# Effect of Vaccination and Culling on the Dynamics of Rabies Transmission from Stray Dogs to Domestic Dogs

Authors: Demsis Dejene Hailemichael; Geremew Kenassa Edessa; Purnachandra Rao Koya
Journal: Journal of Applied Mathematics
 (2022)

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## 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.


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## Body

## 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,Ss; exposed stray dogs Es; infected stray dogs, Is; and recovered stray dogs, Rs. Thus, the total domestic dog population is Ns. Additionally, there are four divisions in the domestic dog population that represent susceptible domestic dogs Sd; exposed domestic dog, Ed; infected domestic dog, Id; and recovered domestic dogs, Rd. Thus, the total population is Nd. 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, Sd, is growing at a rate of Ad whereas the susceptible stray dog population, Ss, is growing at a rate of As through recruitment. The idea is that the infected stray dog meets and spreads at a rate of βs into the stray dog division. Suppose that θs 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 βs. 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 αs,αd. Also, both an exposed stray and domestic dog move to infectious class directly. Strays and domestic dogs’ natural mortality rates are denoted by μs and μd, respectively, and indicate the mortality rate of domestic and stray dogs, which is εs and εd, 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 cs,cd, 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

Tables1 and 2 below, respectively, contain descriptions of the model’s variables and parameters.Table 1
Variables in the model and its explanation.
SymbolInformation about the variablesSst,SdtSusceptible stray and domestic dog population at timet, respectivelyEst,EdtExposed stray and domestic dog population at timet, respectivelyIst, IdtInfectious stray and domestic dog population at timet, respectivelyRst,RdtRecovered stray and domestic dog population at timet, respectivelyTable 2
Details of the model parameters.
ParameterExplanationAdNumber of domestic dog recruitmentAsNumber of stray dog recruitmentβdRate of stray dog transmission to domestic dogsβsRate of transmission from stray to a stray dogδdThe proportion of exposed domestic dogs’ clinical outcomesδsThe proportion of exposed stray dogs’ clinical outcomesθdRate of domestic dog vaccinationθsRate of stray dog vaccinationμdThe natural death rate of domestic dogsμsThe natural death rate of stray dogsαdRate of domestic dog vaccination immunity lossαsRate of stray dog vaccination immunity lossεdThe ratio of rabies-related deaths in domestic dogsεsThe ratio of rabies-related deaths in stray dogscdRabid domestic dog culling ratecsRabid stray dog culling rate


### 2.2. Model Equations

The model is a system of eight ordinary differential equations, as shown by the transmission flowchart in Figure1 and also the statements made on the connections between the variables and parameters.
(1)dSddt=Ad+αdRd−βdSdIs−θd+μdSddEddt=βdSdIs−δd+μdEddIddt=δdEd−εd+cd+μdIddRddt=θdSd−αd+μdRddSsdt=As+αsRs−βsSsIs−θs+μsSsdEsdt=βsSsIs−δs+μsEsdIsdt=δsEs−εs+cs+μsIsdRsdt=θsSs−αs+μsRs.Figure 1
Flowchart illustrating the spread of rabies from stray dogs to domestic dogs.The total population forNdt and Nst is
(2)Ndt=Sdt+Edt+Idt+VdtNst=Sst+Est+Ist+Vst.Due to this, combining the differential equations for the domestic dog and stray dog populations in system (1) will result in
(3)dNddt=Ad−μdNd−cd+εdIddNsdt=As−μsNs−cs+εsIS,where Nd and Ns represent the entire populations of domestic dogs and stray dogs, correspondingly, during time t.


### 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 timet≥0, in order to study the model system in the feasible region. Two regions make up the model system (1); consequently, Ω=Ωd×Ωs. Consider the following lemma.Lemma 1.
The solution setSd,Ed,Id,R_d,Ss,Es,Is,Rs∈R+8 is limited in the possible region of the model system (1) Ω.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:(4)dNddt=dSddt+dEddt+dIddt+dRddt=Ad−μdNd−cd+εdId.If there is no disease-related mortality rate, then it follows that(5)dNddt=Ad−μdNd.Similarly,(6)dNsdt=As−μsNs.SupposedNd/dt≤0 and dNs/dt≤0; we will get Nd≤Ad/μd and Ns≤As/μs, on differentiating inequality results in 0≤Nd≤Ad/μd and 0≤Ns≤As/μs.Therefore, equations (5) and (6) become
(7)dNddt≤Ad−μdNd,(8)dNsdt≤As−μsNs.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).
(9)Ω.d=s.d,E.d,I.d,R.d∈R.+4N.d≤AdμdΩs=Ss,Es,Is,Rs∈R+4,Ns≤Asμs.Consequently, a practical solution is given. Therefore,Ω=Ω.d×Ωs. It follows from the common comparison theorem applied to differential inequality in [1] that from equations (7) and (8), solutions of Nd and Ns become
(10)Nd≤N.d0e−μ.d.t+Adμd1−e−μ.d.t,(11)Ns≤Ns0e−μst+Asμs1−e−μst.Taking the limit ast⟶∞ for equations (10) and (11) becomes
(12)0≤Nd≤Adμd,0≤Ns≤Asμs.The total population of domestic dogNdt and stray dogNst approaches to Ad/μdandAs/μs, respectively, where Ad/μd and As/μs are the upper bounds. This implies that at any time t, each and every solution with initial conditions in Ω is leftovers inΩ[7]. And if Nd>Ad/μd and Ns>As/μs, then the solutions Sd,Ed,Id,R_d,Ss,Es,Is,Rs enter the feasible region Ω at any time t. Therefore, every solution of system (1) is positively invariant, and hence, in region Ω, the model can continue to be mathematically sound and medically relevant. Thus, the investigation of solution Sd,Ed,Id,R_d,Ss,Es,Is,Rs 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 denotedE0=Sd∗,Ed∗,Id∗,Rd∗,Ss∗,Es∗,Ss∗,Is∗,Rs∗. Therefore, the next category will be 0 if there is no rabies. Ed∗=Id∗=Es∗=Is∗=0. Including this in system (1) results in
(13)A.d+αdRd−θd+μdS.d=0θdSd−αd+μdRd=0As+αsRs−θs+μsSs=0θsSs−αs+μsRs=0.The DFEE0 will be obtained by rearranging (13) mathematically.
(14)E0=Adμd+αdμdμd+θd+αd,0,0,Adθdμdμd+θd+αd,Asμs+αsμsμs+θs+αs,0,0,Asθsμsμs+θs+αs.IfRe<1, 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 (Re)

