Vaccination is the main hope to contain the COVID-19 pandemic currently enveloping the world (1, 2). Several vaccines for SARS-CoV-2 are under development, and their potential impact and limitations are hotly debated. It is expected that herd immunity (3) (4-6) will play a role in containing the pandemic once a sufficiently high proportion of the world population gains adaptive immunity. Nevertheless, if high morbidity, mortality, and economic catastrophe are to be avoided, the vast majority of the population should acquire immunity through vaccination rather than infection. While global eradication of COVID-19 could be a difficult goal to achieve (7, 8) , a successful vaccination program may target regional elimination in the short-to medium-term. Hence, vaccination uptake will have a direct and critical influence on the dynamics of the COVID-19 pandemic, and the ability of the healthcare systems to manage it.
In this perspective, we argue that the level of vaccination uptake in populations, the effective prioritisation of potential vaccine recipients, and the efficient use of the resources needed by vaccination administration programs, will be the key determinants in how the COVID-19 pandemic is contained and/or eliminated in populations in the coming years. We posit that without widespread uptake, any vaccination program will fail regardless of the efficacy (9, 10) of the vaccine itself. Therefore, what we describe in this perspective are conditions under which the vaccination programs can achieve their maximum uptake and effectiveness, and the modelling and implementation approaches needed to help the vaccination programs achieve this.
We propose that game theoretic modelling, which is the established theoretical framework for modelling rational decision making (11 -19) , coupled with social network analysis and agent-based techniques for modelling the population and simulating the disease dynamics (4) (20 -24) , will give us the most effective toolset to model the vaccination uptake. This multi-faceted approach will produce a definitive roadmap for the implementation of vaccination programs which will lead to successful long-term management of COVID-19.
1.1 In the coming years, the most important factors that will determine the success of controlling the COVID-19 pandemic will be the level of vaccination uptake by the population, and the effective use of resources to administer the vaccine in highly varied and variable settings.
1.2 Game theory, supplemented by social network analysis and agent-based modelling, should be extensively used by researchers to model vaccination uptake by populations and guide difficult policy decisions regarding vaccination programs and thus maximise containment of the COVID-19 pandemic.
Given the limited availability of vaccines at the initial stages, effective prioritisation and optimal use of resources will be crucial. If the vaccine is of the type which reduces the transmissibility of the SARS-CoV-2 virus, prioritisation should be based on targeting individuals, cities, or states that can act as critical nodes of transmission or superspreaders. Whereas if the vaccine is of the type that reduces symptoms or mortality, prioritisation should be based on targeting individuals, cities, or states that are likely to have poor outcomes if infected.
Game theory, together with social networks and agent-based modeling, can be used as a primary theoretical framework in determining effective prioritisation of scarce resources needed in different vaccination programs, depending on a broad range of bounding conditions for successful implementation.
Effective SARS-CoV-2 vaccine development, production and dissemination are expected to take at least 12 to 18 months (25, 26) . Currently, WHO lists 30 vaccine candidates as under investigation in human clinical trials and 119 in preclinical trials (27) . Efforts to develop vaccines have explored different approaches, ranging from recombinant vaccines and nucleotide-based vaccines to subunit vaccines and others. Most vaccine candidates are aimed at inducing neutralising antibodies against the viral spike (S) protein of SARS-CoV-2, which prevent binding to ACE2 receptors of host cells (2) . Eight vaccine candidates are in phase 3 clinical trials at present, with three other candidates in phase 2 or phase 2/3. Among the vaccine candidates which are currently in phase 3, the leading ones are 1) the ChAdOx1 vaccine developed by the University of Oxford and AstraZeneca, 2) the Moderna RNA vaccine, which is designed to induce antibodies against a portion of the coronavirus S protein, 3) the Sinopharm inactivated SARS-CoV-2 vaccine, and 4) the CanSino Biologics Ad5-nCoV vaccine, which employs an adenovirus vector. Pfizer and BioNTech are currently testing their BNT162 vaccine candidate in a combined phase 2/3 trial structure (28) .
