A Dynamical Study of Stability and Bifurcation in an In-Vitro Bovine Mastitis System Incorporating Nonlinear Incidence and Therapeutic Functions
DOI:
https://doi.org/10.63746/njtd.v23i1.3398Keywords:
Bovine Mastitis, In-vitro Model, Bifurcation Analysis, Non-linear Treatment function, Sensitivity AnalysisAbstract
A mathematical model was developed to study the transmission dynamics of Bovine Mastitis, a prevalent dairy cattle disease caused by bacterial infection, leading to significant economic losses and reduced milk quality. The model incorporated nonlinear transmission and treatment functions, and its epidemiological well-posedness was established. Stability analysis revealed that the disease-free equilibrium is stable if the basic reproduction number Rm is less than unity. However, the model exhibited backward bifurcation, indicating that Rm < 1 is necessary but not sufficient for effective control, and that initial population sizes and disease prevalence also play a crucial role. Sensitivity analysis identified key parameters influencing Rm, including per capita recruitment rate , transmission rates and , and maximum treatment rate . Numerical simulations demonstrated that targeted interventions, such as optimizing recruitment, reducing transmission, and maximizing treatment rates, can effectively control infectious populations and mitigate mastitis incidence, but require careful consideration of the complex disease dynamics.
References
Abebe, R.; A. Markos; M. Abera and B. Mekbib. (2023). Incidence rate, risk factors, and bacterial causes of clinical mastitis on dairy farms in Hawassa City, southern Ethiopia. Scientific Reports, 13(1), 10945. https://doi.org/10.1038/s41598-023-37328-1
Adeyemi, M.O. and Oluyo, T.O. (2023). Mathematical modelling for the Control of Fly Borne Mastitis Disease in Cattle. Front. Appl. Math. Stat. 9:1171157. https://doi.org/10.3389/fams.2023.1171157
Agarwal, M. and Verma, V. (2012). Modeling and Analysis of the Spread of Infectious disease. Cholera with environmental fluctuations. Application and applied Mathematics, 7(1):406-425.
Ahmad, A.; F. Abbas; M. Farman; E. Hincal; A. Ghaffar; A. Akgül and M.K. Hassani. (2024). Flip bifurcation analysis and mathematical modeling of cholera disease by taking control measures. Scientific Reports, 14(1), 10927. https://doi.org/10.1038/s41598-024-59640-0
Aligaz, A.A. and Munganga, J.M.W. (2018). Mathematical Modelling of the Transmission Dynamics of Contagious Bovine Pleuropneumonia with Vaccination and Antibiotic Treatment. Journal of Applied Mathematics. Vol.2019:1-10.
Arnold, M. (2019). Preventing Summer Mastitis in Heifers Begins with Horn Fly Control. UK College of Agriculture, Food and Environment, Cooperative Extension Service. http://entomology.ca.uky.edu/files/recs0/ent12-diary.pdf
Arsenopoulos, K.; E. Triantafillous; G. Filioussis and E. Papadopoulous, E. (2018). Fly Repellency Using Deltamethrin May reduce Intramammary Infections of Dairy Cows under Intensive Management. Comparative Immunology, Microbiology and Infectious Diseases. 61: 16-23.
Ashraf, A. and Imran, M. (2020). Causes, types, etiological agents, prevalence, diagnosis, treatment, prevention, effects on human health and future aspects of bovine mastitis. Animal Health Research Reviews, 21(1), 36–49. doi:10.1017/S1466252319000094.
Barrett, J. (2024). Negotiating Mastitis: Animal Health Assemblages, New Ways of Seeing Disease and Hidden Frictions to Changing Antimicrobial Use Practices (Doctoral dissertation, University of Maryland, Baltimore County). https://www.proquest.com/openview/
Beddington, J.R. and De-Angelis. (1975). Mutual interference between parasites or predators and its effect on searching efficiency. Journal of Animal Ecology, 44:331-340.
Bentout, S.; S. Djilali and B. Ghanbari. (2021). Backward, Hopf bifurcation in a heroin epidemic model with treat age. International Journal of Modeling, Simulation, and Scientific Computing, 12(02), 2150018. https://doi.org/10.1142/S1793962321500185
Blowey, R.W. and Edmondson, P.W. (1995). Mastitis control in dairy herds. Pp. 29. Ipswich, Farming Press.
