Global Sensitivity Analysis of a Rabies Epidemic Model involving Dog Vaccination and Dog Population Management

Rabies is a zoonotic disease which is spread by animals mostly carnivores. Rabies regards as tropic disease. In this article, we construct a mathematical model for rabies involving dog vaccination and dog population management, namely sterilization and monitoring of dog movements. The model has rabies-free equilibrium point and endemic equilibrium point. We determine the effective reproduction ratio using next generation matrix. Our dynamical analysis shows that rabies-free equilibrium point is conditionally stable. A global sensitivity analysis is performed to investigate which intervention is the most crucial among the two interventions considered in the model. We use Latin hypercube sampling method to generate parameter space. To investigate the parameter sensitivity, we calculate the partial rank correlation coefficient. We provide numerical experimental results related to stability and global sensitivity analysis. The results show that the effective reproduction ratio is more sensitive to dog population management than vaccination intervention. This suggests that dog population management intervention significantly reduces the effective reproduction ratio compared to vaccination programs. In addition, the number of infectious dogs has a strong correlation with dog culling actions.


Introduction
Rabies is a neglected zoonotic disease tha can infect all mammals, including humans [1] and spread by animals mostly carnivores [2].Rabies (mad dog disease) is a severe infection of the central nervous system by the rabies virus (family Rhabdoviridae, genus Lyssavirrus) [3].Rabies is a zoonotic disease which is.The more frequent means of contagion is through bites of infected mammals (especially dogs, but also bats, skunks, foxes, raccoons and wolves) [4].Dogs, the main rabies reservoir species, usually infect by an unprovoked bite [5].Rabies is spread on almost all continents except the Antarctic continent.More than 150 countries have been impacted by this disease [6].The incidence is higher in Africa and South East Asia, nearly double in men than in women, with a higher peak in childhood [4].Over 90% of human rabies deaths today occur in Asia and Africa [7].Every year more than 55,000 people die from rabies [6].
Rabies is one of the few communicable diseases that can possibly be controlled by currently available tools for veterinary and public health interventions.ASEAN guides several interventions as actions to eliminate rabies, including vaccination, animal movement control, and dog population management [7].Approaches to manage dog populations include culling, fertility control, and sheltering [8].In this article, a mathematical model of the spread of rabies among dogs that takes into account vaccination and dog population management is discussed.
Mathematical models have long been used to study the dynamics of the spread of rabies [9]- [14].In [14], a rabies epidemic model is constructed involving populations of stray dogs, domestic dogs and humans.The author considered the intervention of vaccination and culling of stray dogs.However, the author only focused on dynamical analysis and does not provide a description of research results regarding the impact of vaccination programs and culling of stray dogs on the spread of rabies.In 2022, Astuti [9] developed a rabies epidemic model that only involved the dog population.Vaccination and quarantine interventions were considered in the model.The findings are that vaccination and quarantine can decrease the number of exposed dogs and infected dogs.Nonetheless, the author does not deliver an explanation concerning which intervention is more sufficient in controlling the spread of rabies in the dog population.Meanwhile, a mathematical model related to the spread of rabies in human and dog populations are discussed in [10].The model constructed involves vaccination interventions in dogs and humans.The authors' findings are that combining vaccination interventions and dog population management is a way to control the spread of rabies.Reducing the stray dog population can make it easier to achieve vaccination coverage targets.
Broadly speaking, the articles mentioned above have considered the vaccination and culling of dogs.However, there are still few authors who have studied in more depth what interventions are most effective in controlling the spread of rabies.One method that can be used to identify which interventions are most effective is to carry out sensitivity analysis.Sensitivity analysis is divided into two, namely local and global sensitivity analysis.Global sensitivity analysis is usually performed by calculating the partial rank correlation coefficient (PRCC) from the parameter space generated using the Latin hypercube sampling (LHS) technique as done in [15]- [22].LHS permits an unbiased estimate of the average model output, with the advantage that it needs fewer samples than simple random sampling to achieve the same accuracy [23].In this article, we develop a model of a rabies epidemic in a dog population.We considered dog vaccination and dog population management interventions (sterilization, culling of infectious dogs, and monitoring of dog movements).To examine which interventions were most effective, we conduct a global sensitivity analysis.

