IMPACT OF VACCINATION AND STERILIZATION ON THE TRANSMISSION DYNAMICS OF RABIES

. We have constructed a mathematical model by dividing the dog and human populations into eight compartments as the rabies virus is likely to spread in both populations. In the model, disease-controlling strategies such as vaccination, sterilization and culling are taken into consideration, and their impact is studied. The current study assumes that dogs can transmit rabies among dogs as well as to human population. We have applied the next-generation matrix technique to compute the basic reproduction number. Also, each parameters involved are subjected to sensitivity analysis using the approach of normalized sensitivity index. The disease-free (or rabies-free) and endemic-equilibrium points are discovered analytically. The endemic equilibrium point is shown to be locally asymptotically stable. The numerical simulations, which use approximations for parameter values, shows that effective method for controlling rabies transmission is a combination of vaccination, sterilization and culling of infected dogs. The findings indicate that the annual dog birth rate is also a critical factor in affecting the rabies virus spread.

Rabies is a viral zoonotic infectious disease in warm-blooded animals that has huge public significance [26].The etiological agent is a virus belonging to the genus Lyssavirus.Canine rabies is the form carried by domestic dogs that is overwhelmingly responsible for approximately 59,000 human deaths per year, mostly in Asia and Africa [22].Among the total death due to the rabies in the world more than 95% of which occur in Asia and Africa [20].Among the total human death 45% prevails in south Asian Association for Regional Cooperation (SAARC) countries [24].Despite being a preventable disease, impact of rabies is increasing day by day, which is a worrisome issue in developing and developed countries [16].The main reservoir hosts for rabies are domestic dogs in low-and middle-income countries, but wild animals including foxes, skunks and raccoon dogs also maintain rabies in some parts of the world [28].All species of mammals are susceptible to rabies virus infection, but dogs remain the main carrier of rabies and are responsible for most of the human rabies death world wise [6].
Rabies is an invariably fatal, highly pathogenic zoonotic, yet non notifiable viral disease in Nepal [27,33].It is one of the major zoonotic threats in Nepal.Each year, several outbreaks and deaths associated with rabies are reported in the animal sector [25].Likewise, animal bites, more importantly, dog bites are reported in thousands of human individuals leading to the consumption of a large number of pre-and post-prophylactic measures.Rabies has been diagnosed in Nepal for a long time, but the information on its epidemiology, impact, and control remains scattered [32].Official reports show that each year 100-300 livestock and 10 to 100 humans die of rabies in Nepal, but these numbers very likely underestimate the actual rabies burden [14,27].
Mathematical models offer a relatively inexpensive way to predict and understand future dynamics of the disease and help to predict whether it becomes an epidemic or not [2,3].Leung and Davis studied mathematically the rabies vaccination target for stray dog populations.They presented a method to estimate vaccination target for stray dogs when the dog population is made up of stray, free-roaming and confined dogs [22].Islam et al. derived a high association between the basic reproduction number with environmental carrying capacity and vice versa.They compared different types of control strategies implemented in Dhaka, Bangladesh [18].Shigui Ruan has constructed a SEIR type model for the spread of rabies virus among dogs and from dogs to humans and use rabies data in China from 1996 to 2010 for the estimations of parameter values [29].Zhang et al. proposed a deterministic model to study the transmission dynamics of rabies in China and explored effective control and prevention measure [35].Bornaa et al. proposed mathematical model to study the dynamics of the transmission of rabies, incorporating predation of dogs by humans [5].Huang (2019) studied transmission dynamics of rabies for dog, Chinese ferret badger and human interactions in Zhejiang province and found that transmission between dogs and Chinese ferret badger, the quantity of dogs, and the vaccination rate of dogs play important roles in the transmission of rabies [17] .Abdulmajid and Hassan formulated and analyzed a delay differential equations model for assessing the effects of controls and time delay as incubation period on the transmission dynamics of rabies in human and dog population [1].Hailemichael et al. (2022) have developed a mathematical model by dividing the dog population into two categories : stray dogs and domestic dogs and studied the effect of vaccination and culling on the stray dogs to domestic dogs [15].Our model focuses the transmission of rabies within dogs where the interventions such as vaccination and sterilizations of dogs are not fully on the action [15].Kunwar and Verma (2022) proposed an SEIV deterministic mathematical model to describe the rabies transmission dynamics within household and stray dogs in the context of Nepal.Their simulation results concluded that the transmission of rabies virus will be controlled effectively when both vaccination and sterilization of dogs are implememted among household and stray dogs [21].
In this study, we construct and analyze a deterministic model to examine the transmission dynamics of dog-to-human and dog-to-dog rabies infection in Nepal .

