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On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model

Author

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  • Vassili N. Kolokoltsov

    (Faculty of Computation Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia
    Faculty of Economics, National Research University Higher School of Economics, 101000 Moscow, Russia
    Department of Statistics, University of Warwick, Coventry CV4 7AL, UK)

Abstract

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 to modeling the propagation of ticks continues to grow. To date, the majority of the mathematical studies have been devoted to models based on ordinary differential equations, where spatial variability was taken into account by a discrete parameter. Only a 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. 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. From the mathematical point of view, we give criteria for global stability of the vanishing solution to certain spatially heterogeneous birth and death processes with diffusion.

Suggested Citation

  • Vassili N. Kolokoltsov, 2023. "On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model," Mathematics, MDPI, vol. 11(2), pages 1-25, January.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:478-:d:1037350
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    References listed on IDEAS

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    1. José Paulo Carvalho dos Santos & Lislaine Cristina Cardoso & Evandro Monteiro & Nelson H. T. Lemes, 2015. "A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-10, July.
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