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A Mathematical Model of Anthrax Transmission in Animal Populations

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Abstract

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number \(\mathcal {R}_0\) is calculated, and existence of a unique endemic equilibrium is established for \(\mathcal {R}_0\) above the threshold value 1. Using data from the literature, elasticity indices for \(\mathcal {R}_0\) and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if \(\mathcal {R}_0 < 1\). For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with \(\mathcal {R}_0>1\) and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.

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Acknowledgements

P.vdD. and A.-A.Y. would like to acknowledge the 2nd UNISA-UP workshop where the idea for this model arose. This research was partially supported by NSERC, through a USRA (C.M.S.-R.) and a Discovery Grant (P.vdD.). A.-A.Y. was partially supported by DHS Center Of Excellence for Command, Control and Interoperability at Rutgers University and NSF Computational Sustainability Grant # CCF - 1522054. The authors thank two anonymous reviewers for careful reading and good suggestions, which have improved our exposition.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada

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

  2. Department of Mathematics, Howard University, Washington, DC, 20059, USA

    Abdul-Aziz Yakubu

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  1. C. M. Saad-Roy
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  2. P. van den Driessche
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  3. Abdul-Aziz Yakubu
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Correspondence to C. M. Saad-Roy.

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Saad-Roy, C.M., van den Driessche, P. & Yakubu, AA. A Mathematical Model of Anthrax Transmission in Animal Populations. Bull Math Biol 79, 303–324 (2017). https://doi.org/10.1007/s11538-016-0238-1

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  • DOI: https://doi.org/10.1007/s11538-016-0238-1

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