Infectious disease, in particular viral infections, poses a significant public health threat and socio-economic confounding. Currently, an immediate example is the ongoing global coronavirus disease 2019 pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which continues to spread worldwide and ultimately affected the lives of hundreds of millions of people since December 2019. In view of the disease entering into a global exponential growth phase, the COVID-19 outbreak was declared a global pandemic by World Health Organization (WHO) on March 11, 2020 (1).

To combat the COVID-19 pandemic, a range of stringent non-pharmaceutical interventions (NPIs), such as national or state-wide lockdowns, travel restrictions, prohibitions on mass gatherings, and maintaining secure social distancing, has been introduced in many countries (2). These intervention strategies are able to significantly decrease the frequency of contact between infected and susceptible populations and have been shown to be effective measures to slow down the propagation speed of the COVID-19 pandemic (3). However, NPIs against COVID-19 are not a sustainable solution in the long term because of the substantial economic cost and result in a negative impact on normal social activities (4). According to the International Monetary Fund (IMF) reports, the great lockdown to contain the COVID-19 pandemic has triggered the worst economic downturn since the Great Depression (5).

The adequate vaccination provides a promising and sustainable long-term solution to mitigate the spread of COVID-19. Several safe and effective COVID-19 vaccines have been developed, tested, and approved at an unprecedented pace, driven by the joint efforts of drug researchers around the world after the outbreak (6). Unfortunately, however, a supply shortage is inevitable during the early stage of any new vaccine rollout as the limited production and distribution capacity (7). Decision makers face a crucial public health conundrum of how to allocate scarce vaccines in resource-constrained settings to maximize the benefits of vaccination (8).

Determining the optimal vaccine allocation strategy during the COVID-19 pandemic is a challenging problem. On the one hand, the spread of COVID-19 is a highly complex process. In contrast to the previous coronavirus virus, COVID-19 exhibits several peculiar epidemiological features. For example, some studies have demonstrated the substantial contribution of pre-symptomatic and asymptomatic infections to COVID-19 transmission (9, 10). Public health authorities need a better understanding of such inapparent transmission occurring via person-to-person interaction will play major roles in the prevention and control the COVID-19. In addition, COVID-19 transmission can show noticeable group differences. In other words, several risk factors within the population may have contributed to the heterogeneous transmission of COVID-19, such as children and adolescents have been shown to have lower susceptibility to infection compared with elders (11, 12), older adults have a greater risk of becoming symptomatic than younger (13, 14), beyond this, the risk of requiring hospitalization and risk of death also displayed a similar characteristic (15–17). On the other hand, most of the currently available vaccines require two doses given in tandem over a certain time interval (18–20). A two-dose vaccination schedule faces more complicated and realistic logistics challenges than a simple single-dose vaccination rollout. Specifically, in the context of limited vaccine supply, it is necessary to determine not only the optimal allocation strategies with the first and second doses of vaccine in each time period but also to ascertain if reserving the doses for individuals who have received the first dose to avoid the failure of their first dose vaccine due to insufficient supply. Hence, the two-dose vaccination procedure should not be ignored during the COVID-19 vaccine resources allocation process such that all the available vaccine resources can be administered to the people who need them the most (21).

To address the above-mentioned challenges, in this paper, we investigate a multi-period two-dose vaccine resource allocation problem to deal with a pandemic when the vaccine supply is highly limited. First, we constructed a novel age-structured compartmental model tailored to COVID-19 transmission. More specifically, we modified a deterministic susceptible-exposed-infected-recovered (SEIR) compartmental model by introducing the two-dose vaccination procedure and adding additional compartments to capture different states of the virus propagation. In our model, the different infected individuals are distinguished based on whether they presented with symptoms and on the severity of symptoms. In addition to this, our model also takes into account group heterogeneity, encompassing the group difference in terms of susceptibility, symptom rates, severity, etc. Then, we propose a nonlinear programming (NLP) model that integrates the proposed compartmental model to allocate scarce vaccines resource for suppressing the negative effects caused by the virus transmission. The objective of this model is to minimize the total number of potential deaths in the population over a multi-period planning time horizon. Afterward, we conducted a real-world case study regarding the 2021 COVID-19 vaccination campaign in the Midlands of England to verify the performance of the proposed model. Finally, we also give some practical recommendations for a two-dose vaccine allocation strategy in the context of the COVID-19 pandemic.

