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We present a method for optimal path planning of human walking paths inmountainous terrain, using a control theoretic formulation and aHamilton-Jacobi-Bellman equation. Previous models for human navigation wereentirely deterministic, assuming perfect knowledge of the ambient elevationdata and human walking velocity as a function of local slope of the terrain.Our model includes a stochastic component which can account for uncertainty inthe problem, and thus includes a Hamilton-Jacobi-Bellman equation withviscosity. We discuss the model in the presence and absence of stochasticeffects, and suggest numerical methods for simulating the model. We discuss twodifferent notions of an optimal path when there is uncertainty in the problem.Finally, we compare the optimal paths suggested by the model at differentlevels of uncertainty, and observe that as the size of the uncertainty tends tozero (and thus the viscosity in the equation tends to zero), the optimal pathtends toward the deterministic optimal path.

Recent work by public health experts suggests that incorporating human behavior is crucial in faithfully modeling an epidemic. We present a reaction-diffusion partial differential equation SIR-type population model for an epidemic including behavioral concerns. In our model, the disease spreads via mass action, as is customary in compartmental models. However, drawing from social contagion theory, we assume that as the disease spreads and prevention measures are enacted, noncompliance with prevention measures also spreads throughout the population. We prove global existence of classical solutions of our model, and then perform R0-type analysis and determine asymptotic behavior of the model in different parameter regimes. Finally, we simulate the model and discuss the new facets which distinguish our model from basic SIR-type models. [ABSTRACT FROM AUTHOR]