A sharp threshold for disease persistence in host metapopulations

A sharp threshold is established that separates disease persistence from the extinction of small disease outbreaks in an S→E→I→R→S type metapopulation model. The travel rates between patches depend on disease prevalence. The threshold is formulated in terms of a basic replacement ratio (disease reproduction number), ?(0), and, equivalently, in terms of the spectral bound of a transmission and travel matrix. Since frequency-dependent (standard) incidence is assumed, the threshold results do not require knowledge of a disease-free equilibrium. As a trade-off, for ?(0)>1, only uniform weak disease persistence is shown in general, while uniform strong persistence is proved for the special case of constant recruitment of susceptibles into the patch populations. For ?(0)<1, Lyapunov's direct stability method shows that small disease outbreaks do not spread much and eventually die out.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

Oscillations in a patchy environment disease model

For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.     

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.     

Modeling the Geographic Spread of Rabies in China

In order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and vaccinated subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The existence of the disease-free equilibrium is discussed, the basic reproduction number is calculated, and the effect of moving rates of dogs between patches on the basic reproduction number is studied. To investigate the rabies virus clades lineages, the two-patch submodel is used to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. It is found that the basic reproduction number of the two-patch model could be larger than one even if the isolated basic reproduction number of each patch is less than one. This indicates that the immigration of dogs may make the disease endemic even if the disease dies out in each isolated patch when there is no immigration. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog markets and trades needs to be regulated, and transportation of dogs has to be better monitored and under constant surveillance.     

Optimal patch residence time in egg parasitoids: innate versus learned estimate of patch quality

Charnovs marginal value theorem predicts that female parasitoids should exploit patches of their hosts until their instantaneous rate of fitness gain reaches a marginal value. The consequences of this are that: (1) better patches should be exploited for a longer time; (2) as travel time between patches increases, so does the patch residence time; and (3) all exploited patches should be reduced to the same level of profitability. Patch residence time was measured in an egg parasitoid Anaphes victus (Hymenoptera: Mymaridae) when patch quality and travel time, approximated here as an increased delay between emergence and patch exploitation, varied. As predicted, females stayed longer when patch quality and travel time increased. However, the marginal value of fitness gain when females left the patch increased with patch quality and decreased with travel time. A. victus females appear to base their patch quality estimate on the first patch encountered rather than on a fixed innate estimate, as was shown for another egg parasitoid Trichogramma brassicae. Such a strategy could be optimal when inter-generational variability in patch quality is high and within-generational variability is low.     

Malaria Drug Resistance: The Impact of Human Movement and Spatial Heterogeneity

Human habitat connectivity, movement rates, and spatial heterogeneity have tremendous impact on malaria transmission. In this paper, a deterministic system of differential equations for malaria transmission incorporating human movements and the development of drug resistance malaria in an \(n\) patch system is presented. The disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. For a two patch case, the boundary equilibria (drug sensitive-only and drug resistance-only boundary equilibria) when there is no movement between the patches are shown to be locally asymptotically stable when they exist; the co-existence equilibrium is locally asymptotically stable whenever the reproduction number for the drug sensitive malaria is greater than the reproduction number for the resistance malaria. Furthermore, numerical simulations of the connected two patch model (when there is movement between the patches) suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity. With slow movement (or low migration) between the patches, the drug sensitive strain dominates the drug resistance strain. However, with fast movement (or high migration) between the patches, the drug resistance strain dominates the drug sensitive strain.     

An application of optimal diving models to diving behaviour of Brünnich's guillemots

Although theoretical models predict that the quality of foraging patches has little effect on optimal dive time with increasing depth, many empirical studies show that dive time at a given depth may vary. We developed a model that incorporated patch quality as a parameter of energy intake as a nonlinear function of time, and applied it to the diving behaviour of Brünnich's guillemots, Uria lomvia. The model indicated that optimal dive time can vary widely depending on the parameter. It also explained the convergence of observed dive times with travel time. Assuming the birds dived optimally, this parameter can be estimated from travel time and dive time for each dive. Foraging patches with larger estimated parameter values were favoured by the birds, suggesting that the parameter indicated patch quality. We used this parameter to test an optimal patch use model in divers. The results indicate that Brünnich's guillemots adjust their diving behaviour adaptively depending on patch quality, and that the optimal diving model is valid for prediction of observed dive patterns if patch quality is incorporated appropriately. Copyright 2002 The Association for the Study of Animal Behaviour. Published by Elsevier Science Ltd. All rights reserved.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?₀ can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ? _j , j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

A Two-Patch Model of Gambian Sleeping Sickness: Application to Vector Control Strategies in a Village and Plantations

A compartmental model is described for the spread of Gambian sleeping sickness in a spatially heterogeneous environment in which vector and human populations migrate between two "patches": the village and the plantations. The number of equilibrium points depends on two "summary parameters": gr the proportion removed among human infectives, and R0, the basic reproduction number. The origin is stable for R0 <1 and unstable for R0 >1. Control strategies are assessed by studying the mix of vector control between the two patches that bring R0 below 1. The results demonstrate the importance of vector control in the plantations. For example if 20 percent of flies are in the village and the blood meal rate in the village is 10 percent, then a 20 percent added vector mortality in the village must be combined with a 9 percent added mortality in the plantations in order to bring R0 below 1. The results are quite insentive to the blood meal rate in the village. Optimal strategies (that minimize the total number of flies trapped in both patches) are briefly discussed.     

