Research Article  Open Access
Dynamics of Mycobacterium and bovine tuberculosis in a HumanBuffalo Population
Abstract
A new model for the transmission dynamics of Mycobacterium tuberculosis and bovine tuberculosis in a community, consisting of humans and African buffalos, is presented. The buffaloonly component of the model exhibits the phenomenon of backward bifurcation, which arises due to the reinfection of exposed and recovered buffalos, when the associated reproduction number is less than unity. This model has a unique endemic equilibrium, which is globally asymptotically stable for a special case, when the reproduction number exceeds unity. Uncertainty and sensitivity analyses, using data relevant to the dynamics of the two diseases in the Kruger National Park, show that the distribution of the associated reproduction number is less than unity (hence, the diseases would not persist in the community). Crucial parameters that influence the dynamics of the two diseases are also identified. Both the buffaloonly and the buffalohuman model exhibit the same qualitative dynamics with respect to the local and global asymptotic stability of their respective diseasefree equilibrium, as well as with respect to the backward bifurcation phenomenon. Numerical simulations of the buffalohuman model show that the cumulative number of Mycobacterium tuberculosis cases in humans (buffalos) decreases with increasing number of bovine tuberculosis infections in humans (buffalo).
1. Introduction
Mycobacterium tuberculosis (MTB) and bovine tuberculosis (BTB) are chronic bacterial diseases, classified amongst the closely related species that form the M. tuberculosis complex (MTBC) [1]. The human MTB is caused by tubercle bacillus (M. tuberculosis), while BTB is caused by bovine bacillus (M. bovis) [2]. MTB and BTB affect a wide range of hosts, including domestic livestock (such as cattle, goats, sheep, deer, and bison), wildlife (such as badgers, deer, bison, and African buffalo) which can either be reservoir or spillover, and humans [3].
MTB remains a major global health problem affecting millions of people each year [4]. It is ranked second to human immunodeficiency virus (HIV) among the leading causes of death worldwide [4]. For instance, in the year 2012, there were 8.6 million new MTB cases and 1.3 million MTB deaths globally [4]. Similarly, BTB remains a serious problem for animal and human health in many developing countries [5]. Its widespread distribution has drastic negative socioeconomic impact, affecting public health, international trade, tourism, animal mortality, and milk production [6]. For example, in Argentina, the annual loss due to BTB is estimated to be US$ 63 million [7]. A benefit/cost analyses of BTB eradication in the United States showed an actual cost of US$ 538 million between 1917 and 1992 (current programs cost approximately US$ 3.5–4.0 million per year [5]).
The African buffalo transmits BTB to humans, via aerosol or oral (as a result of consuming raw unpasteurized milk) [1]. Furthermore, BTB can be transmitted from human to human by direct contact [1]. As in cattle, the main sources of BTB transmission in buffalo are direct contact, aerosol, oral, through a bite, or contamination of a skin wound [3] (other means of transmission, such as vertical and pseudovertical [8], also occur). Similarly, MTB can be transmitted from human to human, or from human to buffalo, via coughing or sneezing [1]. In humans, MTB is regarded to be an airborne disease [9]. It typically affects the lungs (pulmonary TB), but can affect other parts of the body also (extrapulmonary TB) [3]. Common signs and symptoms of MTB include coughing, chest pain, fever, weakness, and weight loss. The incubation period of MTB is approximately 2 to 12 weeks. African buffalos infected with BTB show clinical signs only when the disease has reached an advanced stage (the clinical signs of BTB in buffalo at such stage include coughing, debilitation, poor body condition or emaciation, and lagging when chased by helicopter [3, 8]). The incubation period for BTB is approximately 9 months to a year, and infections can remain dormant for years (and reactivate during periods of stress or in old age) [6].
