Host patch selection induced by parasitism: basic reproduction ratio r(0) and optimal virulence

The effects of parasites on the behavior of their hosts are well documented. For example, parasites may affect the habitat selection of the host individual. We used variables aggregation methods to investigate the way in which parasites affect the spatial pattern of susceptible hosts. We developed a simple epidemiological model, taking into account both the reproduction processes of hosts (density-dependent birth and death) and infection, considered separately on two different patches, and the migration of susceptible hosts between these two patches. We used the complete model of three equations to generate an aggregated model describing the dynamics of the combined susceptible and infected host populations. We obtained the basic reproduction ratio (R(0)) from the aggregated model, and then studied the effect of the migratory behavior of susceptible hosts on the ability of the parasite to invade the system. We also used the basic reproduction ratio to investigate the evolution of parasite virulence in relation to the migration decisions of susceptible hosts. We found that host investment in avoidance of the infected patch leads to an increase in optimal virulence if host investment is costly.     

一个具有干预策略的肺结核模型的阈值动力学行为（英文）

文献[4]研究了肺结核传播的动力学行为.该文献仅从数值模拟上分析了疾病的传播和不同策略对疾病传播的影响.本文从理论上对疾病传播和不同策略对疾病传播的影响进行了分析.主要结论如下：得到了模型的基本再生数R_0.R_0决定了疾病传播的动力学行为：如果R_0〈1,则模型仅有一个无病平衡点且是局部渐近稳定的,若R_0〉1则模型存在一个地方病平衡点并且疾病是一致持续的.本文还得到了无病平衡点全局渐近稳定的充分条件.     

Competition of pathogen strains leading to infection with variable infectivity and the effect of treatment

A model for the spread of two strains of a pathogen leading to an infection with variable infectivity is considered. The course of infection is described by two stages with different infectivity levels. The model is extended to account for treatment by including a third stage with different infectivity and survival for those treated. The contribution of each stage to incidence and prevalence is investigated and the effect of infectivity and survival on the basic reproduction ratio is examined. Standard equilibrium analysis is performed for both models, revealing that the successful strain is the one with highest reproduction ratio. If therapy, however, is more effective against the strain that wins in the absence of treatment and its reproduction ratio is sufficiently reduced, it might be outcompeted by the other strain after treatment becomes widely available. In this case, early introduction of treatment can prevent a major 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.     

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.     

Influence of spatial structure on the maintenance of sexual reproduction

We include spatial extension into a model for the maintenance of sexual reproduction introduced recently. The model is based on a broad spectrum of resources, which regrow slowly. Other key features of the model are that sexual reproduction sets in when resources become scarce and that only a few genotypes can coexist locally. The extension of the model to several patches in space is done in two different ways. Model A is based on central egg deposition and allows migration of juveniles into all patches. Model B has a one-dimensional array of patches with migration only between neighboring patches. The main findings are that wide dispersal favors asexuals, while for slower migration there is a wide range of parameter values for which sexually reproducing species always win against asexuals. These results are conform with major patterns for the distribution of parthenogenesis in animals and plants, i.e. the prevalence of parthenogenetic reproduction in minute species, which are easily dispersed by physical forces, such as protists and small metazoans including e.g. bdelloid rotifers, tardigrades and nematodes.     

A Theoretical Assessment of the Effects of Smoking on the Transmission Dynamics of Tuberculosis

Smoking has long being associated with tuberculosis. We present a tuberculosis dynamics model taking into account the fact that some people in the population are smoking in order to assess the effects of smoking on tuberculosis transmission. The epidemic thresholds known as the reproduction numbers and equilibria for the model are determined and stabilities analyzed. Qualitative analysis of the model including positivity and persistence of solutions are presented. The model is numerically analyzed to assess the effects of smoking on the transmission dynamics of tuberculosis. Numerical simulations of the model show that smoking enhances tuberculosis transmission, progression to active disease and in a population of smokers, tuberculosis cannot be controlled even when treatment success is assumed to be as high as 88%. Further, analysis of the reproduction numbers indicates that the number of active tuberculosis cases increases as the number of smokers increase.     

A reaction–diffusion malaria model with incubation period in the vector population

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R₀{\mathcal{R}_0} and show that R₀{\mathcal{R}_0} serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.     

