# Dynamical analysis of mathematical model for Bovine Tuberculosis among human and cattle population

## DOI:

https://doi.org/10.5614/cbms.2019.2.1.6## Keywords:

Basic Reproduction Number, Bovine Tuberculosis, Reinfection## Abstract

*Bovine Tuberculosis*(BTB) is a disease that can attack humans through cattle.The process of transmission can occur through the air and cattle products that are not treated properly. When humans are infected with BTB, reinfection, and relapse may occur. This phenomenon is modeled as an eleven-dimension dynamical system. Our aim is to gain insight into the effect of separation of human activity area into the transmission dynamics of BTB. The model incorporates (among many others features) the dynamics of BTB among human and cattle population, density-dependent infection rate, and reinfection, are rigorously analyzed and simulated. The trivial disease-free equilibrium of the model is shown to be locally asymptotically stable when the two associated basic reproduction number are less than unity. Although the non-trivial equilibrium cannot be shown explicitly, for a special case, this equilibrium is still possible to show and discuss further. Our results suggest that controlling BTB in cattle population may indirectly control the spread of BTB in human. An example of controlling the infected population of infected cattle can be done with the annihilation of infected cattle.

## References

World Health Organization (WHO), 2006. The Control of Neglected Zoonotic Diseases.

Drewe, J.A., Pfeiffer, D.U. and Kaneene, J.B. 2014 Epidemiology of Mycobacterium Bovis. In: Thoen, C.O., Steele, J.H. and Kaneene, J.B. eds., 2014. Zoonotic tuberculosis: Mycobacterium bovis and other pathogenic mycobacteria. John Wiley & Sons.

Torgerson, P.R. and Torgerson, D.J., 2010. Public health and bovine tuberculosis: what's all the fuss about?. Trends in microbiology, 18(2), pp.67-72.

Fox, W., Ellard, G.A. and Mitchison, D.A., 1999. Studies on the treatment of tuberculosis undertaken by the British Medical

Research Council tuberculosis units, 19461986, with relevant subsequent publications. The International Journal of Tuberculosis and Lung Disease, 3(10), pp.S231-S279.

Mallela, A., Lenhart, S. and Vaidya, N.K., 2016. HIVTB co-infection treatment: Modeling and optimal control theory perspectives. Journal of Computational and Applied Mathematics, 307, pp.143-161.

Marx, F.M., Dunbar, R., Enarson, D.A., Williams, B.G., Warren, R.M., Van Der Spuy, G.D., Van Helden, P.D. and Beyers, N., 2014. The temporal dynamics of relapse and reinfection tuberculosis after successful treatment: a retrospective cohort study. Clinical infectious diseases, 58(12), pp.1676-1683.

Hassan, A.S., Garba, S.M., Gumel, A.B. and Lubuma, J.S., 2014. Dynamics of Mycobacterium and bovine tuberculosis in a human-buffalo population. Computational and mathematical methods in medicine, 2014.

Abakar, M.F., Azami, H.Y., Bless, P.J., Crump, L., Lohmann, P., Laager, M., Chitnis, N. and Zinsstag, J., 2017. Transmission dynamics and elimination potential of zoonotic tuberculosis in morocco. PLoS neglected tropical diseases, 11(2), p.e0005214.

De Vos, V., Bengis, R.G., Kriek, N.P.J., Keet, D.F., Raath, J.P., Huchzermeyer, H.F. and Michel, A.L., 2001. The epidemiology of tuberculosis in free-ranging African buffalo (Syncerus caffer) in the Kruger National Park, South Africa. The Onderstepoort journal of veterinary research. 68(2):11930. [PubMed]

Rivero, A., Mrquez, M., Santos, J., Pinedo, A., Snchez, M.A., Esteve, A., Samper, S. and Martn, C., 2001. High rate of tuberculosis reinfection during a nosocomial outbreak of multidrug-resistant tuberculosis caused by Mycobacterium bovis strain B. Clinical infectious diseases, 32(1), pp.159-161.

Michel, A.L., Mller, B. and Van Helden, P.D., 2010. Mycobacterium bovis at the animalhuman interface: A problem, or not?. Veterinary microbiology, 140(3-4), pp.371-381.

Shrikrishna, D., De la Rua-Domenech, R., Smith, N.H., Colloff, A. and Coutts, I., 2009. Human and canine pulmonary

Mycobacterium bovis infection in the same household: re-emergence of an old zoonotic threat?. Thorax, 64(1), pp.89-91.

Whang, S., Choi, S. and Jung, E., 2011. A dynamic model for tuberculosis transmission and optimal treatment strategies in South Korea. Journal of Theoretical Biology, 279(1), pp.120-131.

Grange, J.M., 2001. Mycobacterium bovis infection in human beings. Tuberculosis, 81(1-2), pp.71-77.

Edelstein-Keshet, L. (1988). Mathematical models in biology. New York: Random House.

Diekmann, O. and Heesterbeek, J.A.P., 2000. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation (Vol. 5). John Wiley & Sons.

Diekmann, O., Heesterbeek, J.A.P. and Roberts, M.G., 2009. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), pp.873-885.

de-Camino-Beck, T., Lewis, M.A. and van den Driessche, P., 2009. A graph-theoretic method for the basic reproduction number in continuous time epidemiological models. Journal of Mathematical Biology, 59(4), pp.503-516.

Castillo-Chavez, C., Feng, Z. and Huang, W., 2002. On the computation of ro and its role on. Mathematical approaches for emerging and reemerging infectious diseases: an introduction, 1, p.229.

Diekmann, O. and Heesterbeek, J.A.P., 2000. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation (Vol. 5). John Wiley & Sons.

Diekmann, O., Heesterbeek, J.A.P. and Roberts, M.G., 2009. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), pp.873-885

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*Communication in Biomathematical Sciences*,

*2*(1), 55-64. https://doi.org/10.5614/cbms.2019.2.1.6