In this article, a nonlinear mathematical model used for the impact of vaccination on the control of infectious disease, Japanese encephalitis with a standard incidence rate of mosquitoes, pigs and humans has been planned and analyzed. During the modeling process, it is expected that the disease spreads only due to get in touch with the susceptible and infected class only. It is also assumed that due to the effect of vaccination, the total human population forms a separate class and avoids contact with the infection. The dynamical behaviors of the system have been explored by using the stability theory of differential equations and numerical simulations. The local and global stability of the system for both equilibrium states under certain conditions has been studied. We have set up a threshold condition in the language of the vaccineinduced reproduction number R(α 1 ), which is fewer than unity, the disease dies in the absence of the infected population, otherwise, the infection remains in the population. Furthermore, it is found that vaccine coverage has a substantial effect on the basic reproduction number. Also, by continuous efforts and effectiveness of vaccine coverage, the disease can be eradicated. It is also found a more sensitive parameter for the transmission of Japanese encephalitis virus by using sensitivity analysis. In addition, numerical results are used to investigate the effect of some parameters happening the control of JE infection, for justification of analytical results.

Mathematical modeling of Japanese encephalitis (JE) disease in human population with pig and mosquito has been presented in this paper. The proposed model, which involves three compartments of human (Susceptible, Vaccinated, Infected), two compartments of mosquito (Susceptible, Infected) and three compartments of the pig (Susceptible, Vaccinated, Infected). In this work, it is assumed that JE spreads between susceptible class and infected mosquitoes only. Basic results like boundedness of the model, the existence of equilibrium and local stability issues are investigated. Here, to measure the disease transmission potential in the population the basic reproduction number (R0) from the system has been analyzed w.r.t. control parameters both numerically and theoretically. The dynamical behaviors of the system have been analyzed by using the stability theory of differential equations and numerical simulations at equilibrium points. A numerical verification of results is carried out of the model under consideration.

In the manuscript, the influence of time delay on the transmission of Japanese encephalitis model without vaccination model has been studied. The time delay is because of the existence of an incubation period during which the JE virus reproduces enough in the mosquitoes with the goal that it tends to be transmitted by the mosquitoes to people. The motivation behind this manuscript is to assess the influence of the time delay it takes to infect susceptible human populations after interacting with infected mosquitoes. The steady-states and the threshold value R0 of the delay model were resolved. This value assists with setting up the circumstance that ensure the asymptotic stability of relating  equilibrium points. Utilizing the delay as a bifurcation parameter, we built up the circumstance for the presence of a ”Hopf bifurcation”. Moreover, we infer an express equation to decide the stability and direction of ”Hopf bifurcation” at endemic equilibrium by using center manifold theory and normal structure strategy. It has been seen that delay plays a vital role in stability exchanging. In addition, larger values of virus transmission rate from an infected mosquito to  susceptible individuals and the natural mortality of humans of a model affect the existence of ”Hopf bifurcation”. Finally, to understand some analytical outcomes, the delay framework is simulated numerically.

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