Seasonal and spatial heterogeneities in host and vector abundances impact the spatiotemporal spread of bluetongue
© Charron et al.; licensee BioMed Central Ltd. 2013
Received: 6 February 2013
Accepted: 7 June 2013
Published: 19 June 2013
Bluetongue (BT) can cause severe livestock losses and large direct and indirect costs for farmers. To propose targeted control strategies as alternative to massive vaccination, there is a need to better understand how BT virus spread in space and time according to local characteristics of host and vector populations. Our objective was to assess, using a modelling approach, how spatiotemporal heterogeneities in abundance and distribution of hosts and vectors impact the occurrence and amplitude of local and regional BT epidemics. We built a reaction–diffusion model accounting for the seasonality in vector abundance and the active dispersal of vectors. Because of the scale chosen, and movement restrictions imposed during epidemics, host movements and wind-induced passive vector movements were neglected. Four levels of complexity were addressed using a theoretical approach, from a homogeneous to a heterogeneous environment in abundance and distribution of hosts and vectors. These scenarios were illustrated using data on abundance and distribution of hosts and vectors in a real geographical area. We have shown that local epidemics can occur earlier and be larger in scale far from the primary case rather than close to it. Moreover, spatial heterogeneities in hosts and vectors delay the epidemic peak and decrease the infection prevalence. The results obtained on a real area confirmed those obtained on a theoretical domain. Although developed to represent BTV spatiotemporal spread, our model can be used to study other vector-borne diseases of animals with a local to regional spread by vector diffusion.
There is significant concern regarding the resurgence or emergence of vector-borne diseases of animals with serious consequences for animal health and economics [1–3]. Climate change and socio-economic change are both believed to contribute to the emergence and spread of such diseases . A recent example is the unexpected introduction of the serotype 8 of the bluetongue virus (BTV) in northern Europe in 2006. Bluetongue is a non-contagious vector-borne disease affecting domestic and wild ruminants with high direct and indirect economic consequences [5, 6]. It spread for three years with an annual epidemic peak followed by the disappearance of clinical cases.
A better understanding of the temporal and spatial spread of BTV has direct consequences for the disease prevention and control. The recent incursion of BTV in Europe has been controlled using a massive vaccination. To propose alternative to such a massive strategy if BTV incursions were to occur, we need to better identify where and when targeted strategies should be implemented. Therefore, the occurrence and amplitude of both local (a few km2) and regional epidemics should be more precisely predict.
The spatiotemporal heterogeneity in abundance and distribution of hosts and vectors generally has a strong impact on pathogen spread and persistence [7, 8]. In a seasonal environment such as in Europe, bluetongue is characterized by strong seasonal variations in incidence related to the seasonality of the vector population [9, 10], whose lifecycle largely depends on environmental factors, such as humidity and temperature. As a result, clinical cases almost disappear during the unfavourable season for the vector. In addition to the temporal heterogeneity in vector abundance, the heterogeneity in the spatial distribution of vectors and hosts may also impact bluetongue spread [11, 12]. Such heterogeneities in vector and host abundance and distribution can differ between geographic areas. In livestock populations, they relate to the landscape structure as well as to farming practices as animal populations are managed by farmers.
Mathematical modelling is a relevant approach for investigating the spread of vector-borne diseases [8, 13–16]. As hosts and vectors are mobile entities, a spatial component in vector-borne disease models should be taken into account to better consider the evolution of the biological system [17–19]. This spatial component is not only due to space structuring in terms of density and location of host and vector populations, but also to host and vector movements over space. Different methods of various levels of complexity exist to include this spatial component in epidemiological models. To study the spread of vector-borne diseases, spatially explicit models are generally preferred [13, 20]. They permit to take into account both vector active and passive movements and host movements that occur at different scales. Moreover, such models have been used also to describe the velocity of travelling waves of epidemics [8, 14, 16].
