Research Article | Open Access
Joshua Kiddy K. Asamoah, Francis T. Oduro, Ebenezer Bonyah, Baba Seidu, "Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis", Journal of Applied Mathematics, vol. 2017, Article ID 2451237, 23 pages, 2017. https://doi.org/10.1155/2017/2451237
Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis
We examine an optimal way of eradicating rabies transmission from dogs into the human population, using preexposure prophylaxis (vaccination) and postexposure prophylaxis (treatment) due to public education. We obtain the disease-free equilibrium, the endemic equilibrium, the stability, and the sensitivity analysis of the optimal control model. Using the Latin hypercube sampling (LHS), the forward-backward sweep scheme and the fourth-order Range-Kutta numerical method predict that the global alliance for rabies control’s aim of working to eliminate deaths from canine rabies by 2030 is attainable through mass vaccination of susceptible dogs and continuous use of pre- and postexposure prophylaxis in humans.
Rabies is an infection that mostly affects the brain of an infected animal or individual, caused by viruses belonging to the genus Lyssavirus of the family Rhabdoviridae and order Mononegavirales [1, 2]. This disease has become a global threat and it is also estimated that rabies occurs in more than countries and territories . Raccoons, skunks, bats, and foxes are the main animals that transmit the virus in the United States . In Asia, Africa, and Latin America, it is known that dogs are the main source of transmission of the rabies virus into the human population . When the rabies virus enters the human body or that of an animal, the infection (virus) moves rapidly along the neural pathways to the central nervous system; from there the virus continues to spread to other organs and causes injury by interrupting various nerves . The symptoms of rabies are quite similar to those of encephalitis (see ). Due to movement of dogs in homes or the surroundings, the risk of not being infected by a rabid dog can never be guaranteed. Rabies is a major health problem in many populations dense with dogs, especially in areas where there are less or no preventive measures (vaccination and treatment) for dogs and humans. Treatment after exposure to the rabies virus is known as postexposure prophylaxis (PEP) and vaccination before exposure to the infection is known as preexposure prophylaxis.
The study of optimal control analysis in maximizing or minimizing a said target was introduced by Pontryagin and his collaborators around 1950. They developed the key idea of introducing the adjoint function to a differential equation, by forming an objective functional , and since then there has been a considerable study of infectious disease using optimal control analysis (see [4–12]).
Research published by Aubert , on the advancement of the expense of wildlife rabies in France, incorporated various variables. They follow immunization of domestic animals, the reinforcement of epidemiological reconnaissance system and the bolster given to indicative research laboratories, the costs connected with outbreaks of rabies, the clinical perception of those mammals which had bitten humans, the preventive immunization, and postexposure treatment of people. A significant percentage (72%) of the cost was the preventive immunization of local animals. In France, as in other European nations in which the red fox (Vulpes) is the species most affected, two primary procedures for controlling rabies were assessed in  at the repository level to be specific: fox termination and the oral immunization of foxes. The consolidated costs and advantages of both systems were looked at and included either the expenses of fox separation or the cost of oral immunization. The total yearly costs of both techniques stayed practically identical until the fourth year, after which the oral immunization methodology turned out to be more cost effective. This estimate was made in and readjusted in and affirmed by ex-postinvestigation five years later. Accordingly, it was presumed that fox termination brought about a transient diminishment in the event of the infection while oral immunization turned out to be equipped for wiping out rabies even in circumstances in which fox population was growing. Anderson and May  formulated a mathematical model based on each time step dynamic which was calculated independently in every cell. Later, Bohrer et al.  published a paper on the viability of different rabies spatial immunization designs in a simulated host population.
The research presented by Bohrer  stated that, in desert environments, where host population size varies over time, nonuniform spreading of oral rabies vaccination may, under certain circumstances, be more effective than the commonly used uniform spread. The viability of a nonarbitrary spread of the immunization depends, to some extent, on the dispersal behavior of the carriers. The outcomes likewise exhibit that, in a warm domain in a few high-density regions encompassed by populations with densities below the critical threshold for the spread of the disease, the rabies infection can persist.
Levin et al.  also presented a model for the immune responses to rabies virus in bats. Coyne et al.  proposed an SEIR model, which was also used in a study predicting the local dynamics of rabies among raccoons in the United States. Childs et al.  also researched rabies epidemics in raccoons with a seasonal birth pulse, using optimal control of an SEIRS model which describes the population dynamics. Hampson et al.  also noted that rabies epidemic cycles have a period of 3–6 years in dog populations in Africa, so they built a susceptible, exposed, infectious, and vaccinate model with an intervention response variable, which showed significant synchrony.
