How Math Can Help Prevent Outbreaks of Measles and Other Diseases


The measles vaccine is incredibly effective, yet recurrences and outbreaks abound. Samoa is currently undergoing an outbreak with thousands infected, and in 2019, the United States had the most cases reported since 1992. What can we do about it? In this excerpt from his new book, The Math of Life & Death: 7 Mathematical Principles that Shape Our Lives, mathematical biologist Kit Yates explains how everyone has a part to play in the wider context of public health. Increasing the vaccination rate allows vaccines to prevent deadly illnesses. By understanding the mathematical principles behind disease prevention, the public can better protect themselves from disease.

Sometimes math really is a matter of life and death, particularly when it comes to the outbreak of a deadly disease and the strategies to control it. As well as helping us to understand the unusual features of different disease landscapes, mathematical models of epidemics allow us to peer into the future of disease progression and to take proactive preventive measures, rather than always playing reactive games of catch-up.

Mathematical epidemiology helps us answer a number of perplexing questions that surround childhood diseases such as mumps and rubella.

The idea that a large population of immune individuals can slow or even halt the spread of a disease, as with the dormant periods between outbreaks of childhood diseases, is a mathematical concept known as herd immunity. Surprisingly, this community effect does not require everyone to be immune to the disease for the whole population to be protected.

By reducing the effective reproduction number—the average number of secondary cases produced by a typical case—to less than one, the chain of transmission can be broken and the disease stopped in its tracks. Crucially, herd immunity means that those with immune systems too weak to tolerate vaccination—including the elderly, newborns, pregnant women and people with HIV—can still benefit from the protection of vaccinations.

Vaccinating only 80 percent of the susceptible population against smallpox was enough, in 1977, to complete one of the greatest accomplishments of our species—to wipe a human disease clean off the face of the Earth. The feat has never been repeated. The debilitating and dangerous implications of smallpox infection alone made it a suitable target for eradication. Its low critical immunization threshold—the minimum proportion of the population that must be immunized in order for the infection to die out—also made it a relatively easy target.

Many diseases are harder to protect against because they spread more easily. Chicken pox would require 90 percent of the population to be immune before the rest would be effectively protected and the disease wiped out. Measles, by far the most infectious human-to-human disease on Earth would require between 92 percent and 95 percent of the population to be vaccinated. A study that modeled the spread of a 2015 Disneyland measles outbreak suggested vaccination rates among those exposed to the disease may have been as low as 50 percent, way below the threshold required for herd immunity.


Given the astounding success of the smallpox vaccine in eradicating the formerly deadly disease, why are some vaccination rates for other deadly diseases now falling? One cause of this dangerous trend can be tied to the anti-vaxxer campaign, which has its roots in a now debunked study linking the MMR vaccine to autism.

As an example, in 1996, England, whose rate of rate of vaccination against measles (through the combined measles, mumps and rubella [MMR] injection) hit a record high at 91.8 percent—close to the critical immunization threshold for eliminating measles. Then, in 1998, something happened that would derail the vaccination process for years.

This public health disaster was caused by what was later found to be a fraudulent study in the well-respected medical journal the Lancet. In the study, lead author Andrew Wakefield proposed a link between the MMR vaccine and autism-spectrum disorders. On the back of his "findings," Wakefield launched his own personal anti-MMR campaign, stating in a press conference, "I can't support the continued use of these three vaccines given in combination until this issue has been resolved." Most of the mainstream media couldn't resist the bait.

While indulging the fears of many fretful parents, the media's coverage of the story typically failed to mention that Wakefield's study was conducted on just 12 children, an extremely small cohort from which to draw meaningful large-scale conclusions. Any coverage that did sound a note of caution about the study was drowned out by the warning sirens emanating from most news outlets. In the 10 years that followed the publication of the infamous Lancet paper, the MMR uptake rate would drop from above 90 percent to below 80 percent. Confirmed cases of measles would increase from 56 in 1998 to over 1300, 10 years later. Cases of mumps, which had been becoming less prevalent throughout the 1990s, suddenly skyrocketed.

