Dynamical patterns of epidemic outbreaks in complex heterogeneous networks

We present a thorough inspection of the dynamical behavior of epidemic phenomena in populations with complex and heterogeneous connectivity patterns. We show that the growth of the epidemic prevalence is virtually instantaneous in all networks characterized by diverging degree fluctuations, independently of the structure of the connectivity correlation functions characterizing the population network. By means of analytical and numerical results, we show that the outbreak time evolution follows a precise hierarchical dynamics. Once reached the most highly connected hubs, the infection pervades the network in a progressive cascade across smaller degree classes. Finally, we show the influence of the initial conditions and the relevance of statistical results in single case studies concerning heterogeneous networks. The emerging theoretical framework appears of general interest in view of the recently observed abundance of natural networks with complex topological features and might provide useful insights for the development of adaptive strategies aimed at epidemic containment.     

Spatial spread of an epidemic through public transportation systems with a hub

This article investigates an epidemic spreading among several locations through a transportation system, with a hub connecting these locations. Public transportation is not only a bridge through which infections travel from one location to another but also a place where infections occur since individuals are typically in close proximity to each other due to the limited space in these systems. A mathematical model is constructed to study the spread of an infectious disease through such systems. A variant of the next generation method is proposed and used to provide upper and lower bounds of the basic reproduction number for the model. Our investigation indicates that increasing transportation efficiency, and improving sanitation and ventilation of the public transportation system decrease the chance of an outbreak occurring. Moreover, discouraging unnecessary travel during an epidemic also decreases the chance of an outbreak. However, reducing travel by infectives while allowing susceptibles to travel may not be enough to avoid an outbreak.     

Multiple sources and routes of information transmission: Implications for epidemic dynamics

In a recent paper [20], we proposed and analyzed a compartmental ODE-based model describing the dynamics of an infectious disease where the presence of the pathogen also triggers the diffusion of information about the disease. In this paper, we extend this previous work by presenting results based on pairwise and simulation models that are better suited for capturing the population contact structure at a local level. We use the pairwise model to examine the potential of different information generating mechanisms and routes of information transmission to stop disease spread or to minimize the impact of an epidemic. The individual-based simulation is used to better differentiate between the networks of disease and information transmission and to investigate the impact of different basic network topologies and network overlap on epidemic dynamics. The paper concludes with an individual-based semi-analytic calculation of R₀ at the non-trivial disease free equilibrium.     

SIR dynamics in random networks with heterogeneous connectivity

Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE’s. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.      

Heterogeneity in multiple transmission pathways: modelling the spread of cholera and other waterborne disease in networks with a common water source

Many factors influencing disease transmission vary throughout and across populations. For diseases spread through multiple transmission pathways, sources of variation may affect each transmission pathway differently. In this paper we consider a disease that can be spread via direct and indirect transmission, such as the waterborne disease cholera. Specifically, we consider a system of multiple patches with direct transmission occurring entirely within patch and indirect transmission via a single shared water source. We investigate the effect of heterogeneity in dual transmission pathways on the spread of the disease. We first present a 2-patch model for which we examine the effect of variation in each pathway separately and propose a measure of heterogeneity that incorporates both transmission mechanisms and is predictive of R₀. We also explore how heterogeneity affects the final outbreak size and the efficacy of intervention measures. We conclude by extending several results to a more general n-patch setting.     

Infectious disease transmission and contact networks in wildlife and livestock

The use of social and contact networks to answer basic and applied questions about infectious disease transmission in wildlife and livestock is receiving increased attention. Through social network analysis, we understand that wild animal and livestock populations, including farmed fish and poultry, often have a heterogeneous contact structure owing to social structure or trade networks. Network modelling is a flexible tool used to capture the heterogeneous contacts of a population in order to test hypotheses about the mechanisms of disease transmission, simulate and predict disease spread, and test disease control strategies. This review highlights how to use animal contact data, including social networks, for network modelling, and emphasizes that researchers should have a pathogen of interest in mind before collecting or using contact data. This paper describes the rising popularity of network approaches for understanding transmission dynamics in wild animal and livestock populations; discusses the common mismatch between contact networks as measured in animal behaviour and relevant parasites to match those networks; and highlights knowledge gaps in how to collect and analyse contact data. Opportunities for the future include increased attention to experiments, pathogen genetic markers and novel computational tools.     

Social networks and the spread of Salmonella in a sleepy lizard population

Although theoretical models consider social networks as pathways for disease transmission, strong empirical support, particularly for indirectly transmitted parasites, is lacking for many wildlife populations. We found multiple genetic strains of the enteric bacterium Salmonella enterica within a population of Australian sleepy lizards (Tiliqua rugosa), and we found that pairs of lizards that shared bacterial genotypes were more strongly connected in the social network than were pairs of lizards that did not. In contrast, there was no significant association between spatial proximity of lizard pairs and shared bacterial genotypes. These results provide strong correlative evidence that these bacteria are transmitted from host to host around the social network, rather than that adjacent lizards are picking up the same bacterial genotype from some common source.     

