Dynamics of a feline virus with two transmission modes within exponentially growing host populations.

Feline panleucopenia virus (FPLV) was introduced in 1977 on Marion Island (in the southern Indian Ocean) with the aim of eradicating the cat population and provoked a huge decrease in the host population within six years. The virus can be transmitted either directly through contacts between infected and healthy cats or indirectly between a healthy cat and the contaminated environment: a specific feature of the virus is its high rate of survival outside the host. In this paper, a model was designed in order to take these two modes of transmission into account. The results showed that a mass-action incidence assumption was more appropriate than a proportionate mixing one in describing the dynamics of direct transmission. Under certain conditions the virus was able to control the host population at a low density. The indirect transmission acted as a reservoir supplying the host population with a low but sufficient density of infected individuals which allowed the virus to persist. The dynamics of the infection were more affected by the demographic parameters of the healthy hosts than by the epidemiological ones. Thus, demographic parameters should be precisely measured in field studies in order to obtain accurate predictions. The predicted results of our model were in good agreement with observations.     

Subthreshold and superthreshold coexistence of pathogen variants: the impact of host age-structure

It is well known that in the most general epidemic models with multiple pathogen variants a competitive exclusion principle is valid, such that the variant with the highest reproduction number eliminates the rest. Mechanisms such as super-infection, coinfection, and cross-immunity can lead to pathogen polymorphism where multiple strains coexist. It is also known that variability of infectivity with host age can destabilize the endemic equilibrium and cause oscillations. In this article we show that the hosts' chronological age can itself lead to coexistence of microparasites in the most basic model where competitive exclusion will occur without the age structure. Moreover, the host age-structure leads to multiple subthreshold dominance equilibria, and both weakly and strongly subthreshold coexistence. We find that the two pathogens cannot cooperate to persist subthreshold if neither one of them can persist subthreshold by itself. If, however, one of them can persist subthreshold by itself, it can cause the two pathogens to coexist in a strongly subthreshold equilibrium. The second strain that persists subthreshold through the mediation of the first always has a lower virulence. Our results show that age structure in infectivity can permit the coexistence of competing pathogens when the incidence is of proportionate mixing type (frequency-dependent transmission) and at least one of the strains is virulent.     

Epidemic Models with Heterogeneous Mixing and Treatment

We consider a two-group epidemic model with treatment and establish a final size relation that gives the extent of the epidemic. This relation can be established with arbitrary mixing between the groups even though it may not be feasible to determine the reproduction number for the model. If the mixing of the two groups is proportionate, there is an explicit expression for the reproductive number and the final size relation is expressible in terms of the components of the reproduction number. We also extend the results to a two-group influenza model with proportionate mixing. Some numerical simulations suggest that (i) the assumption of no disease deaths is a good approximation if the disease death rate is small and (ii) a one-group model is a close approximation to a two-group model but a two-group model is necessary for comparing targeted treatment strategies. This research was supported by MITACS and an NSERC Research Grant.     

Dynamics of a feline retrovirus (FeLV) in host populations with variable spatial structure.

The predictions of epidemic models are remarkably affected by the underlying assumptions concerning host population dynamics and the relation between host density and disease transmission. Furthermore, hypotheses underlying distinct models are rarely tested. Domestic cats (Felis catus) can be used to compare models and test their predictions, because cat populations show variable spatial structure that probably results in variability in the relation between density and disease transmission. Cat populations also exhibit various dynamics. We compare four epidemiological models of Feline Leukaemia Virus (FeLV). We use two different incidence terms, i.e. proportionate mixing and pseudo-mass action. Population dynamics are modelled as logistic or exponential growth. Compared with proportionate mixing, mass action incidence with logistic growth results in a threshold population size under which the virus cannot persist in the population. Exponential growth of host populations results in systems where FeLV persistence at a steady prevalence and depression of host population growth are biologically unlikely to occur. Predictions of our models account for presently available data on FeLV dynamics in various populations of cats. Thus, host population dynamics and spatial structure can be determinant parameters in parasite transmission, host population depression, and disease control.     

