Journal of Applied Mathematics

Volume 2019, Article ID 2490313, 10 pages

https://doi.org/10.1155/2019/2490313

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

Department of Mathematical Sciences, University of South Africa, South Africa

Correspondence should be addressed to Justin Manango W. Munganga; moc.liamg@agnagnumwmj

Received 18 September 2018; Revised 16 December 2018; Accepted 3 January 2019; Published 3 February 2019

Academic Editor: Urmila Diwekar

Copyright © 2019 Achamyelesh Amare Aligaz and Justin Manango W. Munganga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 and prove that, for , the disease free equilibrium is globally asymptotically stable; thus CBPP dies out, whereas for , the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Thus, acts as a sharp threshold between the disease dying out or causing an epidemic. As a result, the threshold of antibiotic treatment is . Thus, without using vaccination, more than 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 . Therefore, we have to vaccinate at least 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, and is denoted by . Hence, if of infectious cattle receive antibiotic treatment, then at least of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease.

#### 1. Introduction

Contagious Bovine Pleuropneumonia (CBPP) is a major constraint to cattle production in the key pastoral regions of Africa (see [1–3] for more details). It is caused by Mycoplasma mycoides subspecies mycoides (Mmm) that attacks the lungs and the membranes of cattle and water buffalo. It is transmitted by direct contact between an infected and a susceptible animal which becomes infected by inhaling droplets disseminated by coughing. It causes high morbidity and mortality losses to cattle which leads to economic crisis (see [4–7] for more details). Cost of control of CBPP is also a major concern in African countries [6, 8]. Since some animals can carry the disease without showing signs of illness, controlling the spread is more difficult. In many countries in sub-Saharan Africa, CBPP control is based on vaccination alone, but this strategy does not eradicate the disease [9].

In [10] we presented and analysed a five-compartmental mathematical model of the transmission dynamics of CBPP, without any intervention, having the objective of identifying parameters that have significant role in changing the dynamics of the disease. As a result, from elasticity analysis, we found that the effective contact rate and the rate of recovery are the top two parameters that control the dynamics of the disease in such a way that as the value of decreases and the value of increases, decreases and can be made less than one; as a result the disease can be controlled. However, we know that vaccination is one of the ways of reducing the effective contact rate and antibiotic treatment is one way of reducing infection by increasing the recovery rate.

Thus, in this paper we consider vaccination and antibiotic treatment as a controlling tool of CBPP and present a compartmental model for the transmission dynamics of CBPP containing six compartments: susceptible, vaccinated, exposed, infectious, persistently infected, and recovered compartments. Antibiotic treatment is considered in the model by incorporating rate of recovery of treated cattle such that treated cattle move from infectious compartment to recovered compartment at a rate of .

The objective of this paper is to determine the better control method out of vaccination, antibiotic treatment, and a combination of both. We derive the formula for the control reproduction number and determine the number of cattle to be vaccinated and treated independently and in combination, which will enable us to choose the feasible and effective controlling method in our context. Numerical simulations are performed using MATLAB.

This paper is structured as follows. In Section 2, we present a mathematical model of the dynamics of CBPP, with vaccination and antibiotic interventions. In Section 3, we prove the well-posedness of the model. We calculate equilibria of the system and rigorously derive a formula of the control reproduction number , in Section 4. Stability analysis of the DFE and EE is presented in Section 5, we present parameter values and numerical simulations in Section 6, and lastly, we draw the conclusions and remarks in Section 7.

#### 2. Mathematical Model

We model the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP). In this model we assume intervention by vaccination and antibiotic treatment. Thus, the compartmental model is consisting of susceptible, vaccinated, exposed, infectious, persistently infected, and recovered classes, as shown in Figure 1. We assume an open population, with a total number at time , where all newborn animals are born into susceptible class at rate . Susceptible cattle move to vaccinal immune class at a rate . Cattle in vaccinal immune class can lose vaccinal immunity and return back to susceptible class at a rate . Susceptible animals move to the exposed compartment at a rate . Cattle in the exposed compartment move to the infectious compartment at a rate . Natural mortality occurs at a rate and results in losses from all six compartments. However, we assume that death due to the disease does not occur. The infectious cattle either naturally heal or receive antibiotic treatment and enter directly into the recovered compartment at a rate and , respectively; or they pass through a process of sequestration and enter into persistently infected compartment at a rate . Cattle in persistently infected compartment are encapsulated and infected, but not infectious. As sequestra resolve and/or become noninfected, then the animals in persistently infected compartment move to the recovered compartment at a rate . Recovered cattle remain recovered for life time. Infected sequestra can occasionally be reactivated and in this instance the animal will transition from the persistently infected compartment back to the infectious compartment at a rate . We assume random mixing of all individuals in the population. The total number of population at time , , is given by . The flow diagram of the model is shown in Figure 1.