The main contribution of this paper is as follows: (i) We present an age-structured SEIR-type compartment model to describe the course of COVID-19 transmission, which explicitly considers the two-dose vaccination procedure and several important epidemiologic features of COVID-19, such as pre-symptomatic and asymptomatic transmission, group heterogeneity in susceptibility, symptom rates, severity, as well as multiple mechanisms of vaccine action, etc. (ii) We propose a multi-period two-dose vaccine allocation problem to assist public health authorities in making optimal allocation decisions for COVID-19 pandemic control. To this end, a nonlinear programming model is formulated to minimize the total number of deaths under a limited supply of vaccine resources. (iii) We performed a retrospective study based on a real-world COVID-19 vaccination campaign, which aims to provide some important insights into the strategies for allocating scarce vaccine resources.

The rest of this paper is organized as follows. In Section 2, we review the relevant literature. In Section 3, we develop a multi-period vaccine allocation optimal model that integrates a compartmental model tailored to COVID-19 transmission. In Section 4, we elaborate on a case study based on a real-world COVID-19 outbreak to illustrate the performance of the proposed model. The key results and discussion are presented in Section 5, and finally, conclusions and suggestions for future research are summarized in Section 6.

Infectious disease modeling plays a crucial role in understanding the evolution of infectious diseases and planning for public health responses to mitigate an infectious disease outbreak (22). Typically, the compartmental model provides a powerful tool to describe infectious disease transmission dynamics. One of the most common compartmental models was called the Susceptible-Infectious-Removed (SIR) model, which was first proposed by Kermack and Mckendrick (23). The authors classify the population into three mutually exclusive compartments according to its infection status, namely susceptible, infected, and recovered. Individuals in each compartment are assumed to mix homogeneously, and the transition among these compartments can be described by a system of nonlinear differential equations. Despite its structural simplicity, the SIR model is exceedingly useful and able to easily be extended based on the transmission characteristics of different pathogens. For example, the well-known SEIR model, which considers the incubation period by means of introducing a compartment for exposed individuals, has drawn considerable attention (24). After that, the SEIR model and many of its variants were widely applied in studying the transmission of specific infectious diseases such as SARS (25–27), H1N1 (28–30), Smallpox (31–33), Ebola (34, 35), and Cholera (36, 37).

Currently, the classical SEIR model has also been used to model the recent outbreak of COVID-19 (38–40). Furthermore, several more sophisticated models have been developed to incorporate more important epidemiologic features that can more accurately depict the transmission of the COVID-19 pandemic. For example, Hao et al. (41), Aleta et al. (42), and Whittaker et al. (43) extend the standard SEIR model by considering the different categories of infectious individuals to simulate the transmission dynamics of COVID-19. These studies classified infected according to infectious health state, including pre-symptomatic, asymptomatic, and symptomatic individuals. Walker et al. (44) constructed an age-structured SEIR model, which explicitly takes into account disease severity and healthcare levels to explore the spread of COVID-19 under different health capacity conditions. Viana et al. (45) proposed an age-structured SEIR model including vaccination to evaluate the different scenarios for the relaxation of social distancing measures during the vaccination rollout. Yang et al. (46) developed an SEIR-type model incorporating hospitalized in the general ward and hospitalized in the intensive care unit (ICU), aiming to derive optimal switching strategies between different community mitigation stages based on the daily number of admissions. In the studies above, such compartmental models are generally applied to simulate the spread of infectious diseases in human populations, and these studies focus on evaluating the effectiveness of various non-pharmacological intervention strategies to control the epidemic.

