Abstract
A mathematical model is proposed to study the dynamics of the transmission of rabies, incorporating predation of dogs by humans. The model is shown to have a unique disease-free equilibrium which is globally asymptotically stable whenever . Local sensitivity analysis suggests that the disease can be controlled through reducing contact with infected dogs, increasing immunization of dogs, screening recruited dogs, culling of infected dogs, and use of dog meat as a delicacy.
1. Introduction
Rabies is a zoonotic viral disease that is usually transmitted by an infected animal (such as foxes, raccoons, cats, coyotes, bats, skunks, and dogs) through bites or scratch that introduces the virus into the blood of another animal or a human [1, 2]. Once the virus enters the body, it either migrates directly to the brain via the peripheral nervous system to replicate there or stays in the muscles to replicate before migrating to the brain via the neuromuscular junctions. The infection becomes fatal upon reaching the brain since it produces acute inflammations of the brain, leading to coma and eventually death. The infection usually goes through five distinct phases of incubation (3 to 12 weeks), prodome (2 to 10 days, with worsening symptoms over time), acute neurologic period, coma, and death. Early symptoms of rabies infection are general body weakness, fever, and headache, which are also peculiar with common flu and other viral diseases [3]. Rabies is preventable and treatment is possible at early stages of infection. Antirabies vaccination of all pets and domestic animals and humans, screening of imported animals, and awareness creation are some preexposure prophylaxis strategies that governments can embark on to prevent rabies. Individuals can also prevent the disease by vaccinating their pets and keeping away from possible infection sources. Postexposure prophylaxis strategies including washing the bitten area with soapy water for some time and administering a series of shots of rabies immune globulin at the early stages are often effective.
All mammals are possible reservoirs of the rabies virus, but small rodents such as rats and rabbits are less likely to cause transfers. A survey conducted in Ghana by Addy [4] revealed that, of 1,514 exposed domestic animals examined, dogs, cats, and cattle are responsible for 98.7%, 0.07%, and 0.70% infections, respectively. Rural areas of Southeast Asia and Africa bear the brunt of rabies with tens of thousands contracting the disease annually. Most of these infections are caused by dog bites due to the fact that dogs are closest in contact with humans since they are used as pets, as guards, for hunting purposes, and even as delicacies.
The impact of mathematics continues to increase over the years as more evidence is found to support the fact that mathematical modeling helps to increase our understanding of the dynamics of infectious diseases. This has led to increased use of mathematics to model many infectious disease (see [5–18], and references therein).
Zoonotic diseases have received their fair share of attention by mathematicians. Specifically, Wang and Lou [19] developed an ODE model to study the impact of combinations of culling and vaccination on the control of rabies transmission and observed that vaccination alone was a better strategy whilst culling was worst in the control of rabies transmission. The work of Wang and Lou [19] is however contradicted by that of Carroll et al. [20] that suggested that culling is more effective than vaccination, arguing that adding immunocontraceptives on animal populations could help improve the effectiveness of vaccination. To analyze the strategies for optimal distribution of vaccine baits in order to minimize the spread of the disease and the cost of carrying out the control, Ding et al. [21] formulated a model to describe rabies in raccoons with discrete time and spatial features. Hou et al. [12] suggested that the control of rabies needs to include awareness creation about rabies, increased domestic dog vaccination, and reduction in the number of stray dogs. Asamoah et al. [5] also suggested that vaccination of pets and use of preexposure and postexposure prophylaxis could help in the control of rabies spread. In China and some parts of Africa (for example, the Upper East and West regions of Ghana), dog meat is a delicacy. It is noteworthy that while so much research has been carried out on the transmission and control of rabies, the impact of predation of dogs by humans has not been studied in the transmission of rabies. This current work seeks to incorporate consumption of dog meat by humans into a dog-human rabies model.
The rest of the paper is organized as follows. In Section 2, the mathematical model under consideration is derived, while Section 3 discusses some basic qualitative properties of the model. In Section 4, numerical simulation of the model is carried out to study the impact of various factors on the transmission of rabies. The findings and conclusions are presented in Section 5.
