Modelling the transmission dynamics of Contagious Bovine Pleuropneumonia in the presence of antibiotic treatment with limited medical supply
Abstract
We present and analyze a mathematical model of the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) in the presence of antibiotic treatment with limited medical supply. We use a saturated treatment function to model the effect of delayed treatment. We prove that there exist one disease free equilibrium and at most two endemic equilibrium solutions. A backward bifurcation occurs for small values of delay constant such that two endemic equilibriums exist if Rt ∈ (R∗t,1); where, Rt is the treatment reproduction number and R∗t is a threshold such that the disease dies out if and persists in the population if Rt > R∗t. However, when a backward bifurcation occurs, a disease free system may easily be shifted to an epidemic. The bifurcation turns forward when the delay constant increases; thus, the disease free equilibrium becomes globally asymptotically stable if Rt < 1, and there exist unique and globally asymptotically stable endemic equilibrium if Rt > 1. However, the amount of maximal medical resource required to control the disease increases as the value of the delay constant increases. Thus, antibiotic treatment with limited medical supply setting would not successfully control CBPP unless we avoid any delayed treatment, improve the efficacy and availability of medical resources or it is given along with vaccination.
Keyword : contagious bovine pleuropneumonia, backward bifurcation, equilibria, treatment reproduction number, stability and threshold value
How to Cite
Aligaz, A. A., & Munganga, J. M. W. (2021). Modelling the transmission dynamics of Contagious Bovine Pleuropneumonia in the presence of antibiotic treatment with limited medical supply. Mathematical Modelling and Analysis, 26(1), 1-20. https://doi.org/10.3846/mma.2021.11795
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A.A. Aligaz and J.M.W. Munganga. Mathematical modelling of the transmission dynamics of contagious bovine pleuropneumonia with vaccination and antibiotic treatment. Journal of Applied Mathematics, 2019(4):10, 2019. https://doi.org/10.1155/2019/2490313
W. Amanfu. Contagious bovine pleuropneumonia (lung sickness) in Africa. Onderstepoort Journal of veterinary Research, 76(1):13–17, 2009. https://doi.org/10.4102/ojvr.v76i1.55
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R. Naji and B. Abdulateef. The dynamics of model with nonlinear incidence rate and saturated treatment function. Science International, 29:1223–1236, 11 2017.
M. Niamir-Fuller. A review of recent literature on pastoralism and transhumance in Africa. Intermediate Technology publications, London, pp. 18–46, 1999. https://doi.org/10.3362/9781780442761.002
R. Nicholas, R. Ayling and L. McAuliffe. Mycoplasma Diseases of Ruminants. CAB books. CABI, 2008. https://doi.org/10.1079/9780851990125.0000
OIE. Contagious bovine pleuropneumonia. General disease information sheet.
T.T. Olekae and P.R. Naledi. Communicating livestock disease risks in Ngamiland: the case of contagious bovine pleuropneumonia. South African Geographical Journal, 101(2):192–209, 2019. https://doi.org/10.1080/03736245.2019.1581080
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Z. Shuai and P. van den Driessche. Global stability of infectious disease models using Lyapunov functions. SIAM Journal on Applied Mathematics, 73(4):1513– 1532, 2013. https://doi.org/10.1137/120876642
A. Ssematimba, J. Jores and J.C. Mariner. Mathematical modelling of the transmission dynamics of contagious bovine pleuropneumonia reveals minimal target profiles for improved vaccines and diagnostic assays. PLOS ONE, 10(2):e0116730, 2015. https://doi.org/10.1371/journal.pone.0116730
N.E. Tambi, W.O. Maina and C. Ndi. An estimation of the economic impact of contagious bovine pleuropneumonia in Africa. Revue scientifique et technique (International Office of Epizootics), 25(3):999–1011, 2006. https://doi.org/10.20506/rst.25.3.1710
P. van den Driessche and J. Watmough. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29–48, 2002. https://doi.org/10.1016/S0025-5564(02)00108-6
J. Wu, R. Dhingra, M. Gambhir and J.V. Remais. Sensitivity analysis of infectious disease models: methods, advances and their application. Journal of The Royal Society Interface, 10(86), 2013. https://doi.org/10.1098/rsif.2012.1018
J. Zhang, J. Jia and X. Song. Analysis of an SEIR epidemic model with saturated incidence and saturated treatment function. In The Scientific World Journal, 2014. https://doi.org/10.1155/2014/910421