Allee effects can both conserve and create spatial heterogeneity in population densities

In order to determine conditions which allow the Allee effect (caused by biparental reproduction) to conserve and create spatial heterogeneity in population densities, we studied a deterministic model of a symmetric two-patch metapopulation. We proved that under certain conditions there exist stable equilibria with unequal population densities in the two patches, a situation which can be interpreted as conserved heterogeneity. Furthermore, the Allee effect can lead to instability of the equilibrium with equal population densities if some degree of competition is assumed to occur between the subpopulations (non-local competition). This indicates the potential of the Allee effect to create spatial heterogeneity. Neither of these effects appear under biologically realistic parameter values in a model where uniparental reproduction is assumed. We proved that both the between-patch migration intensity and the degree of non-local competition are decisive in determining boundaries between these types of behaviour of the spatial system with Allee effect. Therefore, we propose that the Allee effect, migration intensity, and non-local competition should be considered jointly in studies focusing on problems like pattern formation in space and invasions of spreading species.     

Impact of Travel Between Patches for Spatial Spread of Disease

A multipatch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation is obtained along with the basic reproduction number of the system of patches, ℜ₀. Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of ℜ₀ on the travel rates between the two patches are obtained. For parameter values relevant for influenza, these results show that, while banning travel of infectives from the low to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel of infected individuals from the high to the low prevalence patch could result in the low prevalence patch becoming diseasefree, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in this patch alone exceeding the combined number of infectives in both patches without border control. Under the set of parameter values used, our results demonstrate that if border control is properly implemented, then it could contribute to stopping the spatial spread of disease between patches.     

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.     

Estimates for local and movement-based transmission of bovine tuberculosis in British cattle

Both badgers and livestock movements have been implicated in contributing to the ongoing epidemic of bovine tuberculosis (BTB) in British cattle. However, the relative contributions of these and other causes are not well quantified. We used cattle movement data to construct an individual (premises)-based model of BTB spread within Great Britain, accounting for spread due to recorded cattle movements and other causes. Outbreak data for 2004 were best explained by a model attributing 16% of herd infections directly to cattle movements, and a further 9% unexplained, potentially including spread from unrecorded movements. The best-fit model assumed low levels of cattle-to-cattle transmission. The remaining 75% of infection was attributed to local effects within specific high-risk areas. Annual and biennial testing is mandatory for herds deemed at high risk of infection, as is pre-movement testing from such herds. The herds identified as high risk in 2004 by our model are in broad agreement with those officially designated as such at that time. However, border areas at the edges of high-risk regions are different, suggesting possible areas that should be targeted to prevent further geographical spread of disease. With these areas expanding rapidly over the last decade, their close surveillance is important to both identify infected herds qucikly, and limit their further growth.     

On the role of social clusters in the transmission of infectious diseases

We introduce a spatial stochastic model for the spread of tuberculosis and HIV. We have three parameters: the size of the social cluster for each individual and the infection rates within and outside the social cluster. We show that when the infection rate from outside the cluster is low (this is presumably the case for tuberculosis and HIV) then an epidemic is possible only if the typical social cluster and the within infection rate are large enough. These results may be important in formulating new hypotheses for the transmission of TB and HIV.     

On the role of reinfection in the transmission of infectious diseases

We introduce a spatial stochastic model for the spread of tuberculosis. After a primary infection, an individual may become sick (and infectious) through an endogenous reinfection or through an exogenous reinfection. We show that even in the absence of endogenous reinfection an epidemic is possible if the exogenous reinfection parameter is high enough. This is in sharp contrast with what happens for a mean field model corresponding to our spatial stochastic model.     

一类CD4＋T细胞感染HIV病毒模型的全局稳定性

研究了一类具有标准发生率的CD4＋T细胞感染HIV病毒模型的动力学性质．通过分析,得到了病毒消除与否的阚值一基本再生数．证明了当基本再生数小于1时,未感染病毒平衡点全局渐近稳定,病毒将在宿主体内被清除．当基本再生数大于1时,病毒将在宿主体内持续生存,进一步给出了病毒感染平衡点全局渐近稳定的条件．最后对所得结论进行了数值模拟．     

Can Vaccination Save a Zika Virus Epidemic?

Zika virus (ZIKV) is a vector-borne disease that has rapidly spread during the year 2016 in more than 50 countries around the world. If a woman is infected during pregnancy, the virus can cause severe birth defects and brain damage in their babies. The virus can be transmitted through the bites of infected mosquitoes as well as through direct contact from human to human (e.g., sexual contact and blood transfusions). As an intervention for controlling the spread of the disease, we study a vaccination model for preventing Zika infections. Although there is no formal vaccine for ZIKV, The National Institute of Allergy and Infectious Diseases (part of the National Institutes of Health) has launched a vaccine trial at the beginning of August 2016 to control ZIKV transmission, patients who received the vaccine are expected to return within 44 weeks to determine if the vaccine is safe. Since it is important to understand ZIKV dynamics under vaccination, we formulate a vaccination model for ZIKV spread that includes mosquito as well as sexual transmission. We calculate the basic reproduction number of the model to analyze the impact of relatively, perfect and imperfect vaccination rates. We illustrate several numerical examples of the vaccination model proposed as well as the impact of the basic reproduction numbers of vector and sexual transmission and the effect of vaccination effort on ZIKV spread. Results show that high levels of sexual transmission create larger cases of infection associated with the peak of infected humans arising in a shorter period of time, even when a vaccine is available in the population. However, a high level of transmission of Zika from vectors to humans compared with sexual transmission represents that ZIKV will take longer to invade the population providing a window of opportunities to control its spread, for instance, through vaccination.     

