{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T06:27:50Z","timestamp":1771655270536,"version":"3.50.1"},"reference-count":13,"publisher":"Wiley","license":[{"start":{"date-parts":[[2019,2,3]],"date-time":"2019-02-03T00:00:00Z","timestamp":1549152000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2019,2,3]]},"abstract":"<jats:p>In this paper we present a mathematical model for the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) by considering antibiotic treatment and vaccination. The model is comprised of susceptible, vaccinated, exposed, infectious, persistently infected, and recovered compartments. We analyse the model by deriving a formula for the control reproduction number <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> and prove that, for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>&lt;<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>, the disease free equilibrium is globally asymptotically stable; thus CBPP dies out, whereas for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>&gt;<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Thus, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math> acts as a sharp threshold between the disease dying out or causing an epidemic. As a result, the threshold of antibiotic treatment is <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msubsup><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u204e<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">0.1049<\/mml:mn><\/mml:math>. Thus, without using vaccination, more than <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mn mathvariant=\"normal\">85.45<\/mml:mn><mml:mi mathvariant=\"normal\">%<\/mml:mi><\/mml:math> of the infectious cattle should receive antibiotic treatment or the period of infection should be reduced to less than 8.15 days to control the disease. Similarly, the threshold of vaccination is <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msup><mml:mrow><mml:mi>\u03c1<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u204e<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">0.0084<\/mml:mn><\/mml:math>. Therefore, we have to vaccinate at least <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mn mathvariant=\"normal\">80<\/mml:mn><mml:mi mathvariant=\"normal\">%<\/mml:mi><\/mml:math> of susceptible cattle in less than 49.5 days, to control the disease. Using both vaccination and antibiotic treatment, the threshold value of vaccination depends on the rate of antibiotic treatment, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:msub><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><\/mml:math> and is denoted by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:msub><mml:mrow><mml:mi>\u03c1<\/mml:mi><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>. Hence, if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mn mathvariant=\"normal\">50<\/mml:mn><mml:mi mathvariant=\"normal\">%<\/mml:mi><\/mml:math> of infectious cattle receive antibiotic treatment, then at least <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mn mathvariant=\"normal\">50<\/mml:mn><mml:mi mathvariant=\"normal\">%<\/mml:mi><\/mml:math> of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease.<\/jats:p>","DOI":"10.1155\/2019\/2490313","type":"journal-article","created":{"date-parts":[[2019,2,3]],"date-time":"2019-02-03T18:32:24Z","timestamp":1549218744000},"page":"1-10","source":"Crossref","is-referenced-by-count":8,"title":["Mathematical Modelling of the Transmission Dynamics of Contagious Bovine Pleuropneumonia with Vaccination and Antibiotic Treatment"],"prefix":"10.1155","volume":"2019","author":[{"given":"Achamyelesh Amare","family":"Aligaz","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of South Africa, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5276-2977","authenticated-orcid":true,"given":"Justin Manango W.","family":"Munganga","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of South Africa, South Africa"}]}],"member":"311","reference":[{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.prevetmed.2005.09.002"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.prevetmed.2014.03.022"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1016\/j.mimet.2010.03.025"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.3389\/fvets.2017.00100"},{"issue":"2","key":"11","volume":"10, article e0116730","year":"2015","journal-title":"PLoS ONE"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.20506\/rst.25.3.1710"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1051\/animres:2004026"},{"issue":"10, article e46821","key":"3","volume":"7","year":"2012","journal-title":"PLoS ONE"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.11145\/texts.2017.12.253"},{"key":"4","volume-title":"Consultancy on the dynamics of CBPP endemism and the development of effective control\/ eradication strategies for pastoral comunities: final data collection report","year":"2003"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/S0025-5564(02)00108-6"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1137\/120876642"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.prevetmed.2005.09.001"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2019\/2490313.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2019\/2490313.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2019\/2490313.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,2,3]],"date-time":"2019-02-03T18:32:25Z","timestamp":1549218745000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2019\/2490313\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,3]]},"references-count":13,"alternative-id":["2490313","2490313"],"URL":"https:\/\/doi.org\/10.1155\/2019\/2490313","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,3]]}}}