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.     

A Model of Spatial Epidemic Spread When Individuals Move Within Overlapping Home Ranges

One of the central goals of mathematical epidemiology is to predict disease transmission patterns in populations. Two models are commonly used to predict spatial spread of a disease. The first is the distributed-contacts model, often described by a contact distribution among stationary individuals. The second is the distributed-infectives model, often described by the diffusion of infected individuals. However, neither approach is ideal when individuals move within home ranges. This paper presents a unified modeling hypothesis, called the restricted-movement model. We use this model to predict spatial spread in settings where infected individuals move within overlapping home ranges. Using mathematical and computational approaches, we show that our restricted-movement model has three limits: the distributed-contacts model, the distributed-infectives model, and a third, less studied advective distributed-infectives limit. We also calculate approximate upper bounds for the rates of an epidemic's spatial spread. Guidelines are suggested for determining which limit is most appropriate for a specific disease.     

Replicating disease spread in empirical cattle networks by adjusting the probability of infection in random networks

Comparisons between mass-action or “random” network models and empirical networks have produced mixed results. Here we seek to discover whether a simulated disease spread through randomly constructed networks can be coerced to model the spread in empirical networks by altering a single disease parameter — the probability of infection. A stochastic model for disease spread through herds of cattle is utilised to model the passage of an SEIR (susceptible–latent–infected–resistant) through five networks. The first network is an empirical network of recorded contacts, from four datasets available, and the other four networks are constructed from randomly distributed contacts based on increasing amounts of information from the recorded network. A numerical study on adjusting the value of the probability of infection was conducted for the four random network models. We found that relative percentage reductions in the probability of infection, between 5.6% and 39.4% in the random network models, produced results that most closely mirrored the results from the empirical contact networks. In all cases tested, to reduce the differences between the two models, required a reduction in the probability of infection in the random network.     

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.     

Optimal strategy for controlling the spread of HIV/AIDS disease: a case study of South Africa

HIV/AIDS disease continues to spread alarmingly despite the huge amounts of resources invested in fighting it. There is a need to integrate the series of control measures available to ensure a consistent reduction in the incidence of the disease pending the discovery of its cure. We present a deterministic model for controlling the spread of the disease using change in sexual habits and antiretroviral (ARV) therapy as control measures. We formulate a fixed time optimal control problem subject to the model dynamics with the goal of finding the optimal combination of the two control measures that will minimize the cost of the control efforts as well as the incidence of the disease. We estimate the model state initial conditions and parameter values from the demographic and HIV/AIDS data of South Africa. We use Pontryagin's maximum principle to derive the optimality system and solve the system numerically. Compared with the practice in most resource-limited settings where ARV treatment is given only to patients with full-blown AIDS, our simulation results suggest that starting the treatment as soon as the patients progress to the pre-AIDS stage of the disease coupled with appreciable change in the susceptible individuals' sexual habits reduces both the incidence and prevalence of the disease faster. In fact, the results predict that the implementation of the proposed strategy would drive new cases of the disease towards eradication in 10 years.     

Optimal strategy for controlling the spread of HIV/AIDS disease: a case study of South Africa

HIV/AIDS disease continues to spread alarmingly despite the huge amounts of resources invested in fighting it. There is a need to integrate the series of control measures available to ensure a consistent reduction in the incidence of the disease pending the discovery of its cure. We present a deterministic model for controlling the spread of the disease using change in sexual habits and antiretroviral (ARV) therapy as control measures. We formulate a fixed time optimal control problem subject to the model dynamics with the goal of finding the optimal combination of the two control measures that will minimize the cost of the control efforts as well as the incidence of the disease. We estimate the model state initial conditions and parameter values from the demographic and HIV/AIDS data of South Africa. We use Pontryagin's maximum principle to derive the optimality system and solve the system numerically. Compared with the practice in most resource-limited settings where ARV treatment is given only to patients with full-blown AIDS, our simulation results suggest that starting the treatment as soon as the patients progress to the pre-AIDS stage of the disease coupled with appreciable change in the susceptible individuals’ sexual habits reduces both the incidence and prevalence of the disease faster. In fact, the results predict that the implementation of the proposed strategy would drive new cases of the disease towards eradication in 10 years.     

