AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION WITH PRE-EXPOSURE VACCINATION AND POST-EXPOSURE TREATMENT OF DOMESTIC AND STRAY DOGS

: In this paper, we formulate and analyze a mathematical model to study the effect of pre-exposure vaccination and post-exposure treatment on the spread of rabies in domestic and stray dogs. The effective reproduction number ( ℛ e ) was calculated using the next-generation matrix approach. Using the Castillo-Chavez method, the disease-free equilibrium (DFE) point is proven to be locally asymptotically stable if ℛ e < 1 . Using the quadratic Lyapunov function, the endemic equilibrium (EE) point is determined to be globally asymptotically stable if ℛ e > 1. In addition, sensitivity analysis of model parameters on ℛ e was carried out using the normalized forward sensitivity index method. Optimal control analysis using Pontryagin's minimal principle was carried out to minimize the number of exposed and infected individuals as well as the control costs of vaccinating susceptible individuals and treating exposed individuals. Numerical simulations were carried out to verify the analytical results using MATLAB software. The results of the sensitivity analysis show that the transmission rate in stray dogs and the vaccination rate of stray dogs are the most sensitive parameters and are key factors in reducing the prevalence of rabies. The implementation of a combination of two optimal controls (pre-exposure vaccination and post-exposure treatment)


