Suzanne Lenhart

Modeling optimal intervention strategies for cholera.

While cholera has been a recognized disease for two centuries, there is no strategy for its effective control. We formulate a mathematical model to include essential components such as a hyperinfectious, short-lived bacterial state, a separate class for mild human infections, and waning disease immunity. A new result quantifies contributions to the basic reproductive number from multiple infectious classes. Using optimal control theory, parameter sensitivity analysis, and numerical simulations, a cost-effective balance of multiple intervention methods is compared for two endemic populations. Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods.

Climate, environmental and socio-economic change: weighing up the balance in vector-borne disease transmission

Arguably one of the most important effects of climate change is the potential impact on human health. While this is likely to take many forms, the implications for future transmission of vector-borne diseases (VBDs), given their ongoing contribution to global disease burden, are both extremely important and highly uncertain. In part, this is owing not only to data limitations and methodological challenges when integrating climate-driven VBD models and climate change projections, but also, perhaps most crucially, to the multitude of epidemiological, ecological and socio-economic factors that drive VBD transmission, and this complexity has generated considerable debate over the past 10–15 years. In this review, we seek to elucidate current knowledge around this topic, identify key themes and uncertainties, evaluate ongoing challenges and open research questions and, crucially, offer some solutions for the field. Although many of these challenges are ubiquitous across multiple VBDs, more specific issues also arise in different vector–pathogen systems.

Backward bifurcation and optimal control in transmission dynamics of West Nile virus

The paper considers a deterministic model for the transmission dynamics of West Nile virus (WNV) in the mosquito-bird-human zoonotic cycle. The model, which incorporates density-dependent contact rates between the mosquito population and the hosts (birds and humans), is rigorously analyzed using dynamical systems techniques and theories. These analyses reveal the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity) in WNV transmission dynamics. The epidemiological consequence of backward bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for WNV elimination from the population. It is further shown that the model with constant contact rates can also exhibit this phenomenon if the WNV-induced mortality in the avian population is high enough. The model is extended to assess the impact of some anti-WNV control measures, by re-formulating the model as an optimal control problem with density-dependent demographic parameters. This entails the use of two control functions, one for mosquito-reduction strategies and the other for personal (human) protection, and redefining the demographic parameters as density-dependent rates. Appropriate optimal control methods are used to characterize the optimal levels of the two controls. Numerical simulations of the optimal control problem, using a set of reasonable parameter values, suggest that mosquito reduction controls should be emphasized ahead of personal protection measures.

Global stability of a biological model with time delay

This paper gives necessary and sufficient conditions for global stability of certain logistic delay differential equations for all values of the delay. Biological models frequently lead to delay differential equations and to questions concerning the stability of equilibrium solutions of such models. The monographs by Cushing (6) and MacDonald (16) discuss a number of examples of such models from population dynamics, ecology, and physiology. Much of the work with such models has focused on delay equations which are reducible, through a change of variables, to systems of differential equations without delays (3,4,8,14,18,19). Another line of research is concerned with conditions under which linear retarded functional differential equations of the form L x(t) = ax(t) + £ btx(t - r,), r, > 0, /=i are asymptotically stable for all values of the delay (1,3,5,7,9,11,12,14). We propose to combine these lines of research and examine global stability of the widely used logistic model of population dynamics,

Optimal control of boundary habitat hostility for interacting species

We consider boundary control for a parabolic system describing the evolution of two interacting species in a bounded habitat. The control models the hostility of the boundary environment to the maintenance of the species. The objective functional represents the balance between the ecological benefit (modelled by the size of the two populations) and the economic cost of maintaining an ecologically favorable boundary environment (modelled by the boundary friendliness). The unique optimal control is characterized in terms of the solution of the optimality system, which consists of the state system coupled with an adjoint system. Copyright

APPLICATION OF DISTRIBUTED PARAMETER CONTROL MODEL IN WILDLIFE DAMAGE MANAGEMENT

A bioeconomic model for optimal control of wildlife damage by migratory small mammal populations is developed under the framework of a nonlinear distributed parameter control problem. The model first simulates the spatio-temporal dynamics of dispersal population by parabolic diffusive Volterra-Lotka partial differential equation and then optimizes a criterion function of present value combined costs of wildlife damage and harvesting. The existence of a unique optimal solution for a finite time problem is proved. An iterative procedure for numerical solution of the Optimality System with parabolic equations of opposite orientations is developed. The theoretical model is applied to a real life problem using biological and economic data for beaver populations under certain simplistic assumptions.

Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method

We consider the inverse problem of determining the potential in the one-dimensional Schrodinger equation from dynamical boundary observations, which are the range values of the Neumann-to-Dirichlet map. Dynamical boundary data have not been used in the inverse problem for the Schrodinger equation, since the traditional Gelfand–Levitan–Marchenko approach reconstructs the potential from spectral or scattering data. Here we show that one can completely recover the spectral data from the dynamical boundary data. The construction of the spectral data uses new results on exact and spectral controllability for the Schrodinger equation, which we obtain by using the properties of exponential Riesz bases (nonharmonic Fourier series). From the spectral data, we solve the inverse problem using the boundary control method, which—unlike other identification methods based on control and optimization—is consistently linear and, in principle, independent of dimensionality.

A System of Nonlinear Partial Differential Equations Arising in the Optimal Control of Stochastic Systems with Switching Costs

The problem of optimal switching control of a diffusion process with costly switchings leads to a system of fully nonlinear elliptic partial differential equations with implicit obstacles. We obtain results on the existence, uniqueness and regularity of the solution of this system.

Controlling transboundary wildlife damage: modeling under alternative management scenarios

Abstract The migratory nature of nuisance wildlife populations creates a special management problem by imposing a negative diffusion externality on landowners undertaking control efforts. This paper reviews three cost-minimizing wildlife-control models, each internalizing the diffusion externality under different management scenarios, namely, unilateral management, bilateral management, and centralized management. The three management scenarios lead to different optimal behaviors. Property owners exerting unilateral control must leave some wildlife untrapped to generate sufficient population pressure against the flow of continual immigration from neighboring populations. Analysis of the bilateral model indicates that noncooperating neighboring landowners having varying pay-off functions will end up with leaving all wildlife untraped in their parcels. Under the centralized management scenario, landowners find it most profitable to collectively delegate the control responsibility of an entire watershed to a single manager.

Integro-differential equations associated with optimal stopping time of a piecewise-deterministic process

This paper concerns the optimal stopping time problem for a piecewise deterministic process. The process has deterministic dynamics between random jumps. The as¬sociated dynamic programming equation is a variational inequality with integral and (first order) differential terms. Our main results are W4,00-existence results and probabilistic representations for the solutions of the optimal stopping time problem in bounded domains and in R. We also generalize these results to the case when the state space is “countable folds” of Euclidean space