Abstract
In this paper, the population dynamics of rabies-infected dogs are studied. The mathematical model is constructed by dividing the dog population into two categories: stray dogs and domestic dogs. On the other hand, the rabies virus is likely to spread in both populations. In the current model, disease-controlling strategies such as vaccination and culling are applied, and their impact is studied. Both subpopulations of susceptible individuals are vaccinated to control disease spread. The current study assumes that stray dogs can transmit rabies to domestic dogs but not the other way around. Because domestic dogs are under the control of their owners, they are well vaccinated. The model is medically and analytically correct because the findings are idealistic and limited. The next-generation matrix technique is used to compute the effective reproductive amount, and also, each parameter is subjected to sensitivity analysis. The equilibrium point free from disease is discovered, demonstrating that it was asymptotically steady locally and globally. A conditionally global asymptotically stable point of endemic equilibrium is also discovered using the Lyapunov function method. The numerical simulation, which makes use of approximations for parameter values, shows that the most efficient method for avoiding rabies transmission is a combination of vaccination and the culling of infected stray dogs. Using MATLAB’s ode45, this numerical simulation investigation was carried out. Our early findings indicated that the annual dog birth rate is a critical factor in influencing the occurrence of rabies. In the body of the paper, the findings and discussion are organized logically.
1. Introduction
Infectious disease outbreaks have been ongoing in recent decades, and there are more and more instances when long-standing infections could be eradicated, but human conduct has prevented it. This conversation interaction between infectious illness processes and the dynamics of human nature partially explains the rise in interest in directly adding incorporating human nature into mathematical equations of the spread of infectious diseases [1]. Mathematical models that take into consideration human behavior may provide more clarity and more accurate predictions than models that neglect the significant influence that behavior can have on the dynamics of infectious diseases.
The acute infectious disease rabies, which is fatal, is carried on by lyssaviruses. The lyssavirus known as the "prototype species," or rabies lyssavirus, is the most widespread and the greatest hazard to public health. In mammals, it causes brain inflammation. Fever and tickling at the site of exposure are the first signs of infection, followed by one or more other signs such as frantic movement, inability to control certain body parts, loss of consciousness, confusion, fear of water, and unrestrained excitation. Before symptoms start to show up, it often takes one to three months from the day of introduction to the disease [2]. The duration of the symptoms can range from less than a week to over a year. Once the symptoms started to manifest, almost every instance resulted in death. When one animal or person bites or scratches another animal or person, the infection spreads. Additionally, if the saliva of an infected animal touches the mouth, eyes, or nose of a susceptible animal, it might spread the rabies virus. The dog is typically the animal most frequently associated with the transmission of rabies [3].
The goal of vaccination, which is being implemented on a global scale to stop the spread of infectious diseases that have crippled many human societies in the recent past, is without a doubt to inhibit the spread of transmissible diseases. Significant advancements in public health, particularly in the prevention and treatment of infectious illnesses, were made during the 20th century. Vaccinations were critical to achieving that result [4]. Early research, however, merely assumed that mandatory and/or arbitrary vaccination should be practiced due to a lack of vaccination and information. As of today, we are aware that network vaccination programs are more successful when random vaccination is paired combining targeted vaccination and familiarity immunization [5]. Even though irregular immunization does not need topological knowledge of the network, it does require very broad coverage, which makes it quite expensive [6].
A deterministic model was created to examine the dynamics of dog-to-human and dog-to-dog rabies infection in China [7]. The model identified four groupings within both the dog and human populations: those who were susceptible, those exposed to infection, those infectious, and those who recovered. According to the findings, the most efficient ways to reduce human rabies in China are to decrease the dog fertility rate and increase the coverage amount of dog vaccinations. They suggested that mass vaccinations of susceptible dogs could take the place of large-scale culling. This is due to the possibility of community disruption and an increase in the immigration of diseased dogs during the dog culling operation.