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.     

Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.     

Spatial Heterogeneity,Host Movement and Mosquito-Borne Disease Transmission

Mosquito-borne diseases are a global health priority disproportionately affecting low-income populations in tropical and sub-tropical countries. These pathogens live in mosquitoes and hosts that interact in spatially heterogeneous environments where hosts move between regions of varying transmission intensity. Although there is increasing interest in the implications of spatial processes for mosquito-borne disease dynamics, most of our understanding derives from models that assume spatially homogeneous transmission. Spatial variation in contact rates can influence transmission and the risk of epidemics, yet the interaction between spatial heterogeneity and movement of hosts remains relatively unexplored. Here we explore, analytically and through numerical simulations, how human mobility connects spatially heterogeneous mosquito populations, thereby influencing disease persistence (determined by the basic reproduction number R ₀), prevalence and their relationship. We show that, when local transmission rates are highly heterogeneous, R ₀ declines asymptotically as human mobility increases, but infection prevalence peaks at low to intermediate rates of movement and decreases asymptotically after this peak. Movement can reduce heterogeneity in exposure to mosquito biting. As a result, if biting intensity is high but uneven, infection prevalence increases with mobility despite reductions in R ₀. This increase in prevalence decreases with further increase in mobility because individuals do not spend enough time in high transmission patches, hence decreasing the number of new infections and overall prevalence. These results provide a better basis for understanding the interplay between spatial transmission heterogeneity and human mobility, and their combined influence on prevalence and R ₀.     

Life on the edge: reproductive mode and rate of invasive <Emphasis Type="Italic">Phragmites australis</Emphasis> patch expansion

The dynamics of plant invasions from initial colonization through patch expansion are driven in part by mode of reproduction, i.e., sexual (seed) and asexual (clonal fragments and expansion) means. Expansion of existing patches—both rate and mode of spread into a matrix of varying conditions—is important for predicting potential invader impacts. In this study, we used fine-scale genetic assessments and remote sensing to describe both the rate and mode of expansion for 20 Phragmites australis patches in flooded and unflooded wetland units on the Great Salt Lake, UT. We found that the majority of Phragmites patch expansion occurred via clonal spread but we also documented instances of (potentially episodic) seedling recruitment. The mode of patch expansion, inferred from patch edge genet richness, was unrelated to flooding in the wetland unit in the preceding growing season. The rate of Phragmites patch expansion varied from 0.09 to 0.35 year^?1 and was unrelated to the mode of spread. In six patches monitored across two years, monoclonal patches stayed monoclonal, whereas patches with higher genet richness had a marked increase in diversity in the second year. The findings of the present study suggest how this partially clonal species can exploit the benefits of both sexual (i.e., genetic recombination, widespread dispersal, colonization of new areas) and asexual reproduction (i.e., stability of established clones suited to local environmental conditions) to become one of the most successful wetland plant invaders. To control this species, both forms of reproduction need to be fully addressed through targeted management actions.     

Herbicide resistance and gene flow in wild-oats (Avena fatua and Avena sterilis ssp. ludoviciana)

The process of resistance evolution to fenoxaprop-P-ethyl was investigated in the cereal weeds wild-oats (Avena fatua L. and Avena sterilis ssp. ludoviciana Malzew) at a number of locations in England, including one farm where distinct patches occur within fields. Genetic fingerprints produced using PCR-based techniques provided evidence for hybridisation between the species and that resistance had spread from one patch to others. The proportion of total variation due to differences between populations (G_st) was estimated at 33^2%, and herbicide-resistant patches contained on average less genetic diversity than herbicide-sensitive counterparts: both findings were consistent with a high degree of self-pollination. It was however concluded that cross-pollination occurs both within and possibly between species, and that this can result in the spread of herbicide resistance.     