Such a modelling approach has been used to represent the spatiotemporal BTV spread in specific areas [21–24]. If the published models took into account the heterogeneity in distribution and abundance of livestock populations, they tend to assume uniform densities of vectors (i.e. the same number per farm) or uniform host: vector ratio (i.e. more midges on bigger farms) [21–24]. Recently, it has been shown that these are probably unrealistic . Therefore, models that take full account of vector heterogeneity both in space and time have to be developed.
Our objective was to assess, using a modelling approach, how spatiotemporal heterogeneities in abundance and distribution of hosts and vectors impact the occurrence and amplitude of local and regional BT epidemics. We first studied different hypothetical scenarios of spatial heterogeneity in host and vector populations, and then illustrated the theoretical results in a real geographic area.
Material and methods
Parameters of the model of BTV8 spread in midge and cattle populations
Birth rate (per day)
Exit rate (mortality, selling, culling) (per day)
Duration of viremia (days)
Probability of transmission from vector to host
Probability of transmission from host to vector
Biting rate (per day)
Fertility rate (per day)
Mortality rate (per day)
Density-dependence mortality rate (per day)
(b V -m V )/K(t)
Maximum of K(t)
Duration of favourable period (days)
Number of vectors during the unfavourable period
Duration of extrinsic incubation period (days)
Diffusion coefficient (km2/day)
Let Ω be the square spatial domain. X = X (x, y, t) represents time dependant population densities in (x, y) ∈ Ω. During the epidemic, host movements are controlled, therefore the spatial spread of the epidemic is due to vector movements rather than host movements. Moreover, we focus on a local to regional scale, and therefore assume that the spatial spread of BTV8 is exclusively due to local movements of vectors. BTV8 having no detrimental impact on vectors, thereby the diffusion process is similar whatever the health state. Therefore, the diffusion process follows the first Fick’s law:
To discretize the problem (Eq. 1 and 2) we used the finite difference method in space, and we converted the continuous time model into a discrete time one by using the semi-implicit Euler method, that we implemented in Scilab 5.1.
Hypotheses of spatial heterogeneities in hosts and vectors
Hypotheses of spatial heterogeneities in abundance and distribution of hosts and vectors
Homogeneous in hosts and vectors (4 Scenarios)
500 S/C 1 I in central cell Total number of hosts = 840 501
100 S/C h1=106, or h2=107 or h3=108, or h4= 109
Heterogeneous in hosts Homogeneous in vectors (16 scenarios)
Total number of hosts S≈ 840 501 Four densities of occupied cells (Figure 2): 90% OC: 554 S/OC75% OC: 666 S/OC50% OC: 1000 S/OC25% OC: 2025 S/OC 1 I in central cell
6 and 7
Heterogeneous in vectors Homogeneous in hosts (2 scenarios)
100 S/C Grid divided into four sub-areas of different maximum carrying capacities (h1, h2, h3 and h4) (Figure 3a)
100 S/C 25% C: h1, 25% C: h2, 25% C: h3, 25% C: h4 (Figure 3b)
9 and 10
Heterogeneous in hosts and vectors (6 scenarios)
Crossing hypotheses: H2-H4
10 and 11
Real area (Figure 4b)
Real area (Figure 4a) Multiplication of the number of trapped vectors by 100 or 1000
Thirdly, we considered a heterogeneous distribution of hosts and vectors simultaneously (H5). This hypothesis crosses previous hypotheses H2 and H4.
Fourth, a last hypothesis (H6), based on real data, served to illustrate this theoretical work, in particular hypothesis H5.
The date and the prevalence at the epidemic peak in each cell were analysed, as well as the total prevalence on the grid over time. Thereafter, these three outputs are respectively named peak date, local prevalence and total prevalence. The peak dates were compared among the cells located on the four lines between the central cell (the half-diagonal), i.e. the cell of virus introduction, and the corners of the grid. This enabled us to numerically calculate an effective speed of the virus spread. We calibrated the diffusion coefficient (D) and the initial conditions (SH00 0 and SV 0 ) to have an effective speed of the virus spread similar to the estimated velocity by Pioz et al. , for hypothesis H1 and a maximum of the carrying capacity in vectors equals to 107. The theoretical grid is a 41 × 41 km square, each half-diagonal measuring about 29 km. For hypotheses H3 to H6, peak dates and local prevalences were studied for comparable cells, i.e. equidistant and having the same maximum of the carrying capacity in vectors.
