Back in December, when countries around Europe were ramping up their booster programmes to fight Omicron, the World Health Organisation issued a report. Rather than boosting their own populations, the report said, high-income countries should consider sharing vaccine supplies around the world.

There are obvious moral reasons for sharing vaccines, but there is also a selfish one. Every infection presents an opportunity for the virus to mutate. As long as infections are rife around the world there is a high risk of potentially dangerous, vaccine-evading variants emerging. And as we have seen time and time again, variants don't stop at borders. From that point of view, it appears to make sense to protect as large a proportion of the world's population from infection as possible, and vaccines help us do this.

What's the best way to vaccinate the world?

But even when you take the chance for mutation aside, there's mathematical evidence that sharing vaccines is a good idea. Back in 2011 the epidemiologist Petra Klepac, together with colleagues, decided to look at vaccination strategies in a global setting. The diseases she had in mind back then included measles, mumps, polio and rubella, for which vaccines exist and are likely to be needed in the longer term.

Klepac's work suggests that the result you get when countries decide on their vaccine policy in isolation can be far from what would be best for the world as a whole. And that's despite the fact that a global vaccine strategy might also serve individual countries better than their national strategies do.

### A prisoner's dilemma

This kind of paradox isn't new. In the world of maths it's epitomised by a thought experiment called the *prisoner's dilemma*, which has two imaginary partners in crime being interviewed separately by the police. The best outcome for both suspects would occur if both refused to confess to the crime. But since neither prisoner can be sure the other will remain silent, the optimal strategy for each individual is to talk, knowing that at least that'll reduce their sentence. Thus, both prisoners end up confessing, landing them both with a hefty jail term. (See here for a more detailed description of the prisoner's dilemma.)

See here for all our coverage of the COVID-19 pandemic.

The prisoner's dilemma is a staple of mathematical game theory which seeks to analyse situations where people or organisations interact by describing those situations in clearly defined terms: there are a number of players, who each have a set of strategies available to them, each of which comes with a particular pay-off. Assuming players act rationally to maximise their own pay-offs, it then becomes possible to analyse how a situation might play out.

A situation in which no player has an incentive to unilaterally change their chosen strategy is called a *Nash equilibrium*: it's an equilibrium because nothing changes. (The word “Nash” credits the game theorist John Nash, made famous by the movie *A beautiful mind*.) The Nash equilibrium in the prisoner's dilemma is both prisoners deciding to confess: in the absence of communication between the prisoners it's the strategy that gives the best outcome for the individual, regardless of what the other prisoner does.

### Weighing up costs

When Klepac, who is now a member of the JUNIPER modelling consortium, looked at optimal vaccination strategies with her colleagues, it was game theory she called in to help. Standard epidemiological theory gives us a robust general rule as to what proportion of a population should be vaccinated to achieve herd immunity. Assuming a vaccine is 100% effective, that critical proportion is *p*=1-1/*R*_{0} where *R*_{0} is the *basic reproduction number* of the disease (that's the *R* number we're all now familiar with, as it would be in the absence of any intervention and for a completely susceptible population, see here for more details).

Game theory can help understand situations that involve interaction between people, organisations, or countries.

A country's individual goal might simply be to achieve herd immunity, in which case it is looking to vaccinate the critical proportion p of its population. But Klepac and her colleagues also allowed a more complex situation, where a country weighs the cost of vaccination against the burden of disease — after all, not all diseases are deadly enough to warrant an expensive vaccination programme. The burden of disease is hard to quantify of course, but in their simplified model Klepac and her colleagues assumed it to be proportional to the severity of the disease and to the proportion of the population that is infected.

It's now possible to calculate a new optimal proportion, *q*, of the population that should be vaccinated in order to minimise the combined cost of vaccination and burden of disease. If vaccination is expensive compared to the burden of disease, then the proportion *q* might be less than the herd immunity proportion *p*: it's not worth achieving herd immunity in this case, instead it's sufficient to use vaccination to bring the prevalence down to an acceptable level. If, on the other hand, the disease burden is high compared to the vaccination cost, going for herd immunity may be the optimal option.

Because the cost of vaccination and the burden of disease can vary from country to country, depending, for example, on the infrastructures that are in place, this means that the optimal vaccination strategy in one country can differ from that in another. One country may want to eliminate a disease while another one might want to tolerate it — which means it becomes a source of infection to its neighbours, making it impossible to eliminate the disease in neighbouring countries.