On the spatial spread of rabies among foxes

We present a simple model for the spatial spread of rabies among foxes and use it to quantify its progress in England if rabies were introduced. The model is based on the known ecology of fox behaviour and on the assumption that the main vector for the spread of the disease is the rabid fox. Known data and facts are used to determine real parameter values involved in the model. We calculate the speed of propagation of the epizootic front, the threshold for the existence of an epidemic, the period and distance apart of the subsequent cyclical epidemics which follow the main front, and finally we quantify a means for control of the spatial spread of the disease. By way of illustration we use the model to determine the progress of rabies up through the southern part of England if it were introduced near Southampton. Estimates for the current fox density in England were used in the simulations. These suggest that the disease would reach Manchester within about 3.5 years, moving at speeds as high as 100 km per year in the central region. The model further indicates that although it might seem that the disease had disappeared after the wave had passed it would reappear in the south of England after just over 6 years and at periodic times after that. We consider the possibility of stopping the spread of the disease by creating a rabies 'break' ahead of the front through vaccination to reduce the population to a level below the threshold for an epidemic to exist. Based on parameter values relevant to England, we estimate its minimum width to be about 15 km. The model suggests that vaccination has considerable advantages over severe culling.     

Revealing mechanisms of infectious disease spread through empirical contact networks

The spread of pathogens fundamentally depends on the underlying contacts between individuals. Modeling the dynamics of infectious disease spread through contact networks, however, can be challenging due to limited knowledge of how an infectious disease spreads and its transmission rate. We developed a novel statistical tool, INoDS (Identifying contact Networks of infectious Disease Spread) that estimates the transmission rate of an infectious disease outbreak, establishes epidemiological relevance of a contact network in explaining the observed pattern of infectious disease spread and enables model comparison between different contact network hypotheses. We show that our tool is robust to incomplete data and can be easily applied to datasets where infection timings of individuals are unknown. We tested the reliability of INoDS using simulation experiments of disease spread on a synthetic contact network and find that it is robust to incomplete data and is reliable under different settings of network dynamics and disease contagiousness compared with previous approaches. We demonstrate the applicability of our method in two host-pathogen systems: Crithidia bombi in bumblebee colonies and Salmonella in wild Australian sleepy lizard populations. INoDS thus provides a novel and reliable statistical tool for identifying transmission pathways of infectious disease spread. In addition, application of INoDS extends to understanding the spread of novel or emerging infectious disease, an alternative approach to laboratory transmission experiments, and overcoming common data-collection constraints.     

Interacting Epidemics and Coinfection on Contact Networks

The spread of certain diseases can be promoted, in some cases substantially, by prior infection with another disease. One example is that of HIV, whose immunosuppressant effects significantly increase the chances of infection with other pathogens. Such coinfection processes, when combined with nontrivial structure in the contact networks over which diseases spread, can lead to complex patterns of epidemiological behavior. Here we consider a mathematical model of two diseases spreading through a single population, where infection with one disease is dependent on prior infection with the other. We solve exactly for the sizes of the outbreaks of both diseases in the limit of large population size, along with the complete phase diagram of the system. Among other things, we use our model to demonstrate how diseases can be controlled not only by reducing the rate of their spread, but also by reducing the spread of other infections upon which they depend.     

Time-delayed SIS epidemic model with population awareness

This paper analyses the dynamics of infectious disease with a concurrent spread of disease awareness. The model includes local awareness due to contacts with aware individuals, as well as global awareness due to reported cases of infection and awareness campaigns. We investigate the effects of time delay in response of unaware individuals to available information on the epidemic dynamics by establishing conditions for the Hopf bifurcation of the endemic steady state of the model. Analytical results are supported by numerical bifurcation analysis and simulations.     

A comparative evaluation of modelling strategies for the effect of treatment and host interactions on the spread of drug resistance

The evolutionary responses of infectious pathogens often have ruinous consequences for the control of disease spread in the population. Drug resistance is a well-documented instance that is generally driven by the selective pressure of drugs on both the replication of the pathogen within hosts and its transmission between hosts. Management of drug resistance therefore requires the development of treatment strategies that can impede the emergence and spread of resistance in the population. This study evaluates various treatment strategies for influenza infection as a case study by comparing the long-term epidemiological outcomes predicted by deterministic and stochastic versions of a homogeneously mixing (mean-field) model and those predicted by a heterogeneous model that incorporates spatial pair-wise correlation. We discuss the importance of three major parameters in our evaluation: the basic reproduction number, the population level of treatment, and the degree of clustering as a key parameter determining the structure of heterogeneous interactions. The results show that, as a common feature in all models, high treatment levels during the early stages of disease outset can result in large resistant outbreaks, with the possibility of a second wave of infection appearing in the pair-approximation model. Our simulations demonstrate that, if the basic reproduction number exceeds a threshold value, the population-wide spread of the resistant pathogen emerges more rapidly in the pair-approximation model with significantly lower treatment levels than in the homogeneous models. We tested an antiviral strategy that delays the onset of aggressive treatment for a certain amount of time after the onset of the outbreak. The findings indicate that the overall disease incidence is reduced as the degree of clustering increases, and a longer delay should be considered for implementing the large-scale treatment.     

