The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

The invasion, persistence and spread of infectious diseases within animal and plant communities

Recent theoretical and empirical studies of the population biology of infectious diseases have helped to improve our understanding of the major factors that influence the three phases of a successful invasion, namely initial establishment, persistence in the longer term and spread to other host communities. Of central importance in all three phases is the magnitude of the basic reproductive rate or transmission potential of the parasite. The value of this parameter is determined by a variety of biological properties of the association between an individual parasite and its host and the interaction between their populations. The recent epidemic of acquired immunodeficiency syndrome (AIDS) in North America and Europe is employed to illustrate the factors that promote disease establishment and spread. The frequency distribution of the number of different sexual partners per unit of time within homosexual communities is shown to be of central importance with respect to future trends in the incidence of AIDS. Broader aspects of pathogen invasion are examined by reference to simple mathematical models of three species associations, which mirror parasite introduction into resident predator-prey, host-parasite and competitive interactions. Many outcomes are possible, depending on the values of the numerous parameters that influence multi-species population interactions. Pathogen invasion is shown to have far-reaching implications for the structure and stability of ecological communities.     

A simple persistence condition for structured populations

The fundamental question in both basic and applied population biology of whether a species will increase in numbers is often investigated by finding the population growth rate as the largest eigenvalue of a deterministic matrix model. For a population classified only by age, and not stage or size, a simpler biologically interpretable condition can be used, namely whether R ₀, the mean number of offspring per newborn, is greater than one. However, for the many populations not easily described using only age classes, stage-structured models must be used for which there is currently no quantity like R ₀. We determine analogous quantities that must be greater than one for persistence of a general structured population model that have a similar useful biological interpretation. Our approach can be used immediately to determine the magnitude of changes and interactions that would either allow population persistence or would ensure control of an undesirable species.     

Invasion and persistence of infectious agents in fragmented host populations

One of the important questions in understanding infectious diseases and their prevention and control is how infectious agents can invade and become endemic in a host population. A ubiquitous feature of natural populations is that they are spatially fragmented, resulting in relatively homogeneous local populations inhabiting patches connected by the migration of hosts. Such fragmented population structures are studied extensively with metapopulation models. Being able to define and calculate an indicator for the success of invasion and persistence of an infectious agent is essential for obtaining general qualitative insights into infection dynamics, for the comparison of prevention and control scenarios, and for quantitative insights into specific systems. For homogeneous populations, the basic reproduction ratio R(0) plays this role. For metapopulations, defining such an 'invasion indicator' is not straightforward. Some indicators have been defined for specific situations, e.g., the household reproduction number R*. However, these existing indicators often fail to account for host demography and especially host migration. Here we show how to calculate a more broadly applicable indicator R(m) for the invasion and persistence of infectious agents in a host metapopulation of equally connected patches, for a wide range of possible epidemiological models. A strong feature of our method is that it explicitly accounts for host demography and host migration. Using a simple compartmental system as an example, we illustrate how R(m) can be calculated and expressed in terms of the key determinants of epidemiological dynamics.     

Animal movements and the spread of infectious diseases

Domestic and wild animal population movements are important in the spread of disease. There are many recent examples of disease spread that have occurred as a result of intentional movements of livestock or wildlife. Understanding the volume of these movements and the risks associated with them is fundamental in elucidating the epidemiology of these diseases, some of which might entail zoonotic risks. The importance of the worldwide animal trade is reviewed and the role of the unregulated trade in animals is highlighted. A range of key examples are discussed in which animal movements have resulted in the introduction of pathogens to previously disease-free areas. Measures based on heightened surveillance are proposed that mitigate the risks of new pathogen introductions.     

Disease spread in age structured populations with maternal age effects

Fundamental ecological processes, such as extrinsic mortality, determine population age structure. This influences disease spread when individuals of different ages differ in susceptibility or when maternal age determines offspring susceptibility. We show that Daphnia magna offspring born to young mothers are more susceptible than those born to older mothers, and consider this alongside previous observations that susceptibility declines with age in this system. We used a susceptible‐infected compartmental model to investigate how age‐specific susceptibility and maternal age effects on offspring susceptibility interact with demographic factors affecting disease spread. Our results show a scenario where an increase in extrinsic mortality drives an increase in transmission potential. Thus, we identify a realistic context in which age effects and maternal effects produce conditions favouring disease transmission.     

Eradication of infectious diseases in heterogeneous populations

We present a model of infectious diseases in heterogeneous populations, which allows for variable intra- to intergroup contact ratios. We give necessary and sufficient conditions for disease eradication by means of vaccination. Smallpox is used as an illustrative example.     

Neighbourhood control policies and the spread of infectious diseases

We present a model of a control programme for a disease outbreak in a population of livestock holdings. Control is achieved by culling infectious holdings when they are discovered and by the pre-emptive culling of livestock on holdings deemed to be at enhanced risk of infection. Because the pre-emptive control programme cannot directly identify exposed holdings, its implementation will result in the removal of both infected and uninfected holdings. This leads to a fundamental trade-off: increased levels of control produce a greater reduction in transmission by removing more exposed holdings, but increase the number of uninfected holdings culled. We derive an expression for the total number of holdings culled during the course of an outbreak and demonstrate that there is an optimal control policy, which minimizes this loss. Using a metapopulation model to incorporate local clustering of infection, we examine a neighbourhood control programme in a locally spreading outbreak. We find that there is an optimal level of control, which increases with increasing basic reproduction ratio, R(0); moreover, implementation of control may be optimal even when R(0) < 1. The total loss to the population is relatively insensitive to the level of control as it increases beyond the optimal level, suggesting that over-control is a safer policy than under-control.     

