Description
Lyme disease is the most prevalent tick-borne disease in the United States, which humans acquire from an infected blacklegged tick (Ixodes scapularis). Though Lyme disease is not generally considered to be a deadly affliction, untreated cases often result in chronic joint pain and other crippling symptoms. Early studies of Lyme disease focused on how environmental factors, such as climate and ecosystem type, aided in its geographical spread. One key factor not considered by previous research is the tick’s host preference in the presence of multiple hosts. Our mathematical model is a continuous dynamical system of ordinary differential equations (ODEs) that models the interactions between the primary vectors involved: blacklegged ticks (I. scapularis), white-footed mice (Peromyscus leucopus), and white-tailed deer (Odocoileus virginianus), and includes different stages in tick development. Parameters are estimated from numerous studies and the endemic levels in field studies. Based on our model, we also calculate the basic reproduction number, R0, a threshold value that designates whether a disease exists or dies out. Subsequent extensions of the model consider seasonal and migratory effects on Lyme disease spread. The seasonal extension of our base model incorporates time-varying parameters, including infection rates and death rates of specific vectors. A sensitivity analysis of parameters, specifically the time-varying infection rates, is performed to see how the length of a tick’s peak feeding period affects the long-term dynamics of the system. The results of this analysis suggest that a longer tick peak feeding period results in a higher infection prevalence. Lastly, the base model is extended to account for the migration of deer between two neighboring counties, where one is at an endemic steady state and the other is at a disease-free state. The deer migration contributed to local infectiousness in the disease-free county, which eventually reached the endemic steady state after a long period of time. A sensitivity analysis of the migration parameter demonstrated that increasing migration rates can result in increased infectivity in neighboring counties over a long period of time.
