Abstract
To better understand the dynamics of zoonotic diseases, we propose a deterministic mathematical model to study the dynamics of zoonotic brucellosis with a focus on developing countries. The model contains all the relevant biological details, including indirect transmission by the environment. We analyze the essential dynamic behavior of the model and perform an optimal control study to design effective prevention and intervention strategies. The sensitivity analysis of the model parameters is performed. The aim of the controls is tied to reducing the number of infected humans, through health promotional programs within the affected communities. The Pontryagin’s Maximum Principle is used to characterize the optimal level of the controls, and the resulting optimality system is solved numerically. Overall, the study demonstrates that through health promotional programs on zoonotic diseases among villagers, it is vital that they should be conducted with high efficacy.
1. Introduction
Zoonoses (also known as zoonosis and zoonotic diseases) are infectious diseases caused by bacteria, viruses, and parasites that spread between animals and humans. Some types of zoonotic diseases are African sleeping sickness, anthrax, bird flu, brucellosis, influenza, rabies, Zika, and Ebola [1]. The zoonotic diseases have been categorized into the more common endemic zoonoses such as salmonellosis, brucellosis, and leptospirosis which are responsible for more than 2.2 million human deaths and 2.4 billion cases of illness annually and the less common epidemic and emerging zoonoses such as rift valley, anthrax, valley fever, Ebola, and Zika which either occur in sporadic outbreaks in neglected populations or that are new or reappearing with an increased incidence of geographical range [2]. Zoonotic diseases have several modes of transmission, with the main ones being direct, indirect, and vector-borne transmissions [3]. Direct transmission entails coming into contact with the saliva, blood, urine, mucous, feces, or other bodily fluids. Indirect contact is due to coming into contact with areas where animals live and roam or surfaces that have been contaminated, such as pet habitants, pet food, chicken coops, and aquarium tank water. A vector-borne entails being bitten by a tick or an insect like a mosquito or flea. This class of diseases has been the principal source of emerging health risks, and it is estimated that zoonotic pathogens have accounted for more than 60% of emerging infectious diseases recently [2]. Some of the risk factors associated with the emergence of zoonotic diseases and spill over into humans include human encroachment, population expansion, consumption of exotic food, migratory movements, and ecotourism [4]. Endemic zoonotic diseases have the dual impact of causing illness and death in humans and animals as well as substantial loss in resource-poor societies where livestock farming is a major engine of economic growth at the household and national levels. The World Bank estimated that 6 major zoonotic disease epidemics during 1997-2009 resulted in an economic loss of more than $80 billion [1]. Controlling zoonotic disease outbreaks has become ever more important, it has been estimated that since 1940, about 40% of the emerging infectious diseases affecting humans globally, but mainly in developing countries, have originated from animals, both domestic and wild [5, 6].
The attention given to zoonotic diseases has however focused more on emerging zoonoses that pose global economic and health threats and less on the endemic zoonotic disease which tend to occur among populations with little financial muscle, such as in sub-Saharan Africa. The relative risk for emerging infectious disease vents from wildlife sources has continued to be a challenge in sub-Saharan Africa. An increase in the interaction between wildlife, domestic animals, and humans increases the chance of zoonotic disease transmission. The sub-Saharan region was identified as one of the hotspot regions with a high prevalence of endemic zoonotic diseases and where it has a large rural population that lives in close proximity with livestock and wildlife [6, 7]. In some parts of the sub-Saharan countries such as Zimbabwe, they still have problems regarding the control of zoonoses, mainly due to the lack of enough infrastructure and resources for disease surveillance [7]. Poverty and lack of education towards the zoonotic diseases lead many people, especially those from rural areas, accessing commodities such as fresh unpasteurised milk and uninspected meat from domestic animals on the informal food markets [7]. Some researchers have managed to show that the risk of zoonotic diseases would increase or decrease, in the various keeping systems and to the public as a whole depending on their education levels towards zoonotic diseases. Low levels of education towards zoonotic disease among the rural villagers have been one of the major setbacks in the fight against zoonotic diseases [8]. Historically, infectious disease specialists in collaboration with governmental organizations have attempted to develop effective controls and eradication strategies gradually, using field experience that is unique to the region and disease. A particular challenge in controlling zoonotic infections in this way is to appropriately allocate resources in the multispecies system [8]. Thus, designing effective control strategies requires achieving a proper tradeoff between the costs resulting from disease prevalence and the costs of control.
