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Two Applications of Network and Graph Theory in Infectious Diseases

Herd Immunity

In the study of immunity and disease, you may have seen the term “Herd Immunity” used in reference to the threat of anti-vaccine ideas. While coined in 1923, the term still sees common use today [2]. “Herd Immunity” refers to the principle that, as rates of disease immunization go up, rates of infection among non-immunized individuals drop at an increasing rate. The basic reasoning is as follows:

Assume diseases may only be transmitted through direct contact by two non-immunized persons.
Since the (in-person) social network of a nation is not a connected graph, it is not guaranteed that two particular vulnerable individuals have a connection.
If non-immunized individuals are distributed randomly in a national social-network, then we can expect the chance of VULNERABLE – VULNERABLE edge to decrease exponentially as the number of vulnerable people decreases.
Therefor, a vulnerable person actually becomes safer as more and more of their peers become immune, even if the vulnerable person never becomes immune, themself.

Spread of Information

The second application of network and graph theory on disease immunity has darker implications. The Impact of the Web and Social Networks on Vaccination [4] states that almost 50,000 pieces of online vaccine-related content are produced each month. With the rise of social media, people can quickly spread rumors and information to a large circle of acquaintances. This spreading of information in and of itself would be harmless, however, research continually suggests that individuals in social networks are more likely to share their experiences if they perceive them as exceptional. For example, 2 and 3-star rating on Amazon much less common than 1 and 5-star ones [5]. This implies that individuals are most likely to reach out to their social network regarding vaccines if they feel that their child has been adversely affected by vaccination. This has the effect of spreading anti-vaccination sentiment far faster than the less-viral pro-vaccine sentiment.


If we consider these applications in parallel, we begin to construct an explanation for the rise and risk of Anti-Vaccine “hot spots” in states that allow non-medical exemption from vaccine [3]. Interpersonal social networks will behave much like online ones, quickly spreading anti-vaccine ideas among groups of friends and acquaintances in person. When considering our discussion of “herd immunity,” we can see why these pockets of anti-vaxxers are dangerous – Herd Immunity operates under the assumption that those without immunity are evenly spread among a population. Herd Immunity does not lead to mass-immunity under conditions of concentrated vulnerable people – large fully connected groups of vulnerable people are in fact greatly at risk.

So, what can an individual do to mitigate this problem?
Well, besides being vaccinated yourself, one can use online social networks to help influence public opinion. Social media provides channels through which one can reach individuals several steps removed from their own ego network. Don’t cite studies and statistics at people who are anti-vaxx; this is shown to have little to no effect [1]. Instead, The Association of State and Territorial Health Officials suggests speaking about experience and relating to someone at a personal level.

Sources Cited

“Communicating Effectively About Vaccines.”, Association of State and Territorial Health Officials, 2010,–New-Communication-Resources-for-Health-Officials/.

Fine, Paul, et al. “‘Herd Immunity’: A Rough Guide.” Clinical Infectious Diseases, vol. 52, no. 7, 1 Apr. 2011, pp. 911–916., doi:

Fox, Maggie. “Anti-Vaccine Hot Spots Thrive in States That Make It Easy to Opt Out.”, NBCUniversal News Group, 12 June 2018, 2:31PM,

Stahl, J.-P., et al. “The Impact of the Web and Social Networks on Vaccination. New Challenges and Opportunities Offered to Fight against Vaccine Hesitancy.” Médecine Et Maladies Infectieuses, vol. 46, no. 3, 2016, pp. 117–122. Cornell University Library, doi:10.1016/j.medmal.2016.02.002.

Woolf, Max. “A Statistical Analysis of 1.2 Million Amazon Reviews.” Minimaxir | Max Woolf’s Blog, 17 June 2014,


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