Homogeneous in abundance and distribution of hosts and vectors (H1)
Heterogeneous in abundance and distribution of hosts and homogeneous in abundance and distribution of vectors (H2)
The total prevalence on the grid over time confirms these results. A delay of the epidemic is observed as the density of host-occupied cells decreases. There is also a decrease of the total prevalence compared with H1 (Figure 6).
Heterogeneous in abundance and distribution of vectors and homogeneous in abundance and distribution of hosts (H3, H4)
Hypothesis 3: definition of four subareas
Hypothesis 4: variable maximum of carrying capacities in vectors
Heterogeneous in abundance and distribution of hosts and vectors (H5, H6)
Hypothesis 5: theoretical landscape
The total prevalence on the grid over time confirms these results. A delay in the epidemic is observed as the density of host-occupied cells decreases, as well as a decrease in the total prevalence compared with H1 (Figure 7, Figure 10). A balance is observed between H2-H5 and H4-H5. As for H2 and for a maximum of carrying capacity in vectors of h2, the lower the density of host-occupied cells, the longer the peak date is delayed and the lower is the total prevalence (Figure 7, Figure 10). However, the total prevalence for the highest density of occupied cells is similar with hypothesis H4 (Figure 10). Therefore, as soon as the distribution and the abundance of vectors are heterogeneous, they strongly influence the global epidemic dynamics.
Hypothesis 6: application to a real landscape
In addition, the number of hosts in each cell has an impact on the peak date and the local prevalence. For equidistant cells having the same maximum of the carrying capacity in vectors (e.g. cells A2 and C1; Figure 4a), the cell having the largest number of hosts (C1) shows a later peak date and a lower local prevalence than the cell having the lowest number of hosts (A2; Figure 12).
A mathematical modelling approach allowed us to assess the impact of spatiotemporal heterogeneities in abundance and distribution of hosts and vectors on the spatiotemporal spread of BTV8. Individually and jointly, the heterogeneities in abundance and distribution of hosts and vectors delay the peak date and decrease the total infection prevalence. The different hypotheses of heterogeneity that we have tested allowed us to highlight the importance of the maximum of the carrying capacity in vectors and its influence on the spread of BTV8 within each cell. Indeed, cells next to the primary case can become infected later than more distant cells, if the maximum of carrying capacity is lower. Moreover, the density of cells occupied by hosts plays an important role in cases where the maximum carrying capacity in vectors is low (for homogeneous conditions for vectors) and when hosts and vectors are heterogeneous.
The spatial heterogeneity in host and/or vector abundances influences the infection frequency [7, 17, 29]. In our study, a decrease in the density of cells occupied by hosts results in a delay of the peak date and a decrease in the infection prevalence if vectors are homogeneously distributed and their population is large. In the case where hosts are homogeneously distributed, the same trend is observed for cells where the maximum of carrying capacity in vectors is large. However, for cells where this maximum is the lowest, an epidemic can occur, in contrast to the case with homogeneous vector and host populations. The coupling of heterogeneities in hosts and vectors increases the delay of the epidemic and decreases the prevalence.
Different models of the spatial spread of bluetongue have previously been published for specific geographic areas [21–24]. Szmaragd’s model describes the BTV spread within and between farms in Great Britain via a generic kernel, which includes both animal and vector movements. Ducheyne’s model was calibrated with data from the BTV1 and BTV8 epidemics in Southern France. It describes the spatiotemporal BTV spread between farms assuming that the number of new cases per week is half attributable to local dispersion (active) of vectors, and half to long-distance dispersion (passive) of vectors by the wind. Graesboll’s model describes the BTV spread with a high spatial resolution, which includes both animal and vector movements and the seasonality of vectors. Turner’s model is a network model. It takes into account explicitly the spatial dispersal of both hosts and vectors and a seasonal vector to host ratio . It studies the BTV spread between farms in England. Taking into account climatic and environmental data, all of these models consider the spatial heterogeneity of the landscape. Our model advances the field by representing spatial heterogeneity in both hosts and vectors. Here, we highlighted how such heterogeneities concretely impact BTV spread. Moreover, the seasonality of the vector population is managed by a simple function that can be easily related to observed data of vector abundance.