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 valueRe, [16, 17]. A statistic known as the effective reproduction number (Re) can help predict whether such an illness will spread across a community or go extinct. If Re<1, 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 Re>1, so each infected person often creates more than one new infection, as well as the illness can spread all across the community. If Re=1, this illness gets endemic, which means it spreads from one susceptible dog to another at a constant speed from across the community [3]. Letting f.ixstand the proportion of arrival of new contagions in the compartment (i.), V.i+x remains the transmission of individuals addicted to the compartment (i.) by completely other means; also, V.i−x remains the proportion of transmission of individuals out of the compartment (i.). This one is expected that every function f.i,V.i+,andV.i− is continuously differentiable at a minimum double concerning each variable and V.i=V.i−−V.i+. To calculate the Re, it is done by means of the next-generation method and taking the infectious compartments. ρFV− symbolizes the spectral radius of a matrix F.V−, whereve F=∂f.i/∂x.j and  in addition, V=∂V.iE0/∂x.j through i.≥1 aimed at the number of compartments, then 1≤j≤n aimed at the diseased compartments only.
(15)dEddt=βdSdIs−δd+μdEddIddt=δdEd−εd+cd+μdIddEsdt=βsSsIs−δs+μsEsdIsdt=δsEs−εs+cs+μsIs,fi=βdSdIs0βsSsIs0,F=∂fi∂xj=000βdAdμd+αdμdμd+θd+αd0000000βsAsμs+αsμsμs+θs+αs0000,Vi=δd+μdEdεd+cd+μdId−δdEdδs+μsEsεs+cs+μsIs−δsEs,V=∂Vi∂xj=δd+μd000−δdεd+cd+μd0000δs+μs000−δsεs+cs+μs,V−1=1δd+μd000δdδd+μdεd+cd+μd1εd+cd+μd00001δs+μs000δsδs+μsεs+cs+μs1εs+cs+μsRe=ρFV−1.In light of this, the effective reproduction number is provided by(16)Re=βsAsαs+μsδsμsμs+θs+αsμs+δsμs+cs+εs.


### 2.6. Endemic Equilibrium PointsE1

The steady-state conditions where the disease persists in the population are known as endemic equilibrium points.E1=S.d,E.d,I.d,R.d,S.s,E.s,I.s,R.s is the formula for the model’s endemic equilibrium point. We set dSd/dt=dEd/dt=dId/dt=dRd/dt=ds/dt=dEs/dt=dIs/dt=dRs/dt=0. Within system (1) and solving for Sd,Ed,Id,Rd,Ss,Es,Is, and Rs, we get
(17)Ss=μs+δsμs+cs+εsβsδs,Rs=θsμs+δsμs+cs+εsβsδsαs+μs,Is=−μs4+−cs−αs−δs−θs−εsμs3+−cs−αs−θs−εsδs−εs+csθs+αsμs2+δs−εs−csαs+Asβs−θscs−θsεsμs+Asβsδsαsβsδs+μsαs+μsεs+cs+μs,Es=1δsδs+μsαs+μsβs−μs4+−cs−αs−δs−θs−εsμs3+−cs−αs−θs−εsδs−εs+csθs+αsμs2+δs−cs−εsαs+Asβs−θscs−θsεsμs+Asβsδsαs,Ed=−Adαd+μdβdμs4+δs+θs+εs+cs+αsμs3+θs+εs+cs+αsδs+εs+csθs+αsμs2+δs−Asβs+εs+csθs+αsμs−Asβsδsαsδd+μd−βdαd+μdμs4+βsμd2+−δdβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαdδs+θs+εs+cs+αs+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβs−βdεs+csθs+αsμd−βdθs+εs+cs+αsδs+εs+csθs+αsαdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsβd−Asβs+εs+csθs+αsαdμs+αsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαdβsδs,Sd=αd+μdAdβsαs+μsεs+cs+μsδs+μs−βdαd+μdμs4+βsμd2+−δsβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαsδs+εs+cs+αsμs3+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβd−βdεs+csθs+αsμd−αdθs+εs+cs+αsδs+εs+csθs+αsβdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsαd−Asβs+εs+csθs+αsβdμs+δsβsαsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαd,Id=−βdδdαd+μdAdμs4+δs+θs+εs+cs+αsμs3+θs+εs+cs+αsδs+εs+csθs+αsμs2+δs−Asβs+εs+csθs+αsμs−Asβsδsαsεd+cd+μdδd+μd−βdαd+μdμs4+βsμd2+−δsβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαdδs+θs+εs+cs+αsμs3+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβs−βdεs+csθs+αsμd−βdθs+εs+cs+αsδs+εs+csθs+αsαdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsβd−Asβs+εs+csθs+αsαdμs+αsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαdβsδs,Rd=Adθdβsαs+μsεs+cs+μsδs+μs−βdαd+μdμs4+βsμd2+−δsβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαdδs+θs+εs+cs+αsμs3+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβs−βdεs+csθs+αsμd−αdθs+εs+cs+αsδs+εs+csθs+αsβdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsαd−Asβs+εs+csθs+αsβdμs+δsβsαsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαd.