Pharmacologic interventions to deal with an outbreak are another important area of research. In particular, the vaccination campaign has provided a long-term solution, which is regarded as a powerful tool for suppressing infectious disease pathogen transmission (47). Notably, vaccine availability will usually be insufficient when faced with a previously unknown pathogen, especially in the early stage of vaccine rollout (48). The significant challenge facing public health authorities is how to allocate scarce vaccines to mitigate the negative effects of a pandemic. This type of problem is referred to as vaccine allocation for epidemic control, and it has recently received widespread attention from the scientific community attention (37, 47, 49–56).

Several studies have considered using the mathematical programming approach that incorporates compartmental models to address the vaccine allocation optimization problem for epidemic control. For instance, Ren et al. (57) studied vaccine allocation strategy during a smallpox outbreak, where infectious disease spread is described by an SIR compartment model. The authors came up with an approximate method for the representation of the disease dynamics, which aims to formulate this allocation problem as a mixed integer programming model. Furthermore, they also proposed an efficient heuristic algorithm to solve large-scale problems in a reasonable time. Duijzer et al. (58) used a standard SIR model to describe the course of infectious disease transmission in a heterogeneous population. Subsequently, they developed a nonlinear programming model that minimizes the number of vaccine doses used under the effective reproductive number equal to one. The authors provided an efficient solution method based on Perron–Frobenius theory to find the optimal vaccine allocation. In a similar study, Enayati and Ozaltin (53) used the well-known Gini coefficient to determine a balanced influenza vaccine allocation strategy as regards efficiency and equity. They find that group-specific transmission is important in the evolution of the influenza virus and should be taken into account in vaccine allocation decision-making. Ng et al. (59) combined a multi-criterion mathematical programming model with an SIR model to determine the optimal vaccination strategies for seasonal influenza. The proposed multi-criterion optimization problem was solved by the augmented epsilon-constraint method. They showed that the group-targeted vaccination strategy outperforms both the mass and random vaccination strategies. A number of recent studies focused on optimal vaccine allocation strategies for COVID-19 based on age-structured compartmental models. Miura et al. (55) applied an age-structured SIR model to simulate the COVID-19 epidemic trajectories and present a data-driven approach for vaccine allocation. The authors find that optimal vaccine allocation strategies depend on the objective of epidemic control. A similar conclusion has been reached in the work of Matrajt et al. (54) and Molla et al. (60). In another study, Jarumaneeroj et al. (47) proposed an age-structured SIQRV (Susceptible-Infectious-Quarantined-Recovered-Vaccinated) model to delineate the transmission dynamics of COVID-19 and later formulated a nonlinear programming model to obtain the optimal vaccine allocation strategies with the aim to minimize the total weighted burden on the health care system over a multi-period planning horizon. More recently, Tetteh et al. (61) studied the mass and ring vaccination strategies with different vaccine efficacy and population coverage using stochastic network models. González-Parra et al. (62) proposed two nonlinear mathematical models and applied them to explore the optimal vaccine allocation strategy under different scenarios. While these studies have provided valuable management insights regarding the vaccine allocation decisions in the context of an outbreak, they rely on the simplified assumption that the course of vaccination contains only single-dose. However, in the case of some infectious diseases (e.g., COVID-19, Cholera), most approved vaccines require two doses were administered at certain time intervals (6, 37). Obviously, the two-doses vaccination procedure is more complex than the single-dose vaccination, which is often ignored in previous studies on the vaccine allocation optimization problem.

The two-dose vaccine allocation problem to combat a pandemic in the context of limited capacity has received less attention in the literature. Among the few studies on this issue, Matrajt et al. (54) and Leung et al. (37) used mathematical models combined with optimization algorithms to determine the optimal allocation strategy with one and two doses of vaccine under various outbreak settings with different combinations of parameters. There are also some research efforts that have attempted to evaluate the effectiveness of delaying the second dose strategy with the COVID-19 vaccine (6, 63, 64). These studies assume immune protection immediately following vaccination. However, a number of studies have shown that there is a delay between injecting a dose and the onset of dose-specific protection (17, 65), which should not be ignored in the mathematical modeling of vaccination in response to an outbreak.

