain below that threshold. The corresponding epidemiological dynamics under constant transmission are shown in Fig. 4. The detailed scenario tree for this case is provided in Fig. A.3 in Appendix A.

Table 4.

Consolidated summary of the benchmark daily vaccination level K, willingness assumptions W, scenario-specific vaccination rates ν, optimized phase durations ϵ (days), and initiation dates for the high-willingness setting under constant transmission. Phase 0 quantities are common to all scenarios.

Shared input	Value
Benchmark daily vaccination level K	100, 000 doses/day (planning value)
W0 (Phase 0 willingness, fixed)	75%
W1 range (Phase 1)	70%–75%
W2 range (Phase 2)	75%–80%
W3 range (Phase 3)	80%–85%
ν0 (Phase 0 vaccination rate)	0.0109

Quantity (phase-specific)	s1	s2	s3	s4	s5	s6	s7	s8
ν1	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101
ν2	0.0122	0.0122	0.0126	0.0126	0.0122	0.0122	0.0126	0.0126
ν3	0.0165	0.0165	0.0165	0.0165	0.0176	0.0176	0.0176	0.0176
ϵ0 (days)	21.42	21.42	21.42	21.42	21.42	21.42	21.42	21.42
ϵ1 (days)	25.15	22.18	25.15	22.18	25.15	22.18	25.15	22.18
ϵ2 (days)	19.64	20.04	17.29	18.09	19.64	20.04	17.29	18.09
ϵ3 (days)	114.79	117.36	117.15	119.31	114.79	117.36	117.15	119.31
t0	01 Sep
t1	22 Sep
t2	17 Oct	14 Oct	17 Oct	14 Oct	17 Oct	14 Oct	17 Oct	14 Oct
t3	06 Nov	03 Nov	03 Nov	01 Nov	06 Nov	03 Nov	03 Nov	01 Nov

Table 5.

Total vaccinations administered during each phase across eight scenarios. Values are reported as total vaccinations (in millions), followed by the percentage of the population in parentheses.

Scenario	Phase 0	Phase 1	Phase 2	Phase 3	Total
s1	3.06 (19.60%)	2.84 (18.19%)	1.95 (12.48%)	1.84 (11.76%)	9.68 (62.03%)
s2	3.06 (19.59%)	2.61 (16.75%)	2.06 (13.21%)	2.07 (13.29%)	9.81 (62.84%)
s3	3.06 (19.60%)	2.84 (18.19%)	1.83 (11.74%)	2.09 (13.40%)	9.82 (62.93%)
s4	3.06 (19.59%)	2.61 (16.75%)	1.96 (12.57%)	2.34 (14.96%)	9.97 (63.87%)
s5	3.06 (19.60%)	2.84 (18.19%)	1.95 (12.48%)	2.07 (13.26%)	9.92 (63.53%)
s6	3.06 (19.59%)	2.61 (16.75%)	2.09 (13.37%)	2.34 (15.01%)	10.10 (64.72%)
s7	3.06 (19.60%)	2.84 (18.19%)	1.83 (11.74%)	2.34 (14.99%)	10.07 (64.52%)
s8	3.06 (19.59%)	2.61 (16.75%)	1.96 (12.57%)	2.59 (16.60%)	10.22 (65.51%)

Fig. 4.

Epidemiological dynamics under the optimized vaccination pathways for the initially high-willingness setting under constant transmission (β = 0.076398, C = 8.4644). Panels show ICU hospitalizations, cumulative ICU hospitalizations, non-ICU hospitalizations, infections, and vaccine rate over calendar months, together with the no-vaccination reference trajectory.

3.2. COVID-19 outbreak with initially low willingness to vaccinate

In this subsection, we explore a scenario characterized by initially low vaccination willingness combined with higher adherence to protective behaviors. This is modeled through a fixed contact rate C = 10.1202 (Tang et al., 2020) and a lower transmission probability reflecting precautionary behavior and partial PPE compliance. Specifically, the transmission probability decreases across phases, from β₀ = 0.0702 in Phase 0 to β₁ = 0.0520 in Phase 1 and β_r = 0.0468 during the remaining evaluation period. Additional parameter details for this case are provided in Appendix A, Table A.4. All remaining parameters are retained from Table 1. As in Section 3.1, the vaccination program begins on September 1 and continues until March 1. The benchmark daily vaccination level, willingness assumptions, scenario-specific vaccination rates, optimized phase durations, and initiation dates for this setting are summarized in Table 6.