In their paper, modeling the dynamic behavior stability analysis and the prevention of rabies spread with immunization [8], the researchers created an equation that is deterministic for rabies spread changing aspects popular among humans and animals in the region of Ethiopia’s Addis Ababa. Their model integrates a dog immunization program. They computed the fundamental number of reproduction as well as the effective reproduction number. Their findings are completely reliant on the parameters of a population of dogs, indicating that the dog population is to blame for human and livestock infection. They calculated the specific both the total and an effective number of reproductions from data obtained from Addis Ababa’s Ethiopian Public Health Institute and found them in the range of 2 and 1.6, respectively, representing the endemic nature of the illness. Domestic dogs and stray dogs are two subpopulations of dogs in our model. Domestic dogs are defined as dogs that live closest to humans [9], and stray dogs are defined as dogs that freely move in public [10]. Stray dogs have been related to negative effects on the environment as well as general health. The following sections form the paper: Introduction, Model Formulation, Stability Analysis, and Sensitivity Analysis, are the first four steps Section 5 is the simulations and Section 6 is the concluding remarks
2. Model Formulation
In this study, we develop the SEIR model of rabies for stray dogs and the domestic dog population based on the work obtainable in [11–14]. Each stray dog as well as the domestic dog population is divided into categories according to susceptibility, exposure, infection, and recovery. Susceptible collections are uninfected, but then again, if they make contact with rabid dogs, they are at risk of infection. Individuals who are exposed are those that have been infected with the virus but have not yet developed symptoms. Individuals who become infected develop clinical symptoms and, due to the nature of rabies, are unlikely to recover. The recovered classes are those who were vaccinated and recovered before becoming infectious, whereas the rest became infected and died.
The stray dog population has four compartments representing the susceptible stray dogs, ; exposed stray dogs ; infected stray dogs, ; and recovered stray dogs, . Thus, the total domestic dog population is . Additionally, there are four divisions in the domestic dog population that represent susceptible domestic dogs ; exposed domestic dog, ; infected domestic dog, ; and recovered domestic dogs, . Thus, the total population is . It is expected that there is no domestic dogs’ domestic dog spread of rabies infection in the domestic dog’s submodel [15]. In the stray dog submodel, this is supposed that now, the rabies virus spreads directly through one stray dog to the other as well as from the division of infected stray dogs to the population of susceptible domestic dogs. It is further believed that such susceptible domestic dog group, , is growing at a rate of whereas the susceptible stray dog population, , is growing at a rate of through recruitment. The idea is that the infected stray dog meets and spreads at a rate of into the stray dog division. Suppose that characterizes the switch plan outstanding to immunization in the susceptible stray dog’s section; formerly the spread changing aspects develop, somewhere the no effectiveness (letdown) of the vaccine. Additionally, it is believed that domestic dog groups and infectious dogs come into interaction at a rate of . Similar to how immunization is administered to a susceptible domestic dog, the dog’s growth rate toward exposure changes, where the preexposure prevention indicates the domestic dog division’s failing of preexposure prevention. The amount of down resistance in together divisions is characterized by . Also, both an exposed stray and domestic dog move to infectious class directly. Strays and domestic dogs’ natural mortality rates are denoted by and , respectively, and indicate the mortality rate of domestic and stray dogs, which is and , or the fatality rate related to rabies disease in stray and domestic dogs, respectively. Also, the culling rate of stray and domestic dogs are denoted as , respectively. Rabies transmission among domestic dogs was neglect because of minor situations, since the ownership monitors domestic dogs. A person gets sick after being attacked by a rabid dog. The table below explains every one of the model’s variables, of which all are positive.
2.1. Model Variables and Parameters with Their Descriptions
Tables 1 and 2 below, respectively, contain descriptions of the model’s variables and parameters.
2.2. Model Equations
The model is a system of eight ordinary differential equations, as shown by the transmission flowchart in Figure 1 and also the statements made on the connections between the variables and parameters.
