Rongsong Liu

Media/psychological impact on multiple outbreaks of emerging infectious diseases

We use a compartmental model to illustrate a possible mechanism for multiple outbreaks or even sustained periodic oscillations of emerging infectious diseases due to the psychological impact of the reported numbers of infectious and hospitalized individuals. This impact leads to the change of avoidance and contact patterns at both individual and community levels, and incorporating this impact using a simple nonlinear incidence function into the model shows qualitative differences of the transmission dynamics.

Plant Toxicity, Adaptive Herbivory, and Plant Community Dynamics

We model effects of interspecific plant competition, herbivory, and a plant’s toxic defenses against herbivores on vegetation dynamics. The model predicts that, when a generalist herbivore feeds in the absence of plant toxins, adaptive foraging generally increases the probability of coexistence of plant species populations, because the herbivore switches more of its effort to whichever plant species is more common and accessible. In contrast, toxin-determined selective herbivory can drive plant succession toward dominance by the more toxic species, as previously documented in boreal forests and prairies. When the toxin concentrations in different plant species are similar, but species have different toxins with nonadditive effects, herbivores tend to diversify foraging efforts to avoid high intakes of any one toxin. This diversification leads the herbivore to focus more feeding on the less common plant species. Thus, uncommon plants may experience depensatory mortality from herbivory, reducing local species diversity. The depensatory effect of herbivory may inhibit the invasion of other plant species that are more palatable or have different toxins. These predictions were tested and confirmed in the Alaskan boreal forest.

Spatiotemporal Distributions of Migratory Birds: Patchy Models with Delay ∗

We derive and analyze a mathematical model for the spatiotemporal distribution of a migratory bird species. The birds have specific sites for breeding and winter feeding, and usually several stopover sites along the migration route, and therefore a patch model is the natural choice. However, we also model the journeys of the birds along the flyways, and this is achieved using a continuous space model of reaction-advection type. In this way proper account is taken of flight times and in-flight mortalities which may vary from sector to sector, and this information is featured in the ordinary differential equations for the populations on the patches through the values of the time delays and the model coefficients. The seasonality of the phenomenon is accommodated by having periodic migration and birth rates. The central result of the paper is a very general theorem on the threshold dynamics, obtained using recent results on discrete monotone dynamical systems, for birth functions which are subhomogeneous. For such functions, depending on the spectral radius of a certain operator, either there is a globally attracting periodic solution, or the bird population becomes extinct. Evaluation of the spectral radius is difficult, so we also present, for the particular case of just one stopover site on the migration route, a verifiable sufficient condition for extinction or survival in the form of an attractive periodic solution. This threshold is illustrated numerically using data from the U.S. Geological Survey on the bar-headed goose and its migration to India from its main breeding sites around Lake Qinghai and Mongolia.

Traveling waves of the spread of avian influenza

This paper gives a proof for the existence and nonexistence of traveling wave solutions of a reaction-convection epidemic model for the spatial spread of H5N1 avian influenza involving a wide range of bird species and environmental contamination. The threshold condition for the existence of traveling waves coincides with the basic reproduction number exceeding one. The existence of wave solutions is obtained by constructing an invariant cone of initial functions defined on a large spatial domain, applying a fixed point theorem on this cone and then a limiting argument. The invariant cone is based on the information of initial growth pattern of the epidemic and the final size estimation during the entire course of the outbreak.

SOME VECTOR BORNE DISEASES WITH STRUCTURED HOST POPULATIONS: EXTINCTION AND SPATIAL SPREAD*

We derive from a structured population model a system of delay differential equations describing the interaction of five subpopulations, namely susceptible and infected adult and juvenile reservoirs and infected adult vectors, for a vector borne disease with particular reference to West Nile virus, and we also incorporate spatial movements by considering the analogue reaction‐diffusion systems with nonlocal delayed terms. Specific conditions for the disease eradication and sharp conditions for the local stability of the disease‐free equilibrium are obtained using comparison techniques coupled with the spectral theory of monotone linear semiflows. A formal calculation of the minimal wave speed for the traveling waves is given and compared with field observation data.