BTB is typically controlled using isolation or quarantine of infected herds, testandslaughter policy, and pasteurization of milk [10]. In South Africa’s Kruger National Park (KNP), other control measures, such as culling, vaccination, and their combination, are also used [10] (a demographic map of KNP and a herd of African Buffalos [11] are shown in Figure 1). Similarly, MTB in humans is controlled via standard sixmonth course of four antimicrobial drugs [12–15]. The World Health Organization embarked on numerous global initiatives, such as “Stop TB Partnership,” “International Standards of Tuberculosis Care and Patient’s Care,” and the “Global Plan to Stop TB,” with the hope of minimizing the burden of TB worldwide [12].
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Several mathematical models have been developed and used to gain insight into the transmission dynamics of BTB or MTB in populations (see, for instance, [8, 10, 12, 13, 16–19] and some of the references therein). However, none of these studies incorporate humans in the transmission dynamics of BTB. The purpose of the current study is to design, and analyse, a new realistic model (which extends some of the aforementioned studies in the literature) for BTBMTB transmission dynamics. The objective is to gain insight into the qualitative dynamics of the two diseases in a buffalohuman population.
The paper is organized as follows. The new model for BTB and MTB transmission dynamics in a community consisting of human and buffalo is presented in Section 2. The buffaloonly model is rigorously analysed in Section 3, and the full buffalohuman model is analysed in Section 4. Numerical simulations are also reported.
2. Model Formulation
The model to be designed is based on the transmission dynamics of MTB and BTB in a population consisting of humans and African buffalos. The total human population at time , denoted by , is subdivided into seven mutually exclusive compartments of susceptible humans , exposed humans (who have been infected with MTB but have not yet shown clinical symptoms of the disease) , exposed humans with BTB , infected humans with clinical symptoms of MTB , infected humans with clinical symptoms of BTB , and humans who recovered from MTB or BTB , so that Similarly, the total buffalo population (in the herd) at time , denoted by , is split into susceptible , earlyexposed with BTB , earlyexposed with MTB , advancedexposed with BTB , advancedexposed with MTB , infected with clinical symptoms of BTB , infected with clinical symptoms of MTB , and recovered from BTB or MTB , so that The susceptible human population is increased by the recruitment of people (either by birth or immigration) into the humanbuffalo poulation (at a rate ). The population is decreased by infection with MTB (at a rate ) or BTB (at a rate ), where with In (3) and (4), and represent the effective contact rates (i.e., contacts capable of leading to MTB or BTB infection), respectively. Furthermore, and are modification parameters accounting for the assumed reduction in infectiousness of exposed humans, in comparison to infected humans with clinical symptoms of MTB or BTB, respectively. Similarly, and are modification parameters accounting for the assumed reduction in infectiousness of exposed buffalos, in comparison to infected buffalos with clinical symptoms of BTB. The modification parameter accounts for the assumed reduced likelihood of susceptible humans acquiring BTB infection, in comparison to susceptible buffalos acquiring BTB infection. Natural death is assumed to occur in all human compartments at a rate . Thus, the rate of change of the susceptible human population is given by The population of exposed humans with MTB is generated by the infection of susceptible humans with MTB (at the rate ) and is decreased by the development of clinical symptoms of MTB (at a rate ), exogenous reinfection (at a rate , where accounts for the assumption that reinfection of exposed humans with MTB occurs at a rate lower than primary infection of susceptible humans with MTB) and natural death, so that Similarly, the population of exposed humans with BTB is increased by the infection of susceptible humans with BTB (at the rate ) and is reduced by the development of clinical symptoms of BTB (at a rate ), exogenous reinfection (at a rate , with similarly defined as ) and natural death. Thus, The population of humans with clinical symptoms of MTB increases following the development of clinical symptoms of MTB by exposed humans (at the rate ) and exogenous reinfection of exposed and recovered humans (at the rates and , resp., with ). This population is decreased by recovery (at a rate ), natural death, and MTBinduced death (at a rate ), so that The population of infected humans with clinical symptoms of BTB is generated by the development of clinical symptoms of BTB by exposed humans (at the rate ) and reinfection of exposed and recovered humans (at the rates and , resp., with ). This population is decreased by recovery (at a rate ), natural death, and BTBinduced death (at a rate ). This gives The population of humans who recovered from MTB is generated by the recovery of humans with clinical symptoms of MTB (at the rate ). It is decreased by exogenous reinfection (at the rate ) and natural death. Hence, It should be mentioned that, since MTBinfected humans do not completely eliminate the bacteria from their body (usually the bacteria hide in the bone marrow), “recovery” in this case implies (or represents) a long period of latency (which could last for a lifetime) [19, 30].
