Numerical Simulations of Vaccination and Wolbachia on Dengue Transmission Dynamics in the Nonlinear Model

In this study, it is indicated that the world can get rid of the dengue virus by using vaccines and Wolbachia. In many findings, it is observed that Wolbachia therapy is efficacious in those regions that display the minimal to moderate the transmission level. On the contrary, vaccination is highly successful when used in serologically persons and places with large transmission levels. The resilience of stochastic methodology based on the numerical computing schemes will be used to exploit the artificial neural networks (ANNs) modelling legacy, as well as the metaheuristic intelligence using the hybrid of global and local search schemes thru genetic algorithms (GAs) and active-set method (ASA). The combination of both strategies is used to manage the numerical therapies of the mathematical form of the dengue model. The optimal control results through GA-ASA can be retrieved by offering an error-based fitness function generated for dengue model represented via nonlinear systems of equations. The acquired findings are compared to the Adams numerical results to ensure that the suggested stochastic system is accurate. For determining convergence, the training contours are based on various contact rate values. Furthermore, the statistical achievements of the suggested stochastic scheme to solve the novel developed dengue model, which demonstrate the stability and dependability of the dynamical system scheme.

They work principally by boosting immunogenicity to the E protein and non-structural protein 1 (NS1) of the dengue virus (DENV) [7]. Researching the immunogenicity to DENV can aid in the development of an efficient dengue vaccination strategy [8]. However, the prevalence of four immunologically different dengue virus serotypes, all potentially of generating cross-reactive and disease-enhancing antibody responses against with the three remaining sterotypes, has made the production of a dengue vaccine a difficult process [9]. Scientifically, theoretical calculation of mosquito density reduction is vital, and mosquito density decreases by half on the borders in the region of operations. Ross [10] proposed the concept of mosquito density reduction through the construction of mathematical models several decades earlier.
In recent years, the application of mathematical models to the study of infectious disease epidemic spread has benefited the field of public health in general. For dengue virus propagation, mathematical epidemiological investigations of interaction models involving host-vector and human populations have been developed [11]- [14]. Many dengue mathematical models have been constructed to evaluate the efficiency of Vaccine and Wolbachia intervention in lowering dengue transmission. They discovered that Vaccine and Wolbachia could lower the frequency of dengue cases by up to 80%, especially in areas where propagation is low to moderate [15]- [17]. To observe the heterogeneity and dispersing effects of dengue, mathematical models based on the life cycle and diffusion of mosquitos, as well as geographic heterogeneity of mosquitos, have been constructed. Furthermore, diffusion approaches are used when space is treated as a continual component [18]- [20].
Even though various mathematical models have been developed to examine dengue groove in the presence of vaccination and Wolbachia, rare have considered the combination of the two techniques. It is critical to analyze the results of both approaches independently and collectively. The objective of this research is to explore a mathematical model of vaccination and Wolbachia used for dengue transmission and dispersal and its analysis in the form of simulation solutions to assist in understanding dynamic behavior using a stochastic technique. Exploring the prospect of fixing linear/nonlinear systems using the high predictive capabilities of feedforward artificial neural networks (ANNs) optimized with the combined capabilities of local/global search methods is a significant potential of meta-heuristic computing paradigm based on stochastic approach [21]- [23]. Soft computing techniques reported in different diseases such as Convolution neural networks for analysis of plant diseases [24], [25], for COVID-19 disease [26]- [29], artificial neural networks for tuberculosis [30], for forecasting disease [31], for skin diseases [32], chest disease [33], for diagnosis of kidney stone diseases [34], respiratory disease [35], for HIV infection [36], to analyze influenza disease model [37] and for stomach model [38]. In the field of fluid mechanic, ANNs successfully tightened the claws as these techniques have been proved best for nonlinear complicated flow system [39]- [43]. Forecasting and finance require soft computing for rapid marketing [44], [45].
It is investigated that the vaccination and Wolbachia have been used to prevent the dengue spread. Wolbachia possesses properties that differ from the insecticide-based technique, which could have an impact on disease transmission dynamics. We shall demonstrate the effectiveness of these tactics separately or jointly after simulating the model. The goal of the research is to find the numerical solutions via artificial neural network understanding of the strategy's effectiveness, hence a single serotype dengue model will suffice.
Key procedure is as under • The effectiveness is validated using statistical evaluations of the dengue nonlinear system based on differential equations on numerous ANN-GA-ASA trials in terms of mean absolute deviation, semi-interquartile range and 'Theil's inequality coefficient.' • Aside from the accurate results for the Dengue nonlinear differential model with initial conditions, additional valuable features include ease of understanding, faster operation, stability, broad applicability, and robustness. The paper is arranged into 4 sections. Section 2 comprises the formulation of dengue model and its parameters along values. Section 3 shows the ANN-GA-ASA methodology for solving the dengue, as well as the mathematical form of the statistical operators. Section 4 contains the summary of the findings.

