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

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 numberRc and prove that, forRc < 1, the disease free equilibrium is globally asymptotically stable; thus CBPP dies out, whereas for Rc > 1, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Thus, Rc = 1 acts as a sharp threshold between the disease dying out or causing an epidemic. As a result, the threshold of antibiotic treatment is α∗ t = 0.1049. Thus, 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. Similarly, the threshold of vaccination is ρ∗ = 0.0084. Therefore, we have to vaccinate at least 80% 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, αt, and is denoted by ραt . Hence, if 50% of infectious cattle receive antibiotic treatment, then at least 50% of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease.


Introduction
Contagious Bovine Pleuropneumonia (CBPP) is a major constraint to cattle production in the key pastoral regions of Africa (see [1][2][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][5][6][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, R 0 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 R  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 R  , 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.

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.

Equilibria of the System
Proposition 2. Model ( 1)-( 6) has at least two equilibrium points, the disease free equilibrium, and at least one endemic equilibrium.

The Control Reproduction Number (R 𝑐
). Due to the presence of control measures, we will use the term control reproduction number (R  ) instead of the commonly used basic reproduction number (R 0 ).As explained in [12], we use the next generation matrix to calculate the control reproduction number.Compartments , , and  are considered to be the disease compartments and , , and  are the nondisease compartments.We set ] and Therefore, R  = ( −1 ) = /2 + √ (/2) 2 − , where  and  are trace and determinant of the matrix  −1 .Since  = 0, Equivalently, where R 0 = /(( −   ) −   ) 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 .

Local Stability Analysis of the DFE
Theorem 3 (see [12]).If  0 is a DFE of the model given by ( 1)- (7), then  0 is locally asymptotically stable if R  < 1, and unstable if R  > 1, where R  is defined by (36).

Global Stability Analysis of the Endemic Equilibrium (EE)
Theorem 5.If R  > 1, then the endemic equilibrium  * of ( 1)-( 7) is unique and globally asymptotically stable in the interior of Ω.
Proof.We use a graph-theoretic method as explained in [13]., where   > 0 is the weight of arc(, ).Thus, the associated weighted digraph (, ) for the model given by system of ( 1)-( 6) is presented in Figure 2.
Thus, proving uniqueness and global asymptotic stability of X * in interior of Ω provided that R  > 1.  1 of [1].We assume that the life expectancy of cattle is in average 5 years, then the value of  and  is taken to be 1/(5 × 365),  = 0.126, the incubation period between 4 and 8 weeks with mean value of 6 weeks yields  = 1/(6 × 7), without applying antibiotic treatment, the infection period is between 6 and 10 weeks with mean value of 8 weeks and   = 3  , then   = 1/(4×56), the persistently infected period given in a range of 18-21 weeks with an average period of 19 weeks with 4 months × 2 reactivations per month for 582 cases gives  = 0.0009 and  = 0.0075, the rate of vaccination,  =  V   /, where  is the efficiency of vaccine,  V is the proportion vaccinated,   is efficacy of the vaccine, and  is the period of vaccination and vaccinal immunity lasts for 3 years which implies that  = 1/(3 × 365).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.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 each compartment at time  is plotted in Figure 7; in this case, R  = 0.99213 for parametric values in Table 1.

Conclusion and Remarks
In this paper we presented compartmental model and differential equations for the transmission dynamics of CBPP with intervention.We calculated the equilibrium of the system and to study the behaviour of the disease, we derived a formula for the control reproduction number R  .We proved that, for R  < 1, the DFE is globally asymptotically stable, thus CBPP dies out, whereas for R  > 1, the EE is globally asymptotically stable and hence the disease persists in all the populations.Hence R  = 1 acts as a sharp threshold between the disease dying out or causing an epidemic.Without considering any intervention, the model in this paper coincides with the model studied in [10]; in both papers, R 0 = 3.9399 when   = 1/56; see Figure 3. Similarly, when   = 1/(4 × 56), which is the right value, and without any intervention, R 0 = 6.7462; see Figure 4. Hence, without any intervention the disease persists in all the population.However, we can control the disease by giving antibiotic treatment to 85.7% of infectious cattle, without vaccinating any of healthy cattle; see Figure 5.As a second option, we can also control the disease by vaccinating 80% of susceptible cattle within a period of 49 days, without treating any of infectious cattle; see Figure 6.Finally, for parametric values in Table 1, with the assumption that 50% of susceptible are vaccinated within  a period of 73 days and 50% of infectious cattle are treated, R  can be made less than one and hence we can control the disease; see Figure 7.In all the above three intervention methods, R  < 1 and hence the disease can be controlled by properly applying the methods as explained above; however, due to lack of awareness, time, and financial and logistic constraint, the first two methods do not look feasible in the context of developing countries.Therefore, we recommend that vaccination with antibiotic treatment is the best way to control the disease which is in agreement with the result of [1].Since the proportion to be vaccinated  V and  are independent variables of , a given value of  can have many practical interpretation.Therefore, practical implementation of the value of  can be adjusted based on availability, cost of control, and time value.

Figure 1 :
Figure 1: A compartmental model for the transmission dynamics of CBPP with antibiotic treatment and vaccination.

Table 1 :
Description of model parameters and their values, indicating baselines, ranges, and references.Units are days −1 unless otherwise defined.* Proportions.Most of the parameter values used in this paper are explained in Table 1, Sections 2.2 and 2.3 of [14], and Table