Similarly, the population of humans who recovered from BTB is generated by the recovery of humans with clinical symptoms of BTB (at the rate ) and is decreased by reinfection (at the rate ) and natural death, so that The population of susceptible buffalos is generated by the recruitment of buffalos (either by birth or restocking from other herds) at a rate . It is assumed that all recruited buffalos are susceptible. The population of susceptible buffalos is decreased by acquisition of BTB infection (following effective contact with a human or buffalo infected with BTB), at the rate (where, , with the modification parameters accounting for the expected reduced likelihood of humans transmitting of BTB to buffalo, in relation to BTB transmission from a human to another human) or MTB (following effective contact with a human infected with MTB) at a reduced rate (where is a modification parameter accounting for the assumed reduction in the transmissibility of MTB from humans to buffalos, in comparison to MTB transmission from humans to humans), and by natural death (at a rate , buffalos in each epidemiological compartment suffer natural death at this rate). Thus, An important feature of BTB transmission within the buffalo population is that an infected buffalo could be in early or advanced stage of infection. This is owing to the fact that the clinical symptoms of BTB usually take months to manifest in buffalos [6]. Thus, BTB infections can remain dormant for years and reactivate during periods of stress or in old age [6]. These (early and advancedexposed stage) features are incorporated in the model being develop. The population of buffalos earlyexposed to BTB is increased by the infection of susceptible buffalos with BTB (at the rate ). This population is decreased by exogenous reinfection with BTB (at a rate , with ), progression to the advancedexposed class (at a rate ), and natural death. This gives
The population of buffalos earlyexposed to MTB is increased by the infection of susceptible buffalos with MTB (at the rate , where is as defined above). The population is decreased by exogenous reinfection (at a rate ), progression to the advancedexposed MTB class (at a rate ), and natural death. This gives
The population of buffalos at advancedexposed BTB class is increased by the progression of buffalos in the earlyexposed BTB class (at the rate ). It is decreased by exogenous reinfection (at a rate ), development of clinical symptoms of BTB (at a rate ), and natural death, so that Similarly, the population of buffalos at advancedexposed MTB class is generated by the progression of buffalos in the earlyexposed MTB class (at the rate ). It is decreased by exogenous reinfection (at a rate ), development of clinical symptoms of MTB (at a rate ), and natural death. Hence, The population of buffalos with clinical symptoms of BTB is increased by the development of clinical symptoms of exposed buffalos with BTB (at the rate ) and by the exogenous reinfection of exposed and recovered buffalos (at the rates and , resp.). It is decreased by recovery (at a rate ), natural death, and BTBinduced mortality (at a rate ). Thus, The population of buffalos with clinical symptoms of MTB is increased by the development of clinical symptoms of exposed buffalos with MTB (at the rate ) and by the exogenous reinfection of exposed and recovered buffalos (at the rates and , resp.). It is decreased by recovery (at a rate ), natural death, and MTBinduced mortality (at a rate ). Thus, The population of buffalos who recovered from BTB is increased following the recovery of buffalos with clinical symptoms of BTB (at the rate ). It is decreased by reinfection (at the rate ) and natural death, so that Finally, the population of buffalos who recovered from MTB is generated by the recovery of buffalos with MTB (at the rate ) and is decreased following reinfection (at the rate ) and natural death. This gives It is assumed that recovered buffalos and humans acquire permanent natural immunity against BTB or MTB infection so that recovered buffalos and humans do not return to their respective susceptible class (albeit buffalos and humans in recovered classes can acquire reinfection).