Invasion by extremes: population spread with variation in dispersal and reproduction

For populations having dispersal described by fat-tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat-tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment-generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment-generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat-tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest-forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as (1/T)(piuR)/2)(1/2) m yr(-1), where T is generation time, and u is a distance parameter (m2) of Clark et al.'s 2Dt model having shape parameter p = 1. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life-history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.     

An outbreak vector-host epidemic model with spatial structure: the 2015–2016 Zika outbreak in Rio De Janeiro

Background

A deterministic model is developed for the spatial spread of an epidemic disease in a geographical setting. The disease is borne by vectors tosusceptible hosts through criss-cross dynamics. The model is focused on an outbreak that arises from a small number of infected hosts imported into a subregion of the geographical setting. The goal is to understand how spatial heterogeneity of the vector and host populations influences the dynamics of the outbreak, in both the geographical spread and the final size of the epidemic.

Methods

Partial differential equations are formulated to describe the spatial interaction of the hosts and vectors. The partial differential equations have reaction-diffusion terms to describe the criss-cross interactions of hosts and vectors. The partial differential equations of the model are analyzed and proven to be well-posed. A local basic reproduction number for the epidemic is analyzed.

Results

The epidemic outcomes of the model are correlated to the spatially dependent parameters and initial conditions of the model. The partial differential equations of the model are adapted to seasonality of the vector population, and applied to the 2015–2016 Zika seasonal outbreak in Rio de Janeiro Municipality in Brazil.

Conclusions

The results for the model simulations of the 2015–2016 Zika seasonal outbreak in Rio de Janeiro Municipality indicate that the spatial distribution and final size of the epidemic at the end of the season are strongly dependent on the location and magnitude of local outbreaks at the beginning of the season. The application of the model to the Rio de Janeiro Municipality Zika 2015–2016 outbreak is limited by incompleteness of the epidemic data and by uncertainties in the parametric assumptions of the model.

     

Analysis of a mathematical model for tuberculosis: What could be done to increase case detection

This paper presents qualitative and quantitative study of a TB mathematical model to test results from a survey carried out in Benin City, Nigeria. The purpose of the survey was to determine factors that could enhance the case detection rate of tuberculosis. Results from the survey identified four key factors that must be combined for an effective control of TB and increase the case detection rate: effective awareness programme, active cough identification, associated cost factor for treatment of identified cases and effective treatment. The overall effect of these factors on the basic reproduction number under treatment, R_T, of the TB model was considered. In all, a serious concentration on tuberculosis awareness programmes and active cough identification as a marker for someone having TB was shown to significantly reduce the value of the reproduction number, hereby reducing the severity of the disease in the presence of treatment.     

Unifying Viral Genetics and Human Transportation Data to Predict the Global Transmission Dynamics of Human Influenza H3N2

Information on global human movement patterns is central to spatial epidemiological models used to predict the behavior of influenza and other infectious diseases. Yet it remains difficult to test which modes of dispersal drive pathogen spread at various geographic scales using standard epidemiological data alone. Evolutionary analyses of pathogen genome sequences increasingly provide insights into the spatial dynamics of influenza viruses, but to date they have largely neglected the wealth of information on human mobility, mainly because no statistical framework exists within which viral gene sequences and empirical data on host movement can be combined. Here, we address this problem by applying a phylogeographic approach to elucidate the global spread of human influenza subtype H3N2 and assess its ability to predict the spatial spread of human influenza A viruses worldwide. Using a framework that estimates the migration history of human influenza while simultaneously testing and quantifying a range of potential predictive variables of spatial spread, we show that the global dynamics of influenza H3N2 are driven by air passenger flows, whereas at more local scales spread is also determined by processes that correlate with geographic distance. Our analyses further confirm a central role for mainland China and Southeast Asia in maintaining a source population for global influenza diversity. By comparing model output with the known pandemic expansion of H1N1 during 2009, we demonstrate that predictions of influenza spatial spread are most accurate when data on human mobility and viral evolution are integrated. In conclusion, the global dynamics of influenza viruses are best explained by combining human mobility data with the spatial information inherent in sampled viral genomes. The integrated approach introduced here offers great potential for epidemiological surveillance through phylogeographic reconstructions and for improving predictive models of disease control.