Table 6.

Consolidated summary of the benchmark daily vaccination level K, willingness assumptions W, scenario-specific vaccination rates ν, optimized phase durations ϵ (days), and initiation dates for the low-willingness setting. Phase 0 quantities are common to all scenarios.

Shared input	Value
Benchmark daily vaccination level K	100, 000 doses/day
W0 (Phase 0 willingness, fixed)	50%
W1 range (Phase 1)	30%–60%
W2 range (Phase 2)	60%–80%
W3 range (Phase 3)	80%–85%
ν0 (Phase 0 vaccination rate)	0.00472

Quantity	s1	s2	s3	s4	s5	s6	s7	s8
ν1	0.0038	0.0045	0.0038	0.0045	0.0038	0.0045	0.0038	0.0045
ν2	0.0078	0.0078	0.0085	0.0085	0.0078	0.0078	0.0085	0.0085
ν3	0.0146	0.0146	0.0146	0.0146	0.0163	0.0163	0.0163	0.0163
ϵ0 (days)	33.28	33.28	33.28	33.28	33.28	33.28	33.28	33.28
ϵ1 (days)	36.17	31.25	36.17	31.25	36.17	31.25	36.17	31.25
ϵ2 (days)	19.99	20.11	16.24	16.72	19.99	20.11	16.24	16.72
ϵ3 (days)	91.56	96.35	95.31	99.74	91.56	96.35	95.31	99.74
t0	01 Sep
t1	04 Oct
t2	09 Nov	04 Nov	09 Nov	04 Nov	09 Nov	04 Nov	09 Nov	04 Nov
t3	29 Nov	24 Nov	25 Nov	21 Nov	29 Nov	24 Nov	25 Nov	21 Nov

The optimized vaccination pathway for the initially low-willingness case is summarized in Table 6. In this setting, Phase 0 lasts approximately 33.28 days across all scenarios. Phase 1 ranges from 31.25 to 36.17 days, while Phase 2 ranges from 16.24 to 20.11 days. Phase 3 remains the longest phase, lasting approximately 91.56–99.74 days, with an optimal start date between 21 and 29 November depending on the realized scenario. The phase-wise vaccination totals are reported in Table 7, with Phase 0 accounting for approximately 2.19 million vaccinations (14.01% of the Ontario population) and total vaccinations ranging from 9.36 to 10.56 million across scenarios. As shown in Fig. 5, despite the lower initial willingness to vaccinate, ICU occupancy remains below the critical threshold of 566 beds under the optimized scenarios. These results suggest that the combination of phased vaccination, reduced transmission, and sustained precautionary behavior can substantially mitigate severe disease burden.

Table 7.

Total vaccinations administered during each phase across eight scenarios. Values are reported as total vaccinations (in millions), followed by the percentage of the population in parentheses.

Scenario	Phase 0	Phase 1	Phase 2	Phase 3	Total
s1	2.19 (14.01%)	1.84 (11.82%)	1.71 (10.93%)	3.63 (23.23%)	9.36 (59.99%)
s2	2.19 (14.01%)	1.94 (12.45%)	1.85 (11.82%)	4.12 (26.38%)	10.09 (64.66%)
s3	2.19 (14.01%)	1.84 (11.81%)	1.65 (10.56%)	3.91 (25.04%)	9.59 (61.42%)
s4	2.19 (14.01%)	1.95 (12.50%)	1.71 (10.97%)	4.52 (28.93%)	10.37 (66.41%)
s5	2.19 (14.01%)	1.84 (11.82%)	1.71 (10.93%)	3.81 (24.42%)	9.55 (61.18%)
s6	2.19 (14.01%)	1.94 (12.45%)	1.84 (11.81%)	4.31 (27.64%)	10.29 (65.91%)
s7	2.19 (14.01%)	1.84 (11.81%)	1.65 (10.56%)	4.10 (26.28%)	9.78 (62.66%)
s8	2.19 (14.01%)	1.95 (12.50%)	1.71 (10.96%)	4.71 (30.17%)	10.56 (67.64%)

Fig. 5.