Mathematical modeling, analysis, and simulation for infectious diseases have proved to be an essential guiding tool that could give a sound direction to policymakers and public health administration on how to effectively prevent and control zoonotic diseases. Mathematical modeling of zoonotic diseases has been a particular area of burgeoning interest over the last few years, see the articles [4, 8–16] for a few representative samples. The current study complements many of the earlier published studies by providing a rigorous qualitative analysis of a mathematical model which seeks to understand the impact of educational campaigns in curtailing the spread of zoonotic diseases. In this paper, we propose a compartmental model for the spread of zoonoses incorporating all the essential biological details. The model incorporates the aspect of educational campaigns in the human-domestic animal interface space and allows optimal control methods to be used. The model also includes direct and indirect modes of transmission. Our study focuses on zoonotic diseases, humans, and domestic animals; hence, we shall make use of brucellosis as a zoonotic disease and cattle as domestic animals, for illustrative purposes. Brucellosis is a zoonotic disease that affects domesticated animals, wildlife, and humans. Animals acquire the infection mainly through direct contact with infected cattle or indirectly from the environment containing large quantities of bacteria discharged by infected individuals [17], whereas in human, common routes of infection include direct inoculation through cuts and abrasions in the skin or inhalation of infectious aerosols and ingestion of infectious unpasteurized milk or other dairy products [18]. Human to human transmission is extremely rare [17, 18].
The structure of the paper is as follows. Section 2 constitutes model formulation, and analytic results are presented in Section 3. The sensitivity analysis of the reproduction number is reported in Section 4, and the optimal control analysis is presented in Section 5. Numerical simulations are presented in Section 6. The paper concludes with a discussion in Section 7.
2. Model Formulation
In this section, we introduce a continuous mathematical model for the transmission dynamics of zoonotic disease in the form of brucellosis in both humans and cattle. Guided by the information on the natural history of brucellosis infection in both humans and cattle populations to determine, the basic plausible assumptions for the model formulation are determined [19, 20]. The human population is divided into four mutually exclusive epidemiological subpopulations consisting of uneducated susceptibles , educated susceptibles , infected , and the recovered humans, so that the total human population is given by where . The cattle population is divided into three mutually exclusive epidemiological subpopulations consisting of the susceptible , infected , and recovered cattle, so that the total cattle population is given by . We denote the quantity of the brucella in the environment by , which is shed off at a rate by the cattle and by humans. We assume that the brucella in the environment decays at the rate of . The natural death rates for the humans and cattle are given by and , respectively. The recruitment of the susceptible humans and the susceptible cattle are through birth, given by and , respectively. We assume homogeneous mixing, that is, all the susceptible humans have the same likelihood to be infected and also the susceptible cattle have the same chance to be infected. The force of infection for humans is given by where is the effective contact rate by the infected cattle towards humans. is the indirect infection from the brucella in the environment to the susceptible humans. denotes the brucella concentrations measured with respect to their infection doses. The force of infection for the cattle is given by where is the effective contact rate by the infected cattle on the susceptible cattle. is the indirect infection from the brucella in the environment to the susceptible cattle. The uneducated susceptible individuals are educated towards zoonotic diseases at a rate of through educational campaigns in the form of health promotional programs. A parameter represents the efficacy of health promotion programs. If , then the health promotion programs are not effective, if corresponds to completely effective health promotion programs, while implies that the health promotion programs will be effective to some degree. Thus, the susceptible uneducated humans are infected with brucellosis at a rate of while the susceptible educated humans are infected at a rate of . We assume that the infected humans recover from the brucellosis at a rate of while the cattle recover at a rate of , both getting permanent immunity [18, 21]. Parameter represents death due to the respective zoonotic disease, and represents the culling rate. We assume that the rate of culling is at its minimal since most of the villagers in sub-Saharan Africa on the human-domestic animal-wildlife interface are not well educated in terms of dealing with zoonotic diseases. We shall refer to the health promotional programs as educational campaigns, throughout the manuscript. Here, we construct a system of nonlinear differential equations to model the disease dynamics of brucellosis.
The model variables and their descriptions are summarised in Table 1.
The model parameters and their possible values are shown in Table 2.
3. Analysis of the Model
3.1. Positivity and Boundedness of Solutions
It can be easily proved that the domain of biological interest is positively invariant and attracting with respect to the model in equation (3).
