Some epidemiological models with nonlinear incidence

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.Research supported in part by Centers for Disease Control Contract 200-87-0515. Support services provided at University House Research Center at the University of IowaResearch supported in part by NSERC A-8965 and the University of Victoria President's Committee on Faculty Research and Travel     

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.Research supported by grant no. 11-7805 from the Danish Natural Science Research Council, by NIH grant GM 28016, by a fellowship from the Research Foundation of Aarhus University, and by a visiting fellowship from the University of New England, N.S.W.     

A predator prey model with age structure

A general predator-prey model is considered in which the predator population is assumed to have an age structure which significantly affects its fecundity. The model equations are derived from the general McKendrick equations for age structured populations. The existence, stability and destabilization of equilibria are studied as they depend on the prey's natural carrying capacity and the maturation periodm of the predator. The main result of the paper is that for a broad class of maturation functions positive equilibria are either unstable for smallm or are destabilized asm decreases to zero. This is in contrast to the usual rule of thumb that increasing (not decreasing) delays in growth rate responses cause instabilities.Research supported by National Science Foundation Grant No. MCS-7901307-01Research supported by National Scholarship for Study Abroad No. EDN/S-59/80 from the government of India     

Selection in complex genetic systems IV. Multiple alleles and interactions between two loci

Summary A symmetric viability model for two loci with two alleles at one locus and m alleles at the other is suggested and analyzed. The analysis of the equilibria is complete if the two loci are absolutely linked, while if recombination is allowed the analysis is incomplete. The dynamics of the mode! resemble those of the two locus two allele model, namely that for loose linkage there will be no correlation between the loci and for tight linkage there may be strong correlation. The major caveats to this are: 1. The equilibria stable for tight linkage may belong to an array of different structures dependent on the selection and the number of alleles. 2. If both loci are overdominant in viability, the stable equilibria always contain all alleles segregating in the population; otherwise, the stable equilibria may only be two locus two allele high complementarity equilibria for tight linkage. 3. For intermediate linkage values and special selection values the boundary two locus two allele high complementarity equilibria may be stable simultaneously with the totally polymorphic central point at which there is no association between the loci.Dedicated to the memory of Ove Frydenberg.Research supported in part by a grant from the Danish Natural Science Research Council, a grant from National Science Foundation, U.S.A., and by USPHS grant NIH 10452-09-11.     

The reinfection threshold

Thresholds in transmission are responsible for critical changes in infectious disease epidemiology. The epidemic threshold indicates whether infection invades a totally susceptible population. The reinfection threshold indicates whether self-sustained transmission occurs in a population that has developed a degree of partial immunity to the pathogen (by previous infection or vaccination). In models that combine susceptible and partially immune individuals, the reinfection threshold is technically not a bifurcation of equilibria as correctly pointed out by Breban and Blower. However, we show that a branch of equilibria to a reinfection submodel bifurcates from the disease-free equilibrium as transmission crosses this threshold. Consequently, the full model indicates that levels of infection increase by two orders of magnitude and the effect of mass vaccination becomes negligible as transmission increases across the reinfection threshold.     

一类具有垂直传播的SEIRS捕食传染病模型的全局分析

通过假设捕食系统中疾病只在食饵种群中传播,被传染的易惑者经过一段潜伏期后才具有传染性,潜伏者与染病者均具有垂直传播能力,染病者恢复后对该病不具有终身免疫力,建立了一类具有垂直传播的SEIRS捕食传染病模型,运用极限系统理论,分两种情形讨论了系统平衡点的存在性及局部稳定性,利用Lyapunov函数和二次复合矩阵等方法,得到了平衡点全局渐近稳定的条件.     

