The epidemic threshold of vector-borne diseases with seasonality

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R ₀ which is adapted to periodic environments. This R ₀ is estimated numerically for the epidemic in Chichaoua; 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor 3.76.     

The effects of human movement on the persistence of vector-borne diseases

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R₀ for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R₀<1) then it is globally stable and if the disease-free equilibrium is unstable (R₀>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.     

Transmission-virulence trade-offs in vector-borne diseases

Though it is commonly supposed that there is a trade-off between virulence and transmission, there is little data and little insight into what it should look like. Here, we consider the specific case of vector-borne parasites (inspired by human malaria) and analyse an embedded model to understand how specific life-cycle aspects may affect this trade-off. First, we find that, for such parasites, the transmission function may have an S-shape. Second, we find that the trade-off obtained for vector-borne parasites is less sensitive to parameter variations than the trade-off obtained for directly transmitted parasites. Third, we find that other parasite traits, such as the conversion from replicative to infective stages, could have important epidemiological implications. Finally, we compare the effect of treatments targeting either the asexual or the sexual parasite life-stage.     

The importance of culling in Johne's disease control

Johne's disease is caused by Mycobacterium avium subsp. paratuberculosis (MAP) infection and results in economic losses in the dairy industry. To control MAP transmission in herds, test-based culling has been recommended and immediate culling of high shedding animals is typically implemented. In this study, we quantified the effects of MAP control in US dairy herds, using the basic reproduction ratio R₀. The effectiveness of culling strategies was evaluated for good and poor herd management (low- and high-transmission rates, respectively) by a phase diagram approach. To establish a quantitative relationship between culling rates and test properties, we defined the average detection times for low and high shedding animals. The effects of various culling strategies and test characteristics, such as test sensitivity, test turnaround time, and testing interval, were analyzed. To understand the overall effect of model parameters on R₀, we performed global uncertainty and sensitivity analyses. We also evaluated the effectiveness of culling only high shedding animals by comparing three test methods (fecal culture, fecal polymerase chain reaction, PCR, and enzyme-linked immunosorbent assay, ELISA). Our study shows that, in the case of good herd management, culling of only high shedding animals may be effective in controlling MAP transmission. However, in the case of poor management, in addition to immediate culling of high shedding animals, culling of low shedding animals (based on the fecal culture test) will be necessary. Culling of low shedding animals may be delayed 6-12 months, however, if a shorter testing interval is applied. This study suggests that if farmers prefer culling only high shedding animals, faster MAP detection tests (such as the fecal PCR and ELISA) of higher sensitivity should be applied with high testing frequency, particularly on farms with poor management. Culling of infectious animals with a longer testing interval is generally not effective to control MAP.     

The control of vector-borne disease epidemics

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, R₀, the measure of long term endemicity. Despite its widespread application, R₀ does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, E₀, which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on E₀ and R₀ may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector-host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index E₀ to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.     

Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia

Cyclical neutropenia (CN) is a rare hematopoietic disorder in which the patient's neutrophil level drops to extremely low levels for a few days approximately every three weeks. CN is effectively treated with granulocyte colony stimulating factor (G-CSF), which is known to interfere with apoptosis in neutrophil precursors and to consequently increase the circulating neutrophil level. However, G-CSF treatment usually fails to eliminate the oscillation. In this study, we establish an age-structured model of hematopoiesis, which reduces to a set of four delay differential equations with specific forms of initial functions. We numerically investigate the possible stable solutions of the model equations with respect to changes in the parameters as well as the initial conditions. The results show that the hematopoietic system possesses multistability for parameters typical of the normal healthy state. From our numerical results, decreasing the proliferation rate of neutrophil precursors or increasing the stem cell death rate are two possible mechanisms to induce cyclical neutropenia, and the periods of the resulting oscillations are independent of the changing parameters. We also discuss the dependence of the model solution on the initial condition at normal parameter values corresponding to a healthy state. Using insight from our results we design a hybrid treatment method that is able to abolish the oscillations in CN.     

Forward hysteresis and backward bifurcation caused by culling in an avian influenza model

The emerging threat of a human pandemic caused by the H5N1 avian influenza virus strain magnifies the need for controlling the incidence of H5N1 infection in domestic bird populations. Culling is one of the most widely used control measures and has proved effective for isolated outbreaks. However, the socio-economic impacts of mass culling, in the face of a disease which has become endemic in many regions of the world, can affect the implementation and success of culling as a control measure. We use mathematical modeling to understand the dynamics of avian influenza under different culling approaches. We incorporate culling into an SI model by considering the per capita culling rates to be general functions of the number of infected birds. Complex dynamics of the system, such as backward bifurcation and forward hysteresis, along with bi-stability, are detected and analyzed for two distinct culling scenarios. In these cases, employing other control measures temporarily can drastically change the dynamics of the solutions to a more favorable outcome for disease control.     