3.2. The Disease-Free Equilibrium and Basic Reproduction Number
It can be established that system (3) always has a disease-free equilibrium (DFE) given by
Denoting the reproduction number by , which is a measure of the average number of secondary infections generated by a single infectious case in a fully susceptible population during its infectious period [26], the reproduction number is commonly regarded as a threshold quantity for the disease dynamics, essential in determining the transmission and spread of the disease. Using the next-generation matrix notations in [26], the nonnegative matrix that denotes the generation of new infections and the nonsingular matrix that denotes the disease transfer among compartments are, respectively, given by
That is, the reproduction number of system (3) is the spectral radius of the next-generation matrix where
Making use of Theorem 2 in Van den Driessche and Watmough [26], we establish the following result.
Theorem 1. If then is locally asymptotically stable and unstable otherwise.
Furthermore, a stronger result regarding the global dynamics of the DFE can be established. We will utilize the approach of Lyapunov functions [27–31] in the analysis of global asymptotic stability.
Theorem 2. If , the DFE is globally asymptotically stable in . If , the system is uniformly persistent.
Proof. Let Since from (3)
it follows that
where and are as defined in equation (6). It is worth noting that and are nonnegative. By the Perron-Frobenius Theorem [32], the nonnegative matrix has a positive left eigenvector with respect to the eigenvalue that is, Motivated by [30], consider a Lyapunov function
Differentiating along with solutions of (3), we have
If , the equality implies that This leads to since denotes a positive left Perron eigenvector. Hence, when , system (3) yields , and . Thus, the invariant set on which contains only the point . If , we also have , and it can also be shown that the invariant set on which contains only the point . Therefore, by LaSalle’s invariance principle [33], is globally asymptotically stable in when
If then by continuity, in a neighbourhood of in the interior of Solutions in the interior of sufficiently close to move away from the DFE implying that the DFE is unstable. This completes the proof.
The result established in Theorem 2 portrays that is a sharp threshold for disease dynamics: the disease will die out when whereas the disease will persist when Biologically, a uniform persistent system shows that the infection persists for a long period of time. Now, we investigate uniform persistence, and we claim the following result.
Theorem 3. If system (3) is uniformly persistent, namely, there exists a constant such that for any initial conditions satisfying
Proof. Let and Hence, Let be a semiflow induced by the solutions of system (3) and By Equation (4), we have and is bounded in . Therefore, there exist a global attractor for . The disease-free equilibrium is the unique equilibrium on the manifold and is globally asymptotically stable on . Moreover, and no subsets of forms a cycle in . Finally, since the disease-free equilibrium is unstable on if , we deduce that system (3) is uniformly persistent by using a result from [34] (Theorem 1.3.1 and Remark 1.3.1). This completes the proof.
3.3. Endemic Equilibrium
System (3) has the endemic equilibrium point given by in terms of the forces of infection and . Next, we present the local stability of when the reproduction number is sufficiently close to 1. We shall make use of the Centre Manifold Theory [35], but firstly, we present the following Lemma.
Lemma 4. Consider the following general system of ordinary differential equations with a parameter : where is an equilibrium of the system, that is , and assume
A1. is the linearisation matrix of system (3) around the equilibrium and evaluated at . Zero is a simple eigenvalue of , and all other eigenvalues of have negative real parts.
A2. Matrix has a right eigenvector and a left eigenvector corresponding to the zero eigenvalue. Let be the component of and The local dynamics of (16) around zero is totally governed by and . (i). When with , is locally asymptotically stable, there exists a positive unstable equilibrium. When , is unstable and there exists a negative and locally asymptotically stable equilibrium(ii). When with , is unstable; when , is locally asymptotically stable, and there exists a positive unstable equilibrium(iii). When with , is unstable, and there exists a locally asymptotically stable negative equilibrium; when , is stable, and a positive unstable equilibrium exists(iv). When changes from negative to positive, changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable
Theorem 5. The endemic equilibrium point is locally asymptotically stable if and sufficiently close to 1.
The proof of Theorem 5 is outlined in the appendix.
4. Sensitivity Analysis
In this section, we shall perform some sensitivity analysis on our reproduction number. Sensitivity analysis tells us how important each parameter is to disease transmission. Such information is crucial not only for experimental design but also for data assimilation and reduction of complex nonlinear models [36]. Sensitivity analysis is commonly used to determine the robustness of model predictions to parameter values since there are usually errors in data collection and presumed parameter changes. It is used to discover parameters that have a high impact on and should be targeted by intervention strategies.
