A novel approach to investigate the stability analysis and the dynamics of reaction–diffusion SVIR epidemic model

Abstract

Understanding the behaviour of nonlinear dynamical systems is crucial in epidemiological modelling. Stability analysis is one of the important concepts in assessing the qualitative behaviour of such systems. This technique has been widely implemented on deterministic models involving ordinary differential equations (ODEs). Nevertheless, the application of stability analysis to distinct complex systems involving partial differential equations (PDEs) has been infrequent. In this paper, an in-depth investigation is conducted on the stability analysis and the dynamics of a reaction–diffusion SVIR model with partial immunity. Firstly, numerical simulation is performed for the proposed PDE model under several realistic scenarios to demonstrate the crucial factors that may affect the qualitative dynamics of the epidemiological model. Our numerical simulation findings highlight the occurrence of disease-free equilibrium when the transmission rates are sufficiently low, preventing the spread of the disease. However, an endemic situation is observed when the transmission rates are moderate (or high) enough to sustain ongoing transmission, leading to a stable endemic steady state. Other epidemiological factors such as vaccination and treatment (or recovery) strategies also play important roles in controlling disease outbreaks. The transition between disease-free equilibrium (DFE) and endemic equilibrium (EE), resulting from switching stability between equilibria, is of particular interest in epidemiology, due to its implications for public health interventions and strategies. This insight helps to inform decision-making processes regarding the control and prevention of diseases. To examine the stability of steady states in this PDE model, we employ a novel technique called numerical range approaches, which relies on the application of operator theory. Typically, discretising differential operators in the PDE model leads to spectral complexity and a wide range of eigenvalues. Our analysis has demonstrated that employing the numerical range (NR) and the cubic numerical range (CNR) techniques make it feasible to estimate each individual eigenvalue in the spectral range. For the DFE (respectively, EE) scenario, the spectrum of eigenvalues for the PDE system agrees with CNR estimates in which the eigenvalues are real-valued and bounded above by zero (respectively, complex conjugates pairs with negative real parts); these numerical estimations have also been verified analytically. Despite the challenges posed, the use of numerical range techniques provides a good estimation of the eigenvalues spectrum, thereby overcoming the difficulties associated with discretisation, and providing a more comprehensive understanding of the dynamics of the epidemiological system.