Interactions between Social Structure,Demography, and Transmission Determine Disease Persistence in Primates

Catastrophic declines in African great ape populations due to disease outbreaks have been reported in recent years, yet we rarely hear of similar disease impacts for the more solitary Asian great apes, or for smaller primates. We used an age-structured model of different primate social systems to illustrate that interactions between social structure and demography create ‘dynamic constraints’ on the pathogens that can establish and persist in primate host species with different social systems. We showed that this varies by disease transmission mode. Sexually transmitted infections (STIs) require high rates of transmissibility to persist within a primate population. In particular, for a unimale social system, STIs require extremely high rates of transmissibility for persistence, and remain at extremely low prevalence in small primates, but this is less constrained in longer-lived, larger-bodied primates. In contrast, aerosol transmitted infections (ATIs) spread and persist at high prevalence in medium and large primates with moderate transmissibility;, establishment and persistence in small-bodied primates require higher relative rates of transmissibility. Intragroup contact structure – the social network - creates different constraints for different transmission modes, and our model underscores the importance of intragroup contacts on infection prior to intergroup movement in a structured population. When alpha males dominate sexual encounters, the resulting disease transmission dynamics differ from when social interactions are dominated by mother-infant grooming events, for example. This has important repercussions for pathogen spread across populations. Our framework reveals essential social and demographic characteristics of primates that predispose them to different disease risks that will be important for disease management and conservation planning for protected primate populations.     

Mixing in age-structured population models of infectious diseases

Infectious diseases are controlled by reducing pathogen replication within or transmission between hosts. Models can reliably evaluate alternative strategies for curtailing transmission, but only if interpersonal mixing is represented realistically. Compartmental modelers commonly use convex combinations of contacts within and among groups of similarly aged individuals, respectively termed preferential and proportionate mixing. Recently published face-to-face conversation and time-use studies suggest that parents and children and co-workers also mix preferentially. As indirect effects arise from the off-diagonal elements of mixing matrices, these observations are exceedingly important. Accordingly, we refined the formula published by Jacquez et al. [19] to account for these newly-observed patterns and estimated age-specific fractions of contacts with each preferred group. As the ages of contemporaries need not be identical nor those of parents and children to differ by exactly the generation time, we also estimated the variances of the Gaussian distributions with which we replaced the Kronecker delta commonly used in theoretical studies. Our formulae reproduce observed patterns and can be used, given contacts, to estimate probabilities of infection on contact, infection rates, and reproduction numbers. As examples, we illustrate these calculations for influenza based on "attack rates" from a prospective household study during the 1957 pandemic and for varicella based on cumulative incidence estimated from a cross-sectional serological survey conducted from 1988-94, together with contact rates from the several face-to-face conversation and time-use studies. Susceptibility to infection on contact generally declines with age, but may be elevated among adolescents and adults with young children.     

The initialization and sensitivity of multigroup models for the transmission of HIV

We study two multigroup mathematical models of the spread of HIV. In the differential infectivity model, the infected population is divided into groups according to their infectiousness, and HIV is primarily spread by a small, highly infectious, group of superspreaders. In the staged-progression model, every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. We demonstrate the importance of choosing appropriate initial conditions, and define a new approach to distributing the initial population among the subgroups so as to minimize the artificial transients in the solutions due to unbalanced initial conditions. We demonstrate that the rate of removal in and out of a population is an important, yet often neglected, effect. We also illustrate the importance of distinguishing between the number of partners a person has and the number of contacts per partner. By assuming that people with many partners have fewer contacts per partner than people with few partners, we found that the epidemic is less sensitive to the partner acquisition rate than one might expect. However, because the probability of transmission of HIV per contact is low, the epidemic is very sensitive to the number of contacts per partner. Modeling this distinction is particularly important when estimating the impact of programs which encourage people to have fewer sexual partners.     

Modeling optimal treatment strategies in a heterogeneous mixing model

Background

Many mathematical models assume random or homogeneous mixing for various infectious diseases. Homogeneous mixing can be generalized to mathematical models with multi-patches or age structure by incorporating contact matrices to capture the dynamics of the heterogeneously mixing populations. Contact or mixing patterns are difficult to measure in many infectious diseases including influenza. Mixing patterns are considered to be one of the critical factors for infectious disease modeling.

Methods

A two-group influenza model is considered to evaluate the impact of heterogeneous mixing on the influenza transmission dynamics. Heterogeneous mixing between two groups with two different activity levels includes proportionate mixing, preferred mixing and like-with-like mixing. Furthermore, the optimal control problem is formulated in this two-group influenza model to identify the group-specific optimal treatment strategies at a minimal cost. We investigate group-specific optimal treatment strategies under various mixing scenarios.