Following Arriola and Hyman [37], we present the normalized forward sensitivity indices of to our model parameters in Table 2. The sensitivity index for for example, is
The detailed sensitivity indices of resulting from the evaluation to other model parameters are shown in Table 3.
The parameters that result in positive index increase the value of whenever they are increased while those with a negative index decrease the value of whenever they are increased. For example, since , increasing the rate at which individuals are educated on zoonotic diseases by results in the decrease of the reproduction number by . Thus, educating individuals on zoonotic diseases would be crucial in curtailing the spread of brucellosis. Similarly, improving the efficacy of the educational campaigns by would also trigger a reduction of the reproduction number by It is worth noting that the culling of the infected animals would be beneficial, since its sensitivity index is given by . Thus, culling and burning of carcasses should be carried out, since it had also been noted by some researchers that individuals eat infected animals due to lack of sufficient and correct information, regarding the spread or infectiousness of zoonotic diseases [7].
5. Optimal Control
We noted that for , brucellosis becomes an endemic epidemic. Thus, it is of our interest to explore effective control strategies against this zoonotic disease. In sub-Saharan Africa, individuals who live around the frontiers of human-domestic animal-wildlife interface areas do not have enough awareness and education on zoonotic diseases like brucellosis and bovine TB [7]. Thus, implementation of the health promotion programs in the form of educational campaigns could play an important role in controlling zoonotic diseases among the human population.
We now reconsider model system (3), and we introduce two time-dependent controls in and . The control effort models optimal educational campaigns, and the control increases the positive impact of the educated individuals from being infected by zoonotic diseases. It is advisable to note that when the educational campaigns are strengthened with higher efficacy, the infection risk will be reduced. At the same time, we hope to minimize the costs of achieving this. Therefore, in this section, we will perform a study on the optimal design of the educational campaigns to control the transmission and the spread of brucellosis, using optimal control theory [38–40]. Educational campaigns, in this case, are aimed at encouraging the uninfected to have some protective behaviors.
Thus, introducing two-time dependent controls and , system (3) now becomes
The term corresponds to the situation which prevents or limits the contacts between the susceptible humans and the infected environments and animals (they are reduced through necessary and efficient education about zoonotic diseases among the susceptible individuals), where is the control. It is worth noting that smaller implies less efficient educational campaign strategies in the communities (possibly due to, for example, individuals not being given enough and necessary information towards zoonotic diseases or not using the required language for that respective region). The ideal case, , corresponds to a situation when the likelihood of being infected by the zoonotic disease is almost zero. Practically, this can be achieved by those who have managed to understand and are following the educational campaign information as per the health caregivers.
The goal is to minimize the number of infected humans over a finite time interval at a minimal cost of effort, where is the final time. Mathematically, we formulate our objective functional as follows:
The control efforts are assumed to be nonlinear. We choose to model the control efforts using the quadratic terms, , where the coefficients and represent weights in the cost of the controls. The weight constant over the prescribed time horizon is a measure of the relative costs of the intervention in connection with the reduction of infectious humans. Our problem is to find the optimal controls, and , such that where subject to the state equations (19) with initial conditions. Given the criterion (20) and the regularity of the system of equation (19), the existence of optimal controls is guaranteed by standard results in control theory [41]. The necessary conditions that optimal solutions must satisfy are derived from Pontryagin’s Maximum Principle [42]. This principle converts the system (19), (20), and (21) into the problem of minimizing the Hamiltonian given by
From this Hamiltonian and Pontryagin’s Maximum Principle [42], we obtain the following theorem.
Theorem 6. There exist optimal controls and corresponding solutions, , and , that minimizes over . In order for the above statement to be true, it is necessary that there exist continuous functions such that with transversality conditions Furthermore, the optimal control is represented by
Proof. The existence of optimal controls follows from Corollary 4.1 of [41] since the integrand of is a convex function of and the state system satisfies the Lipshitz property with respect to the state variables. The following can be derived from the Pontryagin’s Maximum Principle [42]: with for evaluated at the optimal controls and corresponding states, which results in the adjoint system (23). The Hamiltonian is minimized with respect to the controls at the optimal controls, so we differentiate with respect to and on the set , giving the following optimality conditions: Solving for and , we obtain By using the bounds , we have the property (25).
Our optimal control problem thus couples the state system (19), the adjoint system (23), and the optimality condition (25). These equations are solved numerically using the forward-backward sweeping method, based on parameters and initial conditions listed in Table 1.