II. FORMULATION OF MATHEMATICAL MODEL
A predictable mathematical model is constructed for the dengue treatment that covers Wolbachia and vaccinated presented by Nedii et al. [17]. In this model the attacking agent (mosquito) and effected agent human are distributed into two separate compartments. Attacking agent population are distributed into Aquatic A, Susceptible S, Exposed E and Infectious I , whereas human population are distributed into Susceptible S H , Vaccinated V H , Exposed E H , Infectious I H and Recovered R H .
The purpose of this paper is to get a broad understanding of the potential efficacy of vaccines and Wolbachia. The basic assumptions used to build the model are as follows: • Mosquitoes have no recovery class because they are contagious for the remainder of their lives.
• It is sufficient to utilize a single serotype dengue model. When susceptible individuals are attacked by contaminated non-Wolbachia and Wolbachia-carrying mosquitoes at a rate of λ n and λ W , respectively, they become infected. The human species are injected at a rate of V H . Vaccinated people, on the other hand, are exposed to dengue when the vaccine loses its efficiency at a rates (1 − ε) and they are struck by diseased non-Wolbachia and Wolbachia-carrying mosquitoes at rates λ n and λ W , respectively. We consider diminishing immunity, which occurs at a rate of ϕ h , as well as randomized bulk vaccination. The mathematical constrain for disease model  [17]. is given as: , where We calculate the basic reproduction number, which is the average number of new viruses created by one infected adult in a completely vulnerable group, using the notion of the next generation matrix. In the absence of modifications, the basic reproduction number is,

III. METHODOLOGY
For addressing the nonlinear mathematical dengue model, the developed structure of ANNs utilizing GA-ASA optimum is provided in two steps, as follows: • The fitness function design is described for the ANNs parameters.
• The hybrid combination of GA-ASA provides vital settings for optimizing fitness function.  (2), shown at the bottom of the next page. W signifies the unidentified weights in the above system, given as:

A. STRUCTURE OF ANNS
and VOLUME 10, 2022 In the above system, the activation function log-sigmoid g(τ ) = 1/ 1 + E ∧ (−τ ) is applied given as (3), shown at the bottom of the page.
The GA-ASA techniques are used to optimize an errorbased 'fitness function,' which is given as: The quantites e 1 , e 2 , e 3 . . . and e 13 indicate the fitness functions associated to differential system (1), whereas the corresponding initial conditions is represented in e 14 .

B. OPTIMIZATION PERFORMANCES: GA-ASA
Optimal performance of nonlinear mathematical model of dengue virus with treatment procedure is simulated thru the artificial technique GA-ASA. The genetic algorithm (GA) is a paradigm of abiogenesis introduce by John Holland et al. [46]. He was widely credited as being the first one to bring the concept of crossover, mutation and selections in artificial system which are the necessary parts of genetic algorithm arise as problem-solving agent. Since that day, a variety of genetic algorithm versions have now been devised and utilized in a variety of optimization problems, ranging from discrete systems like salesperson travelling problem to continuous systems arise in designation of airfoils in aerospace, from coloring of graph to pattern recognition, and from monetary markets to multi-objective engineering optimization [47]. Many other applications of GA are in biomedical field such as cancer datasets with multi-dimensional [48], categorization of anomalous computed tomography brain tumor images [49], transcriptomic cancer classifier [50], prediction of liver diseases model [51], in transportation field applicable in vehicle routing model and monorail dynamics model [52], [53], in geophysical side for prediction of air blast [54], cloud model [55], and for groundwater flow model [56], and wind power systems [57], etc.
Using the hybridize with the local search technique, quick convergence is achieved by combining global search with any local search approach. As an initial input, the best GA values are assigned. To standardize the variables, the local search active set algorithm is used. In optimization theory, the active set is highly crucial since it decides which constraints will have an impact on the result of optimization. The active set, for instance, specifies the hyperplanes that cross at the solution point while solving a linear programming issue. Here are few contributions that us ASA such that water supply model to manage the flow [58], optimal control issue based on PDE [59], node-based shape optimization [60], and for electrodynamic problems [61].

