Mathematical Model of Vaccine Noncompliance

Vaccine scares can prevent individuals from complying with a vaccination program. When compliance is high, the critical vaccination proportion is close to being met, and herd immunity occurs, bringing the disease incidence to extremely low levels. Thus, the risk to vaccinate may seem greater than the risk of contracting the disease, inciting vaccine noncompliance. A previous behavior-incidence ordinary differential equation model shows both social learning and feedback contributing to changes in vaccinating behavior, where social learning is the perceived risk of vaccinating and feedback repre- sents new cases of the disease. In our study, we compared several candidate models to more simply illustrate both vaccination coverage and incidence through social learn- ing and feedback. The behavior model uses logistic growth and exponential decay to describe the social learning aspect as well as different functional forms of the disease prevalence to represent feedback. Each candidate model was tested by fitting it to data from the pertussis vaccine scare in England and Wales in the 1970s. Our most parsimonious model shows a superior fit to the vaccine coverage curve during the scare

A Sensitivity Matrix Methodology for Inverse Problem Formulation

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate

An Agent-Based Modeling Approach to Determine Winter Survival Rates of American Robins and Eastern Bluebirds

American Robins (Turdus migratorius) and Eastern Bluebirds (Sialia sialis) are two species of migratory thrushes that breed in Northwest Indiana but historically are uncommon during the winter season. These trends have changed recently, and both species are seen more abundantly during the winter. Recently invaded non-native fruiting plants continue to provide nutrients for the birds throughout the winter and may contribute to the increased avian populations during that time. To measure the effect these food sources contribute to thrush wintering habits, we created an agent-based computer model to simulate the birds\u27 movement in Northwest Indiana along with their food consumption over the course of the winter season. The model incorporates availability of food sources, foraging and roosting behavior, bio-energetics, and starvation, with parameter values informed by the literature. We obtained simulated winter survival rates of the birds that could begin to explain the changes in the birds\u27 migratory patterns

Parameter Estimation and Uncertainty Quantication for an Epidemic Model

We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number (R0 )—an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of R0. Situations are highlighted in which this correlation allows R0 to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts

Investigating Anthropogenic Mammoth Extinction with Mathematical Models

One extinction hypothesis of the Columbian mammoth (Mammuthus columbi), called overkill, theorizes that early humans overhunted the animal. We employ two different approaches to test this hypothesis mathematically: analyze the stability of the equilibria of a 2D ordinary differential equations (ODE) system and develop a metapopulation differential equations model. The 2D ODE system is a modified predator-prey model that also includes migration. The metapopulation model is a spatial expansion of the first model on a rectangular grid. Using this metapopulation system, we model the migration of humans into North America and the response in the mammoth population. These approaches show evidence that human-mammoth interaction would have affected the extinction of the Columbian mammoth during the late Pleistocene

Multistrain Infections in Metapopulations

Viruses and bacteria responsible for infectious diseases often mutate and are carried between geographical regions. We consider a mathematical model which begins to account for these factors. We assume two disjoint populations that only occasionally co-mingle, and two strains of a disease present in these populations. Of interest are the equations describing the dynamics of this system, the conditions under which epidemics will occur, and the long term behavior of the system under various initial conditions. We find general conditions under which a state of disease-free equilibrium is stable. Additionally, we find existence of a biologically relevant equilibrium where two disease strains of unequal strength coexist in a two-population system and we demonstrate that it is likely unstable