Results

The characteristics of the two-group influenza dynamics have been investigated in terms of the basic reproduction number and the final epidemic size under various mixing scenarios. As the mixing patterns become proportionate mixing, the basic reproduction number becomes smaller; however, the final epidemic size becomes larger. This is due to the fact that the number of infected people increases only slightly in the higher activity level group, while the number of infected people increases more significantly in the lower activity level group. Our results indicate that more intensive treatment of both groups at the early stage is the most effective treatment regardless of the mixing scenario. However, proportionate mixing requires more treated cases for all combinations of different group activity levels and group population sizes.

Conclusions

Mixing patterns can play a critical role in the effectiveness of optimal treatments. As the mixing becomes more like-with-like mixing, treating the higher activity group in the population is almost as effective as treating the entire populations since it reduces the number of disease cases effectively but only requires similar treatments. The gain becomes more pronounced as the basic reproduction number increases. This can be a critical issue which must be considered for future pandemic influenza interventions, especially when there are limited resources available.

     

Transmission of infectious diseases en route to habitat hotspots

Background

The spread of infectious diseases in wildlife populations is influenced by patterns of between-host contacts. Habitat “hotspots” - places attracting a large numbers of individuals or social groups - can significantly alter contact patterns and, hence, disease propagation. Research on the importance of habitat hotspots in wildlife epidemiology has primarily focused on how inter-individual contacts occurring at the hotspot itself increase disease transmission. However, in territorial animals, epidemiologically important contacts may primarily occur as animals cross through territories of conspecifics en route to habitat hotspots. So far, the phenomenon has received little attention. Here, we investigate the importance of these contacts in the case where infectious individuals keep visiting the hotspots and in the case where these individuals are not able to travel to the hotspot any more.

Methodology and Principal Findings

We developed a simulation epidemiological model to investigate both cases in a scenario when transmission at the hotspot does not occur. We find that (i) hotspots still exacerbate epidemics, (ii) when infectious individuals do not travel to the hotspot, the most vulnerable individuals are those residing at intermediate distances from the hotspot rather than nearby, and (iii) the epidemiological vulnerability of a population is the highest when the number of hotspots is intermediate.

Conclusions and Significance

By altering animal movements in their vicinity, habitat hotspots can thus strongly increase the spread of infectious diseases, even when disease transmission does not occur at the hotspot itself. Interestingly, when animals only visit the nearest hotspot, creating additional artificial hotspots, rather than reducing their number, may be an efficient disease control measure.     

The critical vaccination fraction for heterogeneous epidemic models

Given a population with m heterogeneous subgroups, a method is developed for determining minimal vaccine allocations to prevent an epidemic by setting the reproduction number to 1. The framework is sufficiently general to apply to several epidemic situations, such as SIR, SEIR and SIS models with vital dynamics. The reproduction number is the largest eigenvalue of the linearized system round the local point of equilibrium of the model. Using the Perron-Frobenius theorem, an exact method for generating solutions is given and the threshold surface of critical vaccine allocations is shown to be a compact, connected subset of a regular (m-1)-dimensional manifold. Populations with two subgroups are examined in full. The threshold curves are either hyperbolas or straight lines. Explicit conditions are given as to when threshold elimination is achievable by vaccinating just one or two groups in a multi-group population and expressions for the critical coverage are derived. Specific reference is made to an influenza A model. Separable or proportionate mixing is also treated. Conditions are conjectured for convexity of the threshold surface and the problem of minimizing the amount of vaccine used while remaining on the threshold surface is discussed.     

A Household-Based Study of Contact Networks Relevant for the Spread of Infectious Diseases in the Highlands of Peru

Background

Few studies have quantified social mixing in remote rural areas of developing countries, where the burden of infectious diseases is usually the highest. Understanding social mixing patterns in those settings is crucial to inform the implementation of strategies for disease prevention and control. We characterized contact and social mixing patterns in rural communities of the Peruvian highlands.