Carroll et al.  also used compartmental models to describe rabies epidemiology in dog populations and explored three control methods: vaccination, vaccination pulse fertility control, and culling. An ordinary differential equation model was used to characterize the transmission dynamics of rabies between humans and dogs by [21, 22]. The work by Zinsstag et al.  further extended the existing models on rabies transmission between dogs to include dog-to-human transmission and concluded that human postexposure prophylaxis (PEP) with a dog vaccination campaign was the more cost effective in controlling the disease in the long run. Furthermore, Ding et al.  formulated an epidemic model for rabies in raccoons with discrete time and spatial features. Their goal was to analyze the strategies for optimal distribution of vaccine baits to minimize the spread of the disease and the cost of carrying out the control. Smith and Cheeseman  show that culling could be more effective than vaccination, given the same efficacy of control, but Tchuenche and Bauch suggest that culling could be counterproductive, for some parameter values (see ).
The work in [27, 28] also presented a mathematical model of rabies transmission in dogs and from the dog population to the human population in China. Their study did not consider the use optimal control analysis to the study of the rabies virus in dogs and from the dog population to the human population. Furthermore, the insightful work of Wiraningsih et al.  studied the stability analysis of a rabies model with vaccination effect and culling in dogs, where they introduced postexposure prophylaxis to a rabies transmission model, but the paper did not consider the noneffectiveness of the pre- and postprophylaxis on the susceptible humans and exposed humans and that of the dog population and the use of optimal control analysis. Therefore, motivated by the research predictions of the global alliance of rabies control  and the work mention above, we seek to adjust the model presented in [27–29], by formulating an optimal control model, so as to ascertain an optimal way of controlling rabies transmission in dogs and from the dog population to the human population taking into account the noneffectiveness (failure) of vaccination and treatment.
The paper is petition as follows. Section 2 contains the model formulation, mathematical assumptions, the mathematical flowchart, and the model equations. Section 3 contains the model analysis, invariant region, equilibrium points, basic reproduction number , and the stability analysis of the equilibria. In Section 4 we present the parameter values leading to numerical values of the basic reproduction number , the herd immunity threshold and sensitivity analysis using Latin hypercube sampling (LHS), and some numerical plots. Section 5 contains the objective functional and the optimality system of the model. Finally, Sections 6 and 7 contain discussion and conclusion, respectively.
2. Model Formulation
We present two subpopulation transmission models of rabies virus in dogs and that of the human population (see Figure 1), based on the work presented in [27–29]. The dog population has a total of four compartments. The compartments represent the susceptible dogs, , exposed dogs, , infected dogs, , and partially immune dogs, . Thus, the total dog population is . The human population also has four compartments representing susceptible humans, , exposed humans, , infected humans, , and partially immune humans, . Thus, the total human population is . It is assumed that there is no human to human transmission of the rabies virus in the human submodel (see ). In the dog submodel, it is assumed that there is a direct transmission of the rabies virus from one dog to the other and from the infected dog compartment to the susceptible human population. It is further assumed that the susceptible dog population, , is increased by recruitment at a rate and is the birth or immigration rate into the susceptible human population, . It is assumed that the transmission and contact rate of the rabid dog into the dog compartment is . Suppose that represents the control strategy due to public education and vaccination in the dogs compartment; then the transmission dynamics become , where is the noneffectiveness (failure) of the vaccine. It is also assumed that the contact rate of infectious dogs to the human population is . Similarly, administrating vaccination to the susceptible humans the progression rate of the susceptible humans to the exposed stage becomes , where is the preexposure prophylaxis (vaccination), represents the failure of the preexposure prophylaxis in the human compartment. Furthermore, administrating postexposure prophylaxis (treatment) to affected humans at the rate decreases the progression rate of the rabies virus, at the exposed class to the infectious class as , where is the failure rate of the postexposure prophylaxis and represents the rate at which exposed humans progress to the infected compartment . The rate of losing immunity in both compartments is represented by and , respectively.
The exposed humans without clinical rabies that move back to the susceptible population are denoted by the rate . The natural death rate of dogs is , and denotes the mortality rate of humans (natural death rate), represents the death rate associated with rabies infection in dogs, and represents the disease induce death in humans. The rate at which exposed dogs die due to culling is , and represents the rate at which exposed dogs without clinical rabies move back to the susceptible dog compartment. Subsequently, using the idea presented in , we assumed that the exposed dogs are treated or quarantined by their owners at the rate ; this implies that is the progression rate of the exposed dogs to the infectious compartment, where is the failure of the treatment or quarantined strategy, and denotes those exposed dogs that develop clinical rabies . Figure 1 shows the mathematical dynamics of the rabies virus in both compartments.
From Figure 1 transmission flowchart and assumptions give the disease pathways as
3. Model Analysis
Lemma 1. The solution set of model system (1) is contained in the feasible region .
Proof. Suppose for all We want to show that the region is positively invariant, so that it becomes sufficient to look at the dynamics of model system (1), given thatwhere is the total population of dogs at any time and is total population of humans at any time .
