The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population. Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibrium. The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.     

The effects of population structure on the spread of the HIV infection

A model for the spread of human immunodeficiency virus (HIV) in a population of male homosexuals is presented. The population is divided into five groups on the basis of degree of sexual activity. Within each group, the individuals are classified as 1) susceptible; 2) infective; or 3) removed because of a lack of sexual activity associated with advanced acquired immunodeficiency disease (AIDS). The infective individuals are further subdivided into four stages of infection. Analyses of the model address two questions with regard to the spread of HIV: (1) What is the effect of level of sexual activity on an individual's risk for infection, and (2) What is the effect that assumptions about mixing between groups have on both individual risk and transmission throughout a population? Results from analyses using a number of different parameter estimates show that increased levels of sexual activity increase the likelihood that an individual will become infected. In addition, the initial spread of the disease is markedly affected by variation in the amount of contact among individuals from different subpopulations. The steady-state incidence of the disease is not markedly affected by variation in the contact patterns, but the size of the steady-state population and therefore the proportion of infected individuals in the population does vary significantly with changes in the degree of mixing among subpopulations. These results show clearly the sensitivity of model outcomes to variation in the patterns of contact among individuals and the need for better data on such interactions to aid in understanding and predicting the spread of HIV.     

An SIS epidemiology game with two subpopulations

There is significant current interest in the application of game theory to problems in epidemiology. Most mathematical analyses of epidemiology games have studied populations where all individuals have the same risks and interests. This paper analyses the rational-expectation equilibria in an epidemiology game with two interacting subpopulations of equal size where decisions change the prevalence and transmission patterns of an infectious disease. The transmission dynamics are described by an SIS model and individuals are only allowed to invest in daily prevention measures like hygiene. The analysis shows that disassortative mixing may lead to multiple Nash equilibria when there are two interacting subpopulations affecting disease prevalence. The dynamic stability of these equilibria is analysed under the assumption that strategies change slowly in the direction of self-interest. When mixing is disassortative, interior Nash equilibria are always unstable. When mixing is positively assortative, there is a unique Nash equilibrium that is globally stable.     

An SIS epidemiology game with two subpopulations

There is significant current interest in the application of game theory to problems in epidemiology. Most mathematical analyses of epidemiology games have studied populations where all individuals have the same risks and interests. This paper analyses the rational-expectation equilibria in an epidemiology game with two interacting subpopulations of equal size where decisions change the prevalence and transmission patterns of an infectious disease. The transmission dynamics are described by an SIS model and individuals are only allowed to invest in daily prevention measures like hygiene. The analysis shows that disassortative mixing may lead to multiple Nash equilibria when there are two interacting subpopulations affecting disease prevalence. The dynamic stability of these equilibria is analysed under the assumption that strategies change slowly in the direction of self-interest. When mixing is disassortative, interior Nash equilibria are always unstable. When mixing is positively assortative, there is a unique Nash equilibrium that is globally stable.     

A network-based approach for resistance transmission in bacterial populations

Horizontal transfer of mobile genetic elements (conjugation) is an important mechanism whereby resistance is spread through bacterial populations. The aim of our work is to develop a mathematical model that quantitatively describes this process, and to use this model to optimize antimicrobial dosage regimens to minimize resistance development. The bacterial population is conceptualized as a compartmental mathematical model to describe changes in susceptible, resistant, and transconjugant bacteria over time. This model is combined with a compartmental pharmacokinetic model to explore the effect of different plasma drug concentration profiles. An agent-based simulation tool is used to account for resistance transfer occurring when two bacteria are adjacent or in close proximity. In addition, a non-linear programming optimal control problem is introduced to minimize bacterial populations as well as the drug dose. Simulation and optimization results suggest that the rapid death of susceptible individuals in the population is pivotal in minimizing the number of transconjugants in a population. This supports the use of potent antimicrobials that rapidly kill susceptible individuals and development of dosage regimens that maintain effective antimicrobial drug concentrations for as long as needed to kill off the susceptible population. Suggestions are made for experiments to test the hypotheses generated by these simulations.     

