Epidemics under environmental noise described by stochastic SIS model

Student thesis: Doctoral Thesis

Abstract

In this research, three different stochastic SIS models are concerned with different environmental noises. We firstly introduce two perturbations in the classical deterministic susceptible-infected-susceptible (SIS) epidemic model. Gray et al. [1] in 2011 used a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter µ + ɣ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian motions for the number of infected population.We then prove that our model has a unique and bounded global solution I(t). Also we establish conditions for extinction and persistence of the infected population I(t). Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory. This work is published in JMAA [2].We then introduce the second model replacing two independent Brownian motions in the first model by two correlated Brownian motions. We consider the two same perturbations in the deterministic SIS model and formulate the original model as a stochastic differential equation (SDE) with two correlated Brownian motions for the number of infected population, based on previous work from Gray et al. in 2011 and Hening's work [3] in 2017. Conditions for the solution to become extinct and persistent are then stated, followed by computer simulations to illustrate the results.Compared to the formal model, the conditions of extinction are extended after correlation between two white noises is considered. However, we are not able to compute the mean and variance of the stationary distribution.Note that this section has also been published as an article in Nonlinear Dynamics in 2019 [4].Moreover, we combined the first model with [5] to add telegraph noise by using Markovian switching to generate the third model. Similarly, conditions for extinction and persistence are then given and proved, followed by explanation on the stationary distribution. Computer simulations are clearly illustrated with different sets of parameters, which support our theorems in this chapter. Compared to two previous models, conditions are given based on the overall behaviour of the solution but not separately specified in every state of the Markov chain.
Date of Award11 Mar 2020
Original languageEnglish
Awarding Institution
  • University Of Strathclyde
SponsorsUniversity of Strathclyde
SupervisorXuerong Mao (Supervisor) & Jiazhu Pan (Supervisor)

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