By Debbie Sniderman
Threats as diverse as HIV, malaria, Zika, Ebola, bioterrorism, and antimicrobial resistance have sparked a growing interest in disease dynamics. Epidemic diseases such as measles have an oscillatory behavior closely analogous to predator-prey dynamics in ecology. Bryan Grenfell (Princeton University) examines the dynamics of measles epidemics in space and time, with special emphasis on how external drivers impact epidemic predictability.
Grenfell attempts to understand predictability and optimize control of infectious diseases; in addition to measles, he studies other childhood epidemics like rotavirus, influenza, respiratory syncytial virus (RSV), and rubella (German measles). These systems are challenging to predict, and prediction is further complicated by the exhibition of strong nonlinear feedback, the evolution of pathogens, and the adaptation of people’s behavior to avoid epidemics.
Why Time Matters
Grenfell’s approach underlines that simple models can still reveal the essence of epidemic dynamics, despite many biological and social complexities. His time-series TSIR models adapt standard, nonlinear ordinary differential equation (ODE) state-space SIR models to account for measles’ operation as a forced oscillator, seasonally driven by the aggregation of children in school. Wavelet spectra help tease out the nonstationary cycles . According to Grenfell, this work is relevant across a surprising range of more complex infections and could be helpful for vaccination and other control measures.
Grenfell advocates the convergence of dynamical systems theory, statistical inference, and detailed epidemic data to explore the diversity of epidemic dynamics. “We should model epidemics from as many pathogens as we can, even non-threatening ones,” he says. “We don’t know what the next public health challenges will be. Something we don’t yet understand may become important.”
“There has been a gradual evolution towards synthesizing epidemic dynamics with biology from smaller integrative scales,” Grenfell adds. “Increasingly we have to drill down to the molecular and cellular levels, and then back up to individual, community, and population levels. For example, there’s currently great progress in predicting seasonal influenza evolution, where all these scales can interact. With so many complexities—network dynamics, complex social networks, individual heterogeneities, within-host dynamics, and battles between viruses and the immune system—the challenge is to find the right level at which to model. Math is the key to closing that gap.”
Fitting Decades of Measles Data with SIR Models
Measles is caused by a highly-transmissible, strongly-immunizing virus that infects individuals for a short time and can cause significant morbidity. An inexpensive vaccine created in 1963 significantly decreased measles incidence throughout the United States and other countries. However, the infection is still a significant killer in many settings, notably parts of sub-Saharan Africa.
Figure 1. One and two-year cycles identified by local wavelet power spectra of London measles. Similar non-stationarity effects on cyclicity are seen with vaccination (like reducing birth rate). Image Credit: .
SIR models, by definition, divide the population into compartments: susceptible, infected, and recovered/immune. People infected with measles move unidirectionally through the classes. Simple population-level dynamical models of this system include a mass-action term, bilinear in the sizes of susceptible and infected populations with a coefficient that may depend on population size and time. Before vaccination, repeated measles epidemics occurred in large cities, corresponding remarkably closely to the dynamics of a simple oscillator with peaks and troughs. According to Grenfell, SIR models work reasonably well to fit rich measles data sets spanning many years.
Births increase recruitment to the susceptible population. When measles invades a susceptible population, it has a basic reproduction ratio, or R0, of approximately 18-20 and quickly depletes the susceptible population. An infected individual acquires lifelong immunity after around 10 days of infection.
As the population becomes increasingly immune, more people who do not get measles are indirectly protected. Herd immunity is a key consequence of this nonlinearity – susceptible people are ‘indirectly’ protected by the population’s overall immunity.
Measles can be analyzed at different time scales with wavelet spectra, which allow observation of strong seasonal trends and regular, biennial, or more irregular longer-period epidemics. Wavelet spectra also show dynamic transitions between epidemic periods. In one example, models reveal that when birth rates were high in London, epidemics occurred yearly. As birth rates dropped, a mixture of annual and major biennial patterns emerged (see Figure 1).
Figure 2. Herd immunity in the SIR model. Image Credit: Bryan Grenfell.
Chaos, seasonality, and behavioral dynamics complicate modeling. Behavioral complexities include variable willingness to be vaccinated and self-imposed isolation upon becoming sick. Measles in the pre-vaccination era tended to be less complicated by such behavior. This behavior, as well as the relatively high incidence of measles, make pre-vaccination dynamics an especially good testbed for understanding the epidemic clockwork.
In the pre-vaccination period, measles persisted for a long time in large cities because sufficient transmissions in many settings maintained a chain of transmission in interepidemic troughs. Epidemics faded out in populations below the critical community size of 250-300K. In contrast, as vaccinations reduce measles incidence, epidemics are becoming more irregular, with frequent local stochastic extinctions of infection. High levels of immunization above a ‘herd immunity’ threshold of 90-95% can interrupt measles transmission even in large populations (see Figure 2).
Given moderate seasonality of transmission (as in the pre-vaccination United Kingdom data), measles dynamics in large cities corresponded quite well to a relatively predictable limit cycle. Seasonality in the U.K. and the U.S. was largely due to enhanced transmission when children aggregated in school. Typically, outbreaks were high before Christmas and low during the summer holidays.
But nonlinear dynamics can interact with stronger seasonal forcing to create a more complex picture. For example, measles is still a major threat in Niamey, the capital of Niger, where highly seasonal transmission causes much higher amplitude forcing than in the U.K. This has the same effect as powerfully pushing a pendulum, and leads to much more irregular, apparently chaotic dynamics.
Even a subtle increase in forcing can drive dynamic complexities. For instance, summer vacations are slightly longer in the U.S. than in the U.K, and these small increases drove remarkably irregular dynamics during the 1920s and 30s. While London saw biennial and annual cycles, Spokane and Los Angeles experienced three- and four-year cycles, as seen in Figure 3 .
Figure 3. Epidemic periodicity was longer and more irregular in U.S. cities compared to London. Image Credit: .
“Despite these complexities, we should always try to predict epidemics or understand the limits of predictability where we can’t,” Grenfell says. “There are great lessons to be learned from predictions made in weather forecasting, which is arguably a more complex system. At the population level, there is evidence for emerging simplicity in epidemiological dynamics. At reasonable scales, the SIR model works well, exposing the limits of predictability. Predict when you can, and understand that problems arise when reaching the limits.”
Grenfell believes that in the future a much broader range of expertise will be used to clarify how dynamics of human behaviors, climate change, and pathogen evolution affect epidemic dynamics.
This article is based on an invited lecture by Bryan Grenfell at the SIAM Annual Meeting, which was held in Boston this July.
Acknowledgments: SIAM News thanks Tom Kepler (Boston University) for his help with editing this article.
 Dalziel, B.D., Bjørnstad, O.N., Van Panhuis, W.G., Burke, D.S., Metcalf, C.J.E., & Grenfell, B.T. (2016). Persistent Chaos of Measles Epidemics in the Prevaccination United States Caused by a Small Change in Seasonal Transmission Patterns. PLOS Computational Biology. http://dx.doi.org/10.1371/journal.pcbi.1004655
 Grenfell, B.T., Bjørnstad, O.N., & Kappey, J. (2001). Travelling waves and spatial hierarchies in measles epidemics. Nature, 414, 716-723.
Grenfell, B.T., Bjørnstad, O.N., & Finkenstadt, B.F. (2002). Dynamics of Measles Epidemics: Scaling Noise, Determinism, and Predictability with the TSIR Model. Ecological Monographs, 72(2), 185-202.