INTRODUCTION
Rabies is a zoonotic viral disease that causes progressive and fatal inflammation of the brain and spinal cord.Clinically, it has two forms, namely malignant rabies (characterized by hyperactivity and hallucinations) and paralytic rabies (characterized by paralysis and coma).Although fatal when clinical signs appear, rabies is completely avoidable.To prevent death from rabies, vaccines, medicines and technology have been used.Despite this, rabies still kills tens of thousands of people every year.Of these cases, approximately 99% are from infected dog bites.Rabies is estimated to cause 59,000 human deaths annually in more than 150 countries, with 95% of cases occurring in Africa and Asia.Due to unreported and uncertain estimates, this figure is likely an underestimate.
The burden of the disease is largely borne by poor rural communities, with about half of cases caused by children under 15 years of age [1].Rabies is estimated to cause 59,000 human deaths annually and WHO recommends intradermal administration of rabies vaccines, as this reduces the number of vaccines required and costs by 60-80% without compromising safety or efficacy.To control tropical diseases, rabies is included in the WHO Roadmap 2021-2030, which sets regional and progressive targets for targeted disease eradication.As a zoonotic disease, rabies requires close cross-sectoral coordination at the national, regional and global levels [2].
The transmission of rabies to humans can occur either in stray animals such as stray cats, wolves, coyotes, foxes, raccoons, skunks, bats, and rodents, or domestic animals like dogs and cats, depending on the environmental context.Therefore, it is crucial to prevent its spread among both humans and animals.There are very effective vaccines available to immunize individuals either before or after exposure to rabies, such as Post-exposure prophylaxis (PEP).Pre-exposure prophylaxis (PrEP), is advised for individuals in high-risk occupations (e.g., laboratory workers AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION handling live rabies virus and viruses associated with rabies) and for those whose work or personal activities may put them close to bats or other potential mammals [2].Today, vaccination has been carried out throughout the world to prevent the spread of the infectious disease rabies.In the field of public health, especially in the prevention and treatment of infectious diseases, significant progress occurred in the 20th century.To achieve these results, vaccination is necessary [3].However, early studies only assumed that mandatory and/or voluntary vaccination should be carried out due to a lack of vaccination and knowledge.To date, network vaccination programs have proven more successful when random vaccinations are administered in pairs, combining targeted vaccination with regular immunization [4].
Until now, the mechanism of rabies spread is still being studied for prevention and mitigation purposes.One approach to understanding the dynamics of the spread of infectious diseases is through mathematical modelling.Many epidemic models are based on the classic SEIR (Susceptible-Exposed-Infectious-Recovered) model.Several rabies epidemic models based on the classical SEIR compartment model used to simulate rabies disease dynamics can be found in [5][6][7][8][9][10] and the references therein.A deterministic model was created to examine the dynamics of dogto-human and dog-to-dog rabies transmission in China [5].This model is a SEIR model, which looks at four groups in dog and human populations.The results of this study indicate that an efficient way to reduce human rabies in China is to reduce the fertility rate of dogs and increase dog vaccination coverage.Ruan et al. [10] built a SEIR basic type model for the spread of rabies virus between dogs and from dogs to humans and used the model to simulate human rabies data in China from 1996 to 2010.Subsequently, this basic model was modified to include pet dogs and stray dogs and applied the model to simulate human rabies data from Guangdong Province, China.
Asamoah et al. [11] investigated the best strategy to stop the spread of rabies from dogs to humans, namely by using pre-exposure prophylaxis (vaccine) and post-exposure prophylaxis (treatment) as a result of public awareness.Furthermore, Taib et al. [12] proposed a deterministic compartmental model with the SEIRS framework to fit actual data regarding the number of rabies cases infected in humans in Sarawak from June 2017 to January 2019.The study results show that controlling dog births can prevent the spread of rabies in the state and increasing dog vaccination coverage and reducing the number of newborn dogs would be a more effective strategy to deal with the current rabies outbreak in Sarawak.Hailemhicael et al. [14] constructed a mathematical model by dividing the dog population into two categories, namely: stray dogs and domestic dogs.On the other hand, the rabies virus tends to spread in both populations.In this model, disease control strategies use vaccination and culling of infected dogs, and the impact is studied.
Optimal control theory is an effective mathematical technique for analyzing a variety of epidemiological models to determine the optimal control strategy to minimize the number of infected individuals [15].More studies on the applications of optimal control to infectious diseases can be found in [16][17][18][19][20][21] and the references therein.Additional research on the use of optimum control for infectious diseases, including rabies, may be found in [22][23][24][25] and the reference therein.This research will expand the model [14] by adding treatment control for infected dogs and the method will be expanded by analyzing optimal control in a model of rabies transmission from a stray dog population to domestic dogs.The research aims to examine the optimal control of vaccination interventions for susceptible individuals (pre-exposure vaccination) and treatment of exposed individuals (post-exposure vaccination) in a model of rabies transmission from stray dogs to domestic dogs.The urgency of this research is because currently, the rate of rabies infection is still high, a significant number of deaths have occurred, and the administration of vaccines and treatment has not been optimal.The problem studied is still a real problem faced by people in the world.The facts on the ground also show that transmission of dogs from stray dogs to domestic dogs, a dog care system that is late in providing vaccinations and treatment after dogs are exposed, has an impact on increasing cases of rabies.Based on this, mathematical modelling of rabies transmission from stray dogs to domestic dogs is needed using optimal control, vaccination for susceptible dogs and treatment for exposed dogs to reduce rabies cases.
Our paper is arranged as follows.In section 2, we formulate the rabies transmission model with pre-exposure vaccination and post-exposure vaccination.In section 3, the positivity and boundedness of the solutions, the equilibrium point, the fundamental reproduction number, and a AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION study of the equilibrium point's global stability are all covered in model analysis.Section 4 presents the sensitivity analysis of the effective reproduction number.In Section 5, the optimal control problem is defined, the existence of an optimal control is demonstrated and characterized, and numerical simulations are shown.Several conclusions are presented in Section 6.

MODEL FORMULATION
A SEIR model with pre-exposure vaccination and post-exposure treatment are formulated to study and analyze the dynamics of rabies infection.The total population of domestic dogs is denoted by   which consists of susceptible domestic dogs (  ), exposed domestic dogs (  ), infected domestic dogs (  ), and partially immune domestic dogs (  ).Meanwhile, the total stray dog population is denoted by   which consists of susceptible stray dogs (  ), exposed stray dogs (  ), infected stray dogs (  ), and partially immune stray dogs (  ).The graphical representation of the proposed model is shown in Figure 1.The detailed descriptions of all the parameters of the model are given in Table 1.Several assumptions used in constructing this model are as follows.
1. Susceptible individuals (  and   ) are vaccinated to become recovered subpopulations (  and   ) and can return to susceptible subpopulations if vaccination rates fall.2. The spread of the disease is assumed to mean that stray dogs can transmit rabies to domestic dogs but not vice versa.
3. The exposed individuals (  and   ) received treatment.
4. Infected individuals with reported symptoms will be hospitalized.
5. Natural death occurs in every subpopulation.6. Deaths due to rabies occur in subpopulations infected with   and   .In this subpopulation there is also culling of infected dogs.

