Dynamics of Mycobacterium and bovine tuberculosis in a Human-Buffalo Population

A new model for the transmission dynamics of Mycobacterium tuberculosis and bovine tuberculosis in a community, consisting of humans and African buffalos, is presented. The buffalo-only component of the model exhibits the phenomenon of backward bifurcation, which arises due to the reinfection of exposed and recovered buffalos, when the associated reproduction number is less than unity. This model has a unique endemic equilibrium, which is globally asymptotically stable for a special case, when the reproduction number exceeds unity. Uncertainty and sensitivity analyses, using data relevant to the dynamics of the two diseases in the Kruger National Park, show that the distribution of the associated reproduction number is less than unity (hence, the diseases would not persist in the community). Crucial parameters that influence the dynamics of the two diseases are also identified. Both the buffalo-only and the buffalo-human model exhibit the same qualitative dynamics with respect to the local and global asymptotic stability of their respective disease-free equilibrium, as well as with respect to the backward bifurcation phenomenon. Numerical simulations of the buffalo-human model show that the cumulative number of Mycobacterium tuberculosis cases in humans (buffalos) decreases with increasing number of bovine tuberculosis infections in humans (buffalo).

[1]  Flavie Goutard,et al.  Risk Analysis and Bovine Tuberculosis, a Re‐emerging Zoonosis , 2006, Annals of the New York Academy of Sciences.

[2]  Wayne M. Getz,et al.  Assessing vaccination as a control strategy in an ongoing epidemic: Bovine tuberculosis in African buffalo , 2006 .

[3]  Hadi Dowlatabadi,et al.  Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: an HIV Model, as an Example , 1994 .

[4]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[5]  A F RANNEY,et al.  Bovine tuberculosis eradication. , 1961, Diseases of the chest.

[6]  R. May,et al.  Population Biology of Infectious Diseases , 1982, Dahlem Workshop Reports.

[7]  C. Castillo-Chavez,et al.  A model for tuberculosis with exogenous reinfection. , 2000, Theoretical population biology.

[8]  S. O. ADEWALE,et al.  MATHEMATICAL ANALYSIS OF A TB TRANSMISSION MODEL WITH DOTS , .

[9]  Xingfu Zou,et al.  Modeling diseases with latency and relapse. , 2007, Mathematical biosciences and engineering : MBE.

[10]  Wayne M. Getz,et al.  Disease, predation and demography: Assessing the impacts of bovine tuberculosis on African buffalo by monitoring at individual and population levels , 2009 .

[11]  Baojun Song,et al.  Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. , 2008, Mathematical biosciences and engineering : MBE.

[12]  G. D. de Lisle,et al.  Mycobacterium bovis in free-living and captive wildlife, including farmed deer. , 2001, Revue scientifique et technique.

[13]  D. Vlahov,et al.  Molecular and geographic patterns of tuberculosis transmission after 15 years of directly observed therapy. , 1998, JAMA.

[14]  W H Foege,et al.  Centers for Disease Control , 1981, Journal of public health policy.

[15]  James E. Campbell,et al.  An Approach to Sensitivity Analysis of Computer Models: Part I—Introduction, Input Variable Selection and Preliminary Variable Assessment , 1981 .

[16]  Hassan Fathabadi,et al.  On Stability Analysis of Nonlinear Systems , 2012 .

[17]  C H Collins,et al.  Bovine tubercle bacilli and disease in animals and man , 1987, Epidemiology and Infection.

[18]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[19]  J. Raath,et al.  The epidemiology of tuberculosis in free-ranging African buffalo (Syncerus caffer) in the Kruger National Park, South Africa. , 2001, The Onderstepoort journal of veterinary research.

[20]  Christopher Dye,et al.  Eliminating human tuberculosis in the twenty-first century , 2008, Journal of The Royal Society Interface.

[21]  Milton C Weinstein,et al.  Cost-Effectiveness of Treating Multidrug-Resistant Tuberculosis , 2006, PLoS medicine.

[22]  Brian K. Reilly,et al.  Natural mortality amoung four common ungulate species on Letaba Ranch, Limpopo Province, South Africa , 2002 .

[23]  E O Powell,et al.  Theory of the chemostat. , 1965, Laboratory practice.

[24]  ' RonaldL.Iman,et al.  An Investigation of Uncertainty and Sensitivity Analysis Techniques for Computer Models , 2006 .

[25]  Wessel Christiaan Oosthuizen,et al.  Chemical immobilization of African buffalo (Syncerus caffer) in Kruger National Park: Evaluating effects on survival and reproduction , 2006 .

[26]  Hristo V. Kojouharov,et al.  Continuous Age‐Structured Model for Bovine Tuberculosis in African buffalo , 2009 .

[27]  J. Emile,et al.  Langhans giant cells from M. tuberculosis‐induced human granulomas cannot mediate mycobacterial uptake , 2007, The Journal of pathology.

[28]  P. Cross,et al.  Wildlife tuberculosis in South African conservation areas: implications and challenges. , 2006, Veterinary microbiology.

[29]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

[30]  Robert G. McLeod,et al.  Sensitivity and uncertainty analyses for a sars model with time-varying inputs and outputs. , 2006, Mathematical biosciences and engineering : MBE.

[31]  M. Weiss,et al.  Bovine tuberculosis: an old disease but a new threat to Africa. , 2004, The international journal of tuberculosis and lung disease : the official journal of the International Union against Tuberculosis and Lung Disease.

[32]  R R Kao,et al.  A model of bovine tuberculosis control in domesticated cattle herds , 1997, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[33]  R. Ruth,et al.  Stability of dynamical systems , 1988 .

[34]  Ronald L. Iman,et al.  Risk methodology for geologic disposal of radioactive waste: small sample sensitivity analysis techniques for computer models, with an application to risk assessment , 1980 .

[35]  C. Bhunu,et al.  Tuberculosis Transmission Model with Chemoprophylaxis and Treatment , 2008, Bulletin of mathematical biology.

[36]  Wayne M. Getz,et al.  Integrating association data and disease dynamics in a social ungulate: Bovine tuberculosis in African buffalo in the Kruger National Park , 2004 .

[37]  Ted Cohen,et al.  Exogenous re-infection and the dynamics of tuberculosis epidemics: local effects in a network model of transmission , 2007, Journal of The Royal Society Interface.

[38]  Carlos Castillo-Chavez,et al.  Dynamical models of tuberculosis and their applications. , 2004, Mathematical biosciences and engineering : MBE.

[39]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[40]  Abba B. Gumel,et al.  Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis , 2011 .