Ebola is a rare, life-threatening virus that manifests in humans and primates. Infected individuals have about a 50 percent of survival, with many patients succumbing to severe bleeding, organ failure, and low blood pressure from extreme fluid loss. The disease is spread via direct contact with body fluids, and blood is the most common contaminator. Ebola’s incubation period is between two and 21 days. Symptoms include sudden muscle pain, fever, fatigue, headache, and sore throat, followed by rash, vomiting, diarrhea, impaired liver and kidney function, and possible internal/external bleeding.
The most widespread and deadly outbreak of Ebola took place in West Africa from late 2013 to 2016. The epidemic resulted in significant casualties and severe socioeconomic disruption as researchers and medics worked to contain the virus. During a minisymposium at the 2018 SIAM Annual Meeting, which took place in Portland, Ore., last week, Kyle Gustafson of the U.S. Navy used parsimonious model selection to model the spread of Ebola in Sierra Leone. “The outbreak really started at the border of Sierra Leone and Guinea, jumped into Sierra Leone, and affected it in the worst sense,” he said. “It expanded all the way from the east side of the country to the west.”
During the epidemic, researchers sequenced over 1,000 genomes and made them available to track Ebola’s mobility.
Researchers frequently employ parsimonious models to better understand human mobility, which is notoriously difficult to study. “They think about describing the fact that a large population center might attract and disperse large numbers of people,” Gustafson said. Gravity analogies are effective tools for monitoring human mobility. Lévy flights—random walks comprising step lengths with heavy-tailed probability distributions—also model human mobility and lend themselves to large jumps, like airplane travel or avoidance of disease outbreaks. For example, a 2006 Lévy flight experiment/currency tracking project called “Where’s George?” traced the serial numbers of dollar bills as they became displaced with use.
Although the Sierra Leone outbreak began from just one or two events and remained relatively contained within the country thanks to border security, cases continually spread from the capital city of Freetown to the surrounding providences and districts. Gustafson used a reaction-diffusion model to investigate Ebola’s movement within the closed system. During the epidemic, researchers sequenced over 1,000 genomes and made them available to track Ebola’s mobility. The data experienced a huge spike when the disease reached Freetown before medics brought it under control. Gustafson expanded an existing gravity model to analyze this data. “We studied the probabilities in this gravity model and the Lévy model,” he said. “You can connect these disease cases by their transmissions by seeing how much the genome has changed and the virus has evolved.”
Gustafson also utilized stochastic processes with dynamic human behavior; a partially-observed transmission network accounted for the rate of mutation. “The pairwise likelihood of transmission changes just by looking at the sequence of the genome,” he said. “This describes the best spatial pattern of how the outbreak is evolving in the country.” Gustafson then used Google Maps to compute driving times between the locations of individual Ebola cases and thus estimate the driving distances. However, he acknowledged that the estimate may not accurately account for things like road conditions and blockages.
The genomic data experienced a huge spike when the disease reached Freetown before medics brought it under control.
Next, Gustafson employed a heat map of populations based on the 2010 census and constructed a transmission network with the collected genomes. “There are huge differences in population densities in Sierra Leone, that’s what drives the gravity model,” Gustafson said. “And there were some cases that seemed to cover a lot of time; those were less likely to be transmission events.” While disease mobility models are calibrated to data and normalized by origin, gravity models concentrate probability in population centers. Gustafson used a gravity model to determine the probability of Ebola landing in a given population. “It’s much more likely to land in places with high populations,” he said. “That reflects to some extent the way that people move around.”
Gustafson segregated Ebola transmission in Sierra Leone into three chunks of 150 days each. “The early cases hadn’t migrated to Freetown,” he said. “After the case count in Freetown got really high, most of the connections were in Freetown. That’s when you would expect the gravity model to be the best model.” Because Freetown had the highest number of cases when the outbreak peaked, the gravity model was not preferable early on but ideal for Freetown — before losing its effectiveness with time. “We see a very clear transition from a strong gravity model to a weaker gravity model or even a power law preference,” Gustafson said. This is reflective of changes to human mobility patterns, as people were likely returning to small villages to attend funerals and avoid large, transmissible crowds. The maximum likelihood also shifted down during the three 150-day windows. “The drifting of the exponents is most likely an effect of human behavior change,” he said. “I would argue that the reason we see changes in the exponents is because people change their behaviors.”
Ultimately, Gustafson hopes to use this study to forecast the future movement of contagious diseases, contain outbreaks, and prevent the loss of life and socioeconomic devastation associated with viral illnesses like Ebola.
||Lina Sorg is the associate editor of SIAM News.