Castillo-Chavez, C. and Song, B. (2004). Dynamical models of tuberculosis and their applications. Math. Biosci. Eng, 1(2), 361-404. https://www.aimspress.com/aimspress-data/mbe/2004/2/PDF/1551-0018_2004_2_361.pdf
Castillo-Chavez, C.; S. Blower; P. van den Driessche; D. Kirschner and A.A. Yakubu. (Eds.). (2002). Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory (Vol. 126). Springer Science & Business Media. https://books.google.com.ng/
Castillo-Chavez, C. and Feng, Z. (1996). Mathematical models for the disease dynamics of tuberculosis. https://ecommons.cornell.edu/server/api/core/bitstreams/02abbe97-efbe-40aa-b38c-fed65589ca8d/content
Chitnis, N.; M.H. James and J.M. Cushing. (2008). Determining Important Parameters in the Spread of Malaria through the Sensitivity Analysis of Mathematical Model, Bulletin of Mathematical Biology. 70(5):1272-1296.
Cobirka, M.; T. Vladimir and S. Petr. (2020). Epidemiology and Classification of Mastitis" Animals 10, no. 12: 2212. https://doi.org/10.3390/ani10122212
Diekman, O.; J.A.P. Hesterbeek and J.A.J. Mertz. (1990). On the Definition and the Computation of the Basic Reproduction Ratio R_0in Models for Infectious Diseases in Heterogenous Population, J. Math. Biol. Vol 28, pp. 365.
Dubey P.; B. Dubey and U.S. Dubey. (2016). Dynamics of a SIR model with nonlinear incidence rate and treatment rate. Nonlinear Appl. Maths. 10:718-737.
Dubey, B.; A. Patra; P. K. Srivastava and U. S. Dubey. (2013). Modeling and analysis of an SEIR model with different types of nonlinear treatment rates. Journal of Biological Systems, 21(03), 1350023. https://doi.org/10.1142/S021833901350023X
Dushoff, J.; W. Huang and C. Castillo-Chavez. (1998). Backwards bifurcations and catastrophe in simple models of fatal diseases. Journal of mathematical biology, 36(3), 227-248. https://pdodds.w3.uvm.edu/teaching/courses/2009-08UVM-300/docs/others/1998/dushoff1998a.pdf
Elfouly, M.A. (2024). Improved Mathematical Models of Parkinson's Disease with Hopf Bifurcation and Huntington's Disease with Chaos. Acta Biotheor 72, 11. https://doi.org/10.1007/s10441-024-09485-x
Gomes, F.; M. J. Saavedra and M. Henriques. (2016). An overview of the role of biofilms in Bovine Mastitis infection, Pathogens and disease, 74(3).
Gumel, A. B. (2012). Causes of backward bifurcations in some epidemiological models. Journal of Mathematical Analysis and Applications, 395(1), 355-365.
Haider, A.; M. Ikram; I. Shahzadi and M. Asif Raza. (2023). Bovine Mastitis. In: Polymeric Nanoparticles for Bovine Mastitis Treatment. Springer Series in Biomaterials Science and Engineering, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-031-39947-3_4
Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM review, 42(4), 599-653. https://doi.org/10.1137/S0036144500371907
Hethcote, H.W and van den Driessche, P. (1991). Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287. https://doi.org/10.1007/BF00160539
Hu, Z.; S. Liu and H. Wang. (2008). Backward bifurcation of an epidemic model with standard incidence rate and treatment rate. Nonlinear Analysis: Real World Applications, 9(5), 2302-2312.
Hu, Z.; W. Ma and S. Ruan. (2012). Analysis of SIR epidemic models with nonlinear incidence rate and treatment. Mathematical biosciences, 238(1), 12-20. https://doi.org/10.1016/j.mbs.2012.03.010
Jana S.; S.K. Nandi and T.K. Kar. (2015). Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment. Acta Biotheor 64:65-84.