The most relevant study to our work was conducted by Parino et al. (21). They proposed a nonlinear programming model to optimize a multi-period two-dose vaccine allocation problem during the COVID-19 outbreak. Our work differs from Parino et al. (21) in three main aspects. First, we incorporate the age-structured population in the multi-period two-dose vaccine allocation decision-making process because age is regarded as a significant risk factor for the transmission of COVID-19. Specifically, the study not only allows us to determine the amount of dose used for the first and second vaccination, respectively, but also provides more detailed strategies for two-dose vaccine allocation by age. Second, the explicit modeling of inapparent transmission from pre-symptomatic and asymptomatic infections in our model and the separation of distinct levels of transmissibility for different types of infections. Third, we consider the specific features of the COVID-19 vaccine, including the delay effect between vaccination with a dose and the onset of dose-specific immune responses, and the current vaccine provides multiple mechanisms of action (for example, by vaccination reduction in risk for infection, reduction in risk for developing symptoms after infection, or reduction in risk for severe symptoms).

In this section, the details of the proposed mathematical model are introduced to delineate the course of disease progression and the vaccination process, and we present a mathematical programming formulation of the two-dose vaccine allocation problem. As shown in Figure 1, the proposed model describes the transmission of the viral pathogens as an evolving dynamic among different epidemiological states in each population group stratified according to age and vaccination status for each time period, based on an extension of the discrete-time, age-structured, deterministic SEIR-type compartmental model. Specifically, the total population is partitioned into 16 5-year age groups (from 0 to 79 years with 5-year increments) and one age group over 80 years. Each age group is further divided into five subgroups account for vaccination status: (i) unvaccinated individuals; (ii) individuals vaccinated with the first dose (protection has yet to be realized); (iii) individuals vaccinated with the first dose (protected by the vaccine effect of the first dose); (iv) individuals vaccinated with the second dose (not improve the protection efficiency yet); (v) individuals vaccinated with the second dose (protected by the full vaccine effect of the two doses). For each population class, we tracked ten infection statuses: susceptible (S), exposed (E), asymptomatic infectious (A), pre-symptomatic infectious (P), mildly-symptomatic infectious (ISM), severely-symptomatic infectious (ISS), hospitalized in the general ward bed (H), hospitalized in the ICU (ICU), recovered (R) and deceased (D). In this figure, the compartments correspond to the infection and vaccination status of individuals in a population affected by the pandemic, and the arrows correspond to the flow between different statuses, while the rates of transition are marked next to the arrows. For simplicity of visualization, we only display the vaccination status transition between epidemiological compartments for susceptible, exposed, pre-symptomatic infectious, and mildly-symptomatic infectious. Apparently, the change of the individual status either depends on disease/clinical (black solid arrows) or vaccine allocation decision-making (red dashed arrows).

With the pathogen continuing to transmit within a population, the individual is initially susceptible, who becomes exposed (but not yet infectious) through human-to-human direct contact in the community with infectious individuals. After the latent period, they will either become pre-symptomatic or asymptomatic individuals. After this, those who are asymptomatic individuals will recover naturally without any intervention or hospitalization. Additionally, the pre-symptomatic individuals progress to mildly-symptomatic infectious. They will either transition to recovery by natural healing in the same manner as asymptomatic individuals, or the remaining fraction will become severely-symptomatic infectious requiring hospitalization. Of these, a proportion of severely-symptomatic infectious developed hospitalized in the general ward bed, while the remainder of severely-symptomatic infectious was hospitalized in the ICU because they required ventilator support or ICU care. Finally, the hospitalized individual will receive medical treatment and may recover completely, while they may also deteriorate and subsequently die. Notably, recovered individuals will wane of infection-derived temporary immunity, after which they return to being fully susceptible.

On the other hand, the vaccine allocation decision-making concerns different vaccination statuses of individuals. If available vaccine capacities exist, these can either be used to administer the first dose or second dose of the vaccine. Moreover, since the vaccine protection is not effective immediately, there is a delay between administration (both first and second dose) and the onset of dose-specific immune responses (60). It is to be mentioned that a minimum/maximum vaccination time interval exists between the first and second doses in order to prevent vaccine failure (64). In this research, we consider a leaky vaccine that partially reduces the risk of infection, developing symptoms after infection, and severe symptoms. To facilitate the modeling and interpretation, we summarize the notations used throughout the rest of this paper in Section 3.1.