The total population for and is
Due to this, combining the differential equations for the domestic dog and stray dog populations in system (1) will result in where and represent the entire populations of domestic dogs and stray dogs, correspondingly, during time .
2.3. The Model’s Invariant Region
Here, the total population of domestic dogs and stray dog population needs to be bounded. To put it another way, all state variables and parameters are taken to be positive for all time , in order to study the model system in the feasible region. Two regions make up the model system (1); consequently, Consider the following lemma.
Lemma 1. The solution set is limited in the possible region of the model system (1) .
Proof. The total population of domestic dogs and stray dogs varies as time change. Due to changes in who is included in the susceptible class. The model yields the following rate of change for the entire domestic dog population:
If there is no disease-related mortality rate, then it follows that
Similarly,
Suppose and ; we will get and , on differentiating inequality results in and .
Therefore, equations (5) and (6) become
Integrating (7) and (8), use the integrating factor method and following some mathematical manipulation, the practical answer for the domestic dog and stray dog population in the area of the model system (1).
Consequently, a practical solution is given. Therefore, . It follows from the common comparison theorem applied to differential inequality in [1] that from equations (7) and (8), solutions of and become
Taking the limit as for equations (10) and (11) becomes
The total population of domestic dog and stray dog approaches to and, respectively, where and are the upper bounds. This implies that at any time , each and every solution with initial conditions in is leftovers in[7]. And if and , then the solutions enter the feasible region at any time . Therefore, every solution of system (1) is positively invariant, and hence, in region , the model can continue to be mathematically sound and medically relevant. Thus, the investigation of solution is restricted to the region.
2.4. Disease-Free Equilibrium Points (DFE)
The disease-free equilibrium point of the model provided by the system is denoted . Therefore, the next category will be 0 if there is no rabies. . Including this in system (1) results in
The DFE will be obtained by rearranging (13) mathematically.
If , the infection will not spread because, in general, throughout its contagious phase, an infected individual creates less than one new infectious disease.
2.5. The Effective Reproduction Number ()
The average amount of reinfection brought on as a pathogen delivered into a group of susceptible domestic dogs and stray dogs is the effective reproduction ratio, the threshold value , [16, 17]. A statistic known as the effective reproduction number () can help predict whether such an illness will spread across a community or go extinct. If , the infection will not spread because, in general, throughout its contagious phase, an infected individual creates less than one new infectious disease. On the other hand, if , so each infected person often creates more than one new infection, as well as the illness can spread all across the community. If , this illness gets endemic, which means it spreads from one susceptible dog to another at a constant speed from across the community [3]. Letting stand the proportion of arrival of new contagions in the compartment (i.), remains the transmission of individuals addicted to the compartment () by completely other means; also, remains the proportion of transmission of individuals out of the compartment (). This one is expected that every function is continuously differentiable at a minimum double concerning each variable and . To calculate the , it is done by means of the next-generation method and taking the infectious compartments. symbolizes the spectral radius of a matrix , whereve and in addition, through aimed at the number of compartments, then aimed at the diseased compartments only.
In light of this, the effective reproduction number is provided by
2.6. Endemic Equilibrium Points
The steady-state conditions where the disease persists in the population are known as endemic equilibrium points. is the formula for the model’s endemic equilibrium point. We set . Within system (1) and solving for and , we get
3. Stability Analysis
3.1. Local Stability at the DFE Point
Toward analyzing local steadiness on the uninfected steadiness point, the Jacobian matrix of the model system (1) at is evaluated. Then, stability is established based on the trace’s indication and the determinant of the Jacobian matrix.
Theorem 2. If then the disease-free equilibrium of a system (1) is locally asymptotically stable or unstable otherwise.