Jin Y.; W. Wang; and S. Xiao. (2007). An SIR model with a non-linear incidence rate. Chaos Solitions Fractals 34: 1482-1497. https://doi.org/10.1016/J.CHAOS.2006.04.022
Kenneth H. M. N.; M. P. Flemming and T.O. Johnny. (2014). Bifurcation analysis of an existing mathematical model reveals novel treatment strategies and suggests potential cure for type 1 diabetes, Mathematical Medicine and Biology: A Journal of the IMA, Volume 31, Issue 3, September 2014, Pages 205–225, https://doi.org/10.1093/imammb/dqt006
Kiseleva, O.; S. Yakovlev; D. Chumachenko and O. Kuzenkov. (2024). Exploring Bifurcation in the Compartmental Mathematical Model of COVID-19 Transmission. Computation, 12(9), 186. https://doi.org/10.3390/computation12090186
Kribs-Zaleta, C. M. and Velasco-Hernández, J. X. (2000). A simple vaccination model with multiple endemic states. Mathematical biosciences, 164(2), 183-201. https://doi.org/10.1016/S0025-5564(00)00003-1
Lakhani, P.; R. Ankita; S.P. Pooja and P.S. Rahul. (2025). Impact of mastitis on the composition and quality of milk. Handbook of Milk Production, Quality and Nutrition: 627-639. https://doi.org/10.1016/B978-0-443-24820-7.00040-7
Lan, G.; S. Yuan and B. Song. (2021). The impact of hospital resources and environmental perturbations to the dynamics of SIRS model. Journal of the Franklin Institute, 358(4), 2405-2433. https://doi.org/10.1016/j.jfranklin.2021.01.015
LaSalle, J.P. (1976). The stability of dynamical systems, regional conference series in Applied Mathematics. SIAM, Philadelphia,1976.
Liu, W.M.; H.W. Hethcote and S.A. Levin. (1987). Dynamical approach of epidemiological models with nonlinear incidence rate. Journal of Math Biol., 25: pp 359-380.
National Mastitis Council (NMC). (2011). Current Concept of Bovine Mastitis, Fifth edition, National Mastitis council’ Verona.
Neculai, V.; S. Andra and M.A. Adina. (2022). Udder Health Monitoring for Prevention of Bovine Mastitis and Improvement of Milk Quality Bioengineering 9, no. 11: 608. https://doi.org/10.3390/bioengineering9110608
Odebiyi, O.A.; J.K. Oladejo; A.A. Yahaya and E.O. Elijah. (2024). Analysis of HIV/AIDS Model with Nonlinear Incidence Function. International Journal of Research and Innovation in Applied Science, 9(4):317-341. Doi: https://doi.org/10.51584/IJRIAS.2024.904023
Odebiyi, O.A.; W.O. Salahu; J.K. Oladejo; A.O. Areo; O.A. Olajide and S.O. Sangoniyi. (2025). Analyzing the influence of Screening and Therapy Compliance on HIV/AIDS Dynamics Using Nonlinear Incidence. Asian Journal of Pure and Applied Mathematics, 7(1):593-610. Doi: https://doi.org/10.56557/ajpam/2025/v7i1226
Omede, B. I.; O.J. Peter; W. Atokolo; B. Bolaji and T.A. Ayoola. (2023). A mathematical analysis of the two-strain tuberculosis model dynamics with exogenous re-infection. Healthcare Analytics, 4, 100266. https://doi.org/10.1016/j.health.2023.100266
Otieno, O.J.; J. Mugisha and J. Odahiambo. (2012). Mathematical Model for Pneumonia Dynamics among Children. Africa Mathematical Sciences Association Conference (AMSA, 2012) 26th -29th Nov.2012.
Paramasivam, R.; D. R. Gopal; R. Dhandapani; R. Subbarayalu; M.P. Elangovan; B. Prabhu and S. Muthupandian. (2023). Is AMR in Dairy Products a Threat to Human Health? An Updated Review on the Origin, Prevention, Treatment, and Economic Impacts of Subclinical Mastitis. Infection and Drug Resistance, 16, 155–178. https://doi.org/10.2147/IDR.S384776
Ruegg, P.L. (2012). New Perspectives in Udder Health Management, Vet. Clin. Food. Anim. 28:149-163.