## 2.1. Model Variables and Parameters with Their Descriptions

Tables1 and 2 below, respectively, contain descriptions of the model’s variables and parameters.Table 1
Variables in the model and its explanation.
SymbolInformation about the variablesSst,SdtSusceptible stray and domestic dog population at timet, respectivelyEst,EdtExposed stray and domestic dog population at timet, respectivelyIst, IdtInfectious stray and domestic dog population at timet, respectivelyRst,RdtRecovered stray and domestic dog population at timet, respectivelyTable 2
Details of the model parameters.
ParameterExplanationAdNumber of domestic dog recruitmentAsNumber of stray dog recruitmentβdRate of stray dog transmission to domestic dogsβsRate of transmission from stray to a stray dogδdThe proportion of exposed domestic dogs’ clinical outcomesδsThe proportion of exposed stray dogs’ clinical outcomesθdRate of domestic dog vaccinationθsRate of stray dog vaccinationμdThe natural death rate of domestic dogsμsThe natural death rate of stray dogsαdRate of domestic dog vaccination immunity lossαsRate of stray dog vaccination immunity lossεdThe ratio of rabies-related deaths in domestic dogsεsThe ratio of rabies-related deaths in stray dogscdRabid domestic dog culling ratecsRabid stray dog culling rate


## 2.2. Model Equations

The model is a system of eight ordinary differential equations, as shown by the transmission flowchart in Figure1 and also the statements made on the connections between the variables and parameters.
(1)dSddt=Ad+αdRd−βdSdIs−θd+μdSddEddt=βdSdIs−δd+μdEddIddt=δdEd−εd+cd+μdIddRddt=θdSd−αd+μdRddSsdt=As+αsRs−βsSsIs−θs+μsSsdEsdt=βsSsIs−δs+μsEsdIsdt=δsEs−εs+cs+μsIsdRsdt=θsSs−αs+μsRs.Figure 1
Flowchart illustrating the spread of rabies from stray dogs to domestic dogs.The total population forNdt and Nst is
(2)Ndt=Sdt+Edt+Idt+VdtNst=Sst+Est+Ist+Vst.Due to this, combining the differential equations for the domestic dog and stray dog populations in system (1) will result in
(3)dNddt=Ad−μdNd−cd+εdIddNsdt=As−μsNs−cs+εsIS,where Nd and Ns represent the entire populations of domestic dogs and stray dogs, correspondingly, during time t.


## 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 timet≥0, in order to study the model system in the feasible region. Two regions make up the model system (1); consequently, Ω=Ωd×Ωs. Consider the following lemma.Lemma 1.
The solution setSd,Ed,Id,R_d,Ss,Es,Is,Rs∈R+8 is limited in the possible region of the model system (1) Ω.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:(4)dNddt=dSddt+dEddt+dIddt+dRddt=Ad−μdNd−cd+εdId.If there is no disease-related mortality rate, then it follows that(5)dNddt=Ad−μdNd.Similarly,(6)dNsdt=As−μsNs.SupposedNd/dt≤0 and dNs/dt≤0; we will get Nd≤Ad/μd and Ns≤As/μs, on differentiating inequality results in 0≤Nd≤Ad/μd and 0≤Ns≤As/μs.Therefore, equations (5) and (6) become
(7)dNddt≤Ad−μdNd,(8)dNsdt≤As−μsNs.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).
(9)Ω.d=s.d,E.d,I.d,R.d∈R.+4N.d≤AdμdΩs=Ss,Es,Is,Rs∈R+4,Ns≤Asμs.Consequently, a practical solution is given. Therefore,Ω=Ω.d×Ωs. It follows from the common comparison theorem applied to differential inequality in [1] that from equations (7) and (8), solutions of Nd and Ns become
(10)Nd≤N.d0e−μ.d.t+Adμd1−e−μ.d.t,(11)Ns≤Ns0e−μst+Asμs1−e−μst.Taking the limit ast⟶∞ for equations (10) and (11) becomes
(12)0≤Nd≤Adμd,0≤Ns≤Asμs.The total population of domestic dogNdt and stray dogNst approaches to Ad/μdandAs/μs, respectively, where Ad/μd and As/μs are the upper bounds. This implies that at any time t, each and every solution with initial conditions in Ω is leftovers inΩ[7]. And if Nd>Ad/μd and Ns>As/μs, then the solutions Sd,Ed,Id,R_d,Ss,Es,Is,Rs enter the feasible region Ω at any time t. Therefore, every solution of system (1) is positively invariant, and hence, in region Ω, the model can continue to be mathematically sound and medically relevant. Thus, the investigation of solution Sd,Ed,Id,R_d,Ss,Es,Is,Rs 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 denotedE0=Sd∗,Ed∗,Id∗,Rd∗,Ss∗,Es∗,Ss∗,Is∗,Rs∗. Therefore, the next category will be 0 if there is no rabies. Ed∗=Id∗=Es∗=Is∗=0. Including this in system (1) results in
(13)A.d+αdRd−θd+μdS.d=0θdSd−αd+μdRd=0As+αsRs−θs+μsSs=0θsSs−αs+μsRs=0.The DFEE0 will be obtained by rearranging (13) mathematically.
(14)E0=Adμd+αdμdμd+θd+αd,0,0,Adθdμdμd+θd+αd,Asμs+αsμsμs+θs+αs,0,0,Asθsμsμs+θs+αs.IfRe<1, 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 (Re)