We chose to model the spatiotemporal BTV8 spread by a reaction–diffusion model. Several shapes of the transmission kernel are possible, but it is difficult to choose the most appropriate one to describe the dynamics of a pathogen spread. Indeed, Szmaragd et al. showed that a Gaussian kernel was the most appropriate to describe the BTV8 spread in northern Europe during 2006 . If this kernel shape, comparable to reaction diffusion models, has been shown to underestimate the probability of the long-distance transmission, and thus is not appropriate to describe the pathogen spread on a larger scale , it can be used on a smaller scale. Graesboll et al. used a Gaussian kernel too, but coupled this approach with the wind dispersion . In our study, the theoretical spatial domain is a 41 × 41 km square. The primary case is always located in the centre of the square, i.e. at 29 km from the domain edges. One limitation is that long-distance dispersal has been neglected; the wind dispersal responsible for the passive movements of vectors generating dispersal up to several hundred kilometres [31, 32]. Coupling short and long-distance dispersal is necessary to study arbovirus spread in animal populations once the spatial scale is large enough that host movements and passive movements of vectors cannot be neglected anymore [33–35].
Observational studies have been conducted to assess risk areas and to predict the spatiotemporal spread of BTV in BTV-free areas [11, 36, 37]. These studies have shown that the landscape heterogeneity, climatic conditions, the distribution and the density of host populations and the abundance of vectors were linked and influenced BTV spatiotemporal spread. Our model highlights similar findings but also allows us to distinguish between the impact of vector versus host heterogeneity. Maps of the basic reproduction number have highlighted the link between vector abundance and BTV spread . However, the vector abundance is difficult to quantify. Hartemink et al. have used trapping data, multiplying the number of trapped Culicoides by 100 to obtain a local density of vectors . Our real geographic area illustrates these differences in local abundance. On a small scale, large differences may exist between cells, whether they are occupied by hosts or not. Entomological studies identify and quantify the different vector species present in different geographic locations [9, 10, 38]. However, the real number of vectors remains difficult to approximate. As shown in our results, the abundance in vectors has a significant impact on the date and on the observed prevalence at the epidemic peak. However, by multiplying the number of vectors by 100 or 1000, we obtained the same qualitative findings on the real geographic area.
Modelling is a relevant approach to investigate the concept of spatiotemporal heterogeneities on the dynamics of virus spread. The distribution of hosts and vectors, and vector abundance strongly influence the dynamics of BTV spread. The application of our model on a real geographic area, although of limited size, allowed us to illustrate the conclusions drawn from a theoretical domain. The reaction–diffusion models are classically used in plant epidemiology [39–41], with the modelled movements generally being for highly volatile entities, and the short versus long-distance movements being taken into account via different diffusion coefficients [40, 42]. Although developed to represent BTV8 spatiotemporal spread at a local to regional scale, our model can be used to study other vector-borne diseases of animals and its extension to a larger area remains possible.
Financial support for this research was provided by INRA, IRSTEA and Basse-Normandie, Bretagne, Pays de la Loire and Poitou-Charentes Regional Councils under SANCRE project, in the framework of “For and About Regional Development” programs. GK was supported by a BBSRC DTG-funded PhD studentship awarded to MB and Dr Jon Read.
