Health

# The link between imperfect vaccines and disease

New mathematical models explore why some vaccines are more effective than others.

(Inside Science) -- A new mathematical model that tests the effectiveness of different vaccine types could help explain why certain diseases are still prevalent despite mass vaccination programs.

The model, which is detailed in a study published in a recent issue of the SIAM Journal on Applied Mathematics, examined the effects that different vaccine types have on individuals compared to the population as a whole.

Vaccines are designed to elicit an immune response that is similar to what would be triggered by a natural infection, but without causing the actual disease.

"An ideal vaccine would provide perfect protection that lasts forever," the study's first author Felicia Magpantay, a postdoctoral fellow at the University of Michigan in Ann Arbor, said in an email.

In real life, however, no vaccine is perfect. They all fail in some way. Magpantay and her colleagues looked at the individual and population-level effects of three types of imperfect vaccines. The first was a "leaky" vaccine, whereby vaccination reduces, but does not eliminate, the chances of infection upon exposure to a disease agent. They also modeled "all-or-nothing" vaccines, which grant perfect lifetime immunity for some individuals, but zero protection for others. Lastly, they looked at "waning" vaccines, which provide perfect protection for only a limited amount of time.

The team took a mathematical model commonly used in epidemiological studies and adapted it so that they could calculate the number of susceptible, infected, and recovered individuals in a population for the three different vaccine types. They also looked at a scenario in which all three vaccine types were circulating in a population. The model factored in infection and recovery rates, as well as contact between susceptible and infected individuals. Most importantly, the model looked at a population made up of groups of individuals of varying ages. This is important because it's been shown that mixing between people of different ages can affect the transmission of diseases.

Using their model, the team determined how many individuals in the model population had to be vaccinated in order to eliminate a disease. They found that this "critical ratio" was the same for all three types of imperfect vaccines. When vaccination coverage was less than this critical ratio–which can change depending on the protection level provided by the vaccine and how a disease spreads–the disease remained at large. Surprisingly, however, the model found that leaky vaccines led to higher infection rates in the population than all-or-nothing or waning vaccines.

"If vaccination coverage is maintained below [the critical ratio] then the disease is not eliminated and the purely leaky vaccine would result in the highest level of infection in the long run," Magpantay said.

Furthermore, the model also suggests that leaky and waning vaccines can lead to a temporary lull in disease propagation following a mass vaccination program. But afterward, the disease surges again within a population, years or even decades later. The authors speculate that a possible "honeymoon" effect, which will be investigated more fully in future research, could explain why certain diseases such as pertussis, or whooping cough, manage to persist in regions that maintain high vaccination coverage.

"The honeymoon period and the transient oscillations are shown in many models of vaccination … [It will vary] depending on the disease, but it is 'long enough' for a population to kind of forget about the severity of the disease because people rarely see it. This can lead to vaccine hesitancy, where people decide not to have their kids vaccinated," said Jane Hefernan, the director of the Center for Disease Modeling at York University in Toronto, Canada. "What is interesting in this study is that the honeymoon period is quite different for each type of vaccine."

Hefernan said the new study provides a good example of how a simple mathematical model study can shed light on questions that would be hard to study otherwise.

"The mathematical modeling community is growing, especially in studies of infectious diseases," said Hefernan, who did not participate in the research. "Public health programs can be better informed from such studies."

The new findings could also have implications for the design of vaccination programs, she added.

"[In the model] the effects of the vaccines are different depending on the type of vaccine being considered. The leaky vaccine acts quite different compared to the all-or-nothing and waning vaccines," Hefernan said. "This means that a public health program should be different if a leaky vaccine is being considered."

The team says they plan to extend their model to examine the role of seasonality on transmission rates and also whether imperfect vaccines change how infectious a person is.

Reprinted with permission from Inside Science, an editorially independent news product of the American institute of Physics, a nonprofit organization dedicated to advancing, promoting and serving the physical sciences.