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 valueRe, [16, 17]. A statistic known as the effective reproduction number (Re) can help predict whether such an illness will spread across a community or go extinct. If Re<1, 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 Re>1, so each infected person often creates more than one new infection, as well as the illness can spread all across the community. If Re=1, this illness gets endemic, which means it spreads from one susceptible dog to another at a constant speed from across the community [3]. Letting f.ixstand the proportion of arrival of new contagions in the compartment (i.), V.i+x remains the transmission of individuals addicted to the compartment (i.) by completely other means; also, V.i−x remains the proportion of transmission of individuals out of the compartment (i.). This one is expected that every function f.i,V.i+,andV.i− is continuously differentiable at a minimum double concerning each variable and V.i=V.i−−V.i+. To calculate the Re, it is done by means of the next-generation method and taking the infectious compartments. ρFV− symbolizes the spectral radius of a matrix F.V−, whereve F=∂f.i/∂x.j and  in addition, V=∂V.iE0/∂x.j through i.≥1 aimed at the number of compartments, then 1≤j≤n aimed at the diseased compartments only.
(15)dEddt=βdSdIs−δd+μdEddIddt=δdEd−εd+cd+μdIddEsdt=βsSsIs−δs+μsEsdIsdt=δsEs−εs+cs+μsIs,fi=βdSdIs0βsSsIs0,F=∂fi∂xj=000βdAdμd+αdμdμd+θd+αd0000000βsAsμs+αsμsμs+θs+αs0000,Vi=δd+μdEdεd+cd+μdId−δdEdδs+μsEsεs+cs+μsIs−δsEs,V=∂Vi∂xj=δd+μd000−δdεd+cd+μd0000δs+μs000−δsεs+cs+μs,V−1=1δd+μd000δdδd+μdεd+cd+μd1εd+cd+μd00001δs+μs000δsδs+μsεs+cs+μs1εs+cs+μsRe=ρFV−1.In light of this, the effective reproduction number is provided by(16)Re=βsAsαs+μsδsμsμs+θs+αsμs+δsμs+cs+εs.


## 2.6. Endemic Equilibrium PointsE1

The steady-state conditions where the disease persists in the population are known as endemic equilibrium points.E1=S.d,E.d,I.d,R.d,S.s,E.s,I.s,R.s is the formula for the model’s endemic equilibrium point. We set dSd/dt=dEd/dt=dId/dt=dRd/dt=ds/dt=dEs/dt=dIs/dt=dRs/dt=0. Within system (1) and solving for Sd,Ed,Id,Rd,Ss,Es,Is, and Rs, we get
(17)Ss=μs+δsμs+cs+εsβsδs,Rs=θsμs+δsμs+cs+εsβsδsαs+μs,Is=−μs4+−cs−αs−δs−θs−εsμs3+−cs−αs−θs−εsδs−εs+csθs+αsμs2+δs−εs−csαs+Asβs−θscs−θsεsμs+Asβsδsαsβsδs+μsαs+μsεs+cs+μs,Es=1δsδs+μsαs+μsβs−μs4+−cs−αs−δs−θs−εsμs3+−cs−αs−θs−εsδs−εs+csθs+αsμs2+δs−cs−εsαs+Asβs−θscs−θsεsμs+Asβsδsαs,Ed=−Adαd+μdβdμs4+δs+θs+εs+cs+αsμs3+θs+εs+cs+αsδs+εs+csθs+αsμs2+δs−Asβs+εs+csθs+αsμs−Asβsδsαsδd+μd−βdαd+μdμs4+βsμd2+−δdβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαdδs+θs+εs+cs+αs+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβs−βdεs+csθs+αsμd−βdθs+εs+cs+αsδs+εs+csθs+αsαdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsβd−Asβs+εs+csθs+αsαdμs+αsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαdβsδs,Sd=αd+μdAdβsαs+μsεs+cs+μsδs+μs−βdαd+μdμs4+βsμd2+−δsβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαsδs+εs+cs+αsμs3+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβd−βdεs+csθs+αsμd−αdθs+εs+cs+αsδs+εs+csθs+αsβdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsαd−Asβs+εs+csθs+αsβdμs+δsβsαsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαd,Id=−βdδdαd+μdAdμs4+δs+θs+εs+cs+αsμs3+θs+εs+cs+αsδs+εs+csθs+αsμs2+δs−Asβs+εs+csθs+αsμs−Asβsδsαsεd+cd+μdδd+μd−βdαd+μdμs4+βsμd2+−δsβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαdδs+θs+εs+cs+αsμs3+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβs−βdεs+csθs+αsμd−βdθs+εs+cs+αsδs+εs+csθs+αsαdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsβd−Asβs+εs+csθs+αsαdμs+αsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαdβsδs,Rd=Adθdβsαs+μsεs+cs+μsδs+μs−βdαd+μdμs4+βsμd2+−δsβd+θd+αdβs−βdθs+εs+cs+αsμd−βdαdδs+θs+εs+cs+αsμs3+βsδs+εs+cs+αsμd2+θd+αdβs−βdθs+εs+cs+αsδs+εs+cs+αsθd+αdβs−βdεs+csθs+αsμd−αdθs+εs+cs+αsδs+εs+csθs+αsβdμs2+βsεs+cs+αsδs+αsεs+csμd2+Asβd+εs+cs+αsθd+αdβs−βdεs+csθs+αsδs+βsαsεs+csθd+αdμd−δsαd−Asβs+εs+csθs+αsβdμs+δsβsαsεs+csμd2+Asβd+εs+csθd+αdμd+Asβdαd.