- Chomel BB, Belotto A, Meslin FX: Wildlife, exotic pets, and emerging zoonoses. Emerg Infect Dis. 2007, 13: 6-11. 10.3201/eid1301.060480.PubMed CentralView ArticlePubMed
- Jones KE, Patel NG, Levy MA, Storeygard A, Balk D, Gittleman JL, Daszak P: Global trends in emerging infectious diseases. Nature. 2008, 451: 990-993. 10.1038/nature06536.View ArticlePubMed
- Roche B, Guegan JF: Ecosystem dynamics, biological diversity and emerging infectious diseases. C R Biol. 2011, 334: 385-392. 10.1016/j.crvi.2011.02.008.View ArticlePubMed
- Gould EA, Higgs S: Impact of climate change and other factors on emerging arbovirus diseases. Trans R Soc Trop Med Hyg. 2009, 103: 109-121. 10.1016/j.trstmh.2008.07.025.PubMed CentralView ArticlePubMed
- Velthuis AG, Saatkamp HW, Mourits MC, de Koeijer AA, Elbers AR: Financial consequences of the Dutch bluetongue serotype 8 epidemics of 2006 and 2007. Prev Vet Med. 2009, 93: 294-304.View ArticlePubMed
- Velthuis AGJ, Mourits MCM, Saatkamp HW, de Koeijer AA, Elbers ARW: Financial evaluation of different vaccination strategies for controlling the bluetongue virus serotype 8 epidemic in The Netherlands in 2008. PLoS One. 2011, 6: e19612-10.1371/journal.pone.0019612.PubMed CentralView ArticlePubMed
- Smith DL, Dushoff J, McKenzie FE: The risk of a mosquito-borne infection in a heterogeneous environment. PLoS Biol. 2004, 2: 1957-1964.View Article
- Backer JA, Nodelijk G: Transmission and control of African horse sickness in The Netherlands: a model analysis. PLoS One. 2011, 6: e23066-10.1371/journal.pone.0023066.PubMed CentralView ArticlePubMed
- Foxi C, Delrio G: Larval habitats and seasonal abundance of Culicoides biting midges found in association with sheep in northern Sardinia, Italy. Med Vet Entomol. 2010, 24: 199-209. 10.1111/j.1365-2915.2010.00861.x.View ArticlePubMed
- Ander M, Meiswinkel R, Chirico J: Seasonal dynamics of biting midges (Diptera: Ceratopogonidae: Culicoides), the potential vectors of bluetongue virus, in Sweden. Vet Parasitol. 2012, 184: 59-67. 10.1016/j.vetpar.2011.08.009.View ArticlePubMed
- Guis H, Tran A, de la Rocque S, Baldet T, Gerbier G, Barragué B, Biteau-Coroller F, Roger F, Viel J-F, Mauny F: Use of high spatial resolution satellite imagery to characterize landscapes at risk for bluetongue. Vet Res. 2007, 38: 669-683. 10.1051/vetres:2007025.View ArticlePubMed
- Hartemink NA, Purse BV, Meiswinkel R, Brown HE, de Koeijer A, Elbers ARW, Boender GJ, Rogers DJ, Heesterbeek JAP: Mapping the basic reproduction number (R0) for vector-borne diseases: a case study on bluetongue virus. Epidemics. 2009, 1: 153-161. 10.1016/j.epidem.2009.05.004.View ArticlePubMed
- Raffy M, Tran A: On the dynamics of flying insects populations controlled by large scale information. Theor Popul Biol. 2005, 68: 91-104. 10.1016/j.tpb.2005.03.005.View ArticlePubMed
- Tran A, Raffy M: On the dynamics of dengue epidemics from large-scale information. Theor Popul Biol. 2006, 69: 3-12. 10.1016/j.tpb.2005.06.008.View ArticlePubMed
- Lewis M, Rencławowicz J, van den Driessche P: Traveling waves and spread rates for a West Nile virus model. Bull Math Biol. 2006, 68: 3-23. 10.1007/s11538-005-9018-z.View ArticlePubMed