Localized reactive badger culling increases risk of bovine tuberculosis in nearby cattle herds

Human and livestock diseases can be difficult to control where infection persists in wildlife populations. Control of bovine tuberculosis (bTB) in British cattle is complicated by the maintenance of Mycobacterium bovis (the causative agent of bTB) in badgers, acting as reservoirs of infection. Although over 20 000 badgers were culled to control bTB between 1975 and 1997, the incidence of bTB in cattle has substantially increased in parts of Great Britain in recent decades. Our case-control study, involving 1208 cattle herds, provides further evidence of the detrimental effect of localized reactive badger culling in response to the disclosure of a confirmed bTB herd breakdown in cattle. The presence of any reactive badger culling activity and increased numbers of badgers culled in the vicinity of a herd were associated with significantly increased bTB risk, even after adjusting for other important local risk factors. Such findings may partly explain why some earlier localized approaches to bTB control were ineffective.     

具有时滞和自食的非自治阶段结构竞争系统的周期解

研究了一类具有时滞和自食的非自治阶段结构竞争系统的动力学行为,其中一种群分幼年和成年两个阶段.当发生自食现象时,利用重合度理论中的延拓定理,得到该系统正周期解存在的充分条件.     

Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever

We present two ordinary differential equation models for Rift Valley fever (RVF) transmission in cattle and mosquitoes. We extend existing models for vector-borne diseases to include an asymptomatic host class and vertical transmission in vectors. We define the basic reproductive number, ₀, and analyse the existence and stability of equilibrium points. We compute sensitivity indices of ₀ and a reactivity index (that measures epidemicity) to parameters for baseline wet and dry season values. ₀ is most sensitive to the mosquito biting and death rates. The reactivity index is most sensitive to the mosquito biting rate and the infectivity of hosts to vectors. Numerical simulations show that even with low equilibrium prevalence, increases in mosquito densities through higher rainfall, in the presence of vertical transmission, can result in large epidemics. This suggests that vertical transmission is an important factor in the size and persistence of RVF epidemics.     

Modeling relapse in infectious diseases

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.     

Bovine virus diarrhea and the vector-borne diseases Anaplasmosis and Bluetongue: a sero-surveillance in free-ranging red deer (<Emphasis Type="Italic">Cervus elaphus</Emphasis>) in selected areas of Switzerland

Due to climate changes, diseases emerging from southerly adjacent areas (Mediterranean countries) are likely to spread northward. Expanded migration of red deer harbors the risk of introducing new pathogens into a naive population of either wild or domestic animals. Little is known about the importance of red deer as a potential reservoir for diseases of domestic ruminants in Switzerland. Deer is susceptible for all three agents that were selected in this study: bovine virus diarrhea virus (BVDV), Anaplasma marginale (AM), and Bluetongue virus (BTV). The goal of this project was to establish the serological status of red deer in Switzerland concerning BVDV, AM, and BTV, and to assess the possible impact of disease dynamics with a focus on potential transmission of these diseases from red deer to cattle or vice versa. Sampling areas were selected according the following criteria: abundance of red deer, potential insect vector distribution due to climatic conditions, and traditional alpine pasture husbandry along with known migration routes of red deer. Blood samples were collected during the regular hunting season 2004 and 2005 by hunters and gamekeepers. There was no serological evidence for the presence of the vector-borne diseases AM and BT in red deer in Switzerland. Four out of 234 sera showed a positive result for BVD, corresponding to a sero-prevalence of 1.7% (95% CI 0.46–4.38). Facing the fact of the high sero-prevalence for BVD in Swiss cattle (60–80%) disease transmission from red deer to cattle in these areas under investigation is rather unlikely.     

Modeling Tick-Borne Disease: A Metapopulation Model

Recent increases in reported outbreaks of tick-borne diseases have led to increased interest in understanding and controlling epidemics involving these transmission vectors. Mathematical disease models typically assume constant population size and spatial homogeneity. For tick-borne diseases, these assumptions are not always valid. The disease model presented here incorporates non-constant population sizes and spatial heterogeneity utilizing a system of differential equations that may be applied to a variety of spatial patches. We present analytical results for the one patch version and find parameter restrictions under which the populations and infected densities reach equilibrium. We then numerically explore disease dynamics when parameters are allowed to vary spatially and temporally and consider the effectiveness of various tick-control strategies.     

具有脉冲效应的时滞捕食系统的正周期解

提出了一个具有脉冲效应的周期时滞捕食系统,运用叠合的方法研究了该系统的周期解的存在性．结论表明在一定的条件下捕食者和食饵周期变化．     

具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性

主要研究具有脉冲和时滞的Lotka-Volterra系统的正周期解的存在性和全局渐近稳定性．     

Evidence for transmission of the zoonotic apicomplexan parasite Babesia duncani by the tick Dermacentor albipictus

Babesiosis is a potentially fatal tick-borne zoonotic disease caused by a species complex of blood parasites that can infect a variety of vertebrates, particularly dogs, cattle, and humans. In the United States, human babesiosis is caused by two distinct parasites, Babesia microti and Babesia duncani. The enzootic cycle of B. microti, endemic in the northeastern and upper midwestern regions, has been well characterised. In the western United States, however, the natural reservoir host and tick vector have not been identified for B. duncani, greatly impeding efforts to understand and manage this zoonotic disease. Two and a half decades after B. duncani was first described in a human patient in Washington State, USA, we provide evidence that the enzootic tick vector is the winter tick, Dermacentor albipictus, and the reservoir host is likely the mule deer, Odocoileus hemionus. The broad, overlapping ranges of these two species covers a large portion of far-western North America, and is consistent with confirmed cases of B. duncani in the far-western United States.