This section contains thorough reviews of the acquired findings for the dengue treatment in the presence of Wolbachia and non-Wolbachia (1). Equation (1) comprises the 3 cases of dengue reduction. (i) Vaccination (ii) Wolbachia treatment and (iii) Vaccination and Wolbachia combine. The Genetic
Algorithm and active set algorithm is adopted for 0-1 inputs along 0.1 step size of the dengue mathematical model in equation (1). The findings of the Adams comparison show that the dengue modeling approach is valid. Moreover, statistical findings are provided to ensure that the planned procedure is precise and accurate.
VOLUME 10, 2022 Using the above system, the fitness function may be written as (24), shown at the bottom of the next page. The dengue mathematical system is solved utilizing the hybrid GA-ASA for 100 iterations with 120 parameters by optimizing the above fitness function. The proposed solutions of the dengue mathematical system are represented by the best weight vectors, which are as follows: VOLUME 10, 2022 Optimization of equation (1) is made through hybrid computing technique GA-ASA by setting 100 runs along 5 neurons. The balance weight of dengue model by GA-ASA are performed and then trained weight along 5 neurons of dengue model is set in Figure 2   in range10 −6 to 10 −3 , for A N in range 10 −6 to 10 −3 , for S N in interval 10 −6 to 10 −2 , E N ranging 10 −8 to 10 −4 , for I N in range 10 −7 to 10 −5 , A W in interval 10 −8 to 10 −4 , S W in 10 −7 -10 −5 , E W in the range 10 −7 to 10 −5 , and I W in interval 10 −7 to 10 −5 proven the resemblance of GA-ASA with numerical Adam method. Figure 5 (a-c) -7 (a-c) signifies the concert operators built on E-VAF, TIC and MAD to decipher the dengue mathematical model. It is noted that the finest results of S H , V H , E H , and I H classes with E-VAF, MAD and TIC lie about 10 −10 to 10 −2 , 10 −6 to 10 −04 and 10 −10 -10 −8 , respectively. The good result of R H , A N , S N , and E N lie around 10 −8 − 10 −6 , 10 −6 − 10 −2 and 10 −6 − 10 −4 , for these operators, respectively. While, another appropriate results of I N , A W , S W , E W , and I W lie around 10 −9 to 10 −4 , 10 −06 to 10 −4 and 10 −11 to 10 −9 , respectively. Since these findings, one might conclude that the proposed strategy is accurate. Figures 8 (a-c)

V. CONCLUSION
This study is related to design a new coding scheme for tackling the nonlinear dengue mathematical model concern with treatment process. The treatment process is studied for three different situations such as vaccination, Wolbachia therapy, and combine both therapies. Artificial neural networks are used in conjunction with the features of the Genetic algorithm's global and Active-set algorithm as local search methodologies. The dengue nonlinear model is properly tested by manipulating the GA-ASA with neural networks layer structure for 5 number of neurons. The suggested model is tested through overlapping results with Adams numerical technique in the appropriate precision level while simulate the dengue model. Statistical tools in the form of ''Mean'', ''Median'', ''Semi inter quartile range'', ''coefficient of Theil's inequality'', ''Mean absolute deviation'', and ''Variance account for'' reveal the exactness and worth of adopted algorithm. ''Mean'', and ''Semi inter quartile range'' operators are used for global significance of nonlinear mathematical dengue model. The MAD and TIC operators are used to calculate the satisfactory values of the performance indices.