ИСТИНА

Ticks and tick-borne diseases present a well-known threat to the health of people in many parts of the globe. The scientific literature devoted both to field observations and modelling the propagation of ticks continues to grow. So far, the majority of the mathematical studies were devoted to models based on ordinary differential equations, where spatial variability was taken into account by a discrete parameter. Only few papers use spatially nontrivial diffusion models, and they are devoted mostly to spatially homogeneous equilibria. Here we develop diffusion models for the propagation of ticks stressing spatial heterogeneity. This allows us to assess the sizes of control zones that can be created (using various available techniques) to produce a patchy territory, on which ticks will be eventually eradicated. The well-known local techniques for fighting ticks include the treatment of soil (planned burns, cleaning and decreasing humidity levels, chemical treatment), relatively new tick-targeted strategies such as TickBots, specific actions on hosts, like dipping of cattle with acaricide, and biological methods like infesting woods with ants. Mathematical content of the theory lies in the study of the diffusive limit of the evolutionary model of the propagation and the related criteria for the dying out of the solutions. Using averaged parameters taken from various field observations, we apply our theoretical results to the concrete cases of the lone star ticks of North America and of the taiga ticks of Russia. We shall discuss open questions and further perspectives of research in this direction.