Equation (2) giveswhich yieldsSimilarly (3) givesNow, assuming that there are no disease induced death rate and culling effect in the dogs’ compartment, it implies that (5) and (6) becomeSuppose , , , and , and then imposing the theorem proposed in  on differential inequality results in and Therefore (7) becomesSolve (8) and (9) using the integrating factor (IF) method. Thus , After some algebraic manipulation the feasible solution of the dogs’ population in model system (1) is in the regionSimilarly the human population follows suit, and from (9) this implies that the feasible solution of the human population of model system (1) is in the region Therefore, the feasible solutions are contained in . Thus . From the standard comparison theorem used on differential inequality in , it implies thatHence, the total dog population size as Similarly, the total human population size as . This means that the infected state variables of the two populations tend to zero as time goes to infinity. Therefore, the region is pulling (attracting) all the solutions in This gives the feasible solution set of model system (1) as
Hence, (1) is mathematically well posed and epidemiologically meaningful.
3.1. Disease-Free Equilibrium
Suppose there is no infection of rabies in both compartments; then . Incorporating this into (1) leads to
After some algebraic manipulation of (14), the disease-free equilibrium point becomes with
3.2. Basic Reproduction Number
Here, the basic reproduction number measures the average number of new infections produced by one infected dog in a completely susceptible (dog and human) population (see also ). Now taking , and as our infected compartments giveswhere , and
Now, using the next generation matrix operator and the Jacobian matrix as described in , results in
Using the fact that gives and evaluated at aswhere the element in matrix constitutes the new infection terms, while that of matrix constitutes the new transfer of infection terms from one compartment to another. Now, splitting matrix into four submatrices and finding its corresponding inverses result in , given byLettingimpliesFinding the matrix determinant of (22) and denoting it by give the expression , where is the identity matrix of a matrix; thus
This gives a characteristic equation of the form ; solving the characteristic polynomial results in the following eigenvalues: The basic reproduction number is the spectral radius (largest eigenvalue) , also defined as the dominant eigenvalue of .
Remark 2. contains the secondary infection produced by the infectious compartment of dogs (in the presence of preexposure prophylaxis (vaccination), postexposure prophylaxis (treatment/quarantine), and culling of exposed dogs). When , the infection gradually leaves the dog compartment, but when , the rabies virus remains in the dog compartments for a longer time, thereby increasing the rate at which the susceptible dogs and humans get infected by a rabid dog.
3.3. Endemic Equilibrium
The endemic equilibrium is given as
Note that if , it results in the disease-free equilibrium; if , then there exists a unique endemic equilibrium; if , then there exist two endemic equilibriums.
3.4. Stability Analysis of
Linearizing (1) at and subtracting eigenvalue along the main diagonal yieldwhere
Simplifying matrix giveswhere
From (28) the four characteristic factors that are negative are where , , , and . The other four characteristic factors can be obtained using the Routh-Hurwitz criterion. Routh-Hurwitz stability criterion is a test to ascertain the nature of the eigenvalues. If the roots of the polynomial are all positive, then the polynomial has a negative real part [35, 36]. The remaining four characteristic eigenvalues are obtained as follows:
Hence, simplifying the coefficient of the above characteristic polynomial in (31) yields
Therefore, from the Routh-Hurwitz criterion of order four, it implies that the conditions, , , , , and , are satisfied if . Hence, the disease-free equilibrium is locally asymptotically stable when (see ).
3.4.1. Global Stability of
Theorem 3. The disease-free equilibrium of model (1) is globally asymptotically stable if and unstable if .
Proof. Let be a Lyapunov function with positive constants , , , and such thatTaken the derivative of the Lyapunov function with respect to time givesPlugging (1) into (34) results inNow, after forming the Lyapunov function on the space of the eight state variables, thus , and introducing the idea from , it is clear that if , , , and at the disease-free equilibrium are globally stable (thus, , , , and ), then , , , and as .
Therefore, it can be assumed that(see ) and replacing it into (35) yieldsThis implies thatEquating the coefficient of , , , and in (38) to zero gives and we obtain Additionally if and only if . Therefore, for it shows that , , , and as . Hence, the largest compact invariant set in is the singleton set . Therefore, from La Salle’s invariance principle, we conclude that is globally asymptotically stable in if (see also [38, 39]).
3.5. Global Stability of Endemic Equilibrium
Theorem 4. The endemic equilibrium of model (1) is globally asymptotically stable whenever .
Proof. Suppose ; then the existence of the endemic equilibrium point is assured. Using the common quadratic Lyapunov function as illustrated in , we consider a Lyapunov function with the following candidate:Now, differentiating (42) along the solution curve of (1) givesFrom (1) it implies that and , which when plugged into (43) givesNow assuming and substituting it into (44), we have