Methods
The study is conducted by doing the following steps: ❖ Formulating a mathematical model of rabies spread among dogs incorporating vaccination and dog population management.❖ Determining the equilibrium points of the model and the conditions for its existence.❖ Determining the effective reproduction ratio employing the next-generation matrix [24].❖ Analyzing the stability of equilibrium point.❖ Performing global sensitivity analysis as conducted in [16], [17], [20], [25], [26].

Model Formulation
In this section, we describe the model construction procedure.The assumptions used in building the model are as follows: ▪ The dog population is divided into four subpopulations, namely the susceptible dog ( ) ▪ The recruitment, vaccination, transmission, and natural death rate is constant; ▪ The dog population is not closed.Now we will explain the model construction process based on the above assumptions.The susceptible dogs is decreased due to dog vaccination which results in vaccinated dogs leaving the susceptible dog class and moving into the class of vaccinated dogs with a constant vaccination rate, namely  .In addition, the subpopulation of susceptible dogs is also reducing due to new infections with constant transmission rates ( )  .Afterward, the dog subpopulation was reduced due to natural mortality with constant natural death rate ( )  .In contrast, the subpopulation of susceptible dogs grows with a constant recruitment rate ( )  and the presence of vaccinated dogs leaving the vaccinated class and entering the susceptible class due to loss of immunity at rate of  .The dynamics of the susceptible dogs is represented by the following equation.
( ) where [0,1) u  is a parameter related to the control of the presence of new dogs (sterilization and monitoring dog movement).
The latent dog subpopulation increases due to the presence of newly infected dogs leaving the susceptible class and entering the latent class.The subpopulation of latent dogs decreases due to natural death and there are dogs that have passed the latent period and thus move to the infectious class at rate of  .The dynamics of the latent dogs is represented by the following equation.
The subpopulation of infectious dogs rises due to the presence of dogs that have passed the latent period leaving the latent class and entering the infectious class.In contrast, the infectious dog subpopulation declined due to natural deaths, deaths due to rabies at rate of  , and deaths due to culling intervention at rate of  .The subpopulation of vaccinated dogs increases due to vaccination and conversely decreases due to natural death and loss of immunity at a rate of  .The following equations represent the dynamics of the infectious dog and vaccinated dog subpopulation, respectively.
The flow diagram of the constructed model can be seen in Figure 1. ( )

Equilibrium Points and Effective Reproduction Ratio
After solving system equation (4) when 0 ,0,0, ) Now, we determine the effective reproduction ratio.Based on model (4), we get the following regeneration and transition matrix, respectively ( ) Therefore, the Jacobian matrix of (5) at 0 E are as follows 1 00 Hence, from ( 6) and ( 7), we obtain the following next generation matrix 12 1 0 0 0 0 , 0 0 0 It is easy to see that 0 e RR = .Hence, the equilibrium 1 E exists if 0 1 R  .

Stability Analysis and Numerical Simulation
We now provide the results of stability analysis of the rabies-free equilibrium point.

Global Sensitivity Analysis
In this subsection, we discuss the results of global sensitivity analysis.The baseline values of parameter used for the global sensitivity analysis correspond to those given in Table 1.We obtain the parameter space using the LHS technique.10,000 models are simulated by generating the value of ,,    and u following a uniform distribution.Parameter   0,1 u  while with minimum and maximum values of ,,    are 10% below and above of the baseline values, respectively.The other parameters are set to be fixed.

