Population size dependent incidence in models for diseases without immunity

Epidemiological models of SIS type are analyzed to determine the thresholds, equilibria, and stability. The incidence term in these models has a contact rate which depends on the total population size. The demographic structures considered are recruitment-death, generalized logistic, decay and growth. The persistence of the disease combined with disease-related deaths and reduced reproduction of infectives can greatly affect the population dynamics. For example, it can cause the population size to decrease to zero or to a new size below its carrying capacity or it can decrease the exponential growth rate constant of the population.     

Analysis of a disease transmission model in a population with varying size

An S I R S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.Partially supported by NSF Grant No. DMS-8703631. This work was done while this author was visiting the University of VictoriaResearch supported in part by NSERC A-8965     

Dynamic models of infectious diseases as regulators of population sizes

Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.Research supported by Centers for Disease Control contract 200-87-0515. Support services provided at the University of Iowa Center for Advanced Studies     

Stability analysis for models of diseases without immunity

Summary A cyclic, constant parameter epidemiological model is described for a closed population divided into susceptible, exposed and infectious classes. Distributed delays are introduced and the model is formulated as two coupled Volterra integral equations. The delays do not change the general nature of thresholds or asymptotic stability; in all cases considered the disease either dies out, or approaches an endemic steady state.This work was partially supported by NIH Grant AI 13233 and NSERC Grant A-4645     

The effect of density-dependent treatment and behavior change on the dynamics of HIV transmission

In this work, we propose a model for heterosexual transmission of HIV/AIDS in a population of varying size with an intervention program in which treatment and/or behavior change of the infecteds occur as an increasing function of the density of the infected class in the population. This assumption has socio-economic implications which is important for public health considerations since density-dependent treatment/behavior change may be more cost-saving than a program where treatment/behavior change occurs linearly with respect to the number of infecteds. We will make use of the conservation law of total sexual contacts which enables us to reduce the two-sex model to a simpler one-sex formulation. Analytical results will be given. Unlike a similar model with linear treatment/behavior change in Hsieh (1996) where conditions were obtained for the eradication of disease, we will show that density-dependent treatment/behavior change cannot eradicate the disease if the disease is able to persist without any treatment/behavior change. This work demonstrates the need to further understand how treatment/behavior change occurs in a society with varying population.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

Integral equation models for endemic infectious diseases

Summary Endemic infectious diseases for which infection confers permanent immunity are described by a system of nonlinear Volterra integral equations of convolution type. These constant-parameter models include vital dynamics (births and deaths), immunization and distributed infectious period. The models are shown to be well posed, the threshold criteria are determined and the asymptotic behavior is analysed. It is concluded that distributed delays do not change the thresholds and the asymptotic behaviors of the models.This work was partially supported by NIH Grant AI 13233.     

一类总人口在变化的HIV/AIDS传播的动力学模型

建立了HIV/AIDS传播的具有常数移民和指数出生的SI型模型,其中易感人群按照有无不良行为被分为两组.分别对具双线性传染率和具标准传染率的模型讨论了其无病平衡点和地方病平衡点的存在性,并就某些重要的特殊情况进行了平衡点和稳定性的分析.     

Epidemiological models for sexually transmitted diseases

The classical models for sexually transmitted infections assume homogeneous mixing either between all males and females or between certain subgroups of males and females with heterogeneous contact rates. This implies that everybody is all the time at risk of acquiring an infection. These models ignore the fact that the formation of a pair of two susceptibles renders them in a sense temporarily immune to infection as long as the partners do not separate and have no contacts with other partners. The present paper takes into account the phenomenon of pair formation by introducing explicitly a pairing rate and a separation rate. The infection transmission dynamics depends on the contact rate within a pair and the duration of a partnership. It turns out that endemic equilibria can only exist if the separation rate is sufficiently large in order to ensure the necessary number of sexual partners. The classical models are recovered if one lets the separation rate tend to infinity.This work has been supported by Deutsche Forschungsgemeinschaft     

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     

Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control

SEIR epidemiological models with the inclusion of quarantine and isolation are used to study the control and intervention of infectious diseases. A simple ordinary differential equation (ODE) model that assumes exponential distribution for the latent and infectious stages is shown to be inadequate for assessing disease control strategies. By assuming arbitrarily distributed disease stages, a general integral equation model is developed, of which the simple ODE model is a special case. Analysis of the general model shows that the qualitative disease dynamics are determined by the reproductive number , which is a function of control measures. The integral equation model is shown to reduce to an ODE model when the disease stages are assumed to have a gamma distribution, which is more realistic than the exponential distribution. Outcomes of these models are compared regarding the effectiveness of various intervention policies. Numerical simulations suggest that models that assume exponential and non-exponential stage distribution assumptions can produce inconsistent predictions.     

Monitoring and modeling density-dependent growth of anchorage-dependent cells

The density-dependent growth of Chinese hamster ovary (CHO) cells was monitored on-line by using an inverted microscope. A flow system was employed for cell cultivation so that nutrient concentration could be maintained and metabolic wastes were removed. With the help of video image analysis, local cells density could be accurately calculated and cell motility and exposed cell surface area could be estimated. A computer program which accounted for change of sell size and translocation of cells was developed to stimulate cell growth. The stimulated results of the population dynamics and the variations in cell size showed good agreement with our experimental observations, Cell motility and initial cell distribution on the substratum were found to have strong effect on cell growth.     