Positivity of the solutions
Model system (1) describes the human population, it is very important to prove that all the solution of the system (1) is positive.We stated and proved the following lemma.
Based on Lemma 2 in [27], the invariant region of the model ( 1) is ℝ +0 8 .As a result, the solution of the model ( 1) with initial conditions nonnegative is nonnegative.The proof of Lemma 1 is complete ∎
In the same way, by integrating the second equation of (4) using the integral factor method and applying initial condition, we get for all  ≥ 0. Thus, } are the feasible solutions of the stray dogs.Therefore, the region Ω = Ω 1 × Ω 2 is positively invariant and the model ( 1) is well-posed or biologically and epidemiologically.The proof of Lemma 2 is complete.∎ Next, for convenience in the discussion, we make the following substitutions =   +   +   ,   =   +   ,   =   +   ,   =   +   , and   =   +   .

The effective reproduction number
Using the next-generation matrix method described by [28], we can calculate the effective reproduction number of the model (1).By using the notation as in [28], ℱ  is the rate at which new infections appear in compartment  and   is the rate of transfer of individuals into and out of compartment .Let  = (  ,   ,   ,   )  .The right-hand side of the model ( 1) is written as ̇= ℱ  ()−  (), where Evaluating the Jacobian matrix of ℱ  () and   () at the disease-free equilibrium point  0 , we get, respectively, The next-generation matrix is .
Hence, the effective reproduction number of the model ( 1) is the spectral radius of matrix  −1 , that is, The effective reproduction number, ℛ  , shows the average number of new infections that are caused by a single rabies-infected individual in a population during its infectious period with preexposure vaccination and post-exposure treatment of domestic and stray dogs used to control strategies.

Endemic equilibrium point
When rabies is present in a population, model (1)  .
The effective reproduction number, ℛ  , shows the average number of new infections that are caused by a single rabies-infected individual in a population during its infectious period with pre-

Global stability of DFE
The method of Castillo-Chavez et al. [29] is used to examine the global stability of DFE.Next, the model system (1) can be expressed as follows: Let  stands for the number of the uninfected compartment,  stand for the number of uninfected compartments, and  0 = ( 0 , 0) stands for the disease-free equilibrium point.Then, the model ( 1) can be expressed as follows: To ensure the global asymptotic stability of DFE, the following conditions (H1) and (H2) must be satisfied.
(H1) For Consequently, the following theorem is true if the system satisfies the above conditions (1) and (2).
Next, from the second equation of system ( 9 It is clear that, (, 0) = 0.Then, we get .
Here,  is a Metzler-matrix since all off-diagonal entries of the matrix  are non-negative.
The proof of Theorem 1 is complete.∎ AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION It can be observed from Theorem 1 that the globally asymptotically stable disease-free equilibrium point  0 is if ℛ e < 1.Thus, the infected individuals eventually vanish and the disease dies out.

Global stability of EE
Using the Lyapunov function, the global asymptotic stability of the endemic equilibrium is explored.We will create a Lyapunov function by referring to [30,31].
Substitute ( 11) and ( 12  (3) The right-hand side of the state systems ( 17) is bounded by a linear function in both the state and control variables.
(4) The integrand  in the objective functional ( 18) is convex to control.
(5) There exist constants  1 ≥ 0,  2 ≥ 0, and  3 > 1 that make the integrand  in the objective functional (18) bounded by Proof.We create the proof in the following steps: (1) Using the fact that all model states (  ,   ,   ,   ,   ,   ,   ,   ) ∈ Ω are bounded below and above, any solutions to the state equations are also bounded.Because the state solutions are bounded, the Lipschitz property of the state system with respect to the state variables is satisfied.Hence, condition (1) is met.
( (3) From the system of differential equation ( 13 .Therefore, all solutions of the model ( 13) are bounded.
From the state equation system (13), the state equations are linearly dependent on the controls   ,   ,   , and   .Hence, the right-hand sides of the state systems ( 13) can be AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION written as a linear function of   ,   ,   , and   with coefficients depending on time and state [33].Thus, condition (3) holds.
The proof of Theorem 3 is complete.∎

Characterization of the Optimal Controls
To obtain the necessary conditions for optimal control, we use Pontryagin's maximum [34] principle to the Hamiltonian function  is defined for all  ∈ [0,   ] by The adjoint system ( 13) is generated by partially differentiating the Hamiltonian function (20) to the corresponding state variables   ,   ,   ,   ,   ,   ,   , and   as For  ∈ [0,   ], the optimal control   * ,  }} . ( The proof of Theorem 4 is complete.∎