The current pace of these SARS-CoV-2 vaccine clinical trials and their projected manufacture have raised hopes that the conventional timeline for vaccine development and delivery will be significantly reduced (29, 30) . Moreover, the global collaborative effort, COVAX, has been initiated to negotiate and ensure equitable access of SARS-CoV-2 vaccines to all participating countries regardless of income levels (28) . Several countries have also made alternative arrangements for procuring approved vaccines directly from the manufacturers. Nevertheless, despite unprecedented levels of accelerated research and international cooperation, no country is expected to procure sufficient doses to vaccinate its entire population for many months following the release of first vaccines.
Achieving herd immunity is often a key objective of population-level vaccination coverage (31) . Herd immunity is a population threshold that marks the necessary proportion of the population that needs to be immune to an infection, either through vaccination or through exposure to the pathogen, so that the transmission of the infectious agent is sufficiently disrupted, and the entire population is protected (3). It is not desirable to expose a significant portion of the population to the pathogen in order to acquire herd immunity. Rather the objective should be to achieve herd immunity through vaccination to minimise morbidity and mortality.
The level of herd immunity needed to protect a population can be derived from the basic reproduction number (R0), which is defined as the number of secondary infections produced on average by an infected index case within a completely susceptible population, assuming there is no human intervention (4) (5) (6) . The epidemic threshold is defined as the inverse of the basic reproduction number (5, 6) . The level of herd immunity needed to protect a population from an epidemic is equal to the complement of the epidemic threshold (i.e. herd immunity threshold = 1 -1/R0).
The basic reproduction number R0 for COVID-19 and the corresponding herd immunity threshold are not yet known definitively, with current best estimates of R0 ranging from 2.5-3.0 (14) (32 -35), with the corresponding herd immunity thresholds ranging from 60% -67%. Furthermore, much uncertainty remains regarding the nature of the immunogenicity of the pathogen, with important implications for vaccine candidates. It is currently unknown whether humoral or cell-mediated responses drive neutralising immunity or other correlates of protection, and how long any such protection endures (36 -38) . If vaccination does not generate protective immunity in every member of the population who gets vaccinated, then the number of people requiring vaccination for the population to achieve herd immunity will be higher than the number determined by the herd immunity threshold as defined by R0. Similarly, waning vaccine-induced immunity will require greater population coverage than that derived from R0, and also may necessitate the administration of booster vaccination. Also, the targeting of epidemiologically influential subgroups will be important, and the relative importance of some of these subgroups is disease-specific. For example, healthcare and other essential workers may be important subgroups to target for SARS-CoV-2 vaccination due to their relatively high levels of exposure. According to recent evidence (39, 40) , children also may be more influential to transmission than previously assumed. Therefore, despite the transient immunogenicity of SARS-CoV-2, targeted vaccination delivery may achieve herd immunity at or below the threshold derived from R0 if epidemiologically influential subgroups are prioritised. All these aspects of herd immunity specific to COVID-19 will necessarily be foundational to the modeling of SARS-CoV-2 vaccine effectiveness.
Game theory, which is the study of strategic decision making by rational players, is used to study several phenomena and behavioural patterns in human societies and socioeconomical systems, and is applied in fields ranging from evolutionary biology to computer science and project management (4) (41 -45) . Several previous studies have modelled vaccination uptake using game theory in the context of diseases such as influenza, measles, chickenpox and hepatitis (4) . When modelling vaccination uptake using game theory, players usually represent individuals, and the actions involve taking or not taking a vaccine. The payoff is decided by several factors including the perceived risk of infection (perceived prevalence and transmissibility), severity of the disease, financial and non-financial cost of vaccination, and the perceived uptake of vaccination by other players.
To reach a decision, individuals either try achieving utility maximization or comparing the payoffs of two strategies. The decision-making process is modelled in two ways: (a) Selflearning (Aspiration game) (65) (b) social-learning (Imitation game) (4, 66) . Through selflearning, individuals rely on their knowledge, memory and personal perception, and awareness of the disease, and switch strategies if their own aspiration level is not met, while imitation dynamics put individuals into an environment where personal decisions are influenced by the choices of the population, and individuals update their strategies by comparing their own expected payoffs with others in the population and switch to the strategy which gives the better payoff (63, 65, 66) . The imitation game can be played in a homogenous population structure representing a well-mixed population, or a heterogeneous population structure, where the influence by neighbours is determined by the number and strength of contacts between the individual and his/her neighbours (67).
