Stochastic Differential Equations In Epidemiology

This thesis is devoted to investigating nonlinear dynamical systems that model infectious diseases. Our primary interest lies in the analysis of stochastic epidemic modelling, particularly on the part that focuses on compartmental stochastic models.

These models, with or without time delays, treat the influence of three specific kinds of environmental noise. They are Gaussian white noise, Lévy noise, and telegraph noise. Our aim is to explore the influence these noise process types have on disease dynamics, enhancing our understanding of epidemic modelling.

We begin by exploring a deterministic SAIRS epidemic model and identifying two equilibrium states: disease-free and endemic.We derive the basic reproduction number and analyse the global stability of these equilibria. Further, we examine the SAIR model, a special case of the SAIRS model, and establish the global stability of its endemic equilibria using the Lyapunov function method.

Next, we analyse a generalised stochastic SEIQR epidemic model perturbed by Gaussian white noise. We explore V -geometric ergodicity of the model. Furthermore, we establish that under certain conditions, the disease either becomes extinct or is stochastically permanent.

Moreover, we extend the deterministic SAIRS model discussed earlier into a stochastic model influenced by both Gaussian and Lévy noise. We identify the necessary conditions for the disease’s extinction and for its persistence in mean.

Furthermore, we examine a distributed delayed epidemic model incorporating Lévy noise. By employing stochastic Lyapunov functions, we are able to ascertain the conditions under which the disease becomes extinct or persists.

Subsequently, we explore a novel stochastic delayed SVIQR epidemic model with Markovian switching. We introduce a stochastic threshold, which we utilise to assess the conditions for disease extinction and persistence. The influence of time delay on the model is also investigated.

The thesis concludes with an analysis a triple-delayed stochastic epidemic model with general nonlinear incidence and Lévy noise. We propose a stochastic threshold and use it as a criterion to ascertain whether the disease would vanish or continue to exist within the population.