Dynamical Analysis of an Improved Bidirectional Immunization SIR Model in Complex Network

Abstract

In order to investigate the impact of two immunization strategies—vaccination targeting susceptible individuals to reduce their infection rate and clinical medical interventions targeting infected individuals to enhance their recovery rate—on the spread of infectious diseases in complex networks, this study proposes a bilinear SIR infectious disease model that considers bidirectional immunization. By analyzing the conditions for the existence of endemic equilibrium points, we derive the basic reproduction numbers and outbreak thresholds for both homogeneous and heterogeneous networks. The epidemic model is then reconstructed and extensively analyzed using continuous-time Markov chain (CTMC) methods. This analysis includes the investigation of transition probabilities, transition rate matrices, steady-state distributions, and the transition probability matrix based on the embedded chain. In numerical simulations, a notable concordance exists between the outcomes of CTMC and mean-field (MF) simulations, thereby substantiating the efficacy of the CTMC model. Moreover, the CTMC-based model adeptly captures the inherent stochastic fluctuation in the disease transmission, which is consistent with the mathematical properties of Markov chains. We further analyze the relationship between the system’s steady-state infection density and the immunization rate through MCS. The results suggest that the infection density decreases with an increase in the immunization rate among susceptible individuals. The current research results will enhance our understanding of infectious disease transmission patterns in real-world scenarios, providing valuable theoretical insights for the development of epidemic prevention and control strategies.