The effects of spatial movement and group interactions on disease dynamics of social animals

The effects of spatial movements of infected and susceptible individuals on disease dynamics is not well understood. Empirical studies on the spatial spread of disease and behaviour of infected individuals are few and theoretical studies may be useful to explore different scenarios. Hence due to lack of detail in empirical studies, theoretical models have become necessary tools in investigating the disease influence in host-pathogen systems. In this paper we developed and analysed a spatially explicit model of two interacting social groups of animals of the same species. We investigated how the movement scenarios of susceptible and infected individuals together with the between-group contact parameter affect the survival rate of susceptible individuals in each group. This work can easily be applied to various host-pathogen systems. We define bounds on the number of susceptibles which avoid infection once the disease has died out as a function of the initial conditions and other model parameters. For example, once disease has passed through the populations, a larger diffusion coefficient for each group can result in higher population levels when there is no between-group interaction but in lower levels when there is between-group interaction. Numerical simulations are used to demonstrate these bounds and behaviours and to describe the different outcomes in ecological terms.     

Hitchhiking effect of a beneficial mutation spreading in a subdivided population

A central problem in population genetics is to detect and analyze positive natural selection by which beneficial mutations are driven to fixation. The hitchhiking effect of a rapidly spreading beneficial mutation, which results in local removal of standing genetic variation, allows such an analysis using DNA sequence polymorphism. However, the current mathematical theory that predicts the pattern of genetic hitchhiking relies on the assumption that a beneficial mutation increases to a high frequency in a single random-mating population, which is certainly violated in reality. Individuals in natural populations are distributed over a geographic space. The spread of a beneficial allele can be delayed by limited migration of individuals over the space and its hitchhiking effect can also be affected. To study this effect of geographic structure on genetic hitchhiking, we analyze a simple model of directional selection in a subdivided population. In contrast to previous studies on hitchhiking in subdivided populations, we mainly investigate the range of sufficiently high migration rates that would homogenize genetic variation at neutral loci. We provide a heuristic mathematical analysis that describes how the genealogical structure at a neutral locus linked to the locus under selection is expected to change in a population divided into two demes. Our results indicate that the overall strength of genetic hitchhiking--the degree to which expected heterozygosity decreases--is diminished by population subdivision, mainly because opportunity for the breakdown of hitchhiking by recombination increases as the spread of the beneficial mutation across demes is delayed when migration rate is much smaller than the strength of selection. Furthermore, the amount of genetic variation after a selective sweep is expected to be unequal over demes: a greater reduction in expected heterozygosity occurs in the subpopulation from which the beneficial mutation originates than in its neighboring subpopulations. This raises a possibility of detecting a "hidden" geographic structure of population by carefully analyzing the pattern of a selective sweep.     

A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration

The drive to understand the invasion, spread and fade out of infectious disease in structured populations has produced a variety of mathematical models for pathogen dynamics in metapopulations. Very rarely are these models fully coupled, by which we mean that the spread of an infection within a subpopulation affects the transmission between subpopulations and vice versa. It is also rare that these models are accessible to biologists, in the sense that all parameters have a clear biological meaning and the biological assumptions are explained. Here we present an accessible model that is fully coupled without being an individual-based model. We use the model to show that the duration of an epidemic has a highly non-linear relationship with the movement rate between subpopulations, with a peak in epidemic duration appearing at small movement rates and a global maximum at large movement rates. Intuitively, the first peak is due to asynchrony in the dynamics of infection between subpopulations; we confirm this intuition and also show the peak coincides with successful invasion of the infection into most subpopulations. The global maximum at relatively large movement rates occurs because then the infectious agent perceives the metapopulation as if it is a single well-mixed population wherein the effective population size is greater than the critical community size.     

Epidemics in nonrandomly mixing populations: a simulation

Two stochastic, discrete-time simulation models for the spread of an epidemic through a population are presented. The models explore the effects of nonrandom mixing within the population and are based on an SIR epidemic model without vital statistics. They consider a population of preschool children, some of whom attend child care facilities. Disease transmission occurs both within the home neighborhood and at the child care facility used, if any. The two models differ in population size used, population density, the proportions of children using different kinds of care, and the functions used for calculating the probability of disease transmission. Results are presented for seven different variables--length of the epidemic in weeks, number of cases, number of cases in each kind of care (two day care centers, private homes, and children staying at home), and the number of private home providers affected by the epidemic. In addition, the distribution of total epidemic size and the progress of an epidemic are estimated from 25 epidemic trials. The effects of the location of homes of initial cases, the type of care used by initial cases, and the density of the population are discussed. Results from the simulation confirmed the importance of type of care on the risk for disease transmission. Results from all runs of the simulation showed that children who attended a day care center were most likely to become infected, children who went to a private home were intermediate, and children who did not use any day care facility were at the lowest risk. The size and length of the epidemics were related to the presence of the disease in day care centers, regardless of the location of the initial case, and the time at which the disease entered the center(s). The simulations also showed that the geographical distribution of the homes of children attending a particular center was a critical feature involved in the production of epidemics. The center with more widely distributed homes of students was less likely to experience a major epidemic than the center with clustering of student's homes within a neighborhood. This indicates that it is not simply attendance at a day care center that is critical for disease spread, but that the nature of the population of children attending a center is also of critical importance in the actual risk for disease spread within the center. These results are discussed with reference to the spread of hepatitis A among day care centers in Albuquerque, New Mexico.     