Epidemiological dynamics under the optimized vaccination pathways for the initially low-willingness setting. Panels show ICU hospitalizations, cumulative ICU hospitalizations, non-ICU hospitalizations, infections, and vaccine rate over calendar months, together with the no-vaccination reference trajectory.

4. Discussion

Since the COVID-19 pandemic started, various mathematical models and vaccination strategies have been developed to minimize hospitalizations and deaths under different scenarios. Some of the models developed have shown that utilizing strategic vaccination campaigns is more effective than random mass vaccination. For example, Han et al. (Han et al., 2021) employed a data-driven deterministic compartmental model coupled with optimization theory to identify optimal vaccination priority strategies in China. Their study demonstrated that dynamically adaptive vaccination allocation across different age groups could significantly reduce the COVID-19 burden, mainly when vaccine supplies are limited, compared to traditional random mass vaccination strategies. Similarly (Liu & Lou, 2022), explored optimal vaccination policies using SEIR-type models, highlighting the importance of strategic vaccine timing and coverage in mitigating infection rates and alleviating healthcare system strain. While these studies have provided valuable insights, they typically operate within deterministic frameworks or are confined to specific population contexts, such as China or the United States. Our research extends these findings by addressing uncertainties in vaccine rates due to fluctuating public willingness, particularly within Ontario, Canada. By integrating stochastic elements into our model, we account for real-world uncertainties, such as changes in public behavior and vaccine availability, factors often overlooked in previous studies. This approach allows for a more flexible and adaptive vaccination strategy that can be applied in real-time during potential future outbreaks, ensuring that the healthcare system remains within capacity while optimizing the vaccination rate.

Additionally, the work by (Salcedo-Varela et al., 2024) offers further insights by synchronizing lockdown and vaccination policies using an optimal control approach. Their study underscores the need for coordinated policies to optimize health and economic outcomes. While their study aligns with our objectives to minimize strain on the healthcare system, our research advances this field by incorporating stochastic modeling, which provides a more nuanced response to the inherent uncertainties in pandemic management. Moreover (Zhang et al., 2024), applied a 32-compartment model during the United Kingdom's second COVID-19 wave, emphasizing population-stratified vaccination strategies. They found that prioritizing healthcare workers and middle-aged individuals with high contact rates could significantly reduce deaths and hospitalizations. While their model offered essential insights into the effectiveness of structured age and occupation-based vaccination priorities, it primarily relied on deterministic assumptions without accounting for real-time changes in public behavior or vaccine availability.

Building on the existing research, we utilized a COVID-19 transmission dynamics model combined with a stochastic optimization approach to determine the optimal timing and number of individuals vaccinated in Ontario, Canada, from September 1 to March 1. Our primary objective was to minimize the costs associated with overly optimistic assumptions about the vaccination rate while ensuring the healthcare system was not overwhelmed, particularly by maintaining ICU hospitalizations below a critical threshold. In this framework study, transmission was calibrated to reported case data, whereas hospitalization and ICU trajectories were used as model-generated burden outcomes for policy evaluation. Through stochastic optimization, we identified the optimal start times for the four phases of a six-month COVID-19 vaccination campaign during the fall–winter respiratory virus season. To achieve this, we developed an average vaccination rate function, which provided a range of vaccination rates by fixing daily vaccine capacity and varying the willingness-to-vaccinate percentages. We considered two primary scenarios: one involving a COVID-19 outbreak with high initial willingness among individuals to be vaccinated and another with initially low willingness to vaccinate but high adherence to PPE guidelines. Within the high-willingness scenario, we examined two cases: first, where adherence to PPE guidelines gradually decreased while contact rates were cautiously reduced, and second, where PPE adherence was already at its lowest level.

The results from the high–willingness scenario underscore the importance of initiating vaccination campaigns when public willingness is high, as this significantly influences the campaign's success in managing ICU capacities. The findings suggest that maintaining high vaccination rates in the early phases can prevent ICU overcrowding later in the season, even as adherence to PPE guidelines diminishes. By aligning vaccination efforts with periods when the public is more inclined to get vaccinated, healthcare systems can more effectively manage potential surges in COVID-19 cases, thereby reducing the strain on ICU capacities. In the low-willingness scenario, our analysis showed that even with lower initial vaccination uptake, the combination of sustained PPE and a controlled contact rate effectively mitigated the spread of COVID-19, keeping ICU bed usage below the critical threshold throughout the evaluation period.

