Journal of Applied Mathematics

Volume 2017, Article ID 2451237, 23 pages

https://doi.org/10.1155/2017/2451237

## Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis

^{1}Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana^{2}Department of Statistics and Mathematics, Kumasi Technical University, Kumasi, Ghana^{3}Department of Mathematics, University for Development Studies, Navrongo, Ghana

Correspondence should be addressed to Joshua Kiddy K. Asamoah; hg.ude.smia@haomasaj

Received 1 January 2017; Revised 13 March 2017; Accepted 19 April 2017; Published 16 July 2017

Academic Editor: Zhen Jin

Copyright © 2017 Joshua Kiddy K. Asamoah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [2]. Raccoons, skunks, bats, and foxes are the main animals that transmit the virus in the United States [2]. 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 [2]. 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 [2]. The symptoms of rabies are quite similar to those of encephalitis (see [3]). 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 [4], and since then there has been a considerable study of infectious disease using optimal control analysis (see [4–12]).

Research published by Aubert [13], 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 [13] 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 [14] formulated a mathematical model based on each time step dynamic which was calculated independently in every cell. Later, Bohrer et al. [15] published a paper on the viability of different rabies spatial immunization designs in a simulated host population.

The research presented by Bohrer [15] 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. [16] also presented a model for the immune responses to rabies virus in bats. Coyne et al. [17] 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. [18] 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. [19] 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. [20] 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. [23] 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. [24] 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 [25] 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 [26]).

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. [29] 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 [30] 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 [29]). 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 [27]. The rate of losing immunity in both compartments is represented by and , respectively.