Thus, based on the above assumptions and formulations, the model for the BTBMTB transmission dynamics in a humanbuffalo population is given by the following deterministic system of nonlinear differential equations (a flow diagram of the model is depicted in Figure 2, and the associated variables and parameters are described in Tables 1 and 2, resp.):


The model (21) is, to the authors’ knowledge, the first to incorporate humans and MTB dynamics in the transmission dynamics of BTB in a humanbuffalo community. Furthermore, it extends numerous models for BTB transmission in the literature, such as those in [8, 12, 13, 16–19], by, inter alia,(i)including the dynamics of early and advancedexposed buffalos (exposed buffalo classes were not considered in the models in [8, 12, 16–18]),(ii)allowing for BTB and MTB transmission by exposed buffalos and humans (this was not considered in [8, 12, 16–19]),(iii)including the dynamics of humans (this was not considered in [8, 13, 18, 19]),(iv)allowing for the reinfection of exposed and recovered buffalos and humans (this was not considered in [8, 12, 13, 16, 18]),(v)allowing for the transmission of both BTB and MTB in both the buffalo and human populations (this was not considered in the models in [8, 12, 13, 16–18]).
The model (21) will now be rigorously analysed to gain insight into its dynamical features. Before doing so, it is instructive, however, to consider the dynamics within the buffalo population only as below.
3. Analysis of BuffaloOnly Model
Consider the model (21) in the absence of humans (buffaloonly model), obtained by setting the human components to zero (i.e., setting in (21)), given by where, now, The buffaloonly model (22) is fitted using data obtained from South Africa’s Kruger National Park [29], as shown in Figure 3 (from which it is evident that the model mimics the data reasonably well).
It is worth stating that since there are no humans in the dynamics of the buffaloonly model (22), MTB is not transmitted to susceptible buffalos. Furthermore, it is clear from the third equation in (22) that Substituting (24) in the fifth equation in (22) shows that Similarly, by substituting into the equations for and in (22), it follows that Thus, the buffaloonly model reduces to the following (limited) model at steadystate:
Lemma 1. The following biologically feasible region of the buffaloonly model (27) is positively invariant and attracting.
Proof. Adding the equations in the buffaloonly model system (27) gives so that It follows from (30) and the Gronwall inequality that In particular, if . Thus, is positively invariant. Hence, it is sufficient to consider the dynamics of the buffaloonly model (27) in (where the model can be considered to be epidemiologically and mathematically wellposed [31]).
Theorem 2. Let the initial data , , , , . Then, the solutions , , , , and of the buffaloonly model (27) are positive for all .
Proof. It is clear from the first equation of the buffaloonly model (27) that so that Using similar approach, it can be shown that , , , and , for all .
3.1. Asymptotic Stability of DiseaseFree Equilibrium (DFE)
3.1.1. Local Asymptotic Stability
The DFE of the buffaloonly model (27) is given by The linear stability of can be established using the next generation operator method on the system (22) [32, 33]. The matrices (for the new infection terms) and (of the transition terms) associated with the system (27) are given, respectively, by where , , and . It follows that the basic reproduction number of the buffaloonly model (27), denoted by , is given by Hence, using Theorem 2 of [33], the following result is established.
Lemma 3. The DFE, , of the buffaloonly model (27), is locally asymptotically stable (LAS) if and unstable if .
The threshold quantity, , represents the average number of secondary cases of BTB in the buffalo population that one BTBinfected buffalo can generate if introduced into a completely susceptible buffalo population [31, 34, 35].