England is among the countries most severely affected by the COVID-19 pandemic since the first cases of COVID-19 were reported on January 31, 2020 (14). Although some interventions, such as maintaining safe social distancing, contact tracing, and testing of suspect cases were implemented to suppress the early spread of the virus, it was insufficient to control the epidemic in England. In response to this public health emergency, several COVID-19 vaccines have been fast developed and approved through a global collaborative effort between many scientists and were deployed first in England on December 8, 2020 (17). According to the National Health Service (NHS) division, England is composed of seven NHS regions. We select the Midlands of England that the most populous for our case study. Here, we focused on the application of the NLP model proposed in Section 3 to determine the optimal allocation of a limited vaccine resource and to minimize the total number of deaths in the Midlands of England.

We provide the case study data used in the model, including epidemiological parameters, population, initial conditions data, and vaccine efficacy data. The corresponding epidemiological parameters are summarized in Supplementary Tables 1, 2. These parameters are primarily from a series of references and public sources, as cited in the table. Supplementary Table 3 presents the contact matrix between different age groups, which was derived from the POLYMOD survey (72) about contacts relevant for the transmission of diseases for the United Kingdom using the ‘socialmixr’ R package (73), scaling the Midlands of England population demography data to obtain the required daily age-group-specific contact matrix. Furthermore, the time-varying transmission rate 𝛽 _t that captures the transmission efficiency as a function of time due to changes in COVID-19 policy. In our case study, the time-varying transmission rate was obtained from Sonabend et al. (17), which mainly focused on the key epidemiological drivers of COVID-19 in each NHS England region.

Supplementary Table 4 presents information about the population size and age distribution from the Office for National Statistics (ONS) mid-2020 population estimates for the Midlands of England (74). In our case study, we consider a 25 weeks vaccination campaign that began on December 8, 2020, that is, the start time of the vaccination campaign in the Midlands of England. An important data set is the initial condition of the epidemic at the start of the planning horizon. We set the initially infected individuals to 1.21% of the total population in the region, which can be obtained from the ONS coronavirus infection survey regarding the number of people testing positive for COVID-19 within the community population during December 6, 2020, to December 12, 2020, in the Midlands of England (75). We estimated the initial number of infected individuals from each age group by multiplying the total infected individuals by the proportion initially of different age groups from the official United Kingdom government website for data on coronavirus (76). Of these, the number of severely-symptomatic infectious individuals is directly proportional to the number of new infected-hospitalized cases at that time. Therefore, the new infected-hospitalized cases data from GOV.UK can be used to get an approximation of the initial number of severely-symptomatic infectious individuals. In addition, we assume the initial number of remaining infected individuals is assigned homogeneously among the different types of infections, including exposed, asymptomatic infectious, pre-symptomatic infectious, and mildly-symptomatic infectious individuals. Similarly, we set the initially recovered individuals to 9.03% of the total population in the region, which is available from the ONS an estimated of people would have tested positive for antibodies against COVID-19 on a blood test in the Midlands of England in early December 2020 (77). We estimated the initial number of recovered individuals from each age group by multiplying the total recovered individuals by the proportion initially of different age groups (60). Besides this, we initialized the number of deceased individuals, infected-hospitalized cases in the general ward, and infected-hospitalized cases in the ICU using data reported from the GOV.UK on December 8, 2020. On this basis, we estimated the initial number of individuals of these three populations from each age group based on the corresponding proportion initially of different age groups (see Supplementary Table 4). Subsequently, the initial number of susceptible individuals was obtained by subtracting the initial number of infected, infected-hospitalized cases, recovered, and deceased individuals from the population size in the region. Finally, Supplementary Table 5 summarizes the vaccine efficacy against infection, symptomatic disease, and severe disease based mostly on previously published literature.