Levine (46) describes six key drivers that may influence vaccine uptake. These include the epidemic potential of the pathogen, localised transmission potential of the pathogen, safety concerns with the vaccine, strength and flexibility of public health delivery systems, public investment in resources for immunization, and local ownership and individual normative behaviors. The level of vaccine hesitancy by the population (47) , and the local context and its multifactorial determinants also need to be considered.
Assuming that safe and effective SARS-CoV-2 vaccines will be available after timely regulatory approvals, countries will still require enormous resources and systems in place to address vaccination program implementation challenges. The challenges in vaccination program implementation include vaccine procurement and supply chain management, developing and deploying vaccine delivery platforms, developing vaccine delivery strategies including identification of eligible/target subpopulations for vaccination, training of frontline workers, and social mobilization (1). These barriers and challenges must also be viewed in light of the shortages of health workers that exist in many parts of the world. The High Level Commission on Health, Employment and Economic Growth of WHO (48) noted in 2016 that there was a global shortage of 180 million health workers to meet the prescribed minimum threshold of 44.5 health workers per 10, 000 individuals. Given these pre-COVID-19 deficits, the optimal management of human resources to meet the increased demands of the pandemic, manage the normal caseloads from other diseases, and still allocate sufficient human resources for the urgent task of vaccination will be a significant challenge to many countries.
We posit that strategic vaccine delivery and its uptake comprise the most important determinants of containing COVID-19 successfully, and compare favourably against considerations related to vaccine efficacy. Vaccine efficacy is defined as the percentage reduction of disease in a vaccinated group of people compared to an unvaccinated group, using the most favorable conditions (9) (10). Modelling suggests that a vaccine with partial efficacy may have significant impact and cost-effectiveness with maximal gains achieved with earlier introduction (1). Therefore, we argue that any vaccine candidate which has met the minimum endpoints explicitly defined in their phase 3 protocols must be deployed immediately, and the efficacy of such a vaccine, while important, is a concern secondary to achieving strategically targeted coverage as early as possible.
Similarly, we argue that while Non-Pharmaceutical Interventions (NPIs) will retain their significance in containment efforts in the short to medium-term, vaccination uptake will eventually supplant them as the primary means to containing COVID-19. Several countries have used contact tracing and isolation efforts, coupled with mask-wearing and social distancing measures, to contain COVID-19, as an interim measure until reliable vaccines become available (49) . When implemented consistently, these efforts have enjoyed considerable success, but are also both highly resource-intensive and socio-economically disruptive (49, 50) . As such, they may be unsustainable over the long-term due to population fatigue, economic hardship, and the lack of requisite public health resource capital (51) . Therefore, while they are crucial to the containment of COVID-19 in the short term, the importance of vaccination efforts will surpass them in significance in the longterm.
There are many compelling reasons why game theory should be employed in the modelling and analysis of SARS-CoV-2 vaccine uptake. Firstly, compulsory vaccination is likely to encounter a level of resistance from the public, and there may not be enough vaccines to vaccinate everyone. Therefore, decisions will have to be made by governments and policy makers about who to vaccinate, and by individuals about whether they want to take the vaccine. An individual is likely to get vaccinated only if the policy makers decide to offer vaccination to that individual, and the individual decides to accept it. Game theory can be used to explicitly model the decision making of policy makers (vaccine givers) and individuals (vaccine takers), because it has well-defined branches which model "public good" decision making (cooperative game theory) and selfish decision making (noncooperative game theory) (11, 44) .
Secondly, game theory can also model the evolution of strategies. In the COVID-19 context, strategies will evolve as more information becomes available about the disease itself and about the vaccines. Different strategies will contest for primacy, and evolutionary game theory (12, 64) can explicitly model this.
Thirdly, game theory can explicitly account for "bounded rationality" -the limited ability of people to assess reality. Thus, game theory can model the perceived and real payoffs of vaccination and distinguish between them, which is important with respect to COVID-19 because of the amount of misinformation and conspiracy theories present.
