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.
The differential equation model is given by system (1)–(7)
with initial condition
3. Well-Posedness of the System
Let andprovided that , is a compact subset of such that , and , whereThen, (9)–(14) can be written of the form
Theorem 1. System (1)–(6) has a unique solution X(t) which is positive and bounded if is given by (15) and the initial conditions is nonnegative.
Proof. See [10].
4. Equilibria and Control Reproduction Number
4.1. 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.
Proof. The equilibria of the system are obtained by solving equations: From (9)–(14), we haveFrom (21),Putting (23) into (20) yieldsAnd, putting (23) into (22), we find thatSimilarly, putting (24) into (19) gives Then, we have the following two cases for solution of (27).
Case 1. If , then is the only solution of (27) and
Case 2. If , then or are the solutions of (27).
For Case 1, when , let , and for (17)–(22). Then, from (23)–(25), we haveAnd, from (17) and (18),andTherefore, from (28)–(30), is the disease free equilibrium (DFE).
For Case 2, we are done when And, when , let and for (17)–(22). Then (18) gives thatFinally, putting and into (17), we getHence, is an endemic equilibrium (EE), where, are as in (24), (32), (23), and (25), respectively.
4.2. The Control Reproduction Number
Due to the presence of control measures, we will use the term control reproduction number instead of the commonly used basic reproduction number . 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 , where represents the rate of new infections in the disease compartment, being the transfer rate of individuals into compartment by all other means while represents the transfer rate of individual out of compartment . Assuming to be the DFE, we haveTherefore,Therefore, , where and are trace and determinant of the matrix . Since ,
Equivalently,where 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, , 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 .
5. Stability Analysis
5.1. Stability Analysis of the Disease Free Equilibrium (DFE)
5.1.1. Local Stability Analysis of the DFE
Theorem 3 (see [12]). If is a DFE of the model given by (1)–(7), then is locally asymptotically stable if , and unstable if , where is defined by (36).
Proof. See [12].
5.1.2. Global Stability Analysis of the DFE
Theorem 4. If , then disease free equilibrium is globally asymptotically stable in . If , then the DFE is unstable, the system is uniformly persistent and there is at least one equilibrium in interior of , where .
Proof. We use matrix-theoretic method as explained in [13]. We assume and . And, considering , , and as in Section 4.2, we set where is as in (30).
And,We observe that , , and when and in . Since matrix is reducible, we use Theorem 2.1 of [13] to construct a Lyapunov function. Let be the left eigenvector of nonnegative matrix corresponding to the eigenvalue . Thensuch thatandThus, from (41)-(44), we find that and . Hence, . By Theorem 2.1 of [13], is the Lyapunov function for the system when . Since = 0 implies that and , it follows that is the only invariant set in when and . Thus, by LaSalle’s invariance principle, the DFE is globally asymptotically stable in when . If , then for and . Hence, by continuity, in the neighbourhood of the DFE, implies that the DFE is unstable when . Instability of the DFE implies uniform persistence of (1)–(7). Uniform persistence and the positive invariance of the compact set imply the existence of a unique EE of (1)–(7).
5.2. Global Stability Analysis of the Endemic Equilibrium (EE)
Theorem 5. If , then the endemic equilibrium of (1)–(7) is unique and globally asymptotically stable in the interior of .
Proof. We use a graph-theoretic method as explained in [13]. Thus, for construction of a Lyapunov function, set , , , , , and . And, putting into (17)–(22), we find that , , , , , and . We use these equalities and the inequality in differentiation of , and , with respect to , as follows: where , , , , , , , , and all other such that the weight matrix is , where is the weight of arc. Thus, the associated weighted digraph for the model given by system of (1)–(6) is presented in Figure 2.
Along each directed cycle, , , and . Therefore, by Theorem 3.5 of [13], there exists , such that is a Lyapunov function for (1)–(6), where the relations between ’s can be derived from Theorems 3.3 and 3.4 of [13] that implies , implies , and implies . Hence, , , and . And, implies . Hence the largest invariance set for (1)–(7) where is the singleton set .
Thus, proving uniqueness and global asymptotic stability of in interior of provided that .
6. Parameter Values and Numerical Simulations
6.1. Parameter Values
Most of the parameter values used in this paper are explained in Table 1, Sections 2.2 and 2.3 of [14], and Table 1 of [1]. We assume that the life expectancy of cattle is in average years, then the value of and is taken to be , , the incubation period between and weeks with mean value of weeks yields , without applying antibiotic treatment, the infection period is between and weeks with mean value of weeks and , then , the persistently infected period given in a range of – weeks with an average period of weeks with months × 2 reactivations per month for cases gives and , the rate of vaccination, , where is the efficiency of vaccine, 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 . When we introduce antibiotic treatment at a rate of , the period of infection (56 days) will be reduced to some new period such that implies . Since 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 , where is the basic reproduction number as in [10]. This implies that, 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. And, threshold of vaccination is also given by , where is period of vaccination, which can be interpreted that at least 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 depends on , a single value of can have many practical interpretation. For the last option, in applying both vaccination and antibiotic treatment, the threshold value of vaccination depends on the rate of antibiotic treatment, , and is given by . Thus, if we introduce both antibiotic treatment and vaccination in the population such that of infectious cattle receive antibiotic treatment or the period of infection is reduced to 28 days, then at least of susceptible cattle should get vaccination in less than 73.8 days in order to control the disease. Mathematically, it means that, for , we should take to make , 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 instead of .
6.2. Numerical Simulations
Initially we consider a herd size of cattle population which is consisting of an infectious cattle and susceptible cattle with individual animals as the epidemiological units of interest. For the same assumption and parametric values, the result obtained in this paper coincides with the result obtained in [10]; in this case, as shown in Figure 3. Using parameter values in Table 1, model (1)-(6) is numerically solved. If no intervention is considered, the population goes extinct with as shown in Figure 4. Figures 5 and 6 show the number of cattle in each compartment when we consider intervention by antibiotic treatment without vaccination and vice versa, respectively; in both cases, . Lastly, considering both antibiotic treatment and vaccination, the number of cattle in each compartment at time is plotted in Figure 7; in this case, for parametric values in Table 1.
7. 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 . We proved that, for , the DFE is globally asymptotically stable, thus CBPP dies out, whereas for , the EE is globally asymptotically stable and hence the disease persists in all the populations. Hence 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, when ; see Figure 3. Similarly, when , which is the right value, and without any intervention, ; 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 of infectious cattle, without vaccinating any of healthy cattle; see Figure 5. As a second option, we can also control the disease by vaccinating 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 of susceptible are vaccinated within a period of 73 days and of infectious cattle are treated, can be made less than one and hence we can control the disease; see Figure 7. In all the above three intervention methods, 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 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.
Data Availability
Data are available in the literature.
Disclosure
An earlier version of this paper was presented at the International Conference in Mathematical Methods and Models in Biosciences and Young Scientists, Biomath 2017, Skukuza Camp, Kruger Park, South Africa.
Conflicts of Interest
The authors declare that they have no conflicts of interest.