Bulletin of Mathematical Biology

, Volume 79, Issue 2, pp 303–324 | Cite as

A Mathematical Model of Anthrax Transmission in Animal Populations

  • C. M. Saad-Roy
  • P. van den Driessche
  • Abdul-Aziz Yakubu
Original Article

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.

Keywords

Anthrax Hopf bifurcation Global stability Disease control strategy Type reproduction number 

Mathematics Subject Classification

92D30 34D23 

Notes

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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Copyright information

© Society for Mathematical Biology 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of MathematicsHoward UniversityWashingtonUSA

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