Mathematicians Model Chagas Disease

Oakland University offers many research opportunities for undergraduates, some of which are featured on the CBR website. For several summers, the Department of Mathematics and Statistics has administered a Research Experience for Undergraduates (REU) program for Computational and Numerical Statistics and Mathematics (CaNSaM), supported by the National Science Foundation. In the summer of 2009, four undergraduates—Lindsey Kingsland, Andrea Thatcher, Matthew Toeniskoetter, and Benjamin Wood—worked with Profs Anna Spagnuolo and Meir Shillor on a project about Chagas disease.

Chagas disease is a tropical illness caused by a protozoan parasite and transmitted by insects. The disease is typically found in Central and South America. There is no vaccine against Chagas disease, and the best way to prevent the disease is by using sprays to control insect populations. Spagnuolo, Shillor, and their collaborator Gabrielle Stryker described A Model for Chagas Disease With Controlled Spraying in a 2011 paper in the Journal of Biological Dynamics (Volume 5, Pages 299-317). Their abstract states that

“Chagas disease is a vector-borne parasitic disease that infects mammals, including humans, through much of Latin America. This work presents a mathematical model for the dynamics of domestic transmission in the form of four coupled nonlinear differential equations. The four equations model the number of domiciliary vectors, the number of infected domiciliary vectors, the number of infected humans, and the number of infected domestic animals. The main interest of this work lies in its study of the effects of insecticide spraying and of the recovery of vector populations with cessation of spraying. A novel aspect in the model is that yearly spraying, which is currently used to prevent transmission, is taken into account. The model's predictions for a representative village are discussed. In particular, the model predicts that if pesticide use is discontinued, the vector population and the disease can return to their pre-spraying levels in approximately 5–8 years."

The work of the REU students is an extension of this model. Recently, this team published their results, again in the Journal of Biological Dynamics. The abstract of their paper, A Logistic Delay Differential Equation Model for Chagas Disease with Interrupted Spraying Schedules (Volume 6, Page 377-394, 2012), reports that

“This work studies a mathematical model for the dynamics of Chagas disease, a parasitic disease that affects humans and domestic mammals throughout rural areas in Central and South America. It presents a modified version of the model found in Spagnuolo et al. [A model for Chagas disease with controlled spraying, J. Biol. Dyn. 5 (2011), pp. 299-317] with a delayed logistic growth term, which captures an overshoot, beyond the vector carrying capacity, in the total vector population when the blood meal supply is large. It studies the steady states of the system in the case of constant coefficients without spraying, and the analysis shows that for given-averaged parameters, the endemic equilibrium is stable and attracting. The numerical simulations of the model dynamics with time-dependent coefficients are shown when interruptions in the annual insecticide spraying cycles are taken into account. Simulations show that when there are spraying schedule interruptions, spraying may become ineffective when the blood meal supply is large. “

Matthew Toeniskoetter was an Oakland University student who won the Louis R. Bragg Graduating Senior Award from the Department of Mathematics and Statistics. He is now a graduate student in the Mathematics Department at Purdue University. Andrea Thatcher is currently a graduate student in Applied Mathematics at Arizona State University. Benjamin Wood graduated from Michigan State University’s Professional Master of Science Program in Industrial Mathematics in May 2012. He is currently employed at Global Business Services Emerging Leaders Associate at Nielsen in Tampa, Florida. Lindsey Kingsland won the Concordia University - Irvine President’s Academic Showcase award ($1,000) in 2012 for best project, based on the Chagas disease work from the REU program. She used the money to go to Haiti and she is currently a grade 6 – 12 Science and Mathematics teacher at Liberty Academy School in Haiti.