6. Numerical Simulations
In this section, we now present numerical simulations. The existence of optimal control is provided, and the behavior of the optimality system made of 8 ordinary differential equations is evaluated through numerical simulations done with Matlab. The optimality system is solved using an iterative method with Runge-Kutta fourth-order scheme. Starting with a guess for the adjoint variables, the state equations are solved forward in time. Then, these state values are used to solve the adjoint equations backward in time, and the iterations continue until convergence [39].
To illustrate the results of the foregoing analysis, we have simulated system (3) using the parameters in Table 1. We then assume some of the parameters in the realistic range for illustrative purposes. Among the estimated parameters, are the balancing coefficients that have been arbitrarily chosen for illustration purposes. These weight parameters determine the importance of variables in the objective functional (20), thus .
Figure 1(a) shows the impact of the controls on educating susceptible individuals towards zoonotic diseases. Both populations, in the presence and absence of controls, increase in the first 3 years. In the presence of controls, the population begins to drop gradually for the remainder of the period under study. After 3 years in the absence of controls, the population continues to increase steadily for the remainder of the period under review. We can see that in the presence of controls, we have more individuals being educated than in the absence of controls. It is worth noting that the controls are effective from the initial time.
(a)
(b)
(c)
(d)
Figure 1(b) depicts the impact of the controls on the infected human population. We note that when the educational campaign efficacy is low, the controls are not effective.
Figure 1(c) represents the control . The control is initiated from the initial time. After initiation, the control is at the upper bound for the whole period under study. Figure 1(d) represents the control . Control is also initiated from the initial time. After initiation, the control is at the upper bound for approximately 7 years, before returning to the lower bound, 0. Thus, control is feasible for the whole period under review compared to control , which is feasible for only 7 years. In the presence of low educational campaign efficacy, control is unsustainable.
Figure 2(a) illustrates the effects of the controls on educating susceptible individuals towards zoonotic diseases. Both populations, in the presence and absence of controls, increase in the first 3 years. In the presence of controls, the population begins to drop gradually for the remainder of the period under study. After 3 years in the absence of controls, the population continues to increase steadily for the remainder of the period under review. It is worth noting that in the presence of controls, we have more individuals being educated than in the absence of controls. Comparing Figures 1(a) and 2(a), we can see that an increase in from 0.25 to 0.5, does not have much impact on educating more people towards zoonotic diseases.
(a)
(b)
(c)
(d)
Figure 2(b) illustrates the impact of the controls, on the infected human population. The controls start being effective from the initial time. We note that the controls slightly reduce the number of infected human individuals. Thus, increasing the efficacy of the educational campaigns has an impact on reducing the infected population. On comparing Figures 1(b) and 2(b), it is worth noting that increasing the efficacy of the educational campaigns results in the decrease of the infected humans.
Figure 2(c) represents the control , and Figure 2(d) represents the control . The control is initiated from the initial time. After initiation, the control is at the upper bound for the whole period under study. Control is initiated from 0 years and stays at the upper bound for approximately 11 years. Thus, control is feasible for the whole period under review compared to control which is feasible for 11 years. Both controls are crucial.
Figure 3(a) highlights the impact of controls on the population of the educated susceptibles. The controls start being effective from the initial time. The population for the educated susceptibles increases for both cases (with and without controls). It is worth noting that, in the presence of controls, we have more educated individuals as compared to the absence of controls. Comparing Figures 1(a), 2(a), and 3(a), we note that for the scenario when the efficacy of educational campaigns is at 0.75, which is our maximum value, there is no much difference in the number of susceptible individuals who become educated on zoonotic diseases.
(a)
(b)
(c)
(d)
Figure 3(b) highlights the impact of controls on the population of infected individuals. Controls start being effective from the initial time. After 2 years, we can see that we have less infected individuals in the presence of controls as compared to the absence of controls. Thus, for , the cumulative cases for the infected individuals are less, compared to the populations when the values for the efficacy are and
Figure 3(c) represents the control , while Figure 3(d) illustrates the control . The results suggest that more effort should be devoted to both controls since they are both feasible for the whole period under study.