## 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 E0 is evaluated. Then, stability is established based on the trace’s indication and the determinant of the Jacobian matrix.Theorem 2.
IfR0<1 then the disease-free equilibrium E0 of a system (1) is locally asymptotically stable or unstable otherwise.Proof.
The Jacobian matrix of the system (1) at E0. From (14), the disease-free equilibrium point is given by
(18)E0=Sd0,Ed0,Id0,Rd∗,Ss0,Es0,Ss0,Is0,Rs0=Adμd+αdμdμd+θd+αd,0,0,Adθdμdμd+θd+αd,Asμs+αsμsμs+θs+αs,0,0,Asθsμsμs+θs+αs.We determine the system’s Jacobian matrix (1). To accomplish this, system (1)’s individual equations are differentiated by means of the state variable. S.d,E.d,I.d,R.d,S.s,E.s,I.s,andR.s. Let JE0 be the jacobian matrix evaluated at the equilibrium without sickness E0.
(19)JE0=−μd+θd00αd00−βdAdαd+μdμdμd+θd+αd00−μd+δd0000βdAdαd+μdμdμd+θd+αd00δd−μd+cd+εd00000θd00−μd+αd00000000−μs+θs0−βsAsαs+μsμsμs+θs+αsαs00000−μs+δsβsAsαs+μsμsμs+θs+αs000000δs−μs+cs+εs00000θs00−μs+αs.The trace of the above matrix will be(20)TrJE0=−μd+θd−μd+δd−μd+cd+εd−μd+αd−μs+θs−μs+δs−μs+cs+εs−μs+αs<0.Since all parameters are positive, determine the determinant of the Jacobian matrixJE0.Let(21)a1=−μd+θd,a2=−μd+δd,a3=−μd+cd+εd,a4=−μd+αd,a5−μs+θs,a6=−μs+δs,a7=−βdAdαd+μdμdμd+θd+αd,a8=βdAdαd+μdμdμd+θd+αda9=−βsAsαs+μsμsμs+θs+αsa10=βsAsαs+μsμsμs+θs+αsa11=−μs+cs+εsa12=−μs+αs.Then, the above Jacobian matrix will be(22)JE0=a100αd00a700a20000a800δda.300000θd00a.400000000a.50a9αs00000a.6a10000000δsa1100000θs00a12,detJE0=a100αd00a700a20000a800δda300000θd00a400000000a50a9αs00000a6a10000000δsa1100000θs00a12=a3a10αd00a700a2000a80θd0a40000000a50a9αs0000a6a1000000δsa110000θs00a12=a3a2a1αd00a70θda4000000a50a9αs000a6a100000δsa11000θs00a12=a3a2a1a400000a50a9αs00a6a10000δsa1100θs00a12−θdad00a700a50a9as00a6a10000δsa1100θs00a12=a3a2a1a4a50a9αs0a6a.1000δsa.110θs00a.12−θdαda500θs0a.6δs0a9a.10a110αs00a.12,a3a2a1a4−θdαda.500θs0a.6δs0a9a10a110αs00a12=a3a2a1a4−θdαda5a6a100δsa11000a12−θs0a.9αsa6a100δsa110=a3a2a1a4−θdαda5a12a6a11−a10δs−θsαsa6a11−a10δs=a3a2a1a4−θdαda6a11−a10δsa5a12−θsαs.Insert the value ofa.1, a.2, a.3,a.4, a.5, a.6, a.10, a.11, a.12; we will get
(23)detJE0=μd+cd+εdμd+δdμd+θdμd+αd−θdαdμs+δsμs+cs+εs−βsAsαs+μsδsμsμs+θs+αsμs+θsμs+αs−θsαs,detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εs−βsAsαs+μsδsμsμs+θs+αsμsμs+θs+αs.Sinceμd+θdμd+αd−θdαd=μdμd+θd+αd and μs+θsμs+αs−θsαs=μsμs+θs+αs,(24)detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εsμs+δsμs+cs+εsμs+δsμs+cs+εs−βsAsαs+μsδsμsμs+θs+αsμsμs+θs+αs,detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εs1−βsAsαs+μsδsμsμs+θs+αsμs+δsμs+cs+εsμsμs+θs+αs,detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εs1−Reμsμs+θs+αs.From equation (16), Re=βsAsαs+μsδs/μsμs+θs+αsμs+δsμs+cs+εs.Thus, forRe<1, there are the disease-free equilibrium points E0 which are locally asymptotically stable if TrJ.E0<0 and Det.JE0>0, otherwise it is unstable if Re>1. The theorem is proved.