Model formulation
For the mathematical modelling, we divide the population of both animals and humans into four subclasses; susceptible, exposed, infected and vaccinated.The population of dogs in these subclasses at any time t are denoted by S d (t), E d (t), I d (t), and V d (t) respectively.Similarly, the population of human in these subclasses are denoted by S(t), E(t), I(t), and V (t).
When a susceptible human individual is bitten by an infectious animal or contacted with the saliva having rabies virus, this human individual becomes exposed class.The available data indicates that the incubation period ranges from 5 days to 3 years, with a median of 41 days and a mean of 70 days.About 15-20% of those bitten by infected progress to illness and becomes infectious since more and more bitten people are seeking post-exposure prophylaxis (PEP), the recovered rate of infected humans has been increasing in Nepal.
In this model, it is assumed that dogs can transmit the rabies virus to themselves and to each other, also the infected dogs can spread the rabies virus to human via contact, and human do not spread the virus further.The flow diagram of proposed model for dynamics of rabies is illustrated in Figure 1.The following set of eight differential equations captures the dynamics of disease in the proposed model: For human population, we have with initial conditions: Transmission rate of rabies by the interaction between infectious dogs and susceptible dogs.µ d , µ Rate of death due to rabies for dog and human population β dh transmission rate of rabies by the interaction between infectious dogs and susceptible human.

Model analysis
We shall establish here the various properties of the model solution.First, we show the positivity and boundedness of the solution.Next, we show the existence and stability of equilibrium points of the model system (1.1) and (1.2).
2.1.Positivity and boundedness.We assume that the initial populations are chosen such that all the population components remain positive for all time i.e. t ≥ 0. To ensure the positivity of the solution, we have the following theorem.
+ of the model system is positive for all t ≥ 0 in R 8 + .Further, all the solutions of the proposed model with non-negative initial conditions are bounded for all time.
2.2.Invariant region.Here, we demonstrate that the proposed model is correctly laid out biologically and mathematically in the invariant set and establish that the closed region is a positively invariant and the result is stated in the following theorem.
Theorem 2.2.The solution set of the system (1.1) and (1.2) is a feasible region defined as Ω = Ω d × Ω h , where and moreover, the solution set Ω is positively invariant.
In the absence of rabies, we have E d = I d = E = I = V = 0.For the dog population, V d cannot be zero in the case of disease-free equilibrium because susceptible dogs which are vaccinated transfer to vaccinated class.
For the disease-free equiibrium point, we have Solving this equations, we obtain the DFE point , where

