Modelling the Dynamics of Bluetongue Disease and the Effect of Seasonality
We present mathematical models for the midge-borne disease bluetongue, with cattle and sheep as hosts. The models take the form of delay differential equations and incorporate the incubation time of bluetongue in cattle, sheep and midges, and also the larval developmental time of midges. Recovery in cattle and sheep is also included. Both an autonomous and a periodic model are considered, to take account of seasonality. For both models we present conditions for the disease-free state to be linearly stable, and a detailed interpretation of those conditions. The results of simulations are also presented. Important findings include the need for prompt diagnosis of latent infection and appropriate action before the animal turns infectious, and the need for measures that reduce insect bites.
KeywordsBluetongue Delay Stability Periodic Seasonal
Mathematics Subject Classification34K20 34K25 34K60 92D30
We would like to thank the support of the EPSRC and AHVLA for Hayley O’Farrell’s PhD studentship.
- Hendrickx G, Gilbert M, Staubach C, Elbers A, Mintiens K, Gerbier G, Ducheyne E (2006) A wind density model to quantify the airborne spread of Culicoides species during north-western Europe bluetongue epidemic. Prev Vet Med 87(2008):162–181Google Scholar
- Mellor PS (1990) The replication of bluetongue virus in Culicoides vectors. Curr Top Microbiol Immunol 162:143–161Google Scholar
- Szmaragd C, Gunn GJ, Gubbins S (2010a) Assessing the consequences of an incursion of a vector-borne disease. II. Spread of bluetongue in Scotland and impact of vaccination. Epidemics 2(3):139–147Google Scholar
- Szmaragd C, Wilson AJ, Carpenter S, Wood JLN, Mellor PS, Gubbins S (2010b) The spread of bluetongue virus serotype 8 in Great Britain and its control by vaccination. PLoS ONE 5(2):e9353Google Scholar
- Wang H, Kuang Y, Fan M, Periodic solutions of systems of delay differential equations (unpublished manuscript)Google Scholar
- Ward MP, Carpenter TE (1996a) Simulation modeling of the effect of climatic factors on bluetongue virus infection in Australian cattle herds. I. Model formulation, verification and validation. Prev Vet Med 27:1–12Google Scholar
- Ward MP, Carpenter TE (1996b) Simulation modeling of the effect of climatic factors on bluetongue virus infection in Australian cattle herds. II. Model experimentation. Prev Vet Med 27:13–22Google Scholar