A multi-structured epidemic problem with direct and indirect transmission in heterogeneous environments

In this work, we analyse a deterministic epidemic mathematical model motivated by the propagation of a hantavirus (Puumala hantavirus) within a bank vole population (Clethrionomys glareolus). The host population is split into juvenile and adult individuals. A heterogeneous spatial chronological age and infection age structure is considered, and also indirect transmission via the environment. Maturation rates for juvenile individuals are adult density-dependent. For the reaction–diffusion systems with age structures derived, we give global existence, uniqueness and global boundedness results. A model with transmission to humans is also studied here.     

A multi-structured epidemic problem with direct and indirect transmission in heterogeneous environments

In this work, we analyse a deterministic epidemic mathematical model motivated by the propagation of a hantavirus (Puumala hantavirus) within a bank vole population (Clethrionomys glareolus). The host population is split into juvenile and adult individuals. A heterogeneous spatial chronological age and infection age structure is considered, and also indirect transmission via the environment. Maturation rates for juvenile individuals are adult density-dependent. For the reaction-diffusion systems with age structures derived, we give global existence, uniqueness and global boundedness results. A model with transmission to humans is also studied here.     

Reproductive numbers, epidemic spread and control in a community of households

Many of the studies on emerging epidemics (such as SARS and pandemic flu) use mass action models to estimate reproductive numbers and the needed control measures. In reality, transmission patterns are more complex due to the presence of various social networks. One level of complexity can be accommodated by considering a community of households. Our study of transmission dynamics in a community of households emphasizes five types of reproductive numbers for the epidemic spread: household-to-household reproductive number, leaky vaccine-associated reproductive numbers, perfect vaccine reproductive number, growth rate reproductive number, and the individual reproductive number. Each of those carries different information about the transmission dynamics and the required control measures, and often some of those can be estimated from the data while others cannot. Simulations have shown that under certain scenarios there is an ordering for those reproductive numbers. We have proven a number of ordering inequalities under general assumptions about the individual infectiousness profiles. Those inequalities allow, for instance, to estimate the needed vaccine coverage and other control measures without knowing the various transmission parameters in the models. Along the way, we have also shown that in choosing between increasing vaccine efficacy and increasing coverage levels by the same factor, preference should go to efficacy.     

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.     

Using contact networks to explore mechanisms of parasite transmission in wildlife

A hallmark assumption of traditional approaches to disease modelling is that individuals within a given population mix uniformly and at random. However, this assumption does not always hold true; contact heterogeneity or preferential associations can have a substantial impact on the duration, size, and dynamics of epidemics. Contact heterogeneity has been readily adopted in epidemiological studies of humans, but has been less studied in wildlife. While contact network studies are becoming more common for wildlife, their methodologies, fundamental assumptions, host species, and parasites vary widely. The goal of this article is to review how contact networks have been used to study macro‐ and microparasite transmission in wildlife. The review will: (i) explain why contact heterogeneity is relevant for wildlife populations; (ii) explore theoretical and applied questions that contact networks have been used to answer; (iii) give an overview of unresolved methodological issues; and (iv) suggest improvements and future directions for contact network studies in wildlife.     

Modeling the risk of invasion and spread of Tuta absoluta in Africa

Tuta absoluta is an invasive insect that originated from South America and has spread to Europe Africa and Asia. Since its detection in Spain in 2006, the pest is continuing to expand its geographical range, including the recent detection in several Sub-Saharan African countries. The present study proposed a model based on cellular automata to predict year-to-year the risk of the invasion and spread of T. absoluta across Africa. Using, land vegetation cover, temperature, relative humidity and yield of tomato production as key driving factors, we were able to mimic the spreading behavior of the pest, and to understand the role that each of these factors play in the process of propagation of invasion. Simulations by inferring the pest’s natural ability to fly long distance revealed that T. absoluta could reach South of Africa ten years after being detected in Spain (Europe). Findings also reveal that relative humidity and the presence of T. absoluta host plants are important factors for improving the accuracy of the prediction. The study aims to inform stakeholders in plant health, plant quarantine, and pest management on the risks that T. absoluta may cause at local, regional and event global scales. It is suggested that adequate measures should be put in place to stop, control and contain the process used by this pest to expand its range.     