Saha, P. and Ghosh, U. (2021). Global dynamics and control strategies of an epidemic model having logistic growth, non-monotone incidence with the impact of limited hospital beds. Nonlinear Dyn. 2021:971-996. https://doi.org/10.1007/s11071-021-06607-9
Saha, P. and Ghosh, U. (2023). Complex dynamics and conflict analysis of an epidemic-model with non-monotone incidence and saturated treatment. International Journal of Dynamics and Control 11:301-323.
Saha, P.; B. Mondal and U. Ghosh. (2023). Dynamical behaviors of an epidemic model with partial immunity having nonlinear incidence and saturated treatment in deterministic and stochastic environments. Chaos, Solitons & Fractals, 174, 113775. https://doi.org/10.1016/j.chaos.2023.113775
Shafeeq, S. K.; Murtadha M. A; Ahmed A. M; H. F. Al-Husseiny, and Z. Anwar. (2022). Bifurcation analysis of a vaccination mathematical model with application to COVID-19 pandemic. Commun. Math. Biol. Neurosci. 2022: Article-ID. 86. https://scik.org/index.php/cmbn/article/view/7633
Sharomi, O.; C. N. Podder; A. B. Gumel; E.H. Elbasha and J. Watmough. (2007). Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Mathematical Biosciences, 210(2), 436-463.
Sharun, K.; D. Kuldeep; T. Ruchi; B.G. Mudasir; I.Y. Mohd; K.P. Shailesh; P. Mamta, et al. (2021). Advances in Therapeutic and Managemental Approaches of Bovine Mastitis: A Comprehensive Review. Veterinary Quarterly 41 (1): 107–36. doi:10.1080/01652176.2021.1882713.
Smulski, S.; M. Gehrke; K. Libera; A. Cieslak; H. Huang; A.K. Patra and M. Szumacher-Strabel. (2020). Effects of various mastitis treatments on the reproductive performance of cows. BMC veterinary research, 16(1), 99. https://doi.org/10.1186/s12917-020-02305-7
Stanek, P.; P. ?ó?kiewski, and E. Janu?. (2024). A Review on Mastitis in Dairy Cows Research: Current Status and Future Perspectives Agriculture 14, no. 8: 1292. https://doi.org/10.3390/agriculture14081292
Tewari, A. (2014). Bovine Mastitis: An Important Diary Cattle disease, Technical Report. Diaryman, 2014:62-65.
Upadhyay, R.K.; A.K. Pal and S. Kumari. (2019). Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates. Nonlinear Dyn 96, 2351–2368. https://doi.org/10.1007/s11071-019-04926-6
Van den Driessche, P and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1-2), 29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
Wang J.; J. Zhang and Z. Jin. (2010). Analysis of an SIR Model with bilinear incidence rate, Nonlinear Analysis: Real World Applications, 11:2390-2402 https://doi.org/10.1016/j.nonrwa.2009.07.012
Wang, W. (2006). Backward bifurcation of an epidemic model with treatment. Mathematical biosciences, 201(1-2), 58-71.
Wattiaux, M.A. (2021). Mastitis: The Disease and its Transmission. Diary essentials, Babcock Institute for International Diary Research and Development, University of Wisconsin-Madison.
Wei, C. and Chen, L. (2008). Dynamic Analysis of an SIR Epidemic Model with Pulse Vaccination. Discrete Dynamics in nature and society.
Yeruham, I.; Y. Braverman; N.Y. Shipgel; A. Chizov-Ginzburg; A. Seran and M. Winkler. (1996). Mastitis in Diary Caused by Corynebacterium pseudotuberculosis and the Feasibility of Transmission by Housefly I. Veterinary quarterly, 18(3):87-89.
Zhang, J.Z.; Z. Jin; Q.X. Liu and Z.Y Zang. (2008). Analysis of a delayed SIR model with non-linear incidence rate. Discrete dyn, nat, soc., 2008:636153. Pp.1-16. Doi: 10.1155/2008/636153.
Zhang, X and Liu, X. (2008). Backward bifurcation of an epidemic model with saturated treatment function. Journal of mathematical analysis and applications, 348(1), 433-443.
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