Temporal variation in the effects of habitat fragmentation on reproduction of the Mediterranean shrub <Emphasis Type="Italic">Colutea hispanica</Emphasis>

Habitat fragmentation poses a major threat to the viability of plant populations. However, the intensity of fragmentation effects may vary among years. We studied two possible effects of habitat fragmentation (patch size and isolation) on the reproduction and proportion of damaged fruits in 24 patches of the self-compatible shrub Colutea hispanica for three consecutive years with different climate conditions. We also studied the effect of fragmentation on the incidence of two main pre-dispersal seed predators, the butterflies Iolana iolas and Lampides boeticus. High between-year variability was found in number of viable seeds per fruit, number of fruits per plant, total number of viable seeds per plant and proportion of damaged fruits. In 2003, small, isolated patches had a higher fruit set and number of fruits per plant. The proportion of damaged fruits was significantly lower in isolated populations in 2003, while it was very high in all patches in 2004 and 2005. High between-year variability was also found in the proportion of fruits per plant with I. iolas eggs. In 2003 isolated patches had a lower proportion of fruits with I. iolas eggs, but no significant effect of patch size and isolation was found in 2004 or 2005. The proportion of fruits with L. boeticus eggs was similar in the three years of study, although it was slightly higher in large, non-isolated patches in 2003. Thus, the effects of fragmentation on plant reproduction cannot be generalized from one single-year survey. In contrast to the generally accepted idea that fragmentation reduces plant reproduction, plant fitness may increase in isolated patches in years with high fruit production and low seed predation.     

Dispersal, disease and life-history evolution

Discrete-time susceptible-infective-susceptible (S-I-S) disease transmission models can exhibit bistability (alternative stable equilibria) over a wide range of parameter values. We illustrate the richness generated by such 'simple' non-linear systems in the study of two patch epidemic models with disease-enhanced or disease-suppressed dispersal. Dispersal between patches can have a profound impact on local patch disease dynamics. In fact, dispersal between patches may give rise to bistability in parameter regimes without bistability in single patch models.     

Coupled Human-Environment Dynamics of Forest Pest Spread and Control in a Multi-Patch,Stochastic Setting

Background

The transportation of camp firewood infested by non-native forest pests such as Asian long-horned beetle (ALB) and emerald ash borer (EAB) has severe impacts on North American forests. Once invasive forest pests are established, it can be difficult to eradicate them. Hence, preventing the long-distance transport of firewood by individuals is crucial.

Methods

Here we develop a stochastic simulation model that captures the interaction between forest pest infestations and human decisions regarding firewood transportation. The population of trees is distributed across 10 patches (parks) comprising a “low volume” partition of 5 patches that experience a low volume of park visitors, and a “high volume” partition of 5 patches experiencing a high visitor volume. The infestation spreads within a patch—and also between patches—according to the probability of between-patch firewood transportation. Individuals decide to transport firewood or buy it locally based on the costs of locally purchased versus transported firewood, social norms, social learning, and level of concern for observed infestations.

Results

We find that the average time until a patch becomes infested depends nonlinearly on many model parameters. In particular, modest increases in the tree removal rate, modest increases in public concern for infestation, and modest decreases in the cost of locally purchased firewood, relative to baseline (current) values, cause very large increases in the average time until a patch becomes infested due to firewood transport from other patches, thereby better preventing long-distance spread. Patches that experience lower visitor volumes benefit more from firewood movement restrictions than patches that experience higher visitor volumes. Also, cross–patch infestations not only seed new infestations, they can also worsen existing infestations to a surprising extent: long-term infestations are more intense in the high volume patches than the low volume patches, even when infestation is already endemic everywhere.

Conclusions

The success of efforts to prevent long-distance spread of forest pests may depend sensitively on the interaction between outbreak dynamics and human social processes, with similar levels of effort producing very different outcomes depending on where the coupled human and natural system exists in parameter space. Further development of such modeling approaches through better empirical validation should yield more precise recommendations for ways to optimally prevent the long-distance spread of invasive forest pests.     

The basic reproduction number and the probability of extinction for a dynamic epidemic model

We consider the spread of an epidemic through a population divided into n sub-populations, in which individuals move between populations according to a Markov transition matrix Σ and infectives can only make infectious contacts with members of their current population. Expressions for the basic reproduction number, R₀, and the probability of extinction of the epidemic are derived. It is shown that in contrast to contact distribution models, the distribution of the infectious period effects both the basic reproduction number and the probability of extinction of the epidemic in the limit as the total population size N → ∞. The interactions between the infectious period distribution and the transition matrix Σ mean that it is not possible to draw general conclusions about the effects on R₀ and the probability of extinction. However, it is shown that for n = 2, the basic reproduction number, R₀, is maximised by a constant length infectious period and is decreasing in ?, the speed of movement between the two populations.     

Extinction times for closed epidemics: the effects of host spatial structure

Although of practical importance, the relationship between the duration of an epidemic and host spatial structure is poorly understood. Here we use a stochastic metapopulation model for the transmission of infection in a spatially structured host population. There are three qualitatively different regimes for the extinction time, which depend on patch population size, the within‐patch basic reproductive number and the strength of coupling between patches. In the first regime, the extinction time for the metapopulation (i.e. from all patches) is approximately equal to the extinction time for a single patch. In the second regime, the metapopulation extinction time is maximal but also highly variable. In the third regime, the extinction time for the metapopulation (T_E) is given by T_E = a + bn^1/2 where a is the local extinction time (i.e. from last patch), b is the transit time (i.e. the time taken for infection to spread from one patch to another) and n is the total number of patches.