Basic reproduction number.
The basic reproduction number R 0 measures the average number of new rabies infections produced by one rabies infected dog in a completely susceptible (dog and human) population.We have followed the method of Diekmann et al. and Driessche and Watmough to derive the expression for basic reproduction number R 0 [10,11,13].In this model, the compartments E d , I d , E and I are the infected compartments.We use the following notation for derivation of the R 0 : The Jacobian matrix for the system For the next-generation matrix method, the Jacobian matrix at the disease free equilibrium point is put into two submatries F and W , where F is the new nonnegative infectious matrix and W matrix consists of death, increase state, and other transmissions.Thus, we have where The next generation matrix is The basic reproduction number (R 0 ) is the spectral radius, denoted by ρ(F W −1 ), defined as the largest eigenvalue of F W −1 ; which is also called the dominant eigenvalue of F W −1 .Therefore, the spectral radius F W −1 of the next generation matrix is R 0 for the proposed model, and is given by . Therefore, From (2.2), we obtain Therefore, From equation (2.4), we obtain Therefore, Again, using the equilibrium condition of endemic equilibrium point for the human population, equating the left hand side of the equations of the system (1.2) to zero, we find the following expressions: From dS dt E * = 0, we obtain From dE dt E * = 0, we obtain From dI dt E * = 0, we obtain γE * − (m + µ)I * = 0 (2.9) From dV dt E * = 0, we obtain Now, from equation (2.9), we obtain γE * − (m + µ)I * = 0; therefore, Again, from equation (2.10), we obtain (m + σ)V * = κE * , which implies Therefore, Next, from equation (2.8), we obtain β dh S * I * d −(m+κ+γ)E * = 0, which implies Therefore, Again, we derive an expression to calculate I * d and I * using the values of parameters as follows: From dS d dt = 0, we obtain Therefore, By (2.7), we obtain Therefore, where We use the following notation for convenience: Again, using the notation: , and a 7 = (m + σ) > 0, the system of equations (1.1)-(1.2) reduces to the form To point out the stability condition at DEF point, we first find the Jacobian matrix of the system as follows: Now, the Jacobian matrix at disease-free equilibrium point E 0 is Here, the trace of the matrix Now, the determinant of the Jacobian matrix J(E 0 ) after expanding and simplifying yields From (2.15), we note that det[J(E 0 )] > 0 when R 0 < 1.Consequently, when R 0 < 1, then trace[J(E 0 )] < 0 and det[J(E 0 )] > 0, so that the disease-free equilibrium point is locally asymptotically stable if R 0 < 1, otherwise it is unstable.□ 2.5.Stability analysis of endemic equilibrium (EE) point.
Theorem 2.4.The endemic equilibrium point E * of the epidemic model, if exists, it is locally asymptotically stable.
The remaining four characteristic roots are given by: where Thus, one root of the characteristic equation of the Jacobian matrix J(E * ) is λ = −a 6 and other seven characteristic roots are given by the equation which may also be put in the form i=7 i=0 ϕ i λ i = 0, where   2 .
There is currently no fixed database of rabies in Nepal.Therefore, the data in this paper are obtained from previously published literature and reports.We set the beginning of year 2008 as the initial time of our model dynamics.In Nepal, there are approximately 2 million dogs [9].As we do not have dog vaccination data available for the year of 2008, we assume that about 20% (4 × 10 5 ) of the dog population were vaccinated in 2008.In addition for our base case, we assume that about 0.5% dogs were exposed to rabies and 0.01% dogs were infected by rabies.
Number of vials of anti-rabies vaccine consumed in 2008 by human in Nepal is recorded 145,978.Considering the 7 dose per person including wastage and damages, the exposed number of human population is 145978/7 = 20, 854.So, it is reasonable to take the number of exposed human population as 20,000 [31].The human population of Nepal in 2008 is about 26,882,000 (Country Economy, 2018) [8].Based on the data, we assume that about 20,000 human population were exposed, 1000 human were infected and about 15,000 vaccinated.2, we have used model (1.1)-(1.2) to simulate the data and we predicted the trend of exposed to rabies human population in Nepal.For model fitting, we have used exposed rabies cases data of Nepal from 2008 to 2015 for the simulation.However, the availability of observed data of exposed to rabies of human cases is inadequate, the predicted data obtained from the simulation of our model seems to match to the observed data with reasonable parameter values from Figure 3 demonstrates the trend of epidemic (exposed and infected dog and human cases) with the current control and prevention measures for next 30 years.The nature of graph indicates that the human rabies infection will level off in the next couple of years and then with increase to next 8 years under the current measures.Using the simulated parameter values in Table 2, we compute the value of basic reproduction number as R 0 = 1.324 in Nepal.Thus, with the current control and prevention measures, dog and human rabies will persist endemically, which is also justified in Figure 3.