The evolution of parasite virulence and transmission rate in a spatially structured population

If the transmission occurs through local contact of the individuals in a spatially structured population, the evolutionarily stable (ESS) traits of parasite might be quite different from what the classical theory with complete mixing predicts. In this paper, we theoretically study the ESS virulence and transmission rate of a parasite in a lattice-structured host population, in which the host can send progeny only to its neighboring vacant site, and the transmission occurs only in between the infected and the susceptible in the nearest-neighbor sites. Infected host is assumed to be infertile. The analysis based on the pair approximation and the Monte Carlo simulation reveal that the ESS transmission rate and virulence in a lattice-structured population are greatly reduced from those in completely mixing population. Unlike completely mixing populations, the spread of parasite can drive the host to extinction, because the local density of the susceptible next to the infected can remain high even when the global density of host becomes very low. This demographic viscosity and group selection between self-organized spatial clusters of host individuals then leads to an intermediate ESS transmission rate even if there is no tradeoff between transmission rate and virulence. The ESS transmission rate is below the region of parasite-driven extinction by a finite amount for moderately large reproductive rate of host; whereas, the evolution of transmission rate leads to the fade out of parasite for small reproductive rate, and the extinction of host for very large reproductive rate.     

Population genetic structure of a common host predicts the spread of white‐nose syndrome,an emerging infectious disease in bats

Landscape complexity influences patterns of animal dispersal, which in turn may affect both gene flow and the spread of pathogens. White‐nose syndrome (WNS) is an introduced fungal disease that has spread rapidly throughout eastern North America, causing massive mortality in bat populations. We tested for a relationship between the population genetic structure of the most common host, the little brown myotis (Myotis lucifugus), and the geographic spread of WNS to date by evaluating logistic regression models of WNS risk among hibernating colonies in eastern North America. We hypothesized that risk of WNS to susceptible host colonies should increase with both geographic proximity and genetic similarity, reflecting historical connectivity, to infected colonies. Consistent with this hypothesis, inclusion of genetic distance between infected and susceptible colonies significantly improved models of disease spread, capturing heterogeneity in the spatial expansion of WNS despite low levels of genetic differentiation among eastern populations. Expanding our genetic analysis to the continental range of little brown myotis reveals strongly contrasting patterns of population structure between eastern and western North America. Genetic structure increases markedly moving westward into the northern Great Plains, beyond the current distribution of WNS. In western North America, genetic differentiation of geographically proximate populations often exceeds levels observed across the entire eastern region, suggesting infrequent and/or locally restricted dispersal, and thus relatively limited opportunities for pathogen introduction in western North America. Taken together, our analyses suggest a possibly slower future rate of spread of the WNS pathogen, at least as mediated by little brown myotis.     

An epidemic model in a patchy environment

An epidemic model is proposed to describe the dynamics of disease spread among patches due to population dispersal. We establish a threshold above which the disease is uniformly persistent and below which disease-free equilibrium is locally attractive, and globally attractive when both susceptible and infective individuals in each patch have the same dispersal rate. Two examples are given to illustrate that the population dispersal plays an important role for the disease spread. The first one shows that the population dispersal can intensify the disease spread if the reproduction number for one patch is large, and can reduce the disease spread if the reproduction numbers for all patches are suitable and the population dispersal rate is strong. The second example indicates that a population dispersal results in the spread of the disease in all patches, even though the disease can not spread in each isolated patch.     

Spatial heterogeneity, social structure and disease dynamics of animal populations

Social groupings, population dynamics and population movements of animals all give rise to spatio-temporal variations in population levels. These variations may be of crucial importance when considering the spread of infectious diseases since infection levels do not increase unless there is a sufficient pool of susceptible individuals. This paper explores the impact of social groupings on the potential for an endemic disease to develop in a spatially explicit model system. Analysis of the model demonstrates that the explicit inclusion of space allows asymmetry between groups to arise when this was not possible in the equivalent spatially homogeneous system. Moreover, differences in movement behaviours for susceptible and infected individuals gives rise to different spatial profiles for the populations. These profiles were not observed in previous work on an epidemic system. The results are discussed in an ecological context with reference to furious and dumb strains of infectious diseases.     

