Mathematical Model Optimizes Preemptive Cholera Vaccination in Cameroon

By Lina Sorg

Cholera is an infectious waterborne disease that occurs when the toxic aquatic bacterium Vibrio cholerae infiltrates the small intestine, usually through contaminated food and drinking water. The illness is rare in industrialized countries due to modern sewage treatment infrastructures and clean water supplies, but outbreaks are common and endemic—transpiring yearly if not seasonally—in third-world countries. The symptomatic diarrhea and vomiting typically manifest anywhere from 12 hours to five days after ingestion of the bacterium, and can lead to fatal dehydration and circulatory collapse if not treated promptly. In the last decade, major outbreaks materialized in India, Zimbabwe, Ghana, Sierra Leone, and Haiti, among other locations. According to the Centers for Disease Control, sub-Saharan Africa hosts the highest concentration of cases in the 21st century.

Several methods of intervention to prevent cholera outbreaks currently exist, including ongoing efforts to improve sanitation and water security in high-risk areas. Vaccination campaigns are also increasingly common in both epidemic and endemic situations. For instance, the World Health Organization recommends oral cholera vaccines (OCVs) as part of a comprehensive approach to cholera prevention and outbreak response. OCVs offer a more immediate type of short-term disease prevention, and an OCV stockpile was created in 2013 with the goal of directing preemptive vaccines to regions with the greatest need. They have proven effective and have thus been distributed in the wake of large-scale natural disasters and in endemic regions, including Haiti, Uganda, Indonesia, and Bangladesh. At the 2018 SIAM Conference on Mathematics of Planet Earth, which took place earlier this month in Philadelphia, Pa., Michael Kelly of Transylvania University presented a mathematical model that considered demographic pathogen movement when optimizing OCV placement in Cameroon.

The growing stockpile and continued success of vaccination efforts have produced new opportunities for preemptive vaccine campaigns, which directly intersect with human mobility. “Human mobility has long been responsible for the geographical spread of many emergent infectious diseases,” Kelly said. “My interest is in how human and environmental mobility affects disease spread.” He sought to formulate a new mathematical approach using a parametrized control problem that incorporates demographic covariate data and network structure; the goal is to inform discrete community networks using demographic information. To emphasize the significance of demographic breakdown, Kelly presented a toy patch model wherein each patch along a river had its own unique dynamics. “Can we use information about a region to decide where to target vaccination campaigns prior to an actual outbreak’s occurrence?” he asked. “A lot of questions come to mind here, and a lot of things will be important.”

While exploring possible sites for targeted OCV campaigns, Kelly prioritized high-risk areas, allocated available resources, balanced cost with vaccination impact, and accounted for natural uncertainty in the prediction of outbreak locations. This yielded a multi-transmission SIWR model — an extension of the traditional three-compartment susceptible-infectious-recovered model; the “W” compartment measures pathogen concentration in water supplies. Kelly also considered the role of movement and incorporated movement matrices for compartments. “When I talk about movement I’m thinking about how populations are connected through their water sources,” he said.

Kelly’s model utilizes a basic reproduction number (\(R_0\)) as a predictor of outbreak risk. He wants to find the vaccination distribution that will drive R0 below 1 while minimizing the associated vaccination costs. This work is based on a 2015 study where researchers used a lower-order approximation to the \(R_0\) and approximated via a Laurent series expansion. “Each community determined by their network structure has a network risk associated with it,” Kelly said. “The more connected you are, the higher your network risk will be.” To preserve the network structure and community characteristics, he simplifies the next-generation matrix approach to a basic calculation. The resulting preemptive vaccination equation includes expected past transmissibility.

Kelly then applied his model to a case study of cholera in Cameroon, a West African country with approximately 19 million people. The first recorded instance of cholera in Cameroon occurred in 1971. The disease has since become endemic, and 2010 saw the largest outbreak to date in terms of duration, range, and number of reported cases. Despite discussions of risk factors for cholera outbreaks in the country, researchers have never systematically evaluated the infection’s prevalence in Cameroon.

Kelly accessed data (from a 2014 demographic and health survey program) pertaining to the following factors: socioeconomic status, access to clean water, education, sanitation, family/household size, and number of children. He broke down the covariates demographically, identifying areas of farmland, regions with higher education levels, districts with better sanitation, etc. For instance, Kelly determined that the capital city of Yaoundé has cleaner water and higher education levels than many rural cities. He also realized that many cholera cases were concentrated in the North, and thus isolated that part of the country during regression analysis. “The ability to target high-risk populations is critically important in maximizing impact of control and prevention,” Kelly said.

Rather than examining the country as a whole, Kelly built a network that began with sub-regions and expanded outwards when conducting regression analysis, He commenced with the North, which hosted a large number of cases during the 2010 outbreak. This area has weak access to health facilities and experiences a lot of migrant activity. “There’s a lot of movement around that region from surrounding countries, especially seasonally for the farming communities,” Kelly said. He then expanded regression analysis to all regions except the Southwest, due to strong evidence that environmental factors contribute to pathogen survival in that area’s aquatic reservoirs.

When he isolated the North, Kelly found that sanitation seemed to play a significant role in cholera outbreaks. “The better the sanitation, the fewer cases we saw,” he said. But as he expanded his analysis down and into the center of Cameroon, outbreak seemed to be connected to education, family size, and number of children in a household, in addition to only sanitation.

After identifying the strongest indicators for cholera risk, Kelly sought to link that information with his initial model. He connected the covariates to community characteristics, used the results from regression analysis to inform the community of disease transmissibilities, and assumed that final outbreak sizes follow a similar distribution as attack rates. To obtain numerical results about OCV concentration and distribution, he ran a control problem—based on sanitation scores—in the North and extended this to the remaining covariates, like family size and level of education.

To summarize, Kelly approximated a challenging real-world vaccination problem with existing techniques and solved that problem via standard optimization methods. The resulting formulations are tractable and offer insight to optimal preemptive vaccine strategies and OCV placement while acknowledging the natural uncertainty that automatically exist in network communities. While Kelly is pleased with his findings, he acknowledged that there is more to be done. “We’re trying to extend this to larger networks incorporating environmental factors,” he said.

Lina Sorg is the associate editor of SIAM News.