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Stochastic Model of Bovine Babesiosis with Juvenile and Adult Cattle

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Abstract

A stochastic model for Bovine Babesiosis (BB) including ticks, and both juvenile and adult cattle is developed. This model is formulated by a system of continuous-time Markov chains (CTMCs) that is derived based on an extension of the deterministic ordinary differential equation model developed by Saad-Roy et al. (Bull Math Biol 77:514–547, 2015). The nonlinear CTMC model is approximated by a multitype branching process, giving a theoretical estimate of the probability of an outbreak of BB. Unlike the deterministic dynamics where the basic reproduction number is a sharp threshold parameter, the stochastic model indicates that there is always a positive probability of disease extinction within the cattle population. For parameter values from Colombia data, conditional probability distributions are numerically obtained for the time to disease extinction or outbreak, and are found to depend on the host type at the initiation of infection. The models with and without the inclusion of juvenile cattle are compared, and our result highlights that neglecting juvenile bovine in the models may lead to faulty predictions of critical disease statistics: particularly, it may underestimate the risk of infection. Endemic disease prevalence in adult cattle is examined for certain parameter values in the corresponding deterministic model. Notably, with long-lasting immunity, increased tick to juvenile infectivity decreases the proportion of infectious adults.

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Acknowledgements

The authors would like to thank the anonymous reviewer and the editor for helpful comments. The research of PvdD is partially supported by NSERC Discovery Grants. This work was supported by a grant from the Simons Foundation (No. 317407, XW). CMSR acknowledges support from a Jamie Cassels Undergraduate Research Award (JCURA) of the University of Victoria; NSERC through a PGS-D; the Bill and Melinda Gates Foundation Grant OPP1091919 (to Bryan Grenfell); the RAPIDD program of the Science and Technology Directorate Department of Homeland Security and the Fogarty International Center, NIH (to Bryan Grenfell); and the James S. McDonnell Foundation 21st Century Science Initiative Collaborative Award in Understanding Dynamic and Multi-scale Systems (to Simon Levin and Ramanan Laxminarayan). XW and PvdD thank the American Institute of Mathematics where some of this research was discussed.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Washington State University, Pullman, USA

    Xueying Wang

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Chadi M. Saad-Roy & P. van den Driessche

  3. Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, USA

    Chadi M. Saad-Roy

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  1. Xueying Wang
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  2. Chadi M. Saad-Roy
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  3. P. van den Driessche
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Correspondence to Xueying Wang.

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Appendix A: Mean and Standard Derivation of Time to Disease Extinction or Outbreak

Appendix A: Mean and Standard Derivation of Time to Disease Extinction or Outbreak

Table 4 Mean (the value on the top) and standard derivation (the value in the parentheses at the bottom) of time to disease extinction (respectively, an outbreak) given the occurrence of a disease extinction (respectively, an outbreak) associated with Fig. 4 (respectively, 5)
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Wang, X., Saad-Roy, C.M. & van den Driessche, P. Stochastic Model of Bovine Babesiosis with Juvenile and Adult Cattle. Bull Math Biol 82, 64 (2020). https://doi.org/10.1007/s11538-020-00734-x

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