3.1.2. Interpretation of
The threshold quantity, , can be interpreted as follows. It is worth recalling, first of all, that susceptible buffalos can acquire BTB infection following effective contact with either earlyexposed buffalo with BTB (), advancedexposed buffalo with BTB (), or infected buffalo with clinical symptoms of BTB (). It follows that the number of BTB infections generated by an earlyexposed buffalo (near the DFE) is given by the product of the infection rate of an earlyexposed buffalo () and the average duration of stay in the earlyexposed class (). Thus, the average number of BTB infections generated by earlyexposed buffalos is given by
Similarly, the number of BTB infections generated by an advancedexposed buffalo (near the DFE) is given by the product of the infection rate of advancedexposed buffalos (), the probability that earlyexposed buffalo survived the earlyexposed class and move to the advancedexposed class (), and the average duration of stay in the advancedexposed class (). Thus, the average number of BTB infections generated by advancedexposed buffalos is given by
Furthermore, the number of BTB infections generated by an infected buffalo with clinical symptoms of BTB (near the DFE) is given by the product of the infection rate of buffalos with clinical symptoms of BTB (), the probability that an advancedexposed buffalo survived the advancedexposed class and move to the symptomatic class (), and the average duration of stay in the symptomatic class (). Thus, the average number of BTB infections generated by advancedexposed buffalos is given by
The sum of the terms in (37), (38), and (39) gives . That is, the average number of new infections generated by infected buffalos (earlyexposed, advancedexposed, or symptomatic) is given by (noting that and )
The epidemiological implication of Lemma 3 is that BTB can be effectively controlled in (or eliminated from) the buffalo population (herd) if the initial sizes of the state variables of the buffaloonly model (27) are in the basin of attraction of the DFE . It is worth mentioning, however, that TB models with exogenous reinfection are often shown to exhibit the phenomenon of backward bifurcation (where the stable DFE coexists with a stable endemic equilibrium when [12, 17, 25, 36]). The epidemiological implication of this phenomenon is that the classical requirement of is, although necessary, no longer sufficient for diseases elimination [12, 36]. Thus, the presence of backward bifurcation in the transmission dynamics of a disease makes its effective control in a population more difficult. Hence, it is instructive to explore the possibility of such phenomenon in the buffaloonly model (22). This is investigated below.
Theorem 4. The buffaloonly model (22) undergoes backward bifurcation at whenever the bifurcation coefficient, , given by (A.9) (in Appendix A), is positive.
The proof of Theorem 4, based on using centre manifold theory [17, 33], is given in Appendix A. It should be noted that, in the absence of reinfection of exposed and recovered buffalos (i.e., the case of the model (22) with ), the backward bifurcation coefficient, , given by (A.9) in Appendix A, reduces to (it should be noted, from Appendix A, that , , and, from Theorem 2, that all parameters of the buffaloonly model (22) are nonnegative): where , and . Since the bifurcation coefficient, , is automatically negative, it follows from the analyses in Appendix A, and Theorem 4.1 of [17], that the buffaloonly model (22) does not undergo backward bifurcation in the absence of reinfection (this result is consistent with that in [12, 17, 36], on the transmission dynamics Mycobacterium tuberculosis in human populations). This result is summarized below.
Lemma 5. The buffaloonly model (22) does not undergo backward bifurcation at in the absence of the reinfection of exposed and recovered buffalos ().
Hence, this study shows that the reinfection of exposed and recovered buffalos causes the phenomenon of backward bifurcation in the transmission dynamics of BTB and MTB in a buffaloonly population. To further confirm the absence of backward bifurcation in this case, a global asymptotic stability result is established for the DFE of the buffaloonly model (27) in the absence of reinfection (i.e., ) below.
3.2. Global Asymptotic Stability of the DFE
Consider the buffaloonly model (27) in the absence of the reinfection of exposed () and recovered () buffalos.
Theorem 6. The DFE, , of the buffaloonly model (27) with is globally asymptotically stable (GAS) in if .
Proof. Consider the buffaloonly model (27) in the absence of reinfection (). Furthermore, let . Consider the linear Lyapunov function , where with Lyapunov derivative given by (where a dot represents differentiation with respect to time ) Since all the parameters and variables of model (27) are nonnegative (Theorem 2), it follows that for with if and only if . Thus, it follows, by LaSalle’s Invariance Principle [37], that Since (from (44)), it follows that, for sufficiently small , there exists a constant , such that for all . Hence, it follows from the fifth equation of the buffaloonly model (27) that, for . Thus, by comparison theorem [38], , so that, by letting, , Similarly, it can be shown that Thus, it follows from (45) and (46), that . Hence, Furthermore, substituting (44) in the first equation of (27) shows that Thus, by combining equations (44), (47), and (48), it follows that every solution of the equations of the buffaloonly model (27), with and initial conditions in , approaches as (whenever ).