2. Model Formulation
We consider the transmission of rabies from a dog population to a dog-predator human population. Each of the dog and human populations is divided into three compartments of Susceptibles (), Exposed (), and Infected (). We use subscripts and to denote dogs and humans, respectively, so that Susceptible, Exposed, and Infected dogs are represented by , , and , respectively, while Susceptible, Exposed, and Infected humans are represented by , , and , respectively. The total dog and human populations are therefore given by and . Human-to-human transmission of the rabies virus is assumed to be nonexistent or insignificant so that only direct dog-to-dog and dog-to-human transmission is considered. We assume that all recruitments of dogs and humans are made into the susceptible classes and , at constant rates and , respectively. Dogs and humans contract the disease through effective contact with infected dogs, with transmission probabilities and , respectively. Rabies transmission is assumed to be reduced through vaccination of susceptible dogs and humans at rates and , respectively. Upon effective contact with infected dogs, dogs and humans are moved to the exposed classes before being moved (if postexposure prophylaxis is ineffective) to the infected classes at rates and for dogs and humans, respectively. A successful treatment at rate, , of exposed and infected humans is assumed while successful culling at rate is applied to exposed and infected dogs. Dogs and humans are assumed to suffer rabies-induced death at rates and , respectively. Parameters and are taken to be the natural death rates in dogs and humans, respectively. We assume that dogs suffer predation from humans at rate . Since dogs in the infected class are assumed to be culled, we exclude the infected dog population from the predation.
With these assumptions, the mathematical model describing the dynamics of dog-human rabies transmission is represented by the set of differential equations in equation (1). Table 1 shows the descriptions of the parameters used in the model.
For convenience, we make the following substitutions
In the next section, we discuss some basic properties of the model.
3. Basic Properties of the Model
3.1. Positivity of Solutions
Model (1) is an epidemiological model, and hence, it is necessary that the associated population sizes be positive. Model (1) should be considered in a feasible region where such property (nonnegative) is preserved. This is provided in Theorem 1.
Theorem 1. If positive conditions initial conditions are provided for (1), then all its solutions remain positive for .
Proof. From the first equation, if at some point , we have and then, we have . This shows that .
Similar arguments can be used to show that the other state variables have nonnegative solutions for all , hence completing the proof.
Theorem 1 indicates that the model is epidemiologically meaningful and mathematically well posed.
3.2. Boundedness of Solution of the Model
Theorem 2 (Boundedness of the Model). All solutions of system (1) starting in are bounded.
Proof. Consider and .
Then,From the above equations, we haveIf we set and , then,so thatThe feasible region for the model is thus given bySystem (1) can therefore be conveniently studied in .
3.3. Equilibrium Points of the Model
The model exhibits a biologically reasonable disease-free equilibrium , where and satisfy the following quadratic equation:
Hence,
Remark 1. Model (1) has a unique realistic DFE.
Proof. Since the coefficient of and the constant term in (8) have different signs, Descartes’ rule of sign shows that equation (8) has only one positive solution. This concludes the proof of the remark.
The basic reproduction ratio, , is the average number of secondary infections that are caused by one infectious individual, which is introduced into an initially disease-free population, during its infectious period. Using the next-generation method of Diekmann et al. [22], we compute asIn the presence of Infectives, model (1) is said to be exhibiting an endemic equilibrium point , where , and are given byand and satisfy the following set of equations (see Appendix for derivation):Here,wherewhere and can be found by solving equation (12). Even though the solution can be done numerically, the nature of the problem suggests that an algebraic solution may be obtainable. The problem falls in the category of Open Problem 1.
Open Problem 1.
Let , ,, and and be dimensional lower triangular square matrices. Is there an algebraic technique that be used to determine and satisfying the following equations?In the next section, the local stability of the disease-free equilibrium is discussed.
3.4. Local Stability of the Disease-Free Equilibrium Point
We use the Lyapunov indirect method to study the local stability of the disease-free equilibrium point of the model. An equilibrium point is said to be locally asymptotically stable if all eigenvalues of the Jacobian model evaluated at the given equilibrium point have negative real parts.
The Jacobian matrix of the model evaluated at is given by
Two of the eigenvalues ( and ) of are clearly negative and the remaining eigenvalues are the zeros of , where
The second equation of (8) has both zeros being negative if (remembering that )
These conditions are easy to establish using (8).
Also, the first equation in (17) has zeros with negative real parts if
The first condition is true whenever whilst the second condition can be written as , which holds if . The following result is therefore established.
