The rapid, geographic spread of infectious disease presents a continuous threat to daily life. Because many serious infections and vector-borne diseases—notably, malaria and dengue fever—lack effective vaccines, disease control measures instead aim to directly reduce transmission, thus minimizing the need for treatment. For example, mosquito management (through insecticide sprays) is the main control measure for the mosquito-borne dengue virus, as it reduces the number of adult insects able to spread the disease.
In a minisymposium about infectious diseases at the 2017 SIAM Conference on Applications of Dynamical Systems, Alun Lloyd of North Carolina State University modeled the surprisingly perverse consequences of disease control measures for an endemic infection. His research pertains to standardized infection ratio (SIR)-type infections, in which patients are permanently immune after recovery. Lloyd’s model incorporates the basic reproductive number (for the susceptible population), effective reproductive number (for the partially-susceptible population), transmission parameters, and recovery rate. “An absolutely key point to emphasize is that if the system has an endemic equilibrium—a positive steady state—the effective reproductive number at that state is 1,” he said.
Mosquito management (insecticide spraying or bug nets, for example) is the main control measure for malaria and dengue, both of which are transmitted by mosquitos. Many vector-borne diseases lack effective vaccines. Image courtesy of Wikimedia Commons.
Lloyd began by outlining various types of disease control measures, including vaccines. “Vaccinations permanently makes those vaccinees immune, as opposed to a drug treatment,” he said. Drug treatments either reduce susceptibly or eliminate it temporarily, but lack the effectiveness of most vaccines. Lloyd then spoke of the so-called “honeymoon period,” based on the research of Angela McLean. “If you’re at an endemic equilibrium, even modestly effective controls can have a major transient impact,” he said. “This is a consequence of having the effective reproductive number as 1 at the endemic equilibrium.” The resulting sense of false security is the honeymoon effect.
However, susceptibles begin to build up in the body as transmission declines during this period. “You’re storing up some sort of trouble for yourself down the line,” Lloyd said. “A large outbreak may often be seen after the honeymoon. And some control measures can cause transient oscillations whose peaks exceed the pre-control endemic level.” Essentially, if one chooses poor control measures, a damaging post-control outbreak can occur. The incidence of large peaks is particularly troubling in certain infections, such as dengue fever. “Increasing that maximum peak can be a big problem,” Lloyd said. “It overwhelms the local health systems.”
Additionally, epidemiological systems are often subject to seasonal forcing, caused by climatic conditions or school-based transmissions, for example. In these seasonal systems, the introduction of vaccinations can cause large spikes, during which the instantaneous prevalence of infection surpasses pre-control levels. Additional troublesome control measures include vector control, bed nets, and transgenic mosquitos. “An appropriate measure of the impact of control involves looking at the total number of cases that occur from the start of control to a given point in time,” Lloyd said. “These control measures do not generate human immunity, so their use leads to the build-up of susceptible people.” Because they do not create immunity, the controls can break down, for example, if a local population decides to stop spraying insecticide.
Alun Lloyd presented a standard SIR model of the so-called divorce effect. A post-control outbreak drives the relative cumulative incidence above 1, negating the benefits of the transient control period. Image credit: Alun Lloyd.
A previous model by Kenichi Okamoto yields a surprising outcome — an instance in which the relative cumulative incidence (RCI) exceeds 1. This reveals a troubling epidemiological result: time windows can exist during which the total number of disease cases is more than the projected number of cases if no interventive measures were taken. “The post-control outbreak is so large that we lose all benefits and more from the control period,” Lloyd said. He then expanded the Okamoto model into a standard SIR model. Although the RCI falls initially, the modeled post-control outbreak is so large that it drives the RCI above 1; the timing of this outbreak depends on the accumulation of susceptibles. This is called the divorce effect, and it pertains to transient control measures and endemic infections that do not engender the community. “It’s like the honeymoon effect but worse, and it comes from splitting up from your control measure, Lloyd said. “We end up doing worse than if we had done nothing.”
Next, Lloyd explored the consequences of seasonality in a seasonally-forced SIR model. He incorporated weak seasonality, in the form of passive annual oscillations, to avoid the dynamical complexities associated with strong seasonality. The seasonally-forced model yields larger peak RCI values than the standard SIR model. “We see the effects even more strongly here,” he said. “The dynamics and magnitude of the divorce effect depend on timing.”
The results of Lloyd’s model have valuable implications for the control of dengue and other mosquito-borne illnesses in clinical trials. Because weak control can significantly reduce the incidence of an endemic infection, researchers must take great care when analyzing the results of short-term trials. A post-control outbreak could occur several years after the implementation and subsequent termination of a control measure, long after researchers have stopped observing the effects. “Vector control trials are more complicated than a standard clinical trial because people move in and out of the intervention area,” Lloyd said. Thus, it is essential to monitor the dynamics of susceptibles over a lengthy period of time before drawing any conclusions.
Ultimately, Lloyd’s SIR-based approach indicates that certain transient control measures have counterintuitive consequences that could actually exceed the number of disease cases that would have occurred had no intervention been employed. He is currently conducting an empirical investigation of the divorce effect in Zika virus and dengue fever, and searching for mitigation methods to increase the likelihood of transient control success.
|| Lina Sorg is the associate editor of SIAM News.