### A vaccinator's dilemma

You can now ask yourself how communal action, where countries decide on the optimal vaccination coverage for the entire region, compares to individual action, where each country decides on an optimal vaccination strategy for its own region without talking to others. In their paper published in 2011 Klepac and her colleagues imagined a simplified situation with only two countries, with the spread of the disease described by a standard epidemiological model that rests on various assumptions (you can see the details in the paper).

The result is similar to the prisoner's dilemma. There's a Nash equilibrium where each country has chosen a vaccination coverage that's optimal from its own point of view: it can't find a coverage that will come at a lower cost, unless it talks to its neighbour and comes to an agreement. However, had the countries agreed to look at the problem jointly, the optimal strategy may well have turned out different.

Travel restrictions have been in force during the COVID-19 pandemic and also play a role in Klepac's game theoretical analysis.

"This difference between local optima and the global optimum is key here,'" says Klepac. "The paradox is that everyone could be better off at a global optimum because it's cheaper to achieve things together — either because it leads to fewer infections, or because the cost of care is cheaper, or because you have more negotiating power when buying resources — but it's hard to get to the global optimum because countries are tempted to focus on themselves only."

Time can also play an important role here. When Klepac and her colleagues looked at how limited resources should best be allocated to achieve herd immunity in their model world, they found that the best short-term strategy of a country might be to plough its resources into its own vaccination effort. In the longer term, though, sharing resources with the other country would have been better — you can't achieve herd immunity on your own if infected people keep crossing the border.

This is reminiscent of other problems that don't stop at borders, such as climate change or the management of water resources. In a follow-up paper published in 2016 Klepac and colleagues took inspiration from a theory developed to understand international environmental agreements. Again combining game theory with epidemiology and economics, they imagined a world with more than two countries which have the option to form coalitions.

The benefits of cooperation again became clear, and the work also showed that coalitions can be stable in the sense that members gain no advantage by leaving, and non-members don't have an incentive to pile in. Sanctions such as travel restrictions might be necessary, though, to stop non-members free-riding on high vaccination coverage in member countries, which would result in fewer infections arriving at their shores.

### The real world

Klepac's work is highly theoretical, so we can't draw direct conclusions about the real world. "This is maths and it's abstract, so can you apply it to the real world?" she says. "I think it's possible, but the caveat is that these are models based on certain assumptions." Some of these assumptions don't apply to the current pandemic where the virus constantly mutates, and the vaccines' efficacy varies and wanes.

What we can do, however, is regard this work as a first proof of concept showing that cooperation can pay when it comes to vaccination, just as it pays in other situations too.

"In the current context there hasn't been much cooperation," says Klepac. "If you look at the number of boosters administered in high-income countries compared to the number of first doses in low-income countries it's clear that we are focussing on local hoarding of vaccines. I think we haven't used cooperation to the global benefit as much as we could have done. And cooperation and sharing of resources doesn't just apply to sharing of vaccine doses — it also goes for sharing of information and for sharing of viral sequencing data to identify variants of concern.”

It will be interesting to see what lessons we learn, individually and as countries and governments, from experiencing a global pandemic. Perhaps the value of cooperating in vaccination strategies might prove a selfish, as well as a moral advantage.

### Further reading

To find out more about the topics mentioned in this article read

- Maths in a minute: The prisoner's dilemma
- Why do mathematicians play games?
*R*_{0}and herd immunity- How can maths fight a pandemic?

### About this article

Petra Klepac

This article was inspired by a workshop on optimal vaccination strategies, which was hosted virtually by the Newton Gateway to Mathematics in December 2021.

Petra Klepac is Assistant Professor in Infectious Disease Modelling at the London School of Hygiene and Tropical Medicine and a visiting researcher at the University of Cambridge. She was interviewed by Marianne Freiberger, Editor of *Plus*, in January 2021.

*This article was produced as part of our collaborations with JUNIPER, the Joint UNIversity Pandemic and Epidemic Response modelling consortium, and the Isaac Newton Institute for Mathematical Sciences (INI).*

* JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing question about the control of COVID-19. You can see more content produced with JUNIPER
here.*

*The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more. *