Stochastic differential equations to model influenza transmission with continuous and discrete-time Markov chains

This study take account into the modeling, mathematical analysis, developing theories and numerical simulation of Influenza virus transmission together with stochastic behavior. We consider the modified five-compartment SEIRT mathematical model to estimate the recent trends and future prediction of basic reproduction number and infection case. Since the basic reproduction number R0 is not sufficient to predict the outbreak, we apply the discrete-time Markov chain (DTMC), continuous-time Markov chain (CTMC), and Itô stochastic differential equations (SDEs) to calculate the probability of the disease outbreak. Since the H1N1 influenza is a contagious respiratory illness caused by a specific strain of the influenza A virus; which particular strain was designated as H1N1pdm09, with pdm09. We show the positivity and boundedness of solutions and discussed disease-free equilibrium and endemic equilibrium points along with their stability. Based on the results, it reflects that contact patterns affect the dynamics of disease outbreaks. For all the models, the threshold quantity, R0 is calculated which is the key reason to prove the global and local stability analysis of disease-free equilibrium and endemic equilibrium points. A multitude of numerical outcomes are achieved in order to observe the analytical study. In this study, a more realistic representation of infection dynamics, intervention effects, and transmission behaviors is offered by the convergence of deterministic, CTMC, DTMC, and SDE models. Computational methods are used to validate theoretical results, improving the trustworthiness of the model. The stochastic model allows for an examination of the stochastic character of influenza, which advances our understanding of influenza epidemiology, preventative tactics, and potential treatment results under various circumstances. In the present or future outbreak, such analysis is significant to understand the impact of various parameters and their stochastic behavior in the contagious model to prevent such type disease outbreaks.