Finally, game theory can be used in conjunction with other tools, such as prospect theory (52) , Monte-Carlo simulation (53) , and agent based models (54) , which will be useful in modelling the COVID-19 dynamics, and computing and predicting epidemic parameters which can then be used as input to model decision making, completing the feedback loop. It can also employ supercomputing resources in calculating equilibrium solutions (68, 69).
A complex array of factors, parameters, drivers and attributes, which influence the decision by individuals to take the vaccine, the decision by governments and policy makers to provide a vaccine to certain people, or both, need to be considered in modelling SARS-CoV-2 vaccination uptake. The age and gender distributions of people will influence vaccination uptake, since older people are more likely to be adversely affected by COVID-19, and likely to have higher mortality rates (13, 14) , and thus may have more incentive to take the vaccination. Similarly, some preliminary studies suggest that (15, 16) men compared to women are more likely to be symptomatic with COVID-19 or to have higher morbidity, thus men may have comparatively more incentive to take vaccination. The cost and accessibility of the vaccine will obviously influence the levels of uptake. In some countries, governments may make the vaccination compulsory for certain demographics, such as people above a certain age, and this may reduce the incentive for others to vaccinate voluntarily.
The type of vaccine will influence the uptake. We need to model scenarios where the vaccine will be one-off, seasonal, or a chemoprophylaxis. We also need to consider whether the potential vaccine will reduce transmissibility or will reduce the severity of the symptoms and the mortality rate. If the vaccine will reduce transmissibility, then relatively young people, who are more likely to travel and interact with others, should be given preference by policymakers in vaccine access. On the other hand, if the vaccine primarily reduces symptoms or mortality, then older people, and people who have illnesses that increase the likelihood of a poor outcome, will have a higher incentive to take the vaccine, and policy makers will have a higher incentive to give the vaccine to them.
Other influential factors are epidemiological metrics, such as COVID-19 incidence, prevalence, and cumulative incidence, and these need to be considered at suburb, city, state and country levels, creating a complex array of parameters. Furthermore, the perceived epidemiological parameters can differ from real parameters if misinformation is being spread, and this difference between perceived and real parameters can be correlated to the 'bounded rationality' (18, 19, 55, 56) of the potential vaccinees, or the level of 'noise' present in the information. All such context and nuance will need to be modelled.
The level of interaction a person expects to have with the community in general, and other SARS-CoV-2-infected people in particular, will influence vaccination decisions, and will have to be modelled. For example, a person who travels to work by train, or is employed in a people-facing job such as teaching or food service work, may be more likely to vaccinate compared to a person working from home. Similarly, health workers are more likely to vaccinate.
Other factors involve the health of the potential vaccinees. In particular, the perceived and real levels of immunity of people have to be modelled, because people who have relatively higher levels of immunity have less incentive to take vaccination. Similarly, the overall health of the vaccinee, including the presence of chronic diseases, such as hypertension, diabetes, and chronic heart diseases, and the perceived correlation of these conditions with severe COVID-19 (17, 57) will need to be modelled, as will the perceived and real likelihood of adverse side-effects from the vaccine (especially in individuals with other chronic diseases).
Finally, the logistic and human resource management challenges in distributing and administering the vaccine, such as transport of vaccine, storage of vaccine, availability of clinical and support staff to administer the vaccine, and the durability of vaccine after manufacture are factors that need to be considered and modelled in understanding the SARS-CoV-2 vaccination uptake.
Usually, when the prevalence of a disease decreases, the perceived risk decreases, resulting in a smaller perceived payoff for taking the vaccine, which will in turn result in less vaccine uptake. This may over time result in the re-emergence of disease, which will in turn increase the payoff, so more people will take the vaccine. Therefore, there is a risk that COVID-19 will become endemic, and go through endemic stable cycles (and vaccine uptake will also go through corresponding cycles), if the perceived 'payoff' for taking the vaccine is not high enough on average when the prevalence becomes relatively low (4). It will be important to establish the conditions under which the SARS-CoV-2 vaccination uptake will go through such cycles, in different countries and sub-demographics, so that policy decisions can be made that ensure the payoff for the vaccine is always high enough to avoid such endemic stable cycles. As such, game theoretic modelling of vaccination uptake has to be tightly coupled with high fidelity simulation modelling (20, 21) of population demographics and disease dynamics, using techniques such as agent-based modelling, to gain a holistic picture about vaccination uptake.