Sensitivity analysis of the effective reproduction number
The sensitivity analysis discusses how the model parameters affect the effective reproduction number ℛ e as well as the transmission of the disease.The purpose of the sensitivity index is to quantify the initial disease's spread as well as the relative change in ℛ e when one parameter changes while the others stay the same.The applications of a sensitivity index on parameters that have a high influence can help target interventions to control the spread of disease.
We perform the analysis by applying the method of [35] to determine the sensitivity index of the model parameters.The normalized forward sensitivity index of the variable ℛ e , that depends on the differentiability of a parameter , is defined as, where Υ  ℛ e represent the sensitivity index and  is the parameter in the effective reproduction number.
Using parameter values in Table 1, we have the sensitivity indices o ℛ e (Table 2).The sensitivity index is from the most sensitive to the least sensitive.AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION The effective reproduction number of the model (1), ℛ e , is determined by the nine parameters and the sensitivity indices of ℛ e are presented in Table 2 (arrange from the most sensitive to the least) and graph of sensitivity indices of ℛ e with respect to the model parameters can be seen in   It can see that   ,   , and   have a high impact on ℛ e .On the other hand, parameters that have a high sensitivity index but cannot be controlled are   and Λ  .The parameter has negative impacts and positive impacts on ℛ e .The positive sign of sensitivity indices of ℛ e to the model parameters indicates that a decrease (or increase) in the value of each of the parameters in this case, leads to a decrease (or increase) in ℛ e .On the contrary, the negative sign of the sensitivity indices of ℛ e to the model parameters indicates that an increase (or decrease) in the value of each of the parameters, in this case, leads to a decrease (or increase) in ℛ e .To illustrate, the index of treatment rate for exposed stray dogs (  ) is Υ   ℛ e = +0.1242.This implies that an increase (or decrease) by 10% in   while other parameters remain constant, will be followed by an increase (or decrease) in the effective reproduction number ( ℛ e ) by 12.42 %.On the other hand, the index of vaccination rate for exposed stray dogs (  ) has a negative sensitivity index (-0.8650).This implies that the effective reproduction number ( ℛ e ) will immediately decrease (or increase) by 86.5% upon a 10% increase (or decrease) in   while all other parameters stay constant.Consequently, the indices for the remaining parameters are shown in Table 2.
First, we choose   = 0.0017.The numerical simulation of the model (1) shows that the disease-free equilibrium (DFE) point is globally stable for some other parameter values in Table 1.The corresponding effective reproduction number is equal to ℛ e = 0.301326.Figure 3 illustrates the global stability of the disease-free equilibrium point proved in Theorem 1. Figure 3(a) shows the dynamics of the population of susceptible domestic (  ), susceptible stray (  ), and infected domestic (  ).Second, the numerical simulation of the model (1) shows that the endemic equilibrium point is globally stable for   = 0.0087 and some other parameter values in Table 1.The corresponding effective reproduction number is equal to ℛ e = 1.54289 > 1.This implies that the rabies infection will persist in the population.Theorem 2 is numerically illustrated in Figure 4.The numerical result illustrated in Figure 3 confirms that model (1) has only one unique positive endemic equilibrium point when ℛ e > 1.This implies that the rabies infection will persist in the population.Theorem 2 is numerically illustrated in Figure 3.

