Research ArticleNo Access

Global dynamics in a model for anthrax transmission in animal populations

Junli Liu

https://orcid.org/0000-0003-1157-028X

School of Science, Xi’an Polytechnic University, Xi’an 710048, P. R. China

E-mail Address: jlliu2008@126.com

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Mengjie Han

School of Science, Xi’an Polytechnic University, Xi’an 710048, P. R. China

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, and

Tailei Zhang

School of Science, Chang’an University, Xi’an 710064, P. R. China

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https://doi.org/10.1142/S1793524522501194Cited by:1 (Source: Crossref)

Abstract

In this paper, we propose a deterministic model to study the transmission dynamics of anthrax disease, which includes live animals, carcasses, spores in the environment and vectors. We derive three biologically plausible and insightful quantities (reproduction numbers) that determine the stability of the equilibria. We carry out rigorous mathematical analysis on the model dynamics, the global stability of the disease-free and vector-free equilibrium, the disease-free equilibrium and the vector-free disease equilibrium is proved. The global stability of the endemic equilibrium as the basic reproduction number is greater than one is derived in the special case in which the disease-related death rate is zero. The possibility of backward bifurcation is briefly discussed. Numerical analyses are carried out to understand the transmission dynamics of anthrax and investigate effective control strategies for the outbreaks of the disease. Our studies suggest that the larval vector control measure should be taken as early as possible to control the vector population size, a vaccination policy and an animal carcass removal policy are useful methods to control the prevalence of the diseases in infected animal populations, the adult vector control measure is also necessary to prevent the transmission of anthrax.

Keywords:

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References

1. J. L. Aron and R. M. May , The population dynamics of malaria, in The Population Dynamics of Infectious Diseases: Theory and Applications, ed. R. M. Anderson (Chapman and Hall, London, 1982). Crossref, Google Scholar
2. A. Berman and R. J. Plemmons , Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics (Academic Press, New York, London, 1979). Google Scholar
3. K. Blackburn et al., The necrophagous fly anthrax transmission pathway: empirical and genetic evidence from wildlife epizootics, Vector Borne Zoonotic Dis. 14(8) (2014) 576–583. Crossref, Web of Science, Google Scholar
4. C. Castillo-Chavez and B. Song , Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1 (2004) 361–404. Crossref, Web of Science, Google Scholar
5. CDC: Use of anthrax vaccine in the United States recommendations of the Advisory Committee on Immunization Practices (ACIP), 2009, Morb. Mort. Wkly. Rep. 59(6) (2010) 1–36, www.cdc.gov/mmwr. Google Scholar
6. O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts , The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface 5 (2009) 1–13. Google Scholar
7. A. Friedman and A.-A. Yakubu , Anthrax epizootic and migration: Persistence or extinction, Math. Biosci. 241 (2013) 137–144. Crossref, Web of Science, Google Scholar
8. D. J. Ganeva , Analysis of the Bulgarian tabanid fauna with regard to its potential for epidemiological involvement, Bulg. J. Vet. Med. 7(1) (2004) 1–8. Google Scholar
9. B. D. Hahn and P. R. Furniss , A deterministic model of an anthrax epizootic: threshold results, Ecol. Model. 20 (1983) 233–241. Crossref, Web of Science, Google Scholar
10. J. M. Hyman and J. Li , The reproductive number for an HIV model with differential infectivity and staged progression, Linear Algebra Appl. 398 (2005) 101–116. Crossref, Web of Science, Google Scholar
11. J. P. LaSalle , The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, 1976). Crossref, Google Scholar
12. T. Lembo et al., Serologic surveillance of anthrax in the Serengeti ecosystem, Tanzania, 1996–2009, Emerg. Infect. Dis. 17 (2011) 387–394. Crossref, Web of Science, Google Scholar
13. P. Lindeque and P. Turnbull , Ecology and epidemiology of anthrax in the Etosha National Park, Namibia, Onderstepoort J. Vet. Res. 61(1) (1994) 71–84. Web of Science, Google Scholar
14. J. L. Liu, Y. Jia and T. L. Zhang , Analysis of a rabies transmission model with population dispersal, Nonlinear Anal. Real World Appl. 35 (2017) 229–249. Crossref, Web of Science, Google Scholar
15. X. N. Liu and Y. Takeuchi , Spread of disease with transport-related infection and entry screening, J. Theor. Biol. 242 (2006) 517–528. Crossref, Web of Science, Google Scholar
16. Y. Lou and X.-Q. Zhao , Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol. 73 (2011) 2384–2407. Crossref, Web of Science, Google Scholar
17. C. Mackey and C. Kribs , Can scavengers save zebras from anthrax? A modeling study, Infect. Dis. Model. 6 (2021) 56–74. Google Scholar
18. C. Mackey and C. Kribs , Modeling anthrax-rabies interactions in zebra-jackal cycles, J. Theor. Biol. 511 (2021) 110553. Crossref, Web of Science, Google Scholar
19. S. Mushayabasa , Dynamics of an anthrax model with distributed delay, Acta Appl. Math. 144 (2016) 77–86. Crossref, Web of Science, Google Scholar
20. S. Mushayabasa, T. Marijani and M. Masocha , Dynamical analysis and control strategies in modeling anthrax, Comp. Appl. Math. 36 (2017) 1333–1348. Crossref, Web of Science, Google Scholar
21. S. Ruan and J. Wei , On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Med. Biol. 18 (2001) 41–52. Crossref, Google Scholar
22. C. M. Saad-Roy, P. van den Driessche and A.-A. Yakubu , A mathematical model of anthrax transmission in animal populations, Bull. Math. Biol. 79 (2017) 303–324. Crossref, Web of Science, Google Scholar
23. H. L. Smith , Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems (American Mathematical Society, Providence, 1995). Google Scholar
24. A. N. Survely, B. Kvasnicka and R. Torell , Anthrax: A guide for livestock producers, Cattle producer’s Library CL613, Western Beef Res. Committee 613 (2001) 1–3. Google Scholar
25. P. van den Driessche and J. Watmough , Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. Crossref, Web of Science, Google Scholar
26. X.-Q. Zhao and Z. Jing , Global asymptotic behavior in some cooperative systems of functional differential equations, Can. Appl. Math. Quart. 4(4) (1996) 421–444. Google Scholar