- Maidana NA, Yang HM: Spatial spreading of West Nile Virus described by traveling waves. J Theor Biol. 2009, 258: 403-417. 10.1016/j.jtbi.2008.12.032.View ArticlePubMed
- Ostfeld RS, Glass GE, Keesing F: Spatial epidemiology: an emerging (or re-emerging) discipline. Trends Ecol Evol. 2005, 20: 328-336. 10.1016/j.tree.2005.03.009.View ArticlePubMed
- Adams B, Kapan DD: Man bites mosquito: understanding the contribution of human movement to vector-borne disease dynamics. PLoS One. 2009, 4: e6763-10.1371/journal.pone.0006763.PubMed CentralView ArticlePubMed
- Chen Q, Han R, Ye F, Li W: Spatio-temporal ecological models. Ecol Inform. 2011, 6: 37-43. 10.1016/j.ecoinf.2010.07.006.View Article
- Ferreira LC, Udia , Pulino P, Yang , Hyun , Takahashi L:Controlling dispersal dynamics of Aedes aegypti. Math Popul Stud. 2006, 13: 215-236. 10.1080/08898480600950515.View Article
- Szmaragd C, Wilson AJ, Carpenter S, Wood JLN, Mellor PS, Gubbins S: A modeling framework to describe the transmission of bluetongue virus within and between farms in Great Britain. PLoS One. 2009, 4: e7741-10.1371/journal.pone.0007741.PubMed CentralView ArticlePubMed
- Ducheyne E, Lange M, Van der Stede Y, Meroc E, Durand B, Hendrickx G: A stochastic predictive model for the natural spread of bluetongue. Prev Vet Med. 2011, 99: 48-59. 10.1016/j.prevetmed.2011.01.003.View ArticlePubMed
- Graesboll K, Bodker R, Enoe C, Christiansen LE: Simulating spread of Bluetongue Virus by flying vectors between hosts on pasture. Sci Rep. 2012, 2: 863-PubMed CentralView ArticlePubMed
- Turner J, Bowers RG, Baylis M: Modelling bluetongue virus transmission between farms using animal and vector movements. Sci Rep. 2012, 2: 319-PubMed CentralView ArticlePubMed
- Kluiters G, Sugden D, Guis H, McIntyre KM, Labuschagne K, Vilar MJ, Baylis M: Modelling the spatial distribution of Culicoides biting midges at the local scale. J Appl Ecol. 2012, 50: 232-242.View Article
- Charron MVP, Seegers H, Langlais M, Ezanno P: Seasonal spread and control of Bluetongue in cattle. J Theor Biol. 2011, 291: 1-9.View ArticlePubMed
- Van Ark H, Meiswinkel R: Subsampling of large light trap catches of Culicoides (Diptera: Ceratopogonidae). Onderstepoort J Vet Res. 1992, 59: 183-189.PubMed
- Pioz M, Guis H, Calavas D, Durand B, Abrial D, Ducrot C: Estimating front-wave velocity of infectious diseases: a simple, efficient method applied to bluetongue. Vet Res. 2011, 42: 60-10.1186/1297-9716-42-60.PubMed CentralView ArticlePubMed
- Kilpatrick AM, Daszak P, Jones MJ, Marra PP, Kramer LD: Host heterogeneity dominates West Nile virus transmission. Proc Biol Sci. 2006, 273: 2327-2333. 10.1098/rspb.2006.3575.View ArticlePubMed
- De Koeijer A, Boender G, Nodelijk G, Staubach C, Meroc E, Elbers A: Quantitative analysis of transmission parameters for bluetongue virus serotype 8 in Western Europe in 2006. Vet Res. 2011, 42: 53-10.1186/1297-9716-42-53.PubMed CentralView ArticlePubMed
- Sellers RF: Weather, Culicoides, and the distribution and spread of bluetongue and African horse sickness viruses. Bluetongue, African horse sickness, and related orbiviruses: Proceedings of the Second International Symposium. 1992, 284-290.
