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 via 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.
We propose novel estimators for the parameters of an exponential distribution and a normal distribution when the only known information is a sample of sample maxima; i.e., the known information consists of a sample of m values, each of which is the maximum of a sample of n independent random variables drawn from the underlying exponential or normal distribution. We analyze the accuracy and precision of the estimators using extreme value theory, as well as through simulations of the sampling distributions. For the exponential distribution, the estimator of the mean is unbiased and its variance decreases as either m or n increases. Likewise, for the normal distribution, we show that the estimator of the mean has negligible bias and the estimator of the variance is unbiased. While the variance of the estimators for the normal distribution decreases as m , the number of sample maxima, increases, the variance increases as n , the sample size over which the maximum is computed, increases. We apply our method to estimate the mean length of pollen tubes in the flowering plant Arabidopsis thaliana , where the known biological information fits our context of a sample of sample maxima.
Since 2006, the North American bat population has been in rapid decline due to white-nose syndrome (WNS), which is caused by an invasive fungus (Pseudogymnoascus destructans). The little brown bat (Myotis lucifugus) is the species most affected by this emerging disease. We consider how best to prevent local extinctions of this species using mathematical models. Development began in 2017 of a new vaccine for WNS and thus, we analyze the effects of implementing vaccination as a control measure. We create a Susceptible-Exposed-Infectious-Vaccinated hybrid ordinary differential equation and difference equation model informed by the phenology of little brown bats. We compare the effectiveness of annual, biennial, and one-time vaccination programs for multiple durations of immunity length. We also determine the optimal time to vaccinate, if vaccinating only once, as a function of average duration of immunity. Next, we perform a sensitivity analysis to determine the robustness of our results. Finally, we consider other possible control measures together with vaccination to determine the optimal control strategy. We find that if the vaccine offers lifelong immunity, then it will be the most effective control measure considered thus far.
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.
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