Going the Distance for Protein Function Prediction: A New Distance Metric for Protein Interaction Networks

In protein-protein interaction (PPI) networks, functional similarity is often inferred based on the function of directly interacting proteins, or more generally, some notion of interaction network proximity among proteins in a local neighborhood. Prior methods typically measure proximity as the shortest-path distance in the network, but this has only a limited ability to capture fine-grained neighborhood distinctions, because most proteins are close to each other, and there are many ties in proximity. We introduce diffusion state distance (DSD), a new metric based on a graph diffusion property, designed to capture finer-grained distinctions in proximity for transfer of functional annotation in PPI networks. We present a tool that, when input a PPI network, will output the DSD distances between every pair of proteins. We show that replacing the shortest-path metric by DSD improves the performance of classical function prediction methods across the board.     

Impact of group mixing on disease dynamics

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀<1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀>1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.     

Quantifying Inter- and Intra-Population Niche Variability Using Hierarchical Bayesian Stable Isotope Mixing Models

Variability in resource use defines the width of a trophic niche occupied by a population. Intra-population variability in resource use may occur across hierarchical levels of population structure from individuals to subpopulations. Understanding how levels of population organization contribute to population niche width is critical to ecology and evolution. Here we describe a hierarchical stable isotope mixing model that can simultaneously estimate both the prey composition of a consumer diet and the diet variability among individuals and across levels of population organization. By explicitly estimating variance components for multiple scales, the model can deconstruct the niche width of a consumer population into relevant levels of population structure. We apply this new approach to stable isotope data from a population of gray wolves from coastal British Columbia, and show support for extensive intra-population niche variability among individuals, social groups, and geographically isolated subpopulations. The analytic method we describe improves mixing models by accounting for diet variability, and improves isotope niche width analysis by quantitatively assessing the contribution of levels of organization to the niche width of a population.     

How do toxicants affect epidemiological dynamics?

Populations are formed of their constituent interacting individuals, each with their own respective within‐host biological processes. Infection not only spreads within the host organism but also spreads between individuals. Here we propose and study a multilevel model which links the within‐host statuses of immunity and parasite density to population epidemiology under sublethal and lethal toxicant exposure. We analyse this nested model in order to better understand how toxicants impact the spread of disease within populations. We demonstrate that outbreak of infection within a population is completely determined by the level of toxicant exposure, and that it is maximised by intermediate toxicant dosage. We classify the population epidemiology into five phases of increasing toxicant exposure and calculate the conditions under which disease will spread, showing that there exists a threshold toxicant level under which epidemics will not occur. In general, higher toxicant load results in either extinction of the population or outbreak of infection. The within‐host statuses of the individual host also determine the outcome of the epidemic at the population level. We discuss applications of our model in the context of environmental epidemiology, predicting that increased exposure to toxicants could result in greater risk of epidemics within ecological systems. We predict that reducing sublethal toxicant exposure below our predicted safe threshold could contribute to controlling population level disease and infection.     

Calculation of Disease Dynamics in a Population of Households

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-susceptible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks.     

Implications of Tourist–Macaque Interactions for Disease Transmission

During wildlife tourism, proximity or actual contact between people and animals may lead to a significant risk of anthropozoonotic disease transmission. In this paper, we use social network analysis, disease simulation modelling and data on animal health and behaviour to investigate such risks at a site in Morocco, where tourists come to see wild Barbary macaques (Macaca sylvanus). Measures of individual macaques’ network centrality—an index of the strength and distribution of their social relationships and thus potentially their ability to spread disease—did not show clear and consistent relationships with their time spent in close proximity to, or rate of interacting with, tourists. Disease simulation modelling indicated that while higher-ranked animals had a significantly greater ability to spread disease within the group, in absolute terms there was little difference in the size of outbreaks that different individuals were predicted to cause. We observed a high rate of physical contact and close proximity between humans and macaques, including during three periods when the macaques were coughing and sneezing heavily, highlighting the potential risk of disease transmission. We recommend that general disease prevention strategies, such as those aimed at reducing opportunities for contact between tourists and macaques, should be adopted.     

A model for spread of plant disease with periodic removals

A deterministic model for the spread of infectious disease in a plant population consisting of N interacting groups with periodic removals of the infected plants is considered. In the case of two interacting groups with low infection levels, the problem is solved analytically. In the case of N interacting groups arranged in line, where the interaction between the groups decreases exponentially with distance, the mathematical model consists of N nonlinear equations. Numerical solution of these equations for some values of the parameters shows a pattern similar to the solution for the two interacting groups.Contributed from the Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel. No. 1067-E, 1984 series