Methods and Findings

This cross-sectional study was nested in a large prospective household-based study of respiratory infections conducted in the province of San Marcos, Cajamarca-Peru. Members of study households were interviewed using a structured questionnaire of social contacts (conversation or physical interaction) experienced during the last 24 hours. We identified 9015 reported contacts from 588 study household members. The median age of respondents was 17 years (interquartile range [IQR] 4–34 years). The median number of reported contacts was 12 (IQR 8–20) whereas the median number of physical (i.e. skin-to-skin) contacts was 8.5 (IQR 5–14). Study participants had contacts mostly with people of similar age, and with their offspring or parents. The number of reported contacts was mainly determined by the participants’ age, household size and occupation. School-aged children had more contacts than other age groups. Within-household reciprocity of contacts reporting declined with household size (range 70%-100%). Ninety percent of household contact networks were complete, and furthermore, household members'' contacts with non-household members showed significant overlap (range 33%-86%), indicating a high degree of contact clustering. A two-level mixing epidemic model was simulated to compare within-household mixing based on observed contact networks and within-household random mixing. No differences in the size or duration of the simulated epidemics were revealed.

Conclusion

This study of rural low-density communities in the highlands of Peru suggests contact patterns are highly assortative. Study findings support the use of within-household homogenous mixing assumptions for epidemic modeling in this setting.     

Host-parasite population dynamics under combined frequency- and density-dependent transmission

Many host‐parasite models assume that transmission increases linearly with host population density (‘density‐dependent transmission’), but various alternative transmission functions have been proposed in an effort to capture the complexity of real biological systems. The most common alternative (usually applied to sexually transmitted parasites) assumes instead that the rate at which hosts contact one another is independent of population density, leading to ‘frequency‐dependent’ transmission. This straight‐forward distinction generates fundamentally different dynamics (e.g. deterministic, parasite‐driven extinction with frequency‐ but not density‐dependence). Here, we consider the situation where transmission occurs through two different types of contact, one of which is density‐dependent (e.g. social contacts), the other density‐independent (e.g. sexual contacts). Drawing on a range of biological examples, we propose that this type of contact structure may be widespread in natural populations. When our model is characterized mainly by density‐dependent transmission, we find that allowing even small amounts of transmission to occur through density‐independent contacts leads to the possibility of deterministic, parasite‐driven extinction (and lowers the threshold for parasite persistence). Contrastingly, allowing some density‐dependent transmission to occur in a model characterized mainly by density‐independent contacts (i.e. by frequency‐dependent transmission) does not affect the extinction threshold, but does increase the likelihood of parasite persistence. The idea that directly transmitted parasites exploit different types of host contact is not new, but here we show that the impact on dynamics can be fundamental even in the simplest cases. For example, in systems where density‐dependent transmission is normally assumed de facto, we show that parasite‐driven extinction can occur if a small amount of transmission occurs through density‐independent contacts. Many empirical studies are still guided by the traditional density/frequency dichotomy, but our combined transmission function may provide a better model for systems in which both types of transmission occur.     

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.     

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.     

A generalized chain binomial model with application to HIV infection

The original Reed-Frost formulation of the chain binomial model is mathematically equivalent to a stochastic model allowing a Poisson number of effective contacts in a time interval. Their formulation cannot accommodate survey data that necessarily correspond to more complex distributions of partners or contacts, or to large populations where complete random mixing is unlikely. This paper generalizes the Reed-Frost model to accommodate these situations in both the one- and two-population settings. The extension to multiple populations is also outlined. Using the model to predict HIV incidence in San Francisco's homosexual population, we show that the total number of contacts over all partners is more important than the distribution of contacts among partners in determining the number of infected.     

Costs and benefits of group living with disease: a case study of pneumonia in bighorn lambs (Ovis canadensis)

Group living facilitates pathogen transmission among social hosts, yet temporally stable host social organizations can actually limit transmission of some pathogens. When there are few between-subpopulation contacts for the duration of a disease event, transmission becomes localized to subpopulations. The number of per capita infectious contacts approaches the subpopulation size as pathogen infectiousness increases. Here, we illustrate that this is the case during epidemics of highly infectious pneumonia in bighorn lambs (Ovis canadensis). We classified individually marked bighorn ewes into disjoint seasonal subpopulations, and decomposed the variance in lamb survival to weaning into components associated with individual ewes, subpopulations, populations and years. During epidemics, lamb survival varied substantially more between ewe-subpopulations than across populations or years, suggesting localized pathogen transmission. This pattern of lamb survival was not observed during years when disease was absent. Additionally, group sizes in ewe-subpopulations were independent of population size, but the number of ewe-subpopulations increased with population size. Consequently, although one might reasonably assume that force of infection for this highly communicable disease scales with population size, in fact, host social behaviour modulates transmission such that disease is frequency-dependent within populations, and some groups remain protected during epidemic events.     