In conclusion, our study highlights the importance of considering public behavior, adherence to PPE, and implementing certain restrictions on contact rates in the strategic planning of vaccination campaigns. The stochastic framework we developed allows for flexibility in decision-making, enabling policymakers to adjust strategies dynamically as conditions evolve. This model provides a valuable tool for public health planning, offering insights that can guide vaccination campaigns not only for COVID-19 but also for future pandemics where vaccine allocation and public behavior are critical factors.

Our results indicate that public willingness to get vaccinated significantly impacts daily vaccination rates, highlighting the importance of this factor in increasing vaccine uptake. However, willingness alone does not fully explain the variations in daily vaccination numbers, as other factors also play a role. To address these complexities, we employed a stochastic approach to account for uncertainties. In our study, we assumed specific percentages of willingness to vaccinate during different phases. However, these assumptions may only capture some factors that influence public behavior. To obtain more realistic data on willingness, future research could incorporate surveys or analyze social media content, such as Reddit discussions, to observe public opinion in real-time. This approach would allow researchers to consider the effects of various factors on public willingness and track how individual behavior changes over time. By incorporating such dynamic data into the analysis, it would be possible to better understand and anticipate shifts in vaccination behavior across different phases.

CRediT authorship contribution statement

Farah Al Hashimi: Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Data curation. Shengyuan Chen: Writing – review & editing, Validation, Supervision, Investigation. Jianhong Wu: Writing – review & editing, Validation, Supervision, Funding acquisition, Conceptualization.

Code availability

The MATLAB scripts used for simulation of the COVID-19 transmission model and for solving the scenario-based stochastic optimization problem, are available upon request to the authors.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Jianhong Wu reports a relationship with Infectious Disease Modelling that includes: board membership. Given his role as Editor-in-Chief, he had no involvement in the peer review of this article and had no access to information regarding its peer review. Full responsibility for the editorial process for this article was delegated to another journal editor. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Handling Editor: Dr Yiming Shao

Appendix A. Additional Scenario Tables and Figures

Table A.1.

Parameter list for the COVID-19 transmission dynamics and the associated stochastic optimization model for the initially high-willingness setting.

Parameter	Definition	Value
ν0	Vaccination rate of COVID-19 during Phase 0	0.0109
$\left[\underset{\_}{{\nu }_1},\overline{{\nu }_1}\right]$	Range of vaccination rate of COVID-19 during Phase 1	$\left[0.0089687,0.010846\right]$
$\left[\underset{\_}{{\nu }_2},\overline{{\nu }_2}\right]$	Range of vaccination rate of COVID-19 during Phase 2	$\left[0.010846,0.013702\right]$
$\left[\underset{\_}{{\nu }_3},\overline{{\nu }_3}\right]$	Range of vaccination rate of COVID-19 during Phase 3	$\left[0.013702,0.019888\right]$
β0	Transmission probability during Phase 0	0.0702
β1	Transmission probability during Phase 1	0.0723
βr	Transmission probability during the remaining evaluation period	0.076398
C0	Contact rate during Phase 0	11.5801
Cr	Contact rate during the remaining evaluation period	8.4644

Table A.2.

Detailed optimal vaccination pathway across the eight scenarios for the initially high-willingness setting with phase-varying transmission.

Scenario	s1	s2	s3	s4	s5	s6	s7	s8
ν1	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101
ν2	0.0122	0.0122	0.0126	0.0126	0.0122	0.0122	0.0126	0.0126
ν3	0.0165	0.0165	0.0165	0.0165	0.0176	0.0176	0.0176	0.0176
ϵ0	16.50	16.50	16.50	16.50	16.50	16.50	16.50	16.50
ϵ1	16.99	16.52	16.99	16.52	16.99	16.52	16.99	16.52
ϵ2	16.43	16.14	16.02	15.81	16.43	16.14	16.02	15.81
ϵ3	131.08	131.84	131.49	132.17	131.08	131.84	131.49	132.17
t0	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep
t1	17 Sep	17 Sep	17 Sep	17 Sep	17 Sep	17 Sep	17 Sep	17 Sep
t2	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct
t3	20 Oct	20 Oct	20 Oct	19 Oct	20 Oct	20 Oct	20 Oct	19 Oct

Fig. A.1.