X. Zhang and X. Liu. Backward bifurcation of an epidemic model with saturated treatment function. Journal of Mathematical Analysis and Applications, 348(1):433–443, 2008. https://doi.org/10.1016/j.jmaa.2008.07.042
L. Zhou and M. Fan. Dynamics of an SIR epidemic model with limited medical resources revisited. Nonlinear Analysis: Real World Applications, 13(1):312–324, 2012. https://doi.org/10.1016/j.nonrwa.2011.07.036
A.A. Aligaz and J.M.W. Munganga. Mathematical modelling of the transmission dynamics of contagious bovine pleuropneumonia with vaccination and antibiotic treatment. Journal of Applied Mathematics, 2019(4):10, 2019. https://doi.org/10.1155/2019/2490313
W. Amanfu. Contagious bovine pleuropneumonia (lung sickness) in Africa. Onderstepoort Journal of veterinary Research, 76(1):13–17, 2009. https://doi.org/10.4102/ojvr.v76i1.55
A.D. Campbell and A.W. Turner. Studies of contagious bovine pluerophnuemonia of cattle. an improved complementfixation test. Australian Veterinary Journal, 4(29), 1953. https://doi.org/10.1111/j.1751-0813.1953.tb05259.x
T. Caraballo and X. Han. Applied Nonautonomous and Random Dynamical Systems: Applied Dynamical Systems. Springer, 2016. https://doi.org/10.1007/978-3-319-49247-6
FAO. Towards sustainable CBPP control programmes for Africa. FAO-OIEAU/IBAR-IAEA Consultative Group on Contagious Bovine Pleuropneumonia third meeting, Rome, 2003.
FAO. CBPP control: Antibiotics to the rescue? FAO-OIE-AU/IBAR-IAEA Consultative Group Meeting on CBPP in Africa, Rome, 2006.
FAO. Can contagious bovine pleuropneumonia (CBPP) be eradicated? FAOOIE-AU/IBAR-IAEA Consultative group on CBPP fifth meeting, Rome, 2015.
J.A. Hammond and D. Branagan. Contagious bovine pleuropneumonia in Tanganyika. Bulletin of Epizootic Disease in Africa, 13:121–147, 1965.
O.J. Hubschle, R.D. Ayling, K. Godinho, O.T. Lukhele, G. Ipura-Zaire, T.G. Rowan and R.A.J. Nicholas. Danifloxacine(advocin) reduces the spread of contagious bovine pleuropneumonia to healthy in-contact cattle. Res.Vet.Sc., 81:304– 309, 2009. https://doi.org/10.1016/j.rvsc.2006.02.005
J.E. Huddart. Bovine contagious pleuropneumonia - a new approach to field control in Kenya. Veterinary Record, 72:1253–1254, 1960.
W. Jinliang, L. Shengqiang, Z. Baowen and T. Yasuhiro. Qualitative and bifurcation analysis using an SIR model with a saturated treatment function. Mathematical and Computer Modelling, 55(3):710–722, 2012. https://doi.org/10.1016/j.mcm.2011.08.045
S.W. Kairu-Wanyoike, N.M. Taylor, C. Heffernan and H. Kiara. Micro-economic analysis of the potential impact of contagious bovine pleuropneumonia and its control by vaccination in Narok district of Kenya. Livestock Science, 197:61–72, 2017. https://doi.org/10.1016/j.livsci.2017.01.002
M.J. Keeling and P. Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2008. https://doi.org/10.1515/9781400841035
M. Lesnoff, G. Laval, P. Bonnet and A. Workalemahu. A mathematical model of contagious bovine pleuropneumonia (CBPP) within-herd outbreaks for economic evaluation of local control strategies : an illustration from a mixed crop-livestock system in Ethiopian highlands. Animall Research, EDP Sciences, 53:429–438, 2004. https://doi.org/10.1051/animres:2004026
C.H. Li and A.M. Yousef. Bifurcation analysis of a network-based SIR epidemic model with saturated treatment function. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(3):033129, 2019. https://doi.org/10.1063/1.5079631
M.Y. Li and J.S. Muldowney. On R.A. Smith’s autonomous convergence theorem. Rocky Mountain Journal of Mathematics, 25(1):365–378, 03 1995. https://doi.org/10.1216/rmjm/1181072289
M.Y. Li and J.S. Muldowney. A geometric approach to global-stability problems. SIAM Journal on Mathematical Analysis, 27(4):1070–1083, 1996. https://doi.org/10.1137/S0036141094266449
J.C. Mariner, J. McDermott, J.A.P. Heesterbeek, G. Thomson and S.W. Martin. A model of contagious bovine pleuropneumonia transmission dynamics in East Africa. Preventive veterinary medicine, 73(1):55–74, 2006. https://doi.org/10.1016/j.prevetmed.2005.09.001
J. McDermott, J.C. Mariner, J.A.P. Heesterbeek, G. Thomson, P.L. Roeder and S.W. Martin. A heterogeneous population model for contagious bovine pleuropneumonia transmission and control in pastoral communities of East Africa. Preventive Veterinary Medicine, 73(1):75–91, 2006. https://doi.org/10.1016/j.prevetmed.2005.09.002
R. Naji and B. Abdulateef. The dynamics of model with nonlinear incidence rate and saturated treatment function. Science International, 29:1223–1236, 11 2017.
