# Research

## Mathematical Epidemiology

### Risk Assessment for Emerging Infectious Diseases

Dr. Toth is currently working on a collaborative project for a research center that will contain laboratories authorized to study the most dangerous emerging infectious diseases. His portion of the project involves mathematical modeling and simulation of the potential spread of various pathogens in different communities, should they escape from the facility after an accident or malevolent action. The results of these modeling efforts will be used to assess the risk that the laboratories will pose to individuals living in the surround community. Dr. Toth's collaborators on this project include Adi Gundlapalli and Matthew Samore from the University of Utah School of Medicine, engineers from the firm Tetra Tech, and advisors from the NIH. Dr. Toth is also overseeing several graduate and undergraduate students in mathematics and biology working on aspects of this project.

### Seasonality of Infectious Disease Outbreaks

This project is inspired by several years of data collected by researchers at the University Hospital on cases of Respiratory Syncytial Virus (RSV), which causes respiratory tract infections that can be especially severe in very young children. Like other infectious diseases, RSV case counts exhibit extreme seasonality, with the vast majority of cases occurring to the same months each year during the cold season. Interestingly the Utah data also exhibits a clear ``every-other-year" effect in which larger and smaller outbreaks occur in alternate years. Dr. Toth is analyzing periodically forced "SIRS" models that could help explain why biennial cycles appear in RSV outbreaks. Techniques from perturbations and bifurcation theory can provide insights into the parametric resonance that may be causing the yearly oscillations in the hospital data. The relationship between parameters specific to RSV and possible environmental effects specific to Utah could put the system in a parameter regime where a biennial cycle is stable for the differential equation model.

## Population Dynamics

### Bird Lice

This project is a collaboration with members of Professor Dale Clayton's lab in Utah's Biology department involving the population dynamics of ectoparasites. In particular, Dr. Toth is working with scientists who performed long-term laboratory population experiments on two different species of lice living on rock pigeons. They compared across experimental groups to assess the nature of competition between the species and the effect of the birds' preening behavior on competition. Dr. Toth has been developing mathematical models in an attempt to identify the key mechanisms causing the population trends seen over many generations of lice on individual birds in the data. The modeling work to date has led to hypotheses that long-term oscillations in single-species lice populations (on birds whose preening has been impaired) can be explained by one or more of the following: ambient humidity, feather molt of the pigeons, and death of lice eggs caused by adult lice feeding on feathers to which unhatched eggs are glued. In addition, explanation of different trends seen in populations of each of the two lice species could shed light on the ecological and evolutionary history of the competition between related species and host--parasite co-evolution.

### Chemostat Modeling

Dr. Toth's Ph.D. thesis at University of Washington was an analysis of a series of age-structured models for populations living in a chemostat, a laboratory apparatus that provides a controlled, resource-limited environment for microorganisms. Mathematical models of populations in a chemostat are important to theoretical ecologists because they can provide predictions with broad ecological ramifications that are testable in a laboratory. His work shed light on how populations can be affected by the interacting effects of age structure, density dependence, predator--prey dynamics, and environmental periodicity. The results have been published in three papers.

## Behavioral Ecology

### Optimal Foraging

Dr. Toth's interest in mathematical biology stems from an elective course in behavioral ecology he took as an undergraduate majoring in engineering at Princeton University. He was fascinated to discover that biologist were using the same quantitative tools to study the evolutional of animal behavior that engineers use to study operations research. His undergraduate senior thesis was a theoretical analysis of an optimal foraging problem. He used stochastic dynamic programming to find the optimal patch-choice strategy for an animal choosing how long to stay at each patch of food in different environmental scenarios, comparing the simulated performance of the optimal strategy to the performance of suboptimal "rules of thumb" that animals would be more likely to use, given the limits of their information and decision-making ability.

### Life-History Strategy

In graduate school, Dr. Toth worked on another optimization problem in behavioral ecology for a summer project with Dr. Yoh Iwasa at Kyushu University in Japan. They considered an insect that can produce ``diapaused" eggs as winter approaches, so that the egg delays hatching and remains inert in order to better survive the winter. He analyzed a model for an age-structured population with a periodically varying environment and derived the optimal timing of diapause that balances the costs of delaying reproduction with the risk of producing offspring that don't survive the winter.