3.2.
Sensitivity analysis for R 0 .We study the impact of model parameter values on the output estimation of the basic reproduction number using the sensitivity analysis.Here, the normalized forward sensitivity index of a variable to a parameter is applied [30].The normalized forward sensitivity index of a variable to a parameter is defined as the ratio of relative change in the variable to relative change in the parameter.
The normalized sensitivity indices of R 0 related to the parameters for the rabies model, evaluated at the baseline parameter values are tabulated in Table 3.The parameters are ordered from most sensitive to least.The most sensitive parameters are the transmission rate β d and dog recruitment rate A d of the susceptible in Nepal.These are followed by the natural death rate µ d .The least sensitive parameter is the anti-rabies vaccine inefficiency σ d .In general, from the Table 3, it can be seen, parameters that have positive sensitivity indices, namely β d , A d , σ d has a positive impact on R 0 in the condition that other parameters remain constant.That is, increase in the values of β d and A d can increase in the R 0 value in the same direction or cause an outbreak.In contrast, the increase of parameters whose sensitivity indices is negative γ d , m d , µ d , κ d have negative impact on R 0 , minimizing the effect of spread of disease.Therefore, control strategies must focus on a decrease in the parameters β d and A d .Dog sterilization.Sterilization refers to the surgical removal of a dog's reproductive organs (spaying for females and neutering for males), does not have a direct impact on rabies transmission in the human population but decreases the recruitment rate effectively to reduces transmission of rabies indirectly.The primary method for preventing rabies in humans is through vaccination, not sterilization.By reducing the population of stray and unowned dogs by sterilization, it can reduce the overall risk of rabies transmission because stray dogs are more likely to be unvaccinated and have limited access to medical care.Figure 7 demonstrates the population dynamics of exposed and infected dog and human population for the different levels of sterilization of dog population.The graph indicates that steriization is effective measure to some extent.
Dog vaccination.The model is simulated taking the different vaccination rates for dogs as variable (Figure 8).Trends of the graph shows that increased dog vaccination rate have an important effect in regard to the rate of rabies exposed and infected dogs and human population.Furthermore, increasing the vaccination rate (κ d ) of dogs within the model decreased the number of exposed and infected dogs and human.Hence, coverage of dog culling significantly decreases the rabies dog population as well as human rabies cases.While culling has been practiced as a method of control rabies in Nepal, it is often met with opposition from animal welfare and conservation groups.Additionally, controversy arises due to ethical concerns and the potential ecological impacts of culling.Reducing the population subsequent number of dog population (either stray or pet) can disrupt ecosystems and may not be a humane method of rabies control.So, we perfer to focus on other measure as well simultaneously such as public education and other non-lethal methods to prevent the spread of rabies virus.

