Mathematical Modelling of the Transmission Dynamics of Contagious Bovine Pleuropneumonia with Vaccination and Antibiotic Treatment

Abstract

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 ${\mathcal{R}}_c$ and prove that, for , the disease free equilibrium is globally asymptotically stable; thus CBPP dies out, whereas for ${\mathcal{R}}_c>\mathrm{1}$, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Thus, ${\mathcal{R}}_c=\mathrm{1}$ acts as a sharp threshold between the disease dying out or causing an epidemic. As a result, the threshold of antibiotic treatment is ${\alpha }_t^{*}=\mathrm{0.1049}$. Thus, without using vaccination, more than $\mathrm{85.45}\mathrm{\%}$ 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 ${\rho }^{*}=\mathrm{0.0084}$. Therefore, we have to vaccinate at least $\mathrm{80}\mathrm{\%}$ 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, ${\alpha }_t,$ and is denoted by ${\rho }_{{\alpha }_t}$. Hence, if $\mathrm{50}\mathrm{\%}$ of infectious cattle receive antibiotic treatment, then at least $\mathrm{50}\mathrm{\%}$ of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease.