Analysis of a mathematical model of the dynamics of contagious bovine pleuropneumonia

Abstract

Contagious bovine pleuropneumonia (CBPP) is a disease of cattle and water buffalo caused by Mycoplasma mycoides subspecies mycoides (Mmm). It attacks the lungs and the membranes that line the thoracic cavity. The disease is transmitted by inhaling droplets disseminated through coughing by infected cattle. In this paper a deterministic mathematical model for the transmission of Contagious Bovine plueropnemonia is presented. The model is a five compartmental model consisting of susceptible, Exposed, Infectious, Persistently infected and Recovered compartments. We derived a formula for the basic reproduction number R0. For R0 ≤ 1, the disease free equilibrium is globally asymptotically stable, thus CBPP dies out; whereas for R0 > 1, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Elasticity indices for R0 with respect to different parameters are calculated; indicating parameters that are important for control strategies to bring R0 below 1, the effective contact rate β has the largest elasticity index. As the disease control options are associated to these parameters, for some values of these parameters, R0 < 1, thus the disease can be controlled.

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

“…When we introduce antibiotic treatment at a rate of , the period of infection (56 days) will be reduced to some new period such that + + = 1/ implies = 1/ − 1/56. Since R = 1 acts as a sharp threshold between the disease dying out or causing an epidemic, we find that the threshold of antibiotic treatment is given by * = (R 0 − 1)/( + )R 0 = 0.1049, where R 0 is the basic reproduction number as in [10]. This implies that, without using vaccination, more than 85.45% 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.…”

“…) is the basic reproduction number as derived in [10] and /( + ) is the proportion of cattle that survive the vaccination class and the control reproduction number, R , is the average number of secondary cases caused by an infected individual over the course of infectious period in the presence of vaccination and antibiotic treatment. We observe that R < R 0 .…”

“…And, threshold of vaccination is also given by * = ( × V × )/ = ( + )(R 0 − 1) = 0.0084, where is period of vaccination, which can be interpreted that at least 80% of susceptible cattle should get vaccination in less than 49.5 days in order to control the disease; however, since the proportion to be vaccinate V depends on , a single value of can have many practical interpretation. For the last Table 1 with = 1/56 (as in [10]) and 0 = 1, 0 = 499, 0 = 0 = 0 = 0, giving approximate equilibrium values * = 127, * = 12, * = 9, * = 15, * = 337, and R 0 = 3.9399.…”

“…Mathematically, it means that, for = 1/28 − 1/56 = 1/56, we should take = (0.65 × 0.5 × 0.8)/73 to make R < 1, to control the disease. Parameter values considering both antibiotic treatment and vaccination are summarized in Table 1. (i) All parameter values used in this paper and in [10] are the same except the value of which is taken in [10] as 1/56 instead of 1/(4 × 56).…”

“…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, R 0 decreases and can be made less than one; as a result the disease can be controlled.…”