### 3.2. Global Stability of DFEE0

We used the technique suggested by [18] to examine the global stability of the DFE of the system (1).Theorem 3.
Under the condition thatRe<1, the DEF E0 for model system (1) is globally stable.Proof.
LettingM1, M2, M3, and M4 be positive constants, let L be a Lyapunov function.
(25)L=Ss−Ss0−Ss0lnSsSs0+M1Es+M2Is+Rs−Rs0−Rs0lnRsRs0+Sd−Sd0−Sd0lnSdSd0+M3Ed+M4Id+Rd−Rd0−Rd0lnRdRd0.The derivative ofL with respect to time is
(26)dLdt=1−Ss0SsdSsdt+M1dEsdt+M2dIsdt+1−Rs0RsdRsdt+1−Sd0SddSddt+M3dEddt+M4dIddt+1−Rd0RddRddt.From system (1), substitute dSs/dt, dEs/.dt, dIs/.dt, dRs/.dt, dSd/.dt, dEd/dt, dId/dt, and dRd/.dt into equation (26) to obtain
(27)dLdt=1−Ss0SsAs+αsRs−βsSsIs−θs+μsSs+M1βsSsIs−δs+μsEs+M2δsEs−εs+cs+μsIs+1−Rs0RsθsSs−αs+μsRs+1−Sd0SdAd+αdRd−βdSdIs−θd+μdSd+M3βdSdIs−δd+μdEd+M4δdEd−εd+cd+μdId+1−Rd0RdθdSd−αd+μdRd.With the Lyapunov functionL formed just on space of the state variables Ss,Es,Is, Rs, Sd,Ed, Id, and Rd, it is evident that if Est, Ist, Edt, and Idt at the DFE are globally stable (therefore, Es=Is=Ed=0 and Id=0), then Sst⟶Ss0, Rst⟶Rs0,Sdt⟶Sd0, and Rdt⟶Rd0 as t⟶∞.Hence, it can be considered that(28)Ss≤Ss0=Asμs+αsμsμs+θs+αs,Rs≤Rs0=Asθsμsμs+θs+αsSd≤Sd0=Adμd+αdμdμd+θd+αd,Rd≤Rs0=Adθdμdμd+θd+αd.Inserting equation (28) into (27), we get
(29)dLdt≤1−Ss0Ss0As+αsRs−βsSs0Is−θs+μsSs0+M1βsSs0Is−δs+μsEs+M2δsEs−εs+cs+μsIs+1−Rs0Rs0θsSs0−αs+μsRs0+1−Sd0Sd0Ad+αdRd0−βdSd0Is−θd+μdSd0+M3βdSd0Is−δd+μdEd+M4δdEd−εd+cd+μdId+1−Rd0Rd0θdSd0−αd+μdRd0.At the disease-free equilibrium point,Sst⟶Ss0, Rst⟶Rs0,Sdt⟶Sd0,Rdt⟶Rd0 as t⟶∞.(30)dLdt≤M1βsSs0Is−δs+μsEs+M2δsEs−εs+cs+μsIs+M3βdSd0Is−δd+μdEd+M4δdEd−εd+cd+μdId,(31)dLdt≤M2δs−M1δs+μsEsM1βsSs0+M3βdSd0−M2εs+cs+μsIsM4δd−M3δd+μdEd−M4εd+cd+μdId.When the coefficients ofEs,Is,Ed, and Id in (31) tend to zero, the following results appear:
(32)M4=0,M3=0,M1=δs,M2=μs+δs.Inserting equation (32) into equation (31) yields
(33)dLdt≤μs+δsδs−δsδs+μsEs+δsβsSs0−μs+δsμs+cs+εsIs.From equation (14), Ss0=Asμs+αs/μsμs+θs+αs; then, insert this in equation (33); we get
(34)dLdt≤δsβsAsμs+αsμsμs+θs+αs−μs+δsμs+cs+εsIs,dLdt≤δsβsAsμs+αsμsμs+θs+αs−μs+δsεs+cs+μsIs,dLdt≤μs+δsμs+cs+εsμs+δsμs+cs+εsδsβsAsμs+αsμsμs+θs+αs−μs+δsμs+cs+εsIs,dLdt≤μs+δsμs+cs+εsδsβsAsμs+αsμsμs+θs+αsμs+δsμs+cs+εs−μs+δsμs+cs+εsμs+δsμs+cs+εsIs.Therefore,(35)dLdt≤μs+δsμs+cs+εsRe−1Is≤0,ifR0≤1.Additionally,dL/dt=0 if and only if Is=0.Therefore, for Es=Is=Ed=Id=0, it shows that Sdt⟶Ss0=Asμs+αs/μsμs+θs+αs,Rst⟶Rs0=Asθs/μsμs+θs+αs,Sd⟶Sd0=Adμd+αd/μdμd+θd+αd,Rd⟶Rs0=Adθd/μdμd+θd+αd as t⟶∞.As a result, the singleton setE0 is the biggest compact invariant set in the set Ss,Es,Is,Rs,Sd,Ed,Id,Rd∈Ω:dL/dt≤0. We thus draw the conclusion that E0 is globally asymptotically stable in the case where R0≤1 is based on La Salle’s invariance principle [9, 10, 18].


## 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 E0 is evaluated. Then, stability is established based on the trace’s indication and the determinant of the Jacobian matrix.Theorem 2.
IfR0<1 then the disease-free equilibrium E0 of a system (1) is locally asymptotically stable or unstable otherwise.Proof.
The Jacobian matrix of the system (1) at E0. From (14), the disease-free equilibrium point is given by
(18)E0=Sd0,Ed0,Id0,Rd∗,Ss0,Es0,Ss0,Is0,Rs0=Adμd+αdμdμd+θd+αd,0,0,Adθdμdμd+θd+αd,Asμs+αsμsμs+θs+αs,0,0,Asθsμsμs+θs+αs.We determine the system’s Jacobian matrix (1). To accomplish this, system (1)’s individual equations are differentiated by means of the state variable. S.d,E.d,I.d,R.d,S.s,E.s,I.s,andR.s. Let JE0 be the jacobian matrix evaluated at the equilibrium without sickness E0.
(19)JE0=−μd+θd00αd00−βdAdαd+μdμdμd+θd+αd00−μd+δd0000βdAdαd+μdμdμd+θd+αd00δd−μd+cd+εd00000θd00−μd+αd00000000−μs+θs0−βsAsαs+μsμsμs+θs+αsαs00000−μs+δsβsAsαs+μsμsμs+θs+αs000000δs−μs+cs+εs00000θs00−μs+αs.The trace of the above matrix will be(20)TrJE0=−μd+θd−μd+δd−μd+cd+εd−μd+αd−μs+θs−μs+δs−μs+cs+εs−μs+αs<0.Since all parameters are positive, determine the determinant of the Jacobian matrixJE0.Let(21)a1=−μd+θd,a2=−μd+δd,a3=−μd+cd+εd,a4=−μd+αd,a5−μs+θs,a6=−μs+δs,a7=−βdAdαd+μdμdμd+θd+αd,a8=βdAdαd+μdμdμd+θd+αda9=−βsAsαs+μsμsμs+θs+αsa10=βsAsαs+μsμsμs+θs+αsa11=−μs+cs+εsa12=−μs+αs.Then, the above Jacobian matrix will be(22)JE0=a100αd00a700a20000a800δda.300000θd00a.400000000a.50a9αs00000a.6a10000000δsa1100000θs00a12,detJE0=a100αd00a700a20000a800δda300000θd00a400000000a50a9αs00000a6a10000000δsa1100000θs00a12=a3a10αd00a700a2000a80θd0a40000000a50a9αs0000a6a1000000δsa110000θs00a12=a3a2a1αd00a70θda4000000a50a9αs000a6a100000δsa11000θs00a12=a3a2a1a400000a50a9αs00a6a10000δsa1100θs00a12−θdad00a700a50a9as00a6a10000δsa1100θs00a12=a3a2a1a4a50a9αs0a6a.1000δsa.110θs00a.12−θdαda500θs0a.6δs0a9a.10a110αs00a.12,a3a2a1a4−θdαda.500θs0a.6δs0a9a10a110αs00a12=a3a2a1a4−θdαda5a6a100δsa11000a12−θs0a.9αsa6a100δsa110=a3a2a1a4−θdαda5a12a6a11−a10δs−θsαsa6a11−a10δs=a3a2a1a4−θdαda6a11−a10δsa5a12−θsαs.Insert the value ofa.1, a.2, a.3,a.4, a.5, a.6, a.10, a.11, a.12; we will get
(23)detJE0=μd+cd+εdμd+δdμd+θdμd+αd−θdαdμs+δsμs+cs+εs−βsAsαs+μsδsμsμs+θs+αsμs+θsμs+αs−θsαs,detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εs−βsAsαs+μsδsμsμs+θs+αsμsμs+θs+αs.Sinceμd+θdμd+αd−θdαd=μdμd+θd+αd and μs+θsμs+αs−θsαs=μsμs+θs+αs,(24)detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εsμs+δsμs+cs+εsμs+δsμs+cs+εs−βsAsαs+μsδsμsμs+θs+αsμsμs+θs+αs,detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εs1−βsAsαs+μsδsμsμs+θs+αsμs+δsμs+cs+εsμsμs+θs+αs,detJE0=μd+cd+εdμd+δdμdμd+θd+αdμs+δsμs+cs+εs1−Reμsμs+θs+αs.From equation (16), Re=βsAsαs+μsδs/μsμs+θs+αsμs+δsμs+cs+εs.Thus, forRe<1, there are the disease-free equilibrium points E0 which are locally asymptotically stable if TrJ.E0<0 and Det.JE0>0, otherwise it is unstable if Re>1. The theorem is proved.