Theorem 3. The disease-free equilibrium state, , of model (1) is locally asymptotically stable whenever and unstable otherwise.
3.5. Local Stability Analysis of Endemic Equilibrium
The stability of the endemic equilibriums of the model is similarly established by finding the eigenvalues of the Jacobian matrix of the model evaluated at the positive solutions of (12). The Jacobian model evaluated at if it exists is given bywhere
The Jacobian matrix has a characteristic polynomial given bywhere
Clearly, , and , , and can similarly be shown to be positive. By Descartes’ rule of signs, the following result is thus established.
Theorem 4. The endemic equilibrium (whenever it exists) is locally asymptotically stable.
3.6. Global Stability of Disease-Free Equilibrium Point
To study the global stability of , we employ the technique of Castillo-Chavez et al. [23] as follows:
If and , then model (1) can be written as
By the theorem of Castillo-Chavez et al. [23], is globally asymptotically stable if the following conditions are satisfied: GS1: is locally asymptotically stable GS2: is globally stable for GS3: , , where is an M-Matrix
The first condition GS1 has been established for in Theorem 3.
For the second condition, we note that is a limiting function of . Therefore, the second condition is satisfied.
As for the third condition, we observe that is given bywhich is an matrix (i.e., all nondiagonal entries are nonnegative).
Furthermore, the matrix is given by
From , we havewhere
Hence, . Similarly, and hence, , satisfies condition GS3. Now, since all three conditions are satisfied, the following result is established.
Theorem 5. The disease-free equilibrium point, , is globally asymptotically stable whenever .
For simulation purposes, we use the following parameter values:
3.7. Sensitivity Analysis
To determine the impact of model parameters on the spread of rabies, we employed global sensitivity analysis technique of Marino et al. [24] to calculate the partial rank correlation coefficients of the model parameters on which the basic reproduction number depends. The result of sensitivity analysis informs which model parameters are most important in determining the output variable (in this case, ) and hence should be given much attention in terms of measurement for accuracy. The PRCC of results are shown in Figure 1.
Local sensitivity analysis discusses the impact of small changes in parameter values on the spread of the disease. To study the sensitivity of the disease spread to parameter changes, the basic reproduction number is used, since it determines the persistence or possible eradication of the disease. The normalized forward sensitivity index is used to perform the analysis. It is defined as follows.
Let be a differentiable function . Then, the normalized forward sensitivity index of relative to is given by
Local sensitivity indexes are presented in Table 2. This index measures the relative percentage change in due to a percentage change in . The sensitivity indexes of the parameters determining are calculated and presented in Table 2. A positive sensitivity index of a parameter implies that an increase (decrease) in the value of the parameter will result in an increase (decrease) in . On the other hand, a negative sensitivity index implies that an increase (decrease) in the value of the parameter will result in a decrease (an increase) in .
3.8. Bifurcation Analysis
To study the stability of endemic equilibrium points, the center manifold theory as described in Theorem 4.1 of [25] is used as an alternative to the Lyapunov indirect method.
Choosing as a bifurcation parameter leads the Jacobian model to have a zero simple eigenvalue. The right and left eigenvectors associated with this zero eigenvalue are given by and with , where
The bifurcation coefficients are thus given bywhere
Clearly, is always positive and the sign of is dependent on . The following result therefore easily follows [25].
Theorem 6. Model (1) undergoes a backward bifurcation at when and a forward bifurcation when in which case the endemic equilibrium will be locally asymptotically stable.
We note that since is GAS whenever , then backward bifurcation may only occur whenever approaches unity from above.
4. Numerical Simulation
We performed numerical simulation in this section to study the impact of various parameters on the spread of the rabies infection. To perform these experiments, the following initial values are used: , , , , , and and the baseline parameter values are taken from equation (29).
To illustrate the local stability of the disease-free equilibrium, model (1) is solved for various initial values of the state variables and the results are plotted on the same graph. This procedure is done for the case when and when . The fact that the DFE is LAS whenever as in Theorem 3 is demonstrated in Figure 2.
(a)
(b)
To illustrate the result in Theorem 1, we solved model (1) for two different scenarios: (a) when initial conditions are less than and (b) when initial conditions are greater than . Figure 3 shows that the feasible region is an invariant attractor of system (1).