Scenario tree for the initially high-willingness setting with phase-varying transmission.

Fig. A.2.

Histogram of total vaccinations by phase across the eight scenarios for the initially high-willingness setting with phase-varying transmission.

Table A.3.

Detailed optimal vaccination pathway across the eight scenarios for the initially high-willingness setting under constant transmission.

Scenario	s1	s2	s3	s4	s5	s6	s7	s8
ν1	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101	0.0098	0.0101
ν2	0.0122	0.0122	0.0126	0.0126	0.0122	0.0122	0.0126	0.0126
ν3	0.0165	0.0165	0.0165	0.0165	0.0176	0.0176	0.0176	0.0176
ϵ0	21.42	21.42	21.42	21.42	21.42	21.42	21.42	21.42
ϵ1	25.15	22.18	25.15	22.18	25.15	22.18	25.15	22.18
ϵ2	19.64	20.04	17.29	18.09	19.64	20.04	17.29	18.09
ϵ3	114.79	117.36	117.15	119.31	114.79	117.36	117.15	119.31
t0	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep
t1	22 Sep	22 Sep	22 Sep	22 Sep	22 Sep	22 Sep	22 Sep	22 Sep
t2	17 Oct	14 Oct	17 Oct	14 Oct	17 Oct	14 Oct	17 Oct	14 Oct
t3	06 Nov	03 Nov	03 Nov	01 Nov	06 Nov	03 Nov	03 Nov	01 Nov

Fig. A.3.

Scenario tree for the initially high-willingness setting under constant transmission.

Fig. A.4.

Histogram of total vaccinations by phase across the eight scenarios for the initially high-willingness setting under constant transmission.

Table A.4.

Parameter list for the COVID-19 transmission dynamics and the associated stochastic optimization model for the initially low-willingness setting.

Parameter	Definition	Value
ν0	Vaccination rate of COVID-19 during Phase 0	0.0047211
$\left[\underset{\_}{{\nu }_1},\overline{{\nu }_1}\right]$	Range of vaccination rate of COVID-19 during Phase 1	$\left[0.00234,0.00645\right]$
$\left[\underset{\_}{{\nu }_2},\overline{{\nu }_2}\right]$	Range of vaccination rate of COVID-19 during Phase 2	$\left[0.00645,0.01370\right]$
$\left[\underset{\_}{{\nu }_3},\overline{{\nu }_3}\right]$	Range of vaccination rate of COVID-19 during Phase 3	$\left[0.01370,0.01989\right]$
β0	Probability of COVID-19 transmission during Phase 0	0.0702
β1	Probability of COVID-19 transmission during Phase 1	0.0520
βr	Probability of COVID-19 transmission during the remaining evaluation time	0.0468

Table A.5.

Detailed optimal vaccination pathway across the eight scenarios for the initially low-willingness setting.

Scenario	s1	s2	s3	s4	s5	s6	s7	s8
ν1	0.0038	0.0045	0.0038	0.0045	0.0038	0.0045	0.0038	0.0045
ν2	0.0078	0.0078	0.0085	0.0085	0.0078	0.0078	0.0085	0.0085
ν3	0.0146	0.0146	0.0146	0.0146	0.0163	0.0163	0.0163	0.0163
ϵ0	33.28	33.28	33.28	33.28	33.28	33.28	33.28	33.28
ϵ1	36.17	31.25	36.17	31.25	36.17	31.25	36.17	31.25
ϵ2	19.99	20.11	16.24	16.72	19.99	20.11	16.24	16.72
ϵ3	91.56	96.35	95.31	99.74	91.56	96.35	95.31	99.74
t0	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep	01 Sep
t1	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct	04 Oct
t2	09 Nov	04 Nov	09 Nov	04 Nov	09 Nov	04 Nov	09 Nov	04 Nov
t3	29 Nov	24 Nov	25 Nov	21 Nov	29 Nov	24 Nov	25 Nov	21 Nov

Fig. A.5.

Scenario tree for the initially low-willingness setting.

Fig. A.6.

Histogram of total vaccinations by phase across the eight scenarios for the initially low-willingness setting.