## 3.2. Global Stability of DFEE0

We used the technique suggested by [18] to examine the global stability of the DFE of the system (1).Theorem 3.
Under the condition thatRe<1, the DEF E0 for model system (1) is globally stable.Proof.
LettingM1, M2, M3, and M4 be positive constants, let L be a Lyapunov function.
(25)L=Ss−Ss0−Ss0lnSsSs0+M1Es+M2Is+Rs−Rs0−Rs0lnRsRs0+Sd−Sd0−Sd0lnSdSd0+M3Ed+M4Id+Rd−Rd0−Rd0lnRdRd0.The derivative ofL with respect to time is
(26)dLdt=1−Ss0SsdSsdt+M1dEsdt+M2dIsdt+1−Rs0RsdRsdt+1−Sd0SddSddt+M3dEddt+M4dIddt+1−Rd0RddRddt.From system (1), substitute dSs/dt, dEs/.dt, dIs/.dt, dRs/.dt, dSd/.dt, dEd/dt, dId/dt, and dRd/.dt into equation (26) to obtain
(27)dLdt=1−Ss0SsAs+αsRs−βsSsIs−θs+μsSs+M1βsSsIs−δs+μsEs+M2δsEs−εs+cs+μsIs+1−Rs0RsθsSs−αs+μsRs+1−Sd0SdAd+αdRd−βdSdIs−θd+μdSd+M3βdSdIs−δd+μdEd+M4δdEd−εd+cd+μdId+1−Rd0RdθdSd−αd+μdRd.With the Lyapunov functionL formed just on space of the state variables Ss,Es,Is, Rs, Sd,Ed, Id, and Rd, it is evident that if Est, Ist, Edt, and Idt at the DFE are globally stable (therefore, Es=Is=Ed=0 and Id=0), then Sst⟶Ss0, Rst⟶Rs0,Sdt⟶Sd0, and Rdt⟶Rd0 as t⟶∞.Hence, it can be considered that(28)Ss≤Ss0=Asμs+αsμsμs+θs+αs,Rs≤Rs0=Asθsμsμs+θs+αsSd≤Sd0=Adμd+αdμdμd+θd+αd,Rd≤Rs0=Adθdμdμd+θd+αd.Inserting equation (28) into (27), we get
(29)dLdt≤1−Ss0Ss0As+αsRs−βsSs0Is−θs+μsSs0+M1βsSs0Is−δs+μsEs+M2δsEs−εs+cs+μsIs+1−Rs0Rs0θsSs0−αs+μsRs0+1−Sd0Sd0Ad+αdRd0−βdSd0Is−θd+μdSd0+M3βdSd0Is−δd+μdEd+M4δdEd−εd+cd+μdId+1−Rd0Rd0θdSd0−αd+μdRd0.At the disease-free equilibrium point,Sst⟶Ss0, Rst⟶Rs0,Sdt⟶Sd0,Rdt⟶Rd0 as t⟶∞.(30)dLdt≤M1βsSs0Is−δs+μsEs+M2δsEs−εs+cs+μsIs+M3βdSd0Is−δd+μdEd+M4δdEd−εd+cd+μdId,(31)dLdt≤M2δs−M1δs+μsEsM1βsSs0+M3βdSd0−M2εs+cs+μsIsM4δd−M3δd+μdEd−M4εd+cd+μdId.When the coefficients ofEs,Is,Ed, and Id in (31) tend to zero, the following results appear:
(32)M4=0,M3=0,M1=δs,M2=μs+δs.Inserting equation (32) into equation (31) yields
(33)dLdt≤μs+δsδs−δsδs+μsEs+δsβsSs0−μs+δsμs+cs+εsIs.From equation (14), Ss0=Asμs+αs/μsμs+θs+αs; then, insert this in equation (33); we get
(34)dLdt≤δsβsAsμs+αsμsμs+θs+αs−μs+δsμs+cs+εsIs,dLdt≤δsβsAsμs+αsμsμs+θs+αs−μs+δsεs+cs+μsIs,dLdt≤μs+δsμs+cs+εsμs+δsμs+cs+εsδsβsAsμs+αsμsμs+θs+αs−μs+δsμs+cs+εsIs,dLdt≤μs+δsμs+cs+εsδsβsAsμs+αsμsμs+θs+αsμs+δsμs+cs+εs−μs+δsμs+cs+εsμs+δsμs+cs+εsIs.Therefore,(35)dLdt≤μs+δsμs+cs+εsRe−1Is≤0,ifR0≤1.Additionally,dL/dt=0 if and only if Is=0.Therefore, for Es=Is=Ed=Id=0, it shows that Sdt⟶Ss0=Asμs+αs/μsμs+θs+αs,Rst⟶Rs0=Asθs/μsμs+θs+αs,Sd⟶Sd0=Adμd+αd/μdμd+θd+αd,Rd⟶Rs0=Adθd/μdμd+θd+αd as t⟶∞.As a result, the singleton setE0 is the biggest compact invariant set in the set Ss,Es,Is,Rs,Sd,Ed,Id,Rd∈Ω:dL/dt≤0. We thus draw the conclusion that E0 is globally asymptotically stable in the case where R0≤1 is based on La Salle’s invariance principle [9, 10, 18].