Proof. The Jacobian matrix of the system (1) at . From (14), the disease-free equilibrium point is given by
We determine the system’s Jacobian matrix (1). To accomplish this, system (1)’s individual equations are differentiated by means of the state variable. . Let be the jacobian matrix evaluated at the equilibrium without sickness .
The trace of the above matrix will be
Since all parameters are positive, determine the determinant of the Jacobian matrix .
Let
Then, the above Jacobian matrix will be
Insert the value of , , ,, , , , , ; we will get
Since and
From equation (16),
Thus, for , there are the disease-free equilibrium points which are locally asymptotically stable if and , otherwise it is unstable if . The theorem is proved.
3.2. Global Stability of DFE
We used the technique suggested by [18] to examine the global stability of the DFE of the system (1).
Theorem 3. Under the condition that , the DEF for model system (1) is globally stable.
Proof. Letting , , , and be positive constants, let be a Lyapunov function.
The derivative of with respect to time is
From system (1), substitute , , , , , , , and into equation (26) to obtain
With the Lyapunov function formed just on space of the state variables ,,, , ,, , and , it is evident that if , , , and at the DFE are globally stable (therefore, and ), then , , and as
Hence, it can be considered that
Inserting equation (28) into (27), we get
At the disease-free equilibrium point, , , as
When the coefficients of ,, and in (31) tend to zero, the following results appear:
Inserting equation (32) into equation (31) yields
From equation (14), ; then, insert this in equation (33); we get
Therefore,
Additionally, if and only if .Therefore, for , it shows that as .
As a result, the singleton set is the biggest compact invariant set in the set . We thus draw the conclusion that is globally asymptotically stable in the case where is based on La Salle’s invariance principle [9, 10, 18].
4. Sensitivity Analysis for
The effect of model parameter values on output estimation of is shown in this section. The value of a particular input parameter determines the output value of . Since the model’s parameter values are unknown, changes in the input parameter values can influence the effective reproduction number’s output values. Sensitivity analysis is used to measure this uncertainty to assess whether or not rabies is spreading in the population. The proportion of the relative changes in a variable, and a parameter is known as the normalized forward sensitivity index. Its research is also focused here on parameter values displayed in Table 3 below.
The formulation of the sensitivity index is as such when the variable is a differentiable function of the parameter: the following is the definition of the normalized forward sensitivity index of variable that depends on parameter .
In this instance, we have the number of effective reproductions computed.
From the model, the sensitivity index of concerning is given by
Again, the sensitivity index of concerning is given by
Similarly, the sensitivity index concerning , , and is given by
The sensitivity indices of concerning parameters are obtained and summarized in Table 4.
The infection rates of stray dogs and annual stray dog births are the most sensitive and positive parameters, followed by the proportion of exposed stray dogs’ clinical outcome parameter and the loss rate of vaccination immunity of stray dogs , on the basis of the sensitivity indicators. As a result of increasing these parameters, the effective reproduction number will increase. Furthermore, the most negatively sensitive indicator is the stray dog mortality rate , which is followed by the rabies mortality rate , the stray dog vaccination rate , and stray dog culling rate . As a result, as these parameters are increased, the effective reproduction number decreases.
5. Numerical Simulations
Several findings and their interpretations have been discussed in this section. ODE45 is the default solver in MATLAB for solving ordinary differential equations (ODEs). This function implements the Runge-Kutta process with a configurable time step for effective calculation. The simulation’s main goal is to see how model parameters respond during a rabies epidemic. Its research is also focused on the parameter values in Table 3 and initial condition values shown in Table 5 below.
