## Benefits of Group Cooperation vs. Personal Decisions

Some people prefer not to get vaccinated against certain illnesses, for example, influenza, either from the belief that their immune system can handle it naturally, or if they do happen to get sick, the symptoms would simply pass in a few days. This indifference is perhaps expected given the fact that getting vaccinated has some costs associated with it: traveling to a designated location, time spent waiting in line, risk of possible side effects, etc. For the small benefit of avoiding flu-symptoms, many people choose not to receive vaccination. A recent article by Quanta Magazine, however, highlighted that getting vaccinated benefits more people than just the individual. In the context of the SIR epidemic model, vaccinating an individual removes them from the contact network, preventing them from catching the illness.

Removing people from the contact network lowers the epidemic’s value of $k$, the number of susceptible people an individual comes in contact with. This, in turn, lowers the epidemic’s basic reproduction number, $R_0$. Lowering $R_0$ to or below 1 slows down any present outbreaks and prevents future ones from occurring. If we know the value of $R_0$ precisely enough, we can also determine the fraction of people required to be removed from the contact network in order to bring $R_0$ down to 1 and prevent any outbreaks. This is also referred to as a level of “herd immunity”. Letting this fraction be $r$, for instance, the article tells us that:

$r=1-\frac{1}{R_0}$

It is easy to verify that this is a valid solution. We know that $R_0=kp$, so:

$1=kp \Rightarrow k=\frac{1}{p} = (1-r)k_0$,

where $k_0$ is the initial k value before vaccinations were administered.

$\Rightarrow (1-r)k_0 = k_0(\frac{1}{pk_0}) = \frac{1}{p} = \boxed{k}$

A natural roadblock to achieving this outcome is that the fraction of people who voluntarily choose to be vaccinated is lower than the required $r$. Certain measures can be taken to increase this fraction, such as enabling greater access to vaccines through non-traditional settings, and increasing overall community demand for vaccines through reminders and recall systems. Hence, even if getting vaccinated seems unpleasant to an individual, it has larger overarching benefits including protecting those who cannot be vaccinated for medical reasons.

Source: https://www.quantamagazine.org/the-unforgiving-math-that-stops-epidemics-20171026