The Role of Environmental Transmission in Recurrent Avian Influenza Epidemics

Avian influenza virus (AIV) persists in North American wild waterfowl, exhibiting major outbreaks every 2–4 years. Attempts to explain the patterns of periodicity and persistence using simple direct transmission models are unsuccessful. Motivated by empirical evidence, we examine the contribution of an overlooked AIV transmission mode: environmental transmission. It is known that infectious birds shed large concentrations of virions in the environment, where virions may persist for a long time. We thus propose that, in addition to direct fecal/oral transmission, birds may become infected by ingesting virions that have long persisted in the environment. We design a new host–pathogen model that combines within-season transmission dynamics, between-season migration and reproduction, and environmental variation. Analysis of the model yields three major results. First, environmental transmission provides a persistence mechanism within small communities where epidemics cannot be sustained by direct transmission only (i.e., communities smaller than the critical community size). Second, environmental transmission offers a parsimonious explanation of the 2–4 year periodicity of avian influenza epidemics. Third, very low levels of environmental transmission (i.e., few cases per year) are sufficient for avian influenza to persist in populations where it would otherwise vanish.     

TB-infected deer are more closely related than non-infected deer

Identifying mechanisms of pathogen transmission is critical to controlling disease. Social organization should influence contacts among individuals and thus the distribution and spread of disease within a population. Molecular genetic markers can be used to elucidate mechanisms of disease transmission in wildlife populations without undertaking detailed observational studies to determine probable contact rates. Estimates of genealogical relationships within a bovine tuberculosis-infected white-tailed deer (Odocoileus virginianus) population indicated that infected deer were significantly more closely related than non-infected deer suggesting that contact within family groups was a significant mechanism of disease transmission. Results demonstrate that epidemiological models should incorporate aspects of host ecology likely to affect the probability of disease transmission.     

Multigeneration Reproduction Ratios and the Effects of Clustered Unvaccinated Individuals on Epidemic Outbreak

An SIR epidemiological community-structured model is constructed to investigate the effects of clustered distributions of unvaccinated individuals and the distribution of the primary case relative to vaccination levels. The communities here represent groups such as neighborhoods within a city or cities within a region. The model contains two levels of mixing, where individuals make more intra-group than inter-group contacts. Stochastic simulations and analytical results are utilized to explore the model. An extension of the effective reproduction ratio that incorporates more spatial information by predicting the average number of tertiary infections caused by a single infected individual is introduced to characterize the system. Using these methods, we show that both the vaccination coverage and the variation in vaccination levels among communities affect the likelihood and severity of epidemics. The location of the primary infectious case and the degree of mixing between communities are also important factors in determining the dynamics of outbreaks. In some cases, increasing the efficacy of a vaccine can in fact increase the effective reproduction ratio in early generations, due to the effects of population structure on the likely initial location of an infection.     

More than meets the eye: detecting cryptic microgeographic population structure in a parasite with a complex life cycle

Nonrandom recruitment of parasites among hosts can lead to genetic differentiation among hosts and mating dynamics that promote inbreeding. It has been hypothesized that strictly aquatic parasites with intermediate hosts will behave as panmictic populations among hosts because ample opportunity exists for random mixing of unrelated individuals during transmission to the definitive host. A previous allozyme study on the marine trematode Lecithochirium fusiforme did not support this hypothesis; in that, there was genetic differentiation among, and significant heterozygote deficiencies within, definitive hosts. We revisit this system and use microsatellites to obtain multilocus genotypes. Our goal was to determine whether cryptic subgroups and/or the presence of clones could account for the apparent deviation from 'panmixia'. We find strong evidence for cryptic subdivision (three genetic clusters) that causes the Wahlund effect and differentiation among definitive hosts. After accounting for these cryptic groups, we see panmictic genetic structure among definitive hosts that is consistent with the 'high mixing in aquatic habitats' hypothesis. We see evidence for cotransmission of clones in all three clusters, but this level of clonal structure did not have a major impact in causing deviations from Hardy-Weinberg equilibrium, and only affected genetic differentiation among hosts in one cluster. A cursory examination of the data may have led to incorrect conclusions about nonrandom transmission. However, it is obvious in this system that there is more than meets the eye in relation to the actual make-up of parasite populations. In general, the methods we employ will be useful for elucidating hidden patterns in other organisms where cryptic structure may be common (e.g. those with limited morphology or complex life histories).