## 4. Sensitivity Analysis forRE

The effect of model parameter values on output estimation ofRe is shown in this section. The value of a particular input parameter determines the output value of Re. 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.Table 3
Modeling parameters and their values (unit: month-1).
ParameterRateExplanationBasisAd120Number of domestic dog recruitment[16]As32Number of stray dog recruitment[16]βd0.000004Rate of stray dog transmission to domestic dogsEstimationβs0.000008Rate of transmission from stray to a stray dogEstimationδd0.37The proportion of exposed domestic dogs’ clinical outcomes[8]δs0.35The proportion of exposed stray dogs’ clinical outcomes[8]θd0.54Rate of domestic dog vaccination[16]θs0.54Rate of stray dog vaccination[16]μd0.11The natural death rate of domestic dogs[19]μs0.24The natural death rate of stray dogs[19]αd0.5Rate of domestic dog vaccination immunity loss[19]αs0.5Rate of stray dog vaccination immunity loss[19]εd1The ratio of rabies-related deaths in domestic dogs[19]εs1The ratio of rabies-related deaths in stray dogs[19]cd0.5Rabid domestic dog culling rate[16]cs0.5Rabid stray dog culling rateEstimationThe 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 variableX that depends on parameter ω:SωX=∂X/∂ω×ω/X.In this instance, we have the number of effective reproductionsRe=βsAsαs+μsδs/μsμs+θs+αsμs+δsμs+cs+εs computed.From the model, the sensitivity index ofRe concerning βs is given by
(36)SβsRe=∂Re∂βs×βsRe=+1.Again, the sensitivity index ofRe concerning δs is given by
(37)SβsRe=∂Re∂δs×δsRe=+0.4067796610.Similarly, the sensitivity index concerningθs, cs, and εs is given by
(38)SAsRe=∂Re∂As×AsRe=+1,SαsRe=∂Re∂αs×αsRe=0.2850506757,SμsRe=∂Re∂μs×μsRe=−1.413934758,SθsRe=∂Re∂θs×θsRe=−0.4218750000,SεsRe=∂Re∂εs×εsRe=−0.5747126437,ScsRe=∂Re∂cs×csRe=−0.2873563218.The sensitivity indices ofRe concerning parameters ω are obtained and summarized in Table 4.Table 4
Sensitivities ofRe.
The symbol for the parameterValue for the sensitivity indexβs+1δs+0.4067796610As+1αs+0.2850506757μs−1.413934758θs−0.4218750000εs−0.5747126437cs−0.2873563218The infection rates of stray dogsβs and annual stray dog births As are the most sensitive and positive parameters, followed by the proportion of exposed stray dogs’ clinical outcome parameter δs and the loss rate of vaccination immunity of stray dogs αs, on the basis of the sensitivity indicators. As a result of increasing these parameters, the effective reproduction number Re will increase. Furthermore, the most negatively sensitive indicator is the stray dog mortality rate μs, which is followed by the rabies mortality rate εs, the stray dog vaccination rate θi, and stray dog culling rate cs. As a result, as these parameters are increased, the effective reproduction number Re 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 Table3 and initial condition values shown in Table 5 below.Table 5
Initial conditions used in the rabies model.
State variableSdEdIdRdSsEsIsRsInitial value3200601525280080200Figure2 curves demonstrate that if no intervention was used θd=θs=cd=cs=0, 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 θd=θs=θd=θi=0, 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 θd=θs=50%andcd=cs=50% has the greatest effect on reducing the number of exposed and infected domestic dogs, followed by vaccination only θd=θs=50%andcd=cs=0 and then culling only θd=θs=0andcd=cs=50%. Vaccination alone θd=θs=50%andcd=cs=0is more effective than culling alone θd=θs=0andcd=cs=50% in exposed group populations. However, culling the population of an infected group is more effective than vaccination alone.Figure 2
The effect of vaccination (parameterθd and θs) and culling (parameter cd and cs) on the exposed and infected domestic dog populations.
(a)
Exposed domestic dog(b)
Infected domestic dogAccording to the existence of the graphs in Figure3, when no intervention (θd=θs=cd=cs=0), 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 θd=θs=cd=cs=0 is used, the number of exposed and infected stray dogs decreases faster when one θd=θs=0andcd=cs=50% or θd=θs=50%andcd=cs=0 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 θd=θs=cd=cs=50% has a greater impact than either vaccination or culling only.Figure 3
The effect of vaccinationparameterθdandθs and culling (parametercdandcs) on the exposed and infected stray dog populations.
(a)
Exposed stray dog(b)
Infected stray dogOne of most sensitive factors formulating the characteristics of dog rabies spread seems to be the natural death rate (μ), 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.Figure 4
The impact of natural mortality rate (parameterμ) on the exposed and infected stray dog populations.
(a)
Exposed stray dog(b)
Infected stray dogVariables in the vaccination rateθs for stray dogs were used in the simulation (see Figure 5). Increased stray dog vaccination rates θs 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 θs of stray dogs within the model decreased the number of exposed and infected domestic dog populations.Figure 5
The effect of stray dog’s vaccination (parameterθs) on the exposed and infected domestic dog populations.
(a)
Exposed domestic dog(b)
Infected domestic dogFigure6 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.Figure 6
Effect of stray dog recruitment rate (parameterAs) on the infected domestic dog population.


## 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.



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