Global sensitivity on
As revealed in the previous section, namely in Theorem 1, 0 R can be used to determine whether the disease will disappear from the population or not.The smaller the 0 R , the closer the goal of achieving rabies elimination.Hence, we provide the outcomes of a global sensitivity analysis on 0 R .This investigation is carried out to examine the control parameters that have the most impact on 0 R .The parameters investigated are parameters related to the duration of immunity provided by the vaccine ( )  , vaccination rate ( )  , culling rate of infectious dogs ( )  , and parameters associated to sterilization and monitoring dog movement ( ) Based on the results of the global sensitivity analysis in Figure 3(a), three parameters, namely ,, u  and  have negative PRCC values, while  has positive PRCC value.These results indicate that increasing of ,, u  and  values can reduce 0 R , while decreasing  values can increase 0 R .The effect of changing the value of each control parameter on the 0 R is statistically significant because the p-value of each control parameter is less than 0.01 (see Figure 3(b)).It can be seen that u is the control parameter that has the most influence on the 0 R .These results suggest that the critical key to managing the spread of rabies is controlling dog births and dog movements.One intervention that can be done to control dog births is to enforce a mass sterilization program for dogs.The scatter plot results as shown in Figure 4 very clearly show the relationship between the four control parameters and 0 R .However, the clearest pattern is the figure that shows the relationship between u and 0 R while the other images show more spreading pattern.It is obvious that there is a downhill pattern from left to right in the pictures which show scatter plot ,, u  and vs 0 R .The results indicates that there is a negative correlation between ,, u  and with 0 R .On the contrary, an uphill pattern from left to right can be seen in the image which shows scatter plot  vs 0 R .This indicates that there is a positive correlation between  and 0 R .Based on the distribution of points in each scatter plot, it is clear that u has the strongest correlation with 0 R .This result is in line with the results of the PRCC calculation.The results of the PRC calculation and scatter plot tell us that u is the most influential parameter on the 0 R .

Global Sensitivity on i D
We also perform a global sensitivity analysis on i D .We are interested in studying what control parameter had the greatest effect on the number of infectious dogs at.Here, ()

Conclusions
In this article, we construct a mathematical model for rabies involving dog vaccination and dog population management.The model has a conditionally stable rabies-free equilibrium point.A global sensitivity analysis is performed to investigate which intervention is the most crucial among the two interventions considered in the model.Our results show that the effective reproduction ratio is more sensitive to dog population management than vaccination intervention.This indicates that intervention in

▪
Vaccination for susceptible dog only;

Figure 1 .
Figure 1.Compartment diagram of the model

. 1 P
It is clear that all roots of polynomial ( )  have negative real part.

0 E
results of the dynamical analysis, we conduct numerical simulations.The initial values of the simulation are  in Table1.These parameter values give 0 22.404 1 R = .According to Theorem 1, is unstable.This indicates that i D does not go to 0. This result is in accordance with the solution plot of ( ) i Dt in Figure2(a).The second simulation is carried out by choosing a smaller transmission rate, in 0 0.22404 1 R = .Theorem 1 guarantees 0 E is asymptotically stable which suggests i D goes towards 0. This result agrees with the solution plot of ( ) i Dt given in Figure 2(b).

Figure 3 .Figure 4 .
Figure 3. PRCC value of , , , u    and its corresponding p-value number of infectious dogs and the value of control parameters(  )    , , , u    at time t .Figure5shows the i PRCC D − dynamics in the first 50 days.Most of the control parameters had fluctuating i PRCC D − in the first 50 days.Regardless, the i PRCC D − of  is relatively constant.The parameters that do not have a statistically significant correlation with i D (p-value > 0.01) have i PRCC D − in the range -0.0453 to 0.0453 (see gray shaded area).It can be seen that only  whose i PRCC D − is outside the shaded area.This indicates that only  has a statistically significant correlation with i D .Furthermore, the i PRCC D − of  is negative, indicating that the greater the  , the smaller the i D .