Numerical simulations of the optimal control
In this section, we discuss the numerical results of the system (17) to investigate the effect of the following itemized optimal control strategies on the spread of the disease in a population.With the help of the software Matlab.This section focuses on demonstrating some numerical results of qualitative analysis and optimal control problem ( 17)-( 19) through the forward-backwards Sweep method [10].Using a forward fourth-order Runge-Kutta scheme and the conditions ( 20) and ( 21 We examine and compare two different combinations of control intervention strategies for both domestic and stray dogs.The simulations of the optimal control are divided into three strategies: implementation of post-exposure treatment (  ,  ), implementation of pre-exposure vaccination (  ,  ), and implementation of the combination of pre-exposure vaccination (  ,   ) and post-exposure treatment (  ,   ).
• Strategy 1 (implementation of post-exposure treatment) The combination of control of post-exposure treatment for domestic dogs (  ) and stray dogs (  ) is used to optimize the objective function , whereas we set both domestic and stray dogs' pre-exposure vaccinations to zero (  =   = 0).For both domestic and stray dogs, there is a significant reduction in the number of exposed individuals (  ,  ) and infected individuals (  ,   ) when compared to cases without controls (Figure 5(a-d)).The control profile is shown in Fig. • Strategy 2 (implementation of pre-exposure vaccination) The combination of control of pre-exposure vaccination for domestic dogs (  ) and stray dogs (  ) is used to optimize the objective function , whereas we set both domestic and stray dogs' post-exposure treatment to zero (  =   = 0) .For both domestic and stray dogs, there is a significant reduction in the number of infected individuals (  ,  ) when compared to cases without controls (Fig. 6(c-d).The control profile is shown in Fig. 6(e), and control   is at the upper bound for about 4.7 years and the control of   at the beginning of the period is around 0. The calculations above, it shows that by using parameter values as in Table 1 and using the objective function as in equation ( 18) with weights at the end of the control period (  = 8), the results obtained are that Strategy 3 (combination of pre-exposure vaccination and post-exposure treatment on domestic dogs and stray dogs) is a strategy with minimum objective function value over the 8 year intervention period.

CONCLUSION
In this study, rabies transmission in pet and stray dog populations with pre-exposure vaccination and post-exposure treatment was studied using a nonlinear mathematical model.The model has a disease-free equilibrium point and an endemic equilibrium point.The global dynamics of the model are determined by the effective reproduction number, which is obtained from the next generation matrix method.Using the next-generation matrix approach, the effective reproduction number ℛ e can be determined.It has been demonstrated that, assuming ℛ e < 1, the disease-free equilibrium point is globally asymptotically stable using the Castillo-Chavez method.However, if ℛ e > 1, the endemic equilibrium will be globally asymptotically stable, which is proven using the nonlinear Lyapunov function.
Sensitivity analysis and numerical simulations were carried out on model parameters that influence the spread of rabies in domestic and wild dog populations.The results of the sensitivity analysis shows that the transmission rate in wild dogs (  ) and the death rate for wild dogs due to a disease   are the most sensitive (positive) parameters.This means that it plays an important role in influencing the spread of rabies in a population.On the other hand, the vaccination rate of stray dogs (  ) is the most sensitive (negative) parameter, which means that the vaccine is a key factor in reducing the prevalence of rabies.Next, the model was developed by considering preexposure vaccination and post-exposure treatment in pet dogs and stray dogs as control variables.
In addition, the existence of optimal control has been proven and Pontryagin's minimum principle is used to determine the analytical characterization of optimal control.Numerically, optimal control analysis shows that the application of a combination of two optimal controls (pre-exposure

Figure 1 :
Figure 1: Flow diagram for rabies transmission among domestic and stray dog subgroups.

Figure 2 .
Figure 2. Graph of sensitivity indices of ℛ e with respect to the model parameters.

Figure 3 (Figure 3 .
Figure 3. Simulation of the model (1) showing the global asymptotic stable of the  0 .(a) the dynamics of   ,   , and   (b) the dynamics of   ,   , and   .
), we start with an initial guess for the controls over the time interval [0,   ] and solve the state system(1).Using the new state values, the adjoint system (21) is solved by a backward fourthorder Runge-Kutta scheme.The controls are updated using a convex combination of the previous control values and the new control values from(28).The iterative method is repeated until convergence.Furthermore, in describing the control strategy the parameter values are used in[8,   AN OPTIMAL CONTROL MODEL FOR RABIES TRANSMISSION 9] and weights at the end of the period (  = 8),  1 =  2 =  3 =  4 = 1,  1 = 20,  2 = 10,  3 = 40,  4 = 20.

5Figure 5 .
Figure 5.The simulation results of the model (1) show the effect of post-exposure treatment for domestic dogs (  ) and stray dogs (  ) on the spread of rabies.

Figure 6 .Figure 7 .
Figure 6.The simulation results of the model show the effect of post-exposure vaccination for domestic dogs (  ) and stray dogs (  ) on the spread of rabies.

Table 1 :
The description and numerical values for the model parameters.

TABLE 2 .
The sensitivity indices of ℛ e