The role of incidence rate is very important in the study of epidemiological models. In this article, the analysis of an epidemic problem for the transmission dynamic of HBV with saturated incidence rate is presented, which is more generalize than bilinear incidence rate. After formulating the new mathematical model, the threshold quantity reproduction number $$R_0$$ is investigated by using the well known approach i.e. next generation matrix and investigate the possible equilibriums such as disease free and endemic equilibria. Then for the local and global behavior of the proposed problem, the local asymptotic stability analysis as well as global asymptotic stability analysis are proved. To prove the global asymptotic stability at disease free equilibrium, the classic Lyapunov function theory is considered. Similarly to show global asymptotic stability at endemic equilibria, the geometrical approach is used, which is the generalization of Lyapunov theory. Finally, numeric of the proposed problem are carried out to show the feasibility of the obtained results and the role of saturated incidence rate.

In this article, we extend the concept of triple Laplace transform to the solution of fractional order partial differential equations by using Caputo fractional derivative. The concerned transform is applicable to solve many classes of partial differential equations with fractional order derivatives and integrals. As a consequence, fractional order telegraph equation in two dimensions is investigated in detail and the solution is obtained by using the aforementioned triple Laplace transform, which is the generalization of double Laplace transform. The same problem is also solved by taking into account the Atangana-Baleanu fractional derivative. Numerical plots are provided for the comparison of Caputo and Atangana-Baleanu fractional derivatives.

In this subsection, we presented a mathematical model of Ebola virus (EBOV) proposed by susceptible exposed infected recovered (SEIR) model. In our model, the population is affected by animals. EBOV is an infectious agent causing haemorraghic fever, a severe infectious disease characterised by high fever and bleeding, in humans and some monkeys. Here, we assessed the transmissibility associated with the infection stages of EBOV that generated an epidemic model. In order to do this, in the first step, we formulate the model, and the basic properties of the proposed model are presented. The basic reproductive number is obtained by using the next generation matrix approach. Then, all the endemic equilibrium points related to the disease are derived. We also find the conditions to investigate all possible equilibria of the model in terms of the basic reproduction number (local and global stability). In last, numerical simulation is presented with and without vaccination or control for the proposed model.

Hepatitis B is a global health burden and can persist for years, with nearly two billion infections worldwide, where its spread is influenced by environmental heterogeneity, host-pathogen interactions, and vaccination-induced immune variability. Proper understanding and developing models with a suitable framework is essential to accurately capture the complexity of the hepatitis B virus (HBV) and its transmission. In this work, we present a novel framework of a stochastic model and a forward neural network that combines neural networks and stochastic differential equations to analyze the dynamics of hepatitis B virus transmission, as it is important to capture the inherent uncertainty of disease spread in heterogeneous environments. We formulate the stochastic model with a saturated incidence rate, incorporating the long-term persistence of the disease following key characteristics of the disease transmission. The theoretical analysis of the model is proven to ensure the well-posedness and to determine the conditions for extinction and persistence of the disease. Further, a set of real data of hepatitis B reported cases will be used to produce stochastic simulations, and to train a feed-forward neural network (FFNN), while approximating the model dynamics more effectively. To evaluate the efficacy of the hybrid framework, we demonstrate its performance by the presenting mean squared error (MSE), absolute error (AE), and regression analysis showing strong agreement between the stochastic simulations and neural network predictions.

We study the epidemiology of hepatitis B for different infection routes with hospitalization keeping in view the aesthetic concept to model it with the fractional calculus application. We will use the Caputo–Fabrizio operator for the purposes of fractionation and discuss the existence analyses with uniqueness properties for the newly formulated model. We apply the fixed-point theorem to prove the analysis of existence with uniqueness. It also proves that the model has positive solutions and abounded solutions. We also calculate the reproductive number to discuss steady states, and to perform that the particular epidemic model is stable asymptotically under some constraints. Particularly, we will discuss both the global and local properties of the proposed model by using the mean value theorem, Barbalat's lemma and linearization. We also draw some numerical simulations to verify the theoretical work via some graphical representations.

Hepatitis B virus (HBV) infection is one of the leading causes of death and is a contagious disease that produces chronic liver infection. The infection of hepatitis B has a complex nature involving multiple and long infectious periods, behavioral changes, and healthcare limitations. The multiple infection phases, immune system, behavioral changes, and healthcare saturation have asignificant impact on the dynamics of hepatitis B virus transmission. Since HBV can persist in the body for years, the number of infectious individuals accumulates, and people take precautionary measures as awareness increases. Considering the multiple stages of the disease and the saturation level, we propose an innovative hybrid approach combining a mathematical model with a saturated incidence rate and a forward neural network to represent the transmission dynamics of the hepatitis B virus. First, we prove the biological and mathematical feasibility to show that the model under consideration is well-posed. We also investigate the dynamics of the model using linear stability analysis to derive the stability conditions. In addition, we perform the numerical assessment of the model using a novel hybrid approach of NSFD–FFNN framework to show the accuracy and large-scale numerical simulations.