Figure 2 curves demonstrate that if no intervention was used , the number of exposed domestic dogs increases at first, then starts to decrease. After being bitten by rabid dogs, the susceptible domestic moves to the exposed domestic compartment, causing this. When rabies symptoms appear in the exposed domestic dog’s compartment, the exposed domestic dogs are transferred to the infected compartment, reducing the number of exposed domestic dogs. Figure 2(b) indicates that when no intervention is used , the quantity of domestic dogs with the disease increases before declining, due to the large number of rabies-positive domestic dogs that moved into the infected compartment. Since all infected domestic dogs die and there is no rabies treatment, the number of infected domestic dogs continues to decrease. We can see that intervention is very effective at decreasing the amount of exposed and infected domestic dogs once we compare curves with no intervention with those with intervention for both exposed and infected domestic dogs. The amount of domestic dogs exposed to the infection going to the infected compartment is also decreased. When comparing the two interventions, it appears that a combination of culling and vaccination has the greatest effect on reducing the number of exposed and infected domestic dogs, followed by vaccination only and then culling only . Vaccination alone is more effective than culling alone in exposed group populations. However, culling the population of an infected group is more effective than vaccination alone.
(a) Exposed domestic dog
(b) Infected domestic dog
According to the existence of the graphs in Figure 3, when no intervention (), is made, the number of exposed and infected stray dogs increases at first and then begins to decrease. When infected stray dogs bite other stray dogs, the bitten dogs become rabies-infected and shift from the safe compartment to the exposed compartment. When symptoms appear, exposed stray dogs become infectious and reach the diseased compartment, increasing the amount of infected stray dogs while declining the exposed compartment. In addition, the curves of infected stray dogs are declining because all infected stray dogs are dead. After all, rabies treatment is not available. In comparison to when no intervention is used, the number of exposed and infected stray dogs decreases faster when one or is used. That when susceptible stray dogs are vaccinated, the vaccinated stray dogs are moved to the recover compartment, reducing the number of susceptible stray dogs. It has been discovered that combining vaccination and culling has a greater impact than either vaccination or culling only.
(a) Exposed stray dog
(b) Infected stray dog
One of most sensitive factors formulating the characteristics of dog rabies spread seems to be the natural death rate (), according to the sensitivity report. Figure 4 illustrates how a small rise in the number of stray dogs dying naturally leads toward reduction in the quantity of exposed and stray dogs which can be contagious and conversely. You should keep in mind that preventing rabies deaths by vaccination would cause stray dogs to die naturally.
(a) Exposed stray dog
(b) Infected stray dog
Variables in the vaccination rate for stray dogs were used in the simulation (see Figure 5). Increased stray dog vaccination rates have been shown to have an important effect in regards to the rate of rabies spread in exposed and infected domestic dogs. Furthermore, increasing the vaccination rate of stray dogs within the model decreased the number of exposed and infected domestic dog populations.
(a) Exposed domestic dog
(b) Infected domestic dog
Figure 6 shows how greatly the overall number of infected dogs can be decreased by reducing the number of puppies born to stray dogs each year. This explains the reason that minimizing the annual production of stray dog pups is essential for preventing the spread of rabies.
6. Conclusion
We have settled a mathematical compartmental deterministic analysis of the behavior of rabies spread in this study. It was a model intended to depict the spreading disease rabies from strays to domestic dogs because traditional rabies is widespread in the dog population. In the model, vaccinations were only administered to susceptible households and stray dogs. It is not practicable to vaccine exposed dogs since it is very challenging to identify the exposed dogs who need to be vaccinated. We study the fundamental characteristics of epidemic models in terms of the boundedness and positivity of solutions for our model and find that the model is positive for all positive initial condition values. We carry out stability and equilibrium analyses as well as reproduction number calculations. We demonstrate that both locally and globally, the DFE is asymptotically stable. Our simulation shows that the effective technique for preventing the spread of rabies is vaccination together with the culling of infected dogs and that the yearly dog birth amount has a major influence on the incidence of rabies disease. The model utilized in this study can be properly examined if statistics on the dog population are available. The government might develop rabies-eradication plans using the study’s findings, such as managing the dog population. We also suggest that dog population monitoring be done to estimate the annual dog birth rate.
Data Availability
The data used to support the findings of this manuscript are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.