We also solved the model for different values of the most influential model parameters in order to simulate the effect of those variables. The results are shown in Figures 4–8.
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
From Figure 4, it is observed that increasing the dog-dog rabies transmission rate, leads to a decline (an increase) in the Susceptible dog population (Exposed and Infected dog populations) and a similar (but slight) effect is observed for the Susceptible human population (Exposed and Infected human populations). The smaller impact on human population could be attributable to the fact that infected dogs will often be culled, leading to reduced infection among humans.
From Figure 5, it is observed that increasing the rate of immunization of dogs, , leads to an increase (a decrease) in the Susceptible dog population (Exposed and Infected dog populations) and a similar effect is observed for the Susceptible human population (Exposed and Infected human populations). It is however observed that for higher rates of immunization, the Susceptible dog populations first peak before declining. This could be attributable to the fact that immunization may not confer permanent reduction in transmissibility of rabies from dogs. We also observe that irrespective of the rates of immunization, a steady state is reached, confirming the global stability of the disease-free equilibrium.
5. Findings
The following are the findings of the study:(i)The rabies-free equilibrium point is globally asymptotically stable whenever the basic reproduction number is less than unity.(ii)While the spread of disease is positively correlated with the dog-to-dog transmission rate , recruitment rate of dogs , rate at which exposed dogs develop clinical rabies, and disease-induced death rate among , and natural death rate of humans , it is negatively correlated with the natural death rate of dogs , immunization of dogs , culling rate , rate of consumption of dogs , as indicated in the sensitivity indexes of .(iii)Efforts at controlling the spread of rabies should focus more on dog population than on humans.(iv)Under certain conditions, the model may exhibit either forward or backward bifurcation near . The possibility of forward bifurcation suggests the existence of locally asymptotically stable endemic equilibrium when .
6. Discussion and Conclusion
In this paper, a nonlinear mathematical model has been proposed to study the dynamics of rabies infection among dogs and in a human population that is exposed to dog bites. It is observed that the model has a unique disease-free equilibrium point which is locally asymptotically stable whenever the basic reproduction number is less than unity. The disease-free equilibrium point, , is also globally asymptotically stable whenever . It is also observed that the most influential factors on the spread of the disease are probability of infection upon contact among dogs , rate of immunization of dogs , rate of recruitment of dogs , rate at which exposed dogs develop clinical rabies , disease-induced death rate within the dog population , and natural mortality rate of dogs, . If the rate of immunization in the dog population is increased, it will have an increasing effect on the susceptible human population as shown in Figure 5. If the recruitment rate within the dog population, disease-induced death rate of dogs, rate of contact among dogs, and the rate at which exposed dogs develop clinical rabies each is reduced, with all other factors remaining constant, there will be a positive effect on the susceptible human population as shown in Figures 4 and 6–8. These actions will lead to a reduction in the reproduction number. However, even though it has been accepted that the eradication of infectious diseases in a community should target the reduction of to less than unity, this may not be the only case for this system. It has also been observed that under certain conditions, backward bifurcation may occur around . From the analysis, we noted that since is GAS whenever , then backward bifurcation may only occur whenever approaches unity from above. This has a major implication for the spread of the disease once the basis of reducing below the threshold point, , may not just be enough for effectively controlling the spread of the disease because whenever approaches unity from above, there is the likelihood of coexistence of endemic equilibrium which may result in reinfection of the rabies virus. The conclusions, however, of Asamoah et al. [5]; Hou et al. [12]; and Wang and Lou [19] are that vaccination, which may lead to immunity, as a better control method of rabies infection in dogs will still be relevant in the presence of backward bifurcation.
Appendix
Derivation of Equation (12)
The equilibrium points of the model are found by equating the right-hand sides to zero to obtain
Let the endemic equilibrium be .
From equation (A.1), we have .
Also from (A.3) and (A.5), we get and respectively, which can be used to show that and
From (A.2), we get and upon substituting the expression for , we obtain .
Thus, , , , and have been expressed in terms of and .
Equations (A.1) and (A.2) can then be used to find and .
Substituting , , , and into (A.1) and (A.2) (after some long algebraic simplification) gives the following equations:
Equations (A.7) and (A.8) can then be rewritten as in equation (12).
Conflicts of Interest
The authors declare that they have no conflicts of interest.