Dynamics of a plant–herbivore–predator system with plant-toxicity

A system of ordinary differential equations is considered that models the interactions of two plant species populations, an herbivore population, and a predator population. We use a toxin-determined functional response to describe the interactions between plant species and herbivores and use a Holling Type II functional response to model the interactions between herbivores and predators. In order to study how the predators impact the succession of vegetation, we derive invasion conditions under which a plant species can invade into an environment in which another plant species is co-existing with a herbivore population with or without a predator population. These conditions provide threshold quantities for several parameters that may play a key role in the dynamics of the system. Numerical simulations are conducted to reinforce the analytical results. This model can be applied to a boreal ecosystem trophic chain to examine the possible cascading effects of predator-control actions when plant species differ in their levels of toxic defense.

THE INTERACTION OF MIGRATORY BIRDS AND DOMESTIC POULTRY AND ITS ROLE IN SUSTAINING AVIAN INFLUENZA

We investigate the role of migratory birds in the spread of H5N1 avian influenza, focusing on the interaction of a migratory bird species with nonmigratory poultry. The model is of patch type and is derived with the aid of reaction-advection equations for the migratory birds in the air along the flyways. Poultry may reside at some or all of the four patches of the model, which consist of the breeding patch for the migratory birds, their winter feeding patch, and two stopover patches where birds rest and refuel on their migration. Outward and return migratory routes can be different. The equations for the migratory birds contain time delays which represent the flight times for migratory birds along particular sectors. Delays also appear in the model coefficients via quantities which represent flight survival probabilities for the various sectors. We establish results on positivity, boundedness, global asymptotic stability of the disease-free equilibrium, and the persistence of infection. We also discuss extensions of the model to include the seasonality of the migration phenomenon. Numerical simulations support the analytical findings; here we used data on H5N1 infected ducks in the Poyang Lake region of China.

Slowing the evolution of insecticide resistance in mosquitoes: a mathematical model

A big problem in malaria control is the rapidity with which mosquitoes can develop resistance to insecticides. The possibility of creating evolution-proof insecticides is therefore of considerable interest. Biologists have suggested that effective malaria control, with only weak selection for insecticide resistance, could be achieved if insecticides target only old mosquitoes that have already laid most of their eggs. The strategy aims to exploit the fact that most malarial mosquitoes do not live long enough to transmit the disease. We derive, analyse and compare two mathematical models, one for an insecticide that kills on exposure, and the other for an insecticide that targets only older mosquitoes. Both models predict that insecticide-resistant mosquitoes will become dominant over time but, very importantly, this occurs on a very much slower time scale when the insecticide only affects older mosquitoes. We present analytical results on linear and global stability of the non-trivial equilibrium in which only the resistant mosquito strain is present, together with a theorem comparing the rates of convergence for the two models. Numerical simulations show that the effect of targeting only old mosquitoes on the evolution of resistance is dramatic.

Modeling the dynamics of woody plant-herbivore interactions with age-dependent toxicity

In this paper we study the effects that woody plant chemical defenses may have on interactions between boreal hares that in winter feed almost entirely on twigs. We focus particularly on the fact that toxin concentration often varies with the age of twig segments. The model incorporates the fact that the woody internodes of the youngest segments of the twigs of the deciduous angiosperm species that these hares prefer to eat are more defended by toxins than the woody internodes of the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by hares is much higher than their intake of the biomass of the younger segments of twigs. This age-dependent toxicity of twig segments is modeled using age-structured model equations which are reduced to a system of delay differential equations involving multiple delays in the woody plant–hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems. This showed that age dependence in plant chemical defenses has the capacity to cause hare–plant population cycles, a new result.

Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus

Wolbachia is possibly the most studied reproductive parasite of arthropod species. It appears to be a promising candidate for biocontrol of some mosquito borne diseases. We begin by developing a sex-structured model for a Wolbachia infected mosquito population. Our model incorporates the key effects of Wolbachia infection including cytoplasmic incompatibility and male killing. We also allow the possibility of reduced reproductive output, incomplete maternal transmission, and different mortality rates for uninfected/infected male/female individuals. We study the existence and local stability of equilibria, including the biologically relevant and interesting boundary equilibria. For some biologically relevant parameter regimes there may be multiple coexistence steady states including, very importantly, a coexistence steady state in which Wolbachia infected individuals dominate. We also extend the model to incorporate West Nile virus (WNv) dynamics, using an SEI modelling approach. Recent evidence suggests that a particular strain of Wolbachia infection significantly reduces WNv replication in Aedes aegypti. We model this via increased time spent in the WNv-exposed compartment for Wolbachia infected female mosquitoes. A basic reproduction number