Social contacts and mixing patterns relevant to the spread of infectious diseases

Background

Mathematical modelling of infectious diseases transmitted by the respiratory or close-contact route (e.g., pandemic influenza) is increasingly being used to determine the impact of possible interventions. Although mixing patterns are known to be crucial determinants for model outcome, researchers often rely on a priori contact assumptions with little or no empirical basis. We conducted a population-based prospective survey of mixing patterns in eight European countries using a common paper-diary methodology.

Methods and Findings

7,290 participants recorded characteristics of 97,904 contacts with different individuals during one day, including age, sex, location, duration, frequency, and occurrence of physical contact. We found that mixing patterns and contact characteristics were remarkably similar across different European countries. Contact patterns were highly assortative with age: schoolchildren and young adults in particular tended to mix with people of the same age. Contacts lasting at least one hour or occurring on a daily basis mostly involved physical contact, while short duration and infrequent contacts tended to be nonphysical. Contacts at home, school, or leisure were more likely to be physical than contacts at the workplace or while travelling. Preliminary modelling indicates that 5- to 19-year-olds are expected to suffer the highest incidence during the initial epidemic phase of an emerging infection transmitted through social contacts measured here when the population is completely susceptible.

Conclusions

To our knowledge, our study provides the first large-scale quantitative approach to contact patterns relevant for infections transmitted by the respiratory or close-contact route, and the results should lead to improved parameterisation of mathematical models used to design control strategies.     

The strong-migration limit in geographically structured populations

Summary Some strong-migration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number N _e ( N _T ) appears instead of the actual total population number N _T . 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers N _i are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the N _i are proportional to 1/u, as N _T 8, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection coefficients proportional to 1/N _T , let P _j be the frequency of the allele A _j in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As N _T 8, the deviations of the allelic frequencies in each of the subpopulations from P _j converge to zero; the usual panmictic mutation-selection diffusion is obtained for P _j , with the selection intensities averaged with respect to the stationary distribution of M. In both models, N _e = N _T and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.Supported by the National Science Foundation (Grant No. DEB77-21494)     

A branching model for the spread of infectious animal diseases in varying environments

This paper is concerned with a stochastic model, describing outbreaks of infectious diseases that have potentially great animal or human health consequences, and which can result in such severe economic losses that immediate sets of measures need to be taken to curb the spread. During an outbreak of such a disease, the environment that the infectious agent experiences is therefore changing due to the subsequent control measures taken. In our model, we introduce a general branching process in a changing (but not random) environment. With this branching process, we estimate the probability of extinction and the expected number of infected individuals for different control measures. We also use this branching process to calculate the generating function of the number of infected individuals at any given moment. The model and methods are designed using important infections of farmed animals, such as classical swine fever, foot-and-mouth disease and avian influenza as motivating examples, but have a wider application, for example to emerging human infections that lead to strict quarantine of cases and suspected cases (e.g. SARS) and contact and movement restrictions.     

Perspective: human contact patterns and the spread of airborne infectious diseases.

Networks of social contacts channel the transmission of airborne infections. Emerging insights from fields of science as diverse as mathematics, population biology and the social sciences are beginning to reveal how the contact pattern of the hosts determines the spread and evolution of airborne infectious agents.     

Assessing the ecological risks from the persistence and spread of feral populations of insect-resistant transgenic maize

One source of potential harm from the cultivation of transgenic crops is their dispersal, persistence and spread in non-agricultural land. Ecological damage may result from such spread if the abundance of valued species is reduced. The ability of a plant to spread in non-agricultural habitats is called its invasiveness potential. The risks posed by the invasiveness potential of transgenic crops are assessed by comparing in agronomic field trials the phenotypes of the crops with the phenotypes of genetically similar non-transgenic crops known to have low invasiveness potential. If the transgenic and non-transgenic crops are similar in traits believed to control invasiveness potential, it may be concluded that the transgenic crop has low invasiveness potential and poses negligible ecological risk via persistence and spread in non-agricultural habitats. If the phenotype of the transgenic crop is outside the range of the non-transgenic comparators for the traits controlling invasiveness potential, or if the comparative approach is regarded as inadequate for reasons of risk perception or risk communication, experiments that simulate the dispersal of the crop into non-agricultural habitats may be necessary. We describe such an experiment for several commercial insect-resistant transgenic maize events in conditions similar to those found in maize-growing regions of Mexico. As expected from comparative risk assessments, the transgenic maize was found to behave similarly to non-transgenic maize and to be non-invasive. The value of this experiment in assessing and communicating the negligible ecological risk posed by the low invasiveness potential of insect-resistant transgenic maize in Mexico is discussed.     

Pathogen persistence in infectious pathology

Problems of microorganism's persistence in infectious pathology are discussed in this work. Persistence of bacteria as the form of procaryotic and eucaryotic cells symbiosis unlimitedly long coexistence is considered. Questions of the microbial evolution formed in constant collision of the infective agent with macroorganism defense mechanisms are discussed. The spectrum of known mechanisms bacterial survival in conditions of an infected organism is considered. For discussion the problem of microbial persistence it is offered to include as model alongside with an independent cell, a microbial population as complex self-organizing system--the original "superorganism" having universal chemical regulation, the determining density of a population and equation of some physiological functions. It is offered to consider the host colonization resistance as a phenomenon of general biology directed on maintenance of a microecological homeostasis as a result of symbiotic interactions of an organism and it autochthonous microflora with the "key" kinds of biotope protection. The use of persistence characteristics of microorganisms is proved as a target in conditions of intermicrobial interaction of its allochthonous and autochthonous microflorae. Practical value of such approach in infectious pathology is shown.