In this section, we first validated the proposed model against real-world data from the COVID-19 outbreak in the Midlands of England. Then, we conduct comparative studies to evaluate the performance of our optimal strategy with different strategies, which may be derived from the existing literature. Finally, we elaborate on the numerical results and discussions obtained by solving our model with respect to different settings of vaccine supply levels and the start time of vaccination. Our mathematical formulations were coded in the Julia using the JuMP (78) modeling language and solved using the Interior Point Optimizer (IPOPT) (79) with the MA97 parallel linear solver (80). All computational experiments were run on a desktop computer equipped with LINUX operating system with 8 cores, 1.8 gigahertz CPU, and 32-gigabyte memory.

In this paper, we present a novel multi-period two-dose vaccine allocation model for infectious disease control in the context of limited supply. Our model incorporates the transmission of infectious disease with the allocation and management of scarce vaccine resources with the objective to minimize the total number of deaths in a given population over the finite planning horizon. This model explicitly considers several key realistic features of the vaccine rollout process, such as the minimum and maximum time interval between the administration of the first and second dose should be complied with, the delayed onset of dose-specific immune responses, and multiple mechanisms of action of the vaccine. We have demonstrated the effectiveness of the proposed optimization model on a case study for the 2021 COVID-19 vaccination campaign in the Midlands of England and explored a wide range of plausible scenarios with respect to different levels of vaccine supply and start time of vaccination. We find that it is optimal to allocate vaccines to older age groups first is a robust strategy to avoid more deaths. Moreover, we also observed that releasing more vaccine doses for first-time users would provide an even larger vaccination benefit relative to holding back second doses. Our numerical results underscore the necessity of maintaining appropriate non-pharmaceutical intervention measures during vaccine rollout, particularly in low-resource settings. In addition, it is found that when the vaccine resources are limited but are currently available, starting vaccination as soon as possible provides significant benefits for mitigating the epidemic. The proposed approach is sufficiently generic and flexible and can be easily extended to other countries and regions to identify optimal vaccine allocation strategies for controlling epidemic spreading according to the available data.

Our research has several limitations, which should be considered in the future study. First, our model has only divided the population by age. However, other features, such as sex, occupation, health, geographic region, and race/ethnicity are also essential demographic classification factors in the human social structure. We believe that considering these demographic features will further improve the performance and applicability of the model. Second, we considered all vaccine-eligible individuals were willing to be vaccinated, this is a strong assumption. In the Supplementary Material, we relax this assumption, and a sensitivity analysis was performed to evaluate the effect of vaccine hesitancy. Third, one possible future research direction is to extend the proposed model to take into account the uncertainty of infectious disease transmission, e.g., the rates of symptom development, admission, and mortality. Fourth, although we chose the minimization of deaths as the objective function in this model, other public health objectives of vaccination, such as minimization of years of life lost, hospitalization, or infections are just as important, however, and deserve further exploration. Fifth, the proposed model does not take variants of COVID-19 into consideration. Therefore, our model may need recalibrations to be able to cope with the pandemic caused by new variants or strains, by means of adjusting some epidemiological parameters, which accordingly differ with variants of COVID-19. Sixth, our model has only been validated in a case study regarding the 2021 COVID-19 vaccination campaign in the Midlands of England. In the Supplementary Material, we further explored the application of the proposed model in two distinct National Health Service (NHS) regions. However, the generalization and the external validity of the model to other regions still need further investigation. Furthermore, the proposed vaccine allocation model is able to further generalize from the primary (two doses) to a more realistic scenario that contains the booster (third dose). Lastly, our study only focuses on a single region, it would be interesting to extend our method to incorporate multiple geographical regions.

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding author.

JZ: conceptualization, methodology, software, formal analysis, visualization, and writing—original draft. QW: investigation, and writing—review and editing. MH: conceptualization, investigation, and supervision. All authors contributed to the article and approved the submitted version.

This work is supported by the NSFC Major International (Regional) Joint Research Project Grant No. 71620107003; the Liaoning Revitalizing Talent Program No. XLYC1902010; the Fundamental Research Funds for State Key Laboratory of Synthetical Automation for Process Industries Grant No. 2013ZCX11; the 111 Project 2.0 (No. B08015).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.