Stability and Forward Bifurcation Analysis of an SIPIVR Model for Poliovirus Transmission with Neural Network

The aim of this research is to formulate and analyze a modified SIpIVR mathematical model to study the transmission dynamics of poliovirus and assess the impact of vaccination on disease control. The proposed model extends classical SEIV-type frameworks by incorporating a recovered compartment with long-term immunity and by replacing the traditional exposed class with a pre-infectious compartment (Ip) that captures silent viral shedding during the incubation phase of poliovirus. This modification addresses the critical epidemiological feature that individuals can transmit the virus before showing symptoms while maintaining biological accuracy in compartment definition. Several fundamental analytical properties are rigorously established, including positivity, boundedness, and the existence of a biologically meaningful invariant region. The basic reproduction number R0 is derived using the next-generation matrix approach, and comprehensive stability analysis is carried out. The analysis shows that the DFE is locally and globally asymptotically stable whenever R0<1. Using center manifold theory, a forward bifurcation is rigorously demonstrated, indicating that disease persistence emerges smoothly as R0 crosses unity. Local and global sensitivity analyses of the basic reproduction number R0 identify critical epidemiological parameters, and points to vaccination coverage and transmission rates as key drivers of outbreak dynamics. Numerical simulations confirm the analytical results and illustrates two different epidemiological scenarios, one with R0<1, and another with R0>1 along with neural network analysis by using the same data from both cases in a built-in function package in MATLAB-2020 software. It utilizes all of its hidden layers to check the data used by the model for validation performance and training and to find the absolute and mean squared errors. It also shows how vaccination suppresses the spread of infection. These findings provide a strong mathematical basis for public health policy, offering strategic insight into how vaccination campaigns might be optimized to accelerate progress toward global polio eradication.