离散广义LOGISTIC模型的渐进性态和混沌现象

1引言自Li－Yorke的著名论文《3一周期意味混饨》问世以来,人们对一维离散模型进行了大量细致的研究工作,其中Logistic模型研究的最为彻底,并由此得到一系列普遍适用的理论结果［‘’2］文【3jIw接地对广义LOgistiC模型的稳定性及其吸引域进行了细致的研究,但其结论尚有不完善之处．在研究其周期解及混炖现象时,仅在IUI＜1时对其近似模型进行了初步讨论·这里参数C＞0为种群内禀生长率,U＞一1表示种群对环境（包括营养资源等）利用率程度的参数．显然,当U—0时,模型（2）可化为LOgistiC模型（1）．本文将直接对模型（2）进行…     

Disease propagation in connected host populations with density-dependent dynamics: the case of the Feline Leukemia Virus

Spatial heterogeneity is a strong determinant of host-parasite relationships, however local-scale mechanisms are often not elucidated. Generally speaking, in many circumstances dispersal is expected to increase disease persistence. We consider the case when host populations show density-dependent dynamics and are connected through the dispersal of individuals. Taking the domestic cats (Felis catus)--Feline Leukemia Virus (FeLV) as a toy model of host-microparasite system, we predict the disease dynamics when two host populations with distinct or similar structures are connected together and to the surrounding environment by dispersal. Our model brings qualitatively different predictions from one-population models. First, as expected, biologically realistic rates of dispersal may allow FeLV to persist in sets of populations where the virus would have gone extinct otherwise, but a reverse outcome is also possible: eradication of FeLV from a small population by connexion to a larger population where it is not persistent. Second, overall prevalence as well as depression of host population size due to infection are both enhanced by dispersal, even at low dispersal rates when disease persistence is not achieved in the two populations. This unexpected prediction is probably due to the combination of dispersal with density-dependent population dynamics. Third, the dispersal of non-infectious cats has more influence on virus prevalence than the dispersal of infectious. Finally, prevalence and depression of host population size are both related to the rate of dispersion, to the health status of individuals dispersing and to the dynamics of host populations.     

Optimizing agent-based transmission models for infectious diseases

Background

Infectious disease modeling and computational power have evolved such that large-scale agent-based models (ABMs) have become feasible. However, the increasing hardware complexity requires adapted software designs to achieve the full potential of current high-performance workstations.

Results

We have found large performance differences with a discrete-time ABM for close-contact disease transmission due to data locality. Sorting the population according to the social contact clusters reduced simulation time by a factor of two. Data locality and model performance can also be improved by storing person attributes separately instead of using person objects. Next, decreasing the number of operations by sorting people by health status before processing disease transmission has also a large impact on model performance. Depending of the clinical attack rate, target population and computer hardware, the introduction of the sort phase decreased the run time from 26 % up to more than 70 %. We have investigated the application of parallel programming techniques and found that the speedup is significant but it drops quickly with the number of cores. We observed that the effect of scheduling and workload chunk size is model specific and can make a large difference.

Conclusions

Investment in performance optimization of ABM simulator code can lead to significant run time reductions. The key steps are straightforward: the data structure for the population and sorting people on health status before effecting disease propagation. We believe these conclusions to be valid for a wide range of infectious disease ABMs. We recommend that future studies evaluate the impact of data management, algorithmic procedures and parallelization on model performance.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-015-0612-2) contains supplementary material, which is available to authorized users.     

Susceptible-infected-removed epidemic models with dynamic partnerships

The author extends the classical, stochastic, Susceptible-Infected-Removed (SIR) epidemic model to allow for disease transmission through a dynamic network of partnerships. A new method of analysis allows for a fairly complete understanding of the dynamics of the system for small and large time. The key insight is to analyze the model by tracking the configurations of all possible dyads, rather than individuals. For large populations, the initial dynamics are approximated by a branching process whose threshold for growth determines the epidemic threshold, R ₀, and whose growth rate, , determines the rate at which the number of cases increases. The fraction of the population that is ever infected, , is shown to bear the same relationship to R ₀ as in models without partnerships. Explicit formulas for these three fundamental quantities are obtained for the simplest version of the model, in which the population is treated as homogeneous, and all transitions are Markov. The formulas allow a modeler to determine the error introduced by the usual assumption of instantaneous contacts for any particular set of biological and sociological parameters. The model and the formulas are then generalized to allow for non-Markov partnership dynamics, non-uniform contact rates within partnerships, and variable infectivity. The model and the method of analysis could also be further generalized to allow for demographic effects, recurrent susceptibility and heterogeneous populations, using the same strategies that have been developed for models without partnerships.     

凉拌豆制品中单核细胞增生李斯特菌生长模型建立的研究

单核细胞增生李斯特菌是凉拌豆制品中检出率较高的一种致病菌。本研究考察了温度(4℃、15℃、25℃、30℃和37℃)对凉拌豆制品中单核细胞增生李斯特菌生长的影响,并采用SGompertz和SLogistic模型对不同温度下单核细胞增生李斯特菌的生长数据进行拟合;在此基础上,以拟合度(R2)、准确因子(A_f)和偏差因子(B_f)为指标,比较并建立了凉拌豆制品中单核细胞增生李斯特菌的二级生长模型。结果表明,SGompertz模型能更好的拟合凉拌豆制品中单核细胞增生李斯特菌在不同温度下的生长数据,平方根模型能够较准确地预测凉拌豆制品中单核细胞增生李斯特菌的生长状况,因此依次选择此两种模型作为单核细胞增生李斯特菌在凉拌豆制品中的一级和二级生长模型,且模型具有可靠性。研究结果可为凉拌豆制品中单核细胞增生李斯特菌的定量风险评估提供依据。