Theorem 6 shows that, in the absence of the reinfection of exposed and recovered buffalos (i.e., ), BTB can be eliminated from the buffaloonly population if the reproduction number of the model can be brought to (and maintained at) a value less than unity. Figure 4(a) depicts the solution profiles of the buffaloonly model (27), generated using various initial conditions, showing convergence to the DFE when (in line with Theorem 6).
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3.3. Existence of Endemic Equilibria: Special Case
In this section, the existence of nontrivial (endemic) equilibria (where the components of the infected variables of the model are nonzero) of the buffaloonly model (27) is explored for the special case without reinfection (i.e., ). Solving the equations of the buffaloonly model (27) at steadystate gives the following general form of the endemic equilibrium (denoted by ): where with the force of infection at steadystate given by
Using (50) in the expression for in (51) shows that the nonzero equilibrium of the model (22) satisfies the linear equation: where and . Clearly, the coefficient is always positive, and is positive (negative) if is less than (greater than) unity, respectively. Thus, the linear system (52) has a unique positive solution, given by , whenever . Further, the force of infection for buffalos is negative whenever (which is biologically meaningless). Hence, the buffaloonly model (27) has no positive equilibrium in this case. These results are summarized below.
Theorem 7. The buffaloonly model (27), with , has a unique endemic equilibrium, , whenever , and no endemic equilibrium otherwise.
3.3.1. Global Asymptotic Stability of Endemic Equilibrium
The global asymptotic stability of the unique endemic equilibrium () of the buffaloonly model is explored for the special case without reinfection () and BTBinduced death in buffalos (). It is convenient to define the stable manifold of the DFE of the buffaloonly model (27).
Theorem 8. The unique endemic equilibrium () of the buffaloonly model (27), with , is GAS in if .
Proof. Consider the buffaloonly model (27) with . For this case, it follows from Theorem 7 that the buffaloonly model (27) has a unique endemic equilibrium whenever . Furthermore, setting in model (27) shows that as . Consider the following nonlinear Lyapunov function (of GohVolterra type) for the subsystem of model (27) involving the state variables , , , and (noting that is now replaced by its limiting value ):
where . The Lyapunov derivative of is given by
Using the following steadystate relations (obtained from (27)),
the Lyapunov derivative can be simplified to
Thus,
Finally, since the arithmetic mean exceeds the geometric mean, it follows then that
Furthermore, since all the model parameters are nonnegative, it follows that for . Thus, is a Lyapunov function for the subsystem of model (27) on . Therefore, it follows, by LaSalle’s Invariance Principle [37], that
Since as , it follows from the equation for in (27) that as . The proof is concluded using similar arguments as in the proof of Theorem 6.
The epidemiological implication of Theorem 8 is that BTB will be endemic in the buffalo population if (and ). Figure 4(b) depicts the solutions of model (27) for the case when and , showing convergence of the initial solutions to the unique endemic equilibrium (in line with Theorem 8).
3.4. Sensitivity and Uncertainty Analyses
In this section, sensitivity and uncertainty analyses will be carried out, using Latin hypercube sampling (LHS) and partial correlation coefficient (PRCC) [39–41], to assess the effect of uncertainty in the estimate of the parameter values used to simulate the buffaloonly model (on the simulation results obtained) and to determine the key parameters that drive the dynamics of the disease in the buffalohuman population. The ranges and baseline values of the parameters of the buffaloonly model, given in Table 3 with (i.e., in the absence of backward bifurcation), will be used in these analyses. Each parameter of the buffaloonly model (22) is assumed to obey a uniform distribution [42]. Following [42], a total of 1000 LHS runs () are carried out. Furthermore, the following initial conditions (which are consistent with the dynamics of African buffalo in the Kruger National Park [29]): , , , , , , , , are used in the simulations.
