In Canada, the country and all ten of our provinces have a Department of Public Health and the picture at the top of the post shows many of the leaders of these departments. These are the people who are trying to keep us healthy, in Canada, long enough for treatments and vaccines to be developed. Occupy Math is working with a group that is going to try and build a tool to help these officials; that tool figures out how to best deploy vaccines as they become available. The idea is to look at the network of contacts among people and deploy vaccines so that they do the most to stop the novel corona virus.

This post will look at some standard vaccination strategies and take a shot at explaining how to combine them into a master strategy that can adapt to changing conditions. Occupy Math keeps saying that being mathematically informed makes things better for you and for others. In this post we are explaining a fairly complex mathematical situation with a metaphor — the contact network of people — that will help you think about the situation and deal with it more effectively.

**Vaccinate everyone you can!**

The term *herd immunity* is actually pretty simple. It means the condition in which the average number of people infected by a currently sick person is less than one. The parameter, called R_{0} or *R-naught* is the number of people infected by each person that has the disease. It measures the spread rate of an epidemic. When R_{0} is less than one, an epidemic dies out; when it is more than one, an epidemic spreads. Herd immunity is achieved when R_{0} is less than one. In epidemiology, the question is “how many people do you have to vaccinate to achieve herd immunity?” A couple of European governments stated that they were going to use a “herd immunity” strategy, but because this means “let the disease spread unchecked” if you do not have a vaccine, they have since changed their mind.

Usually the correct strategy is to vaccinate everyone you can; this is made more difficult because of people with religious objections to vaccination, or people who have swallowed whole a pile of pseudo-science about vaccines being dangerous. For Covid-19, however, we cannot do this because there is no vaccine yet. As vaccines are developed and made available, they will be scarce at first, and so we need a strategy to deploy them. Supplying context for those decisions is what most of this post is about.

**Contact Networks**

Occupy Math uses *contact networks*, like the ones shown below, to model epidemics and intervention strategies. The circles are people and lines indicate a form of contact that might spread the disease. The same people are shown, four times, with different levels of distancing and isolation. If we think of the distance between pairs of people as the number of jumps the disease must make to get from one person to another, then the average distance between people is a measure of how spread-out a network is. For each of these networks, the average distance between any two people is shown in green. In the last network one person, in the lower left corner of the network, is completely isolated and is connected to no-one else. This is probably not practical — you gotta eat.

The higher the average distance in the network, the slower the disease will spread. Breaking links in a contact network means practicing proper social distancing, wearing masks, washing your hands to kill the virus you do encounter, and avoiding all but really necessary trips and errands. These networks do not show the actual position of the people — people move around. Rather, the networks show patterns of contact. We are going to color the people in the networks in the rest of this post to show their disease status. There are keys on each diagram for what the colors mean. *Susceptible* people can catch the disease, *infected* people currently have the disease, *removed* people have had the disease, and *vaccinated* people count as removed people because of the vaccine.

**You and your links: stay calm.**

The links in the social networks that are shown all through this post represent contact between people in the network. Inside this mathematical metaphor, you are one of the circles, and you can generate or not generate the links, the connection to other people. If you do a good job of not generating links, you help. If you occasionally screw up, you are human. Another big danger is the deterioration of mental health that comes from isolation, from uncertainty, and from over-thinking every little decision. If you forget a critical item and have to go on a second shopping trip — then, first, make sure it is really a critical item and second, if it is, put on your mask and go get it while obeying all the social distancing rules. One more link, here or there, does not change things too much most of the time. Do your best and try not to get too worked up.

**Random Vaccination**

If we just choose people at random to vaccinate, so that they can no longer spread the disease, then it knocks out nodes in the network, making it harder for the virus to spread. We measure this by looking, again, at the average distance between people. We will apply different strategies to the *same* contact network so we can compare their effect on the average distance. In these diagrams two average distances are given. The first is the distance assuming that the vaccinated people cannot spread the disease; the second, in parentheses, is the original average distance between people, when there was no vaccination.

**Ring Vaccination**

*Ring vaccination* vaccinates people who have contact with an infected person, attempting to wall off the disease. If the disease is easy to detect, this can be a good strategy, but Covid-19 has asymptomatic carriers, meaning that it’s not a practical strategy unless we have really good contact tracing. Since we do not know who has the disease until after they spread it, we do not know who is in the “rings”. Notice that each of the red nodes (infected people) is “ringed” by vaccinated people.

**Block Vaccination**

*Block vaccination* picks connected groups of people, like a neighborhood, and vaccinates all of them. This tends to blow big, connected holes in the contact network. This strategy varies from excellent to useless depending on the exact structure of the network. Vaccinating all heath care workers and grocery store workers, for example, would probably work really well because those are super-high contact areas in the network. The blocks in the picture below have a recovered person in the middle of the block, people we *assume* are infected that have had contact with that person, and then the rest of the block of people is vaccinated.

**High Contact Number Vaccination**

This strategy vaccinates people that have the most contacts first. This will generate some outcomes similar to block vaccination, except that it will vaccinate fewer people in more groups, based on an estimate of the number of people they could spread the disease to. Unlike ring vaccination, this strategy does not erect walls against the spread in the network, but it does spread out the network and slow the disease spread the most effectively (on the one network Occupy Math is using as an example!) The people vaccinated in this example had six or seven contacts — the two highest numbers of contact appear in the network.

**Real Contact Networks are Larger**

The contact networks that Occupy Math used as examples in this post are far smaller than the real ones. They were used to give you a picture of some of the available vaccination strategies (Occupy Math’s list of vaccination strategies is also incomplete). Real contact networks are also inexact — we get them by mining cellphone data, for example. What Occupy Math will be doing with his collaborators is to create an adaptive strategy that takes what we know about the current status of people — susceptible, infected, removed and the areas they are in — and say which basic strategy to follow for the next few days. This master strategy is a type of artificial intelligence that can shift the vaccination technique to match the circumstances. Since no vaccine is available yet, there is some lead time for development.

One big question is *how do you know this will work?* In all honesty, we do not *know* this strategy will work. The idea is to generate lots of different sorts of networks, based on likely real-world situations. In each network, a simulated disease spreads through the network. Each AI strategy generated will then be allowed to manage several simulated epidemics for each contact network. The strategies with better outcomes will be duplicated and variations of them tested again. In the end, we will generate relatively effective strategies — different ones for each type of network. These outcomes will then be analyzed to see if the AI strategies are helpful and to generate advice for public health officials based on the behavior of the effective AI strategies.

The techniques developed will also be handed off to people modeling the real world epidemic if they want them. Occupy Math’s project is likely to generate some good high-level advice. The chance it can actually deal with the real world directly is much smaller. If you have notions for topics to address in future Occupy Math posts, please put them in the comments.

I hope to see you here again,

So wash your hands,

Daniel Ashlock,

University of Guelph,

Department of Mathematics and Statistics