Modelling Cultivar Mixtures Using SEIR Compartmental Models

A SEIR (susceptible, exposed, infectious, removed) compartmental model is constructed to represent disease progress in a two cultivar mixture. The concept of the spore pool is the means by which inoculum exchange between the constituent cultivars is represented. Infection frequencies for each cultivar are permitted to vary with that cultivar's susceptible fraction according to a power-law relationship. For each cultivar, new additions to the susceptible class balance deaths from the removed class so that the total leaf area of all four SEIR classes remains constant. It is shown that an equilibrium with non-zero diseased classes exists in a certain parameter regime. A numerical stability analysis is performed using the model equations linearised about this equilibrium. The effects of changing induced resistance parameters within the model are demonstrated graphically. It is also demonstrated that the proportion of susceptibles as a function of mixture composition has an optimum in the regime where nontrivial equilibria exist, a feature of practical interest provided that equilibrium is reached within a growing season.     

Evolutionary games for spatially dispersed populations

An evolutionary game model is developed that incorporates both spatial dispersion and density effects in the evolutionary dynamic. It is shown that a stable equilibrium (e.g. an evolutionarily stable strategy) of the non-dispersed frequency dynamic becomes a stable equilibrium of the larger system if population density stabilizes at these fixed frequencies. It is also shown, by example, that other equilibria, whose frequencies change from one location to another, may appear when dispersal rates are relatively small.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A6187Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A7822     

An SIR epidemic model with partial temporary immunity modeled with delay

The SIR epidemic model for disease dynamics considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately resusceptible. We study the case of temporary immunity in an SIR-based model with delayed coupling between the susceptible and removed classes, which results in a coupled set of delay differential equations. We find conditions for which the endemic steady state becomes unstable to periodic outbreaks. We then use analytical and numerical bifurcation analysis to describe how the severity and period of the outbreaks depend on the model parameters.      

Disease transmission models with density-dependent demographics

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.Research supported in part by Centers for Disease Control contract 200-87-0515     

Coexistence of pathogens in sexually-transmitted disease models

We present a sexually-transmitted disease (STD) model for two strains of pathogen in a one-sex, heterogeneously-mixing population, where the dynamics are of SIS (susceptible/infected/susceptible) type, and there are two different groups of individuals. We analyze all equilibria for the case where contacts are modeled via proportionate (random) mixing. We find that both strains may under suitable circumstances coexist, and that it is the heterogeneous mixing that creates refuges for each strain as each population group favors one particular strain. This author was partially supported under Chinese NSF grant 19971066.This author was partially supported by The Research Center for Sciences, Xian Jiaotong University, while visiting Xian Jiaotong University, Xian, China.The authors thank two anonymous reviewers for their valuable comments and suggestions.     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

Backward bifurcations and multiple equilibria in epidemic models with structured immunity

Many disease pathogens stimulate immunity in their hosts, which then wanes over time. To better understand the impact of this immunity on epidemiological dynamics, we propose an epidemic model structured according to immunity level that can be applied in many different settings. Under biologically realistic hypotheses, we find that immunity alone never creates a backward bifurcation of the disease-free steady state. This does not rule out the possibility of multiple stable equilibria, but we provide two sufficient conditions for the uniqueness of the endemic equilibrium, and show that these conditions ensure uniqueness in several common special cases. Our results indicate that the within-host dynamics of immunity can, in principle, have important consequences for population-level dynamics, but also suggest that this would require strong non-monotone effects in the immune response to infection. Neutralizing antibody titer data for measles are used to demonstrate the biological application of our theory.     

Mouse model of oropharyngeal candidiasis

Oropharyngeal candidiasis is a frequent cause of morbidity in patients with defects in cell-mediated immunity or saliva production. Animal models of this infection are important for studying disease pathogenesis and evaluating vaccines and antifungal therapies. Here we describe a simple mouse model of oropharyngeal candidiasis. Mice are rendered susceptible to oral infection by injection with cortisone acetate and then inoculated by placing a swab saturated with Candida albicans sublingually. This process results in a reproducible level of infection, the histopathology of which mimics that of pseudomembranous oropharyngeal candidiasis in humans. By using this model, data are obtained after 5-9 d of work.     