Overall, we can see that if the educational campaign’s efficacy is high, , we have more individuals being educated and the maximum reduction of the individuals being infected by brucellosis. Furthermore, we note that the controls are both feasible and can be well implemented when the educational campaign’s efficacy is high. It is also worth noting that, the efficacy of the educational campaigns can be increased without incurring any extra costs. Thus, health promotion programs need to be implemented with high efficacy for them to be effective. Comparing the weights in the cost of the controls, we have ; this implies that the costs incurred for making educational campaigns to have high efficacy are more than just conducting educational campaigns.
7. Discussion
Carrying out educational campaigns on zoonotic diseases within communities could help reduce the public health implications of zoonotic infections in human-domestic animal-wildlife interface areas, although it will come with certain costs. We have formulated and analyzed a differential equation-based deterministic model for zoonotic disease in the form of brucellosis, with the aim of assessing the impact of educational campaigns on curtailing the spread at minimal costs. The reproduction number is computed and shown that it is an important threshold quantity for disease dynamics. Through the construction of suitable Lyapunov functionals, it has been shown that the model has a stable disease-free equilibrium whenever the reproduction number is less than unity. It was shown that if the reproduction number is greater than unity, the disease persists. Further, it was also demonstrated that whenever the reproduction number is greater than unity, then the model has a unique endemic equilibrium point which is globally asymptotically stable. Sensitivity analysis of the model parameters was carried out, and it was noted that educating susceptible individuals towards zoonotic diseases was vital in the reduction of the infection. Also, the educational campaigns employed should have high efficacy. The model is further extended to incorporate two optimal intervention strategies. The major aim of the controls was to reduce the number of infected individuals at minimal costs. The goal of the first optimal intervention strategy aims to increase the rate of educating the susceptibles towards zoonotic diseases. The second control aims at increasing the positive impact of educating the susceptible individuals, that is, increases the efficacy of the educational campaigns. From the illustrations in our study, it is clear that the time-dependent intervention strategies can lead to the reduction of the zoonotic disease in the community. The efficacy of the educational campaigns is varied in the range of 0.25-0.75. Our findings illustrate that the larger the efficacy, the stronger the impact of both control strategies. Furthermore, we noted that for low and high efficacy levels, we obtained different results with the same costs. Thus, educational campaigns should be carried out with high efficacy. It is worth noting that there was no change in the costs, for implementing the controls when . Comparing our results with others in the literature, it is worth noting that most of them focused on the impact of culling in the reduction of brucellosis prevalence. Some noted the importance of the elimination of the disease in domestic ruminants as the best control strategy. The aspect of vaccines was also investigated, and it was noted that vaccinating both the young and the old was vital. Thus, our results also provide an important avenue in the fight against brucellosis.
Our study has a few limitations. We had to base our numerical results from data which has been published in various literatures. More data sets and experimental studies are needed to include more realistic biological processes in the models. We assumed recruitment through birth only for both cattle and humans, and we left out migration; hence, no one joins the model as a susceptible educated individual. Constant populations have been assumed for both humans and cattle . However, just like any other model, we cannot say the model is complete; it can be extended to include the aspect of the seasonality in the spread of the zoonotic disease.
Appendix
A. Proof of Theorem 5
Proof. To apply Lemma 4, the following simplifications and changes of variables are made first. Let , so that and . Further, by using the vector notation , then system (4) can be written in the form , with , such that
The Jacobian matrix of system (A.1) at is given by
from which it can be shown that
with as defined in equation (8).
Now, we consider , , , regardless of whether or . Taking as the bifurcation parameter and considering that and solving for , we have
where
Note that the linearised system of the transformed equation (A.1) with the bifurcation point has a zero eigenvalue. Hence, the Centre Manifold Theory [36] can be used to analyze the dynamics of system (A.1) near . It can be shown that the Jacobian of system (A.1) has a right eigenvector associated with the following zero eigenvalue given by , where
The left eigenvectors of associated with the zero eigenvalue at is given by
where
B. Computation of the Bifurcation Parameters and
For the sign of , it can be shown that the associated nonvanishing partial derivatives of are given by
From (A.9), it follows that
This excludes the possibility of a backward bifurcation since . For the sign of , it is associated with the following nonvanishing partial derivatives of :
It follows from the expressions in (A.11) that
Thus, and and applying Lemma 4 item (iv), we have established the following result.
Theorem 7. The unique endemic equilibrium is locally asymptotically stable for , but close to 1.
Data Availability
The data used to support the findings of this study are included within the article and cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The author did all the work.
Acknowledgments
A. Mhlanga, the author, would also like to acknowledge with thanks the support of the Department of Mathematics, University of Zimbabwe.