On real-valued SDE and nonnegative-valued SDE population models with demographic variability.

Population dynamics with demographic variability is frequently studied using discrete random variables with continuous-time Markov chain (CTMC) models. An approximation of a CTMC model using continuous random variables can be derived in a straightforward manner by applying standard methods based on the reaction rates in the CTMC model. This leads to a system of Itô stochastic differential equations (SDEs) which generally have the form [Formula: see text] where [Formula: see text] is the population vector of random variables, [Formula: see text] is the drift vector, and G is the diffusion matrix. In some problems, the derived SDE model may not have real-valued or nonnegative solutions for all time. For such problems, the SDE model may be declared infeasible. In this investigation, new systems of SDEs are derived with real-valued solutions and with nonnegative solutions. To derive real-valued SDE models, reaction rates are assumed to be nonnegative for all time with negative reaction rates assigned probability zero. This biologically realistic assumption leads to the derivation of real-valued SDE population models. However, small but negative values may still arise for a real-valued SDE model. This is due to the magnitudes of certain problem-dependent diffusion coefficients when population sizes are near zero. A slight modification of the diffusion coefficients when population sizes are near zero ensures that a real-valued SDE model has a nonnegative solution, yet maintains the integrity of the SDE model when sizes are not near zero. Several population dynamic problems are examined to illustrate the methodology.

Separate seasons of infection and reproduction can lead to multi-year population cycles.

Many host-pathogen systems are characterized by a temporal order of disease transmission and host reproduction. For example, this can be due to pathogens infecting certain life cycle stages of insect hosts; transmission occurring during the aggregation of migratory birds; or plant diseases spreading between planting seasons. We develop a simple discrete-time epidemic model with density-dependent transmission and disease affecting host fecundity and survival. The model shows sustained multi-annual cycles in host population abundance and disease prevalence, both in the presence and absence of density dependence in host reproduction, for large horizontal transmissibility, imperfect vertical transmission, high virulence, and high reproductive capability. The multi-annual cycles emerge as invariant curves in a Neimark-Sacker bifurcation. They are caused by a carry-over effect, because the reproductive fitness of an individual can be reduced by virulent effects due to infection in an earlier season. As the infection process is density-dependent but shows an effect only in a later season, this produces delayed density dependence typical for second-order oscillations. The temporal separation between the infection and reproduction season is crucial in driving the cycles; if these processes occur simultaneously as in differential equation models, there are no sustained oscillations. Our model highlights the destabilizing effects of inter-seasonal feedbacks and is one of the simplest epidemic models that can generate population cycles.

Optimal control of a vectored plant disease model for a crop with continuous replanting.

Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato, and cassava. Plant-vector-virus models with continuous replanting are investigated in terms of the effects of selection of cuttings, roguing, and insecticide use on disease prevalence in plants. Previous models are extended to include two replanting strategies: frequencyreplanting and abundance-replanting. In frequency-replanting, replanting of infected cuttings depends on the selection frequency parameter ε, whereas in abundance-replanting, replanting depends on plant abundance via a selection rate parameter also denoted as ε. The two models are analysed and new thresholds for disease elimination are defined for each model. Parameter values for cassava, whiteflies, and African cassava mosaic virus serve as a case study. A numerical sensitivity analysis illustrates how the equilibrium densities of healthy and infected plants vary with parameter values. Optimal control theory is used to investigate the effects of roguing and insecticide use with a goal of maximizing the healthy plants that are harvested. Differences in the control strategies in the two models are seen for large values of ε. Also, the combined strategy of roguing and insecticide use performs better than a single control.

Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models.

Thresholds for disease extinction provide essential information for control, eradication or management of diseases. Through relations between branching process theory and the corresponding deterministic model, it is shown that the deterministic and stochastic thresholds are in agreement for discrete-time and continuous-time infectious disease models with multiple infectious groups. Branching process theory can be applied in conjunction with the deterministic model to give additional information about disease extinction. These relations are illustrated, analytically and numerically, in two settings, a general stage-structured model and a vector-host model applied to West Nile virus in mosquitoes and birds.

Mathematical Modeling of Viral Zoonoses in Wildlife.

Zoonoses are a worldwide public health concern, accounting for approximately 75% of human infectious diseases. In addition, zoonoses adversely affect agricultural production and wildlife. We review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas where further modeling efforts are needed.

Spatial patterns in a discrete-time SIS patch model.

How do spatial heterogeneity, habitat connectivity, and different movement rates among subpopulations combine to influence the observed spatial patterns of an infectious disease? To find out, we formulated and analyzed a discrete-time SIS patch model. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. In low-risk habitats, the disease persists only when the mobility of infected individuals lies below some threshold value, but for high-risk habitats, the disease always persists. When the disease does persist, then there exists an endemic equilibrium (EE) which is unique and positive everywhere. This EE tends to a spatially inhomogeneous disease-free equilibrium (DFE) as the mobility of susceptible individuals tends to zero. The limiting DFE is nonempty on all low-risk patches and it is empty on at least one high-risk patch. Sufficient conditions for the limiting DFE to be empty on other high-risk patches are given in terms of disease transmission and recovery rates, habitat connectivity, and the infected movement rate. These conditions are also illustrated using numerical examples.

Analysis of climatic and geographic factors affecting the presence of chytridiomycosis in Australia.

Chytridiomycosis is an emerging fungal disease that has been implicated in the global decline of amphibian populations. Identifying climatic and geographic factors associated with its presence may be useful in control and prevention measures. Factors such as high altitude, cool temperature, and wet climate have been associated with chytridiomycosis outbreaks. Although some of these factors have been studied in a laboratory setting, there have been few studies in a natural setting. In this investigation, the relationship between altitude, average summer maximum temperature, or the amount of rainfall and the presence or absence of chytridiomycosis are statistically tested using data from 56 study sites in Australia. Currently, in Australia, 48 native species of wild amphibians have been found infected with chytridiomycosis. The 56 sites in the present study, extending along approximately 50% of the coastline of Australia, have been identified as either a chytrid site, where > or = 1 species are infected with chytridiomycosis, or a no-decline site, where none of the species present at the site are experiencing a decline or are known to be infected. The odds-ratio test and two-proportions test applied to this data indicate that the presence of chytridiomycosis in Australia is significantly related to temperature. In particular, the presence of chytridiomycosis is more likely at sites where the average summer maximum temperature is < 30 degrees C. The results of the analyses do not indicate a significant relationship between the presence of chytridiomycosis and altitude or rainfall.