Diffusion-driven instability in a predator-prey system with time-varying diffusivities

We consider an ecological model by Levin and Segel (1976) for predator-prey planktonic species, which consists of two reaction-diffusion equations, and extend it to plankton populations with time-varying diffusivities. The local stability of uniform equilibria is examined both analytically and numerically. It is found that diffusive instability is less likely to occur in systems with time-varying diffusivity than those with constant diffusivity. Contribution No. 803 of the Marine Sciences Research Center, State University of New York, Stony Brook Supported by the Danish Science Research Council (Grant nos. 11-7128, 11-8321), the Danish Research Academy (Grant nos. E-880011, V-890085) and a Travel Grant for Mathematicians (Rejselegat for Matematikere) Supported by Hudson River Foundation, Grant no. 01488AO37     

A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host

The main purpose of this article is to formulate a deterministic mathematical model for the transmission of malaria that considers two host types in the human population. The first type is called "non-immune" comprising all humans who have never acquired immunity against malaria and the second type is called "semi-immune". Non-immune are divided into susceptible, exposed and infectious and semi-immune are divided into susceptible, exposed, infectious and immune. We obtain an explicit formula for the reproductive number, R(0) which is a function of the weight of the transmission semi-immune-mosquito-semi-immune, R(0a), and the weight of the transmission non-immune-mosquito-non-immune, R(0e). Then, we study the existence of endemic equilibria by using bifurcation analysis. We give a simple criterion when R(0) crosses one for forward and backward bifurcation. We explore the possibility of a control for malaria through a specific sub-group such as non-immune or semi-immune or mosquitoes.     

Modelling the effect of a booster vaccination on disease epidemiology

Despite the effectiveness of vaccines in dramatically decreasing the number of new infectious cases and severity of illnesses, imperfect vaccines may not completely prevent infection. This is because the immunity afforded by these vaccines is not complete and may wane with time, leading to resurgence and epidemic outbreaks notwithstanding high levels of primary vaccination. To prevent an endemic spread of disease, and achieve eradication, several countries have introduced booster vaccination programs. The question of whether this strategy could eventually provide the conditions for global eradication is addressed here by developing a seasonally-forced mathematical model. The analysis of the model provides the threshold condition for disease control in terms of four major parameters: coverage of the primary vaccine; efficacy of the vaccine; waning rate; and the rate of booster administration. The results show that if the vaccine provides only temporary immunity, then the infection typically cannot be eradicated by a single vaccination episode. Furthermore, having a booster program does not necessarily guarantee the control of a disease, though the level of epidemicity may be reduced. In addition, these findings strongly suggest that the high coverage of primary vaccination remains crucial to the success of a booster strategy. Simulations using estimated parameters for measles illustrate model predictions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). One of the authors (P.R.) acknowledges the support of the Ellison Medical Foundation.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population

Let birth rates and death rates be constant, birth rates positive, fertilities additive, and each birth rate not larger than twice any other birth rate. Global convergence to equilibria is proved for the model in the title. There is at most one polymorphic equilibrium or there are a continuum of equilibria. The phase portraits are given. If there is a polymorphic equilibrium, then the largest negatively invariant set in the state space is a continuous curve connecting the two fixation equilibria. This curve coincides with the Hardy-Weinberg manifold exactly when the death rate is additive. Disregarding extinction, the polymorphic equilibria are the same for the continuous model as for the corresponding discrete model exactly when the death rate is additive.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Genetic variation in Hawaiian Drosophila. VIII. Heterozygosity and genic changes in isolated populations of D. engyochracea

Drosophila engyochracea, an endemic Hawaiian fly found only in two, finite populations in Volcano National Park, has extensive electrophoretic heterozygosity on a par with that found in species with much wider distributions. A study of six polymorphic loci in both populations over an 18-month period revealed that the population in the more xeric environment is more dynamic genetically as well as more variable. In addition, genetic changes at one locus, Pgm, are correlated to changes in an environmental moisture parameter. These findings confirm that migration is not necessary to maintain genetic variation in isolated populations and demonstrate that D. engyochracea gene pools are susceptible to errors in Hardy-Weinberg equilibria during specific seasonal periods.Portions of this article were submitted in partial fulfillment for a Doctor of Philosophy degree in Genetics awarded to W. W. M. Steiner by the Department of Genetics, University of Hawaii, Honolulu. Research was supported by NSF Grants GB-23230, GM-27586, and GB-29288.