Conclusion
We employed a mathematical compartmental deterministic approach to investigate the dynamics of rabies transmission from dogs to humans and from dogs to dogs.The model was created to illustrate how rabies spreads from dogs to human beings.In this investigation, only susceptible animals and exposed human populations were considered to have undergone immunizations.Because of the challenges involved in identifying affected dogs, it is not currently feasible to immunize them.
We examine the fundamental properties of the epidemic model in terms of boundedness and positivity of the solution of the model, and we conclude that the solution of the state variables is positive at all times for any positive values of the initial condition.Analytically, we determine expressions for both the disease-free and endemic-equilibrium points, and we analyze their stability.The numerical simulation using acceptable parameter values confirmed the results.We demonstrate that the endemic-equilibrium and disease-free equilibrium points are both locally asymptotically stable.
Our research demonstrates that vaccination, along with the culling of infected dogs, sterilization of dogs to reduce the number of puppies born each year, are the most effective methods for reducing the spread of rabies among dogs and humans.Additionally, the dogs may receive vaccination together when they are captured  for sterilization, delivering a dual-pronged approach for rabies management.Based on the results of our research study, we expect that the government may create a program to eradicate rabies.We strongly advise estimating the accurate dog population in order to determine the annual dog birth rate and sterilization levels.The development of a sustainable vaccination, culling, and sterilization plan for the dog population is dependent upon accurate data.
Future modelling exercise may consider the potential role of habitat heterogeneity and its effect on the territorial size and configuration, group size and composition, and the potential indirect effect on transmission dynamics.Our model is limited to the modelling of rabies dynamics regarding vaccination of dog population alone, and not both dog and cattle population.The interventions that proved to reduce rabies prevalence in dog population is not necessarily reduce transmission from dogs to cattle.

Figure 1 .
Figure 1.Compartmental diagram of the rabies dynamics model of dog-human population

d 2 . 4 .Theorem 2 . 3 .
Stability analysis of disease-free equilibrium (DFE) point.The disease-free equilibrium point E 0 of the model represented by (1.1) and (1.2) is locally asymptotically stable if R 0 < 1, otherwise unstable.Proof.For analyzing local stability of the disease-free equilibrium point, the Jacobian matrix of the model system (1.1)-(1.2) at E 0 is evaluated first.Then, stability is ensured based on the trace's indication and the determinant of the Jacobian matrix at the disease-free equilibrium point.For this, we first derive the Jacobian matrix (J) of the system (1.1)-(1.2) by differentiating each of the equations of the system in terms of the state variables S d , E d , I d , V d , S, E, I, V .We denote the right-hand sides of the equations of the system as D S , D E , D I , D V , H S , H E , H I , H V respectively.

3 .
1, 2, . . ., 7, by utilizing the Descartes' rule of sign of roots of polynomial equation, all the roots of the equation (2.16) have negative real parts.Hence, the endemic equilibrium point E * is locally asymptotically stable.□ Numerical simulation, results and discussion

Figure 2 .
Figure 2. Comparison between the reported human rabies exposed cases in Nepal from 2008 to 2015 and the simulation from the model.The value of parameters are given in the table 2.

Figure 3 .
Figure 3. Simulation of the model illustrating the endemic equilibrium for baseline parameter values in table 2 for which R 0 > 1.

5 Figure 4 .
Figure 4. Bar diagram of sensitivity indices taking parameter values in table 2 for R 0 > 1.

Figure 5 .
Figure 5. Influence of parameters on R 0 (a) versus β d ; (b) versus dog recruitment rate A d ; (c) versus β d and σ d ; (d) versus β d and A d .

Figure 6 .
Figure 6.Influence of parameters on R 0 (a) versus m d ; (b) versus µ d ; (c) versus m d and µ d ; (d) versus A d and µ d .

Figure 7 .
Figure 7. Simulation of the effect of dog sterilization on the exposed and infected dog and human population

Figure 8 .
Figure 8. Simulation of effect of vaccination on the exposed and infected dog and human population.

Table 1 .
Description Risk factor of clinical outcome of exposed dogs and human respectively m d , m Natural death rate of dog and human respectively.κ d , κ Vaccination rate to dog and human respectively.β d At the endemic equilibrium point, infection is always present in the system.The endemic equilibrium point is determined by solving the system of equations obtained by equating the right-hand side of the equations in (1.1) and (1.2) to zero.If R 0 is greater than unity, then the system has an endemic infection because of the introduction of those with secondary infection.Let E * (S * 1) 2.3.3.Existence of endemic equilibrium (EE) point.* , E * , I * , V * ) be the d dt E * = 0, we obtain

Table 2 .
Parameters used in the simulation of model for dynamics of rabies

Table 3 .
Sensitivity indices of R 0