How do toxicants affect epidemiological dynamics?

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

A replication-competent, neuronal spread-defective, live attenuated herpes simplex virus type 1 vaccine

Herpes simplex virus type 1 (HSV-1) produces oral lesions, encephalitis, keratitis, and severe infections in the immunocompromised host. HSV-1 is almost as common as HSV-2 in causing first episodes of genital herpes, a disease that is associated with an increased risk of human immunodeficiency virus acquisition and transmission. No approved vaccines are currently available to protect against HSV-1 or HSV-2 infection. We developed a novel HSV vaccine strategy that uses a replication-competent strain of HSV-1, NS-gEnull, which has a defect in anterograde and retrograde directional spread and cell-to-cell spread. Following scratch inoculation on the mouse flank, NS-gEnull replicated at the site of inoculation without causing disease. Importantly, the vaccine strain was not isolated from dorsal root ganglia (DRG). We used the flank model to challenge vaccinated mice and demonstrated that NS-gEnull was highly protective against wild-type HSV-1. The challenge virus replicated to low titers at the site of inoculation; therefore, the vaccine strain did not provide sterilizing immunity. Nevertheless, challenge by HSV-1 or HSV-2 resulted in less-severe disease at the inoculation site, and vaccinated mice were totally protected against zosteriform disease and death. After HSV-1 challenge, latent virus was recovered by DRG explant cocultures from <10% of vaccinated mice compared with 100% of mock-vaccinated mice. The vaccine provided protection against disease and death after intravaginal challenge and markedly lowered the titers of the challenge virus in the vagina. Therefore, the HSV-1 gEnull strain is an excellent candidate for further vaccine development.     

Homogenization,sex, and differential motility predict spread of chronic wasting disease in mule deer in southern Utah

Chronic wasting disease (CWD) is an infectious prion disease that affects mule deer, along with other Cervids. It is a slow-developing, fatal disease which is rare in the free-ranging deer population of Utah. We present a sex-structured, spatial model for the spread of CWD over heterogeneous landscapes, incorporating both horizontal and environmental transmission pathways. To connect the local movement of deer to the regional spread of CWD, we use ecological diffusion with motility coefficients estimated from mule deer movement data. Ecological diffusion allows for aggregation of populations in desirable habitats and therefore allows for an interaction between density dependent disease transmission and landscape structure. The major innovation presented is use of homogenization to accelerate simulations of disease spread in southeastern Utah, from the La Sal Mountains near Moab to the Abajo Mountains near Monticello. The homogenized model provides accuracy while maintaining fidelity to small-scale habitat effects on deer distribution, including differential aggregation in land cover types with high residence times, with errors comparable to the order parameter measuring separation of small and large scales ( \(\epsilon \approx .01\) in this case). We use the averaged coefficients from the homogenized model to explore asymptotic invasion speed and the impact of current population size on disease spread in southeastern Utah.     

Social networks of disease spread in the lower Illinois valley: a simulation approach

This study illustrates the use of disease modeling and simulation techniques to the study of the spread of disease within and between social networks. A Reed-Frost type model of disease spread is used to construct a simulation of the spread of tuberculosis within three prehistoric populations of the Lower Illinois River Valley during Middle Woodland, Late Woodland, and Mississippian times. A high and low population size was modeled for each time period. Late Woodland model 2 (low population estimate) is the only model that experienced pathogen extinction with host survival. The rest of the models experienced rapid and severe host population decline. The results of the simulation suggest that a social network size of between 180 and 440 persons is required under the conditions of this model for host-pathogen coexistence (i.e., endemicity) to occur. The severe population decline experienced by these populations suggests that tuberculosis as modeled here could not have existed in these populations. Future refinements of modeling and simulation techniques can provide additional insights into how disease spreads among social contacts.     