In particular, large scale agent-based simulation methods (54) can be used to model population demographics, including age, gender, and household distributions of people, commuting and travel patterns, workplace locations and size distributions, school location and size distributions, and the mixing patterns of people (21) (See Fig. 1) . These can in turn be used to estimate epidemic parameters such as transmissibility, basic reproduction number, effective reproduction number, incidence, prevalence, and cumulative incidence. An inter-city flux model (4) 
The allocation, prioritisation, and distribution of scarce resources needed for implementation will be an important aspect of any future vaccination program, particularly in countries like India and Brazil with large populations and moderate per-capita spending on health (58) . Such resources include human resources, vaccine resources, rolling stock, and storage facilities. As a result, targeting decisions inevitably will need to be made (59, 60) . Therefore, the vaccination program becomes a resource allocation problem, and modelling optimal vaccination resource allocation is essential.
In scenarios where limited resources have to be optimally distributed and used, cooperative game theory can be applied with maximum benefit. Often, the outcome of a cooperative game played in a system is equivalent to the result of a constrained optimisation process (61) : therefore, such cooperative games are often solved by using a linear programming framework or other optimisation tools. In the context of SARS-CoV-2 vaccination, minimisation of epidemic parameters, such as incidence or prevalence, minimisation of economic costs of the pandemic, minimisation of mortality, and minimisation of disruption to daily lives could be some of the goals of governments and policy makers in deploying vaccines. Therefore, vaccination prioritisation could be modeled as a constrained multiobjective optimisation problem in real time (62) , and cooperative game theory, again coupled with simulation modelling techniques, could be used to solve it.
Clearly, the multi-objective optimisation will have to take into account the nature of the vaccine: If the vaccine reduces the transmissibility of SARS-CoV-2, prioritisation should be given to targeting individuals, cities, or states that can act as critical nodes of transmission or superspreaders. Therefore, the goal will be to reduce transmission, or the number of diagnosed cases. Whereas if the vaccine reduces symptom severity or mortality, prioritisation should be given to individuals, cities, or states that are likely to have poor outcomes. Therefore, the goal will be to reduce mortality and morbidity.
In this perspective, we articulated and discussed the view that vaccination uptake will have the most significant influence in the ultimate control of the COVID-19 pandemic, and effective modelling of uptake using appropriate tools therefore is of paramount importance. We presented the case for game theory, coupled with simulation techniques, social network analysis, and agent-based modelling, to be the most significant mathematical and computational toolset available for this modelling. We highlighted that while the efficacy of the vaccines developed, as well as the efficiency of testing, contact tracing, and isolation procedures, shall remain important factors in containing the COVID-19 pandemic, the effectiveness of the vaccination program and the level of vaccination uptake will surpass these factors in significance in the effort to finally contain, locally eliminate, and globally stabilise the COVID-19 pandemic.
We discussed the modelling of vaccination decision making in detail, and articulated that this has two components: a) the decision making process by individuals to get the vaccine b) the decision making process by governments and policymakers to choose vaccinees, given the reality of limited vaccine doses at the initial stages of vaccination. We argued that the individual decision-making regarding vaccination uptake is influenced by a range of factors including demographics, physical location, level of interaction, the health of the vaccinee, epidemic parameters, and perceptions about the vaccine being introduced. Similarly, the decision making of the government will be influenced by epidemic parameters, the nature of the vaccine being introduced, logistics, management of human resources needed for the vaccination effort, and the amount of vaccine doses available. We explained that noncooperative game theory is ideally suited for modelling individual decision making behaviour regarding vaccination, while cooperative game theory can be used to inform government decisions regarding prioritisation.
We believe that this perspective will stimulate research and deliberation among all stakeholders which will aid the successful implementation of vaccination programs against COVID-19 and its decisive containment in the near future.