- Hendrickx G, Gilbert M, Staubach C, Elbers A, Mintiens K, Gerbier G, Ducheyne E: A wind density model to quantify the airborne spread of Culicoides species during north-western Europe bluetongue epidemic, 2006. Prev Vet Med. 2008, 87: 162-181. 10.1016/j.prevetmed.2008.06.009.View ArticlePubMed
- Murray MD: Akabane epizootics in New South Wales: evidence for long-distance dispersal of the biting midge Culicoides brevitarsis. Aust Vet J. 1987, 64: 305-308. 10.1111/j.1751-0813.1987.tb07332.x.View ArticlePubMed
- Mellor PS, Boorman J, Baylis M: Culicoides biting midges: their role as arbovirus vectors. Annu Rev Entomol. 2000, 45: 307-340. 10.1146/annurev.ento.45.1.307.View ArticlePubMed
- Finlaison DS, Read AJ, Kirkland PD: An epizootic of bovine ephemeral fever in New South Wales in 2008 associated with long-distance dispersal of vectors. Aust Vet J. 2010, 88: 301-306. 10.1111/j.1751-0813.2010.00596.x.View ArticlePubMed
- Calvete C, Estrada R, Miranda MA, Borrás D, Calvo JH, Lucientes J: Ecological correlates of bluetongue virus in Spain: predicted spatial occurrence and its relationship with the observed abundance of the potential Culicoides spp. vector. Vet J. 2009, 182: 235-243. 10.1016/j.tvjl.2008.06.010.View ArticlePubMed
- Sanders CJ, Shortall CR, Gubbins S, Burgin L, Gloster J, Harrington R, Reynolds DR, Mellor PS, Carpenter S: Influence of season and meteorological parameters on flight activity of Culicoides biting midges. J Appl Ecol. 2011, 48: 1355-1364. 10.1111/j.1365-2664.2011.02051.x.View Article
- Balenghien T, Fouque F, Sabatier P, Bicout DJ: Horse-, bird-, and human-seeking behavior and seasonal abundance of mosquitoes in a West Nile virus focus of southern France. J Med Entomol. 2006, 43: 936-946. 10.1603/0022-2585(2006)43[936:HBAHBA]2.0.CO;2.View ArticlePubMed
- Gilligan CA: Modelling soil-borne plant pathogens: reaction–diffusion models. Can J Plant Pathol. 1995, 17: 96-108. 10.1080/07060669509500700.View Article
- Burie J-B, Calonnec A, Langlais M: Mathematical Modeling of Biological Systems. 2008, Boston: Birkhäuser, 2: 11-21. Modeling of the invasion of a fungal disease over a Vineyard, Modeling and Simulation in Science, Engineering and Technology.
- Sapoukhina N, Tyutyunov Y, Sache I, Arditi R: Spatially mixed crops to control the stratified dispersal of airborne fungal diseases. Ecol Model. 2010, 221: 2793-2800. 10.1016/j.ecolmodel.2010.08.020.View Article
- Zawolek MW, Zadoks JC: Studies in focus development - an optimum for the dual dispersal of plant pathogens. Phytopathology. 1992, 82: 1288-1297. 10.1094/Phyto-82-1288.View Article
- Singer RS, MacLachlan NJ, Carpenter TE: Maximal predicted duration of viremia in bluetongue virus-infected cattle. J Vet Diagn Invest. 2001, 13: 43-49. 10.1177/104063870101300109.View ArticlePubMed
- Baylis M, O’Connell L, Mellor PS: Rates of bluetongue virus transmission between Culicoides sonorensis and sheep. Med Vet Entomol. 2008, 22: 228-237. 10.1111/j.1365-2915.2008.00732.x.View ArticlePubMed
- EFSA: Epidemiological analysis of 2006 bluetongue virus serotype 8 epidemic in north-western Europe 2007. 2007, apart from the link http://www.efsa.europa.eu/fr/efsajournal/doc/34r.pdf
- Gerry AC, Mullens BA, Maclachlan NJ, Mecham JO: Seasonal transmission of bluetongue virus by Culicoides sonorensis (Diptera: Ceratopogonidae) at a southern california dairy and evaluation of vectorial capacity as a predictor of bluetongue virus transmission. J Med Entomol. 2001, 38: 197-209. 10.1603/0022-2585-38.2.197.View ArticlePubMed
- Carpenter S, Lunt HL, Arav D, Venter GJ, Mellor PS: Oral susceptibility to bluetongue virus of Culicoides (Diptera: Ceratopogonidae) from the United Kingdom. J Med Entomol. 2006, 43: 73-78. 10.1603/0022-2585(2006)043[0073:OSTBVO]2.0.CO;2.View ArticlePubMed
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