Strategies for introducing Wolbachia to reduce transmission of mosquito-borne diseases

Certain strains of the endosymbiont Wolbachia have the potential to lower the vectorial capacity of mosquito populations and assist in controlling a number of mosquito-borne diseases. An important consideration when introducing Wolbachia-carrying mosquitoes into natural populations is the minimisation of any transient increase in disease risk or biting nuisance. This may be achieved by predominantly releasing male mosquitoes. To explore this, we use a sex-structured model of Wolbachia-mosquito interactions. We first show that Wolbachia spread can be initiated with very few infected females provided the infection frequency in males exceeds a threshold. We then consider realistic introduction scenarios involving the release of batches of infected mosquitoes, incorporating seasonal fluctuations in population size. For a range of assumptions about mosquito population dynamics we find that male-biased releases allow the infection to spread after the introduction of low numbers of females, many fewer than with equal sex-ratio releases. We extend the model to estimate the transmission rate of a mosquito-borne pathogen over the course of Wolbachia establishment. For a range of release strategies we demonstrate that male-biased release of Wolbachia-infected mosquitoes can cause substantial transmission reductions without transiently increasing disease risk. The results show the importance of including mosquito population dynamics in studying Wolbachia spread and that male-biased releases can be an effective and safe way of rapidly establishing the symbiont in mosquito populations.     

Percolation on heterogeneous networks as a model for epidemics

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.     

Effect of systemic diseases on clonal integration: modelling approach

Systemic disease spread has been suggested as a possible disadvantage of clonal plant integration. As connected ramets have higher risk of being infected, disease should cause a selective pressure against clonality. Since experimental tests of this hypothesis are not easy to perform, we chose a modelling approach, by which we could easily separate different factors influencing the process. We used a spatially explicit model of clonal growth with disease spread implemented and we tested the hypothesis that systemic disease decreases the competitive ability of highly integrated clonal plants when compared to less integrated plants with the same parameters. In contrast to our expectations, the integrator was competitively stronger than the splitter in most cases and it lost only when the disease severity and infection rates were very high. We think that the larger the integrated network is, the better the plant utilises its translocation ability. Even a very small amount of resource sharing greatly increased the relative success of the integrator and larger integrators were competitively stronger than the smaller ones. Our results also indicate that although the same infection rate caused more systemic disease in the integrator than in the splitter population, the disease has only a limited potential to select for the splitter strategy. This is caused not only by the advantages of the clonal integration but also by the fact that there is only a small range of infection rates at which there is sufficient difference in disease impact between the strategies.     

Applying a spread model to identify the entry points from which the pine wood nematode, the vector of pine wilt disease, would spread most rapidly across Europe

Pine wilt disease, which can rapidly kill pines, is caused by the pine wood nematode, Bursaphelenchus xylophilus. It is expanding its range in many countries in Asia and measures are being taken at the EU level to prevent its spread from Portugal. Due to the threat to European forests, it is important to prevent additional introductions and target surveillance to the points of entry that pose the greatest risk. In this study, we present a model to identify the European ports from which the nematode can spread most rapidly across Europe. This model describes: (1) the potential spread of the pine wood nematode based on short-distance spread (the active flight of the vector beetles) and long-distance spread (primarily due to human-mediated transportation), and (2) the development of pine wilt disease based on climate suitability and the potential spread of the nematode. Separate introductions at 200 European ports were simulated under various climate change scenarios. We found that the pine wood nematode could invade 19–60% of the study area (30°00 N–72°00 N, 25°00 W–40°00 E) by 2030, with the highest spread from ports located in Eastern and Northern Europe. Based on climate change scenarios, the disease could affect 8–34% of the study area by 2030, with the highest spread from ports located in South-Eastern Europe. This study illustrates how a spread model can be used to determine the critical points of entry for invasive species, so that surveillance can be targeted more accurately and control measures prioritised.     

A network-patch methodology for adapting agent-based models for directly transmitted disease to mosquito-borne disease

Mosquito-borne diseases cause significant public health burden and are widely re-emerging or emerging. Understanding, predicting, and mitigating the spread of mosquito-borne disease in diverse populations and geographies are ongoing modelling challenges. We propose a hybrid network-patch model for the spread of mosquito-borne pathogens that accounts for individual movement through mosquito habitats, extending the capabilities of existing agent-based models (ABMs) to include vector-borne diseases. The ABM are coupled with differential equations representing ‘clouds’ of mosquitoes in patches accounting for mosquito ecology. We adapted an ABM for humans using this method and investigated the importance of heterogeneity in pathogen spread, motivating the utility of models of individual behaviour. We observed that the final epidemic size is greater in patch models with a high risk patch frequently visited than in a homogeneous model. Our hybrid model quantifies the importance of the heterogeneity in the spread of mosquito-borne pathogens, guiding mitigation strategies.