In 2004, investigative journalist Brian Deer discovered fraud in Wakefield's work. Among other claims, Deer claimed to have evidence that Wakefield had manipulated the data in his paper to give the false impression of a link to autism. Deer's evidence of Wakefield's scientific fraud and extreme conflicts of interest eventually led to the offending paper's retraction by the Lancet's editors.
In 2010, Wakefield was struck off the medical register by the General Medical Council. In the 20 years since Wakefield's original paper, at least 14 comprehensive studies on hundreds of thousands of children across the world have found no evidence of a link between MMR and autism. Sadly, though, Wakefield's influence lives on.

Although MMR vaccination in the UK has returned to pre-scare levels, vaccination rates across the developed world as a whole are dropping, and measles cases are increasing. In Europe, 2018 saw more than 60,000 cases of measles, with 72 proving fatal—double the number from the previous year. The United States experienced more measles cases in the first four months of 2019 than in any year for a quarter of a century. The World Health Organization lists what it calls "vaccine hesitancy" as one of 2019's top 10 global health threats. The Washington Post, among other media outlets, attributes the rise of the "anti-vaxxers" directly to Wakefield, describing him as "the founder of the modern anti-vaccination movement." The doctrines of the movement, however, have expanded far beyond Wakefield's now-debunked findings, and anti-vaxxer rhetoric has risen to prominence as a result of support from high-profile celebrities including Jim Carrey, Charlie Sheen and Alicia Silverstone.

Alongside the rise of the celebrity activist has come the emergence of social media, allowing these personalities to promulgate their views directly to their fans on their own terms. With the erosion of trust in the mainstream media, people are increasingly turning to these echo chambers for reassurance. The rise of these alternative platforms has provided a space for the anti-vaccination movement to grow unthreatened and unchallenged by evidence-based science. Wakefield himself even described the emergence of social media as having "evolved beautifully"—for his purposes, perhaps.

Disease Prevention Is in Your Hands

Despite the growing popularity of vaccine hesitancy, the individual decision to get ourselves and our children vaccinated bolsters the herd immunity that keeps whole populations safe. We all have choices to make that affect our likelihood of contracting infectious disease. When we are ill, choices we make affect our likelihood of transmitting disease to others: whether we cancel the much-anticipated get-together with friends; whether we keep our children home from school; whether we cover our mouths when we cough. The crucial decision of whether we vaccinate ourselves and our dependents can only be made ahead of time. It affects our chances not only of catching—but also of transmitting—diseases.

Some of these decisions are inexpensive, making their adoption straightforward. It costs nothing to sneeze into a tissue or a handkerchief. Other decisions provide us with more of a dilemma. It is tempting to send the kids to school even if we know it increases the number of potentially infectious contacts they will make. At the heart of all our choices should be an understanding of the risks and consequences.

Mathematical epidemiology provides a way to assess and understand these decisions. It suggests strategies to tackle disease outbreaks and the preventive measures we can take to avoid them. In conjunction with reliable scientific evidence, mathematical epidemiology demonstrates that vaccination is a no-brainer. The World Health Organization figures show that vaccines prevent millions of deaths every year and could prevent millions more if we could improve global coverage. They are the best way we have of preventing outbreaks of deadly diseases and the only chance we have of terminating their devastating impacts for good. Not only does it protect you, it protects your family, your friends, your neighbors and your colleagues.

→ From The Math of Life and Death by Kit Yates. Copyright © 2019 by Kit Yates. Reprinted by permission of Scribner, an imprint of Simon & Schuster, Inc.

Correction 12/26/2019 11:55 p.m.: Updated to clarify that Deer did not set out to expose Wakefield's work as fraudulent, but rather discovered the fraud in the course of his investigation and reported what he found.