Further studies on cucumber mosaic virus infection of narrow-leafed lupin (Lupinus angustifolius): seed-borne infection, aphid transmission, spread and effects on grain yield

Four field trials were done with narrow-leafed lupins (Lupinus angustifolius) in 1988 - 1989, to examine the effect of sowing seed with 5% and 0.5% cucumber mosaic virus (CMV) infection on subsequent virus spread, grain yield and percentage of infection in harvested seed. A proportion of the CM V-infected seed failed to produce established plants and thus, plots sown with 5% and 0.5% infected seed contained 1.5-2.9% and 0.2-0.3% of seed-infected plants respectively. The rate of virus spread by aphids was faster and resulted in more extensive infection at maturity in plots sown with 5% infected seed than with 0.5% infected seed. In three trials, sowing 5% infected seed resulted in yield losses of 34 - 53% and CMV infection in the seed harvested of 6 - 13%. The spread of CMV infection resulting from sowing 0.5% infected seed did not significantly decrease yield. However, late CMV spread in these plots caused > 1% seed infection. In the fourth trial, which was badly affected by drought, CMV spread only slowly, there was no significant effect of CMV on grain yield and the percentage of infected seed harvested was 3–5 times less than that in the seed sown. When CMV-infected seed was sown at different depths, target depths of 8 and 11 cm decreased the incidence of seed-infected plants by c. 15% and c. 50% respectively compared with sowing at 5 cm. However, in glasshouse tests, treatment with the pre-emergence herbicide simazine failed to selectively cull out seed-infected plants. The field trials were colonised by green peach (Myzus persicae), blue-green (Acyrthosiphon kondoi) and cowpea (Aphis craccivora) aphids. When the abilities of these aphid species and of the turnip aphid (Lipaphis erysimi) in transmitting CMV from lupins to lupins were examined in glasshouse tests, short acquisition access times favoured transmission. With 5–10 min acquisition access times, overall transmission efficiencies were 10.8%, 9.4%, 6.1% and 3.9% for the green peach, cowpea, blue-green and turnip aphids respectively.     

Temporal and spatial spread of Lettuce mosaic virus in lettuce crops in central Spain: factors involved in Lettuce mosaic virus epidemics

Lettuce mosaic virus (LMV) is transmitted by aphid vectors in a nonpersistent manner as well as by seeds. The virus causes severe disease outbreaks in commercial lettuce crops in several regions of Spain. The temporal and spatial patterns of spread of LMV were studied in autumn 2002 in the central region of Spain. Symptomatic lettuce (var. Cazorla) plant samples were collected weekly, first at the seedling stage from the greenhouse nursery and later outdoors after transplantation. The exact position of symptomatic plants sampled in the field was recorded and then material was tested by enzyme‐linked immunosorbent assay to assess virus infection. Cumulative spatial data for infected plants at different growth stages were analysed using spatial analysis by distance indices. For temporal analysis, the monomolecular, Gompertz, logistic and exponential models were evaluated for goodness of fit to the entire set of disease progress data obtained. The results indicated that the disease progress curve of LMV epidemics in the selected area is best described by a Gompertz model and that the epidemic follows a polycyclic disease progression. Our data suggest that secondary cycle of spread occurs when noncolonising aphid species land on the primary infected plants (probably coming from infected seed) and move to adjacent plants before leaving the crop. The role of weeds growing close to lettuce fields as potential inoculum sources of virus and the aphid species most likely involved in the transmission of LMV were also identified.     

A general multi-strain model with environmental transmission: Invasion conditions for the disease-free and endemic states

Although many infectious diseases of humans and wildlife are transmitted via an environmental reservoir, the theory of environmental transmission remains poorly elaborated. Here we introduce an SIR-type multi-strain disease transmission model with perfect cross immunity where environmental transmission is broadly defined by three axioms. We establish the conditions under which a multi-strain endemic state is invaded by another strain which is both directly and environmentally transmitted. We discuss explicit forms for environmental transmission terms and apply our newly derived invasion conditions to a two-strain system. Then, we consider the case of two strains with matching basic reproduction numbers (i.e., R₀), one directly transmitted only and the other both directly and environmentally transmitted, invading each other's endemic state. We find that the strain which is only directly transmitted can invade the endemic state of the strain with mixed transmission. However, the endemic state of the first strain is neutrally stable to invasion by the second strain. Thus, our results suggest that environmental transmission makes the endemic state less resistant to invasion.     

An SIR pairwise epidemic model with infection age and demography

The demography and infection age play an important role in the spread of slowly progressive diseases. To investigate their effects on the disease spreading, we propose a pairwise epidemic model with infection age and demography on dynamic networks. The basic reproduction number of this model is derived. It is proved that there is a disease-free equilibrium which is globally asymptotically stable if the basic reproduction number is less that unity. Besides, sensitivity analysis is performed and shows that increasing the variance in recovery time and decreasing the variance in infection time can effectively control the diseases. The complex interaction between the death rate and equilibrium prevalence suggests that it is imperative to correctly estimate the parameters of demography in order to assess the disease transmission dynamics accurately. Moreover, numerical simulations show that the endemic equilibrium is globally asymptotically stable.