M. Niamir-Fuller. A review of recent literature on pastoralism and transhumance in Africa. Intermediate Technology publications, London, pp. 18–46, 1999. https://doi.org/10.3362/9781780442761.002
R. Nicholas, R. Ayling and L. McAuliffe. Mycoplasma Diseases of Ruminants. CAB books. CABI, 2008. https://doi.org/10.1079/9780851990125.0000
OIE. Contagious bovine pleuropneumonia. General disease information sheet.
T.T. Olekae and P.R. Naledi. Communicating livestock disease risks in Ngamiland: the case of contagious bovine pleuropneumonia. South African Geographical Journal, 101(2):192–209, 2019. https://doi.org/10.1080/03736245.2019.1581080
J.O. Onono, B. Wieland, A. Suleiman and J. Rushton. Policy analysis for delivery of contagious bovine pleuropneumonia control strategies in subSaharan Africa. OIE Revue Scientifique et Technique, 36:195–205, 04 2017. https://doi.org/10.20506/rst.36.1.2621
A. Provost, P. Perreau, A. Breard, C. Le Goff, J.L. Martel, G.S. Cottew and C. Le Goff. Contagious bovine pluerophnuemonia. Revue Scientifique at Technique, Office International des Epizooties, 6:625–679, 1987. https://doi.org/10.20506/rst.6.3.306
H.M. Robert. Logarithmic norms and projections applied to linear differential systems. Journal of Mathematical Analysis and Applications, 45(2):432–454, 1974. https://doi.org/10.1016/0022-247X(74)90084-5
Z. Shuai and P. van den Driessche. Global stability of infectious disease models using Lyapunov functions. SIAM Journal on Applied Mathematics, 73(4):1513– 1532, 2013. https://doi.org/10.1137/120876642
A. Ssematimba, J. Jores and J.C. Mariner. Mathematical modelling of the transmission dynamics of contagious bovine pleuropneumonia reveals minimal target profiles for improved vaccines and diagnostic assays. PLOS ONE, 10(2):e0116730, 2015. https://doi.org/10.1371/journal.pone.0116730
N.E. Tambi, W.O. Maina and C. Ndi. An estimation of the economic impact of contagious bovine pleuropneumonia in Africa. Revue scientifique et technique (International Office of Epizootics), 25(3):999–1011, 2006. https://doi.org/10.20506/rst.25.3.1710
P. van den Driessche and J. Watmough. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29–48, 2002. https://doi.org/10.1016/S0025-5564(02)00108-6
J. Wu, R. Dhingra, M. Gambhir and J.V. Remais. Sensitivity analysis of infectious disease models: methods, advances and their application. Journal of The Royal Society Interface, 10(86), 2013. https://doi.org/10.1098/rsif.2012.1018
J. Zhang, J. Jia and X. Song. Analysis of an SEIR epidemic model with saturated incidence and saturated treatment function. In The Scientific World Journal, 2014. https://doi.org/10.1155/2014/910421
X. Zhang and X. Liu. Backward bifurcation of an epidemic model with saturated treatment function. Journal of Mathematical Analysis and Applications, 348(1):433–443, 2008. https://doi.org/10.1016/j.jmaa.2008.07.042
L. Zhou and M. Fan. Dynamics of an SIR epidemic model with limited medical resources revisited. Nonlinear Analysis: Real World Applications, 13(1):312–324, 2012. https://doi.org/10.1016/j.nonrwa.2011.07.036