Stochastic SIS epidemic models and corresponding statistical inference

Strathclyde theses - ask staff. Thesis no. : T13458This thesis considers the deterministic SIS epidemic model, which has applications to transmission of real-life diseases, such as pneumococcus, gonorrhea and tuberculosis. Environmental noise can a ect the deterministic system signi cantly. There are various types of noise which can be incorporated into the deterministic dynamics according to di erent situations. The e ect of three types of noise on the deterministic SIS epidemic model have been examined in this thesis, which has not been discussed in previous literature. Firstly, assuming that there exists environmental noise in the disease transmission coe cient, we extend the classical SIS epidemic model from a deterministic framework to a stochastic one by incorporating white noise using the parameter perturbation technique, and formulate it as a stochastic di erential equation (SDE) for the number of infectious individuals I(t). For the model to make sense, we then prove that this SDE has a unique global positive solution I(t) and establish conditions for extinction and persistence of I(t) and compare these with the corresponding conditions for the deterministic SIS epidemic model. We also discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. Secondly, assuming that the parameters in the SIS epidemic model experience an abrupt change around the point of threshold value, we incorporate telegraph noise in the deterministic model. We then establish the explicit solution of the stochastic SIS epidemic model, which is useful in performing computer simulations. We also obtain the conditions for extinction and persistence for this model. Afterwards, we take a further step of incorporating both types of the aforementioned noise in the SIS epidemic model. We not only show the existence of a unique global positive solution but also examine the asymptotic properties, including extinction and persistence. The results are illustrated by computer simulations, including examples based on real life diseases for the rst and second stochastic models. Computer simulations based on the explicit solution and the Euler{Maruyama scheme are compared for the SIS model with telegraph noise. Furthermore, statistical inference is always essential in disease analysis. That is the motivation for us to conduct parameter estimation for the SDE SIS model with white noise introduced. Three estimation methods, least squares estimation, the pseudo-Maximum Likelihood Estimation (pseudo-MLE) method and the Bayesian approach are applied to the SDE SIS model. Our main contribution in least squares estimation and pseudo-MLE is variance estimation. We obtain not only the point estimators but also the interval estimators and the joint con dence regions for both estimation techniques. Additionally we investigate the factors which in uence variance in estimation. As for the Bayesian approach, although strong results have been obtained by using the MCMC technique, we use a di erent method where analytic results are obtained without the need to deal with the signi cant computational cost. Computer simulations are performed to illustrate our theory. The three estimation methods are compared both analytically and in the simulation examples.This thesis considers the deterministic SIS epidemic model, which has applications to transmission of real-life diseases, such as pneumococcus, gonorrhea and tuberculosis. Environmental noise can a ect the deterministic system signi cantly. There are various types of noise which can be incorporated into the deterministic dynamics according to di erent situations. The e ect of three types of noise on the deterministic SIS epidemic model have been examined in this thesis, which has not been discussed in previous literature. Firstly, assuming that there exists environmental noise in the disease transmission coe cient, we extend the classical SIS epidemic model from a deterministic framework to a stochastic one by incorporating white noise using the parameter perturbation technique, and formulate it as a stochastic di erential equation (SDE) for the number of infectious individuals I(t). For the model to make sense, we then prove that this SDE has a unique global positive solution I(t) and establish conditions for extinction and persistence of I(t) and compare these with the corresponding conditions for the deterministic SIS epidemic model. We also discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. Secondly, assuming that the parameters in the SIS epidemic model experience an abrupt change around the point of threshold value, we incorporate telegraph noise in the deterministic model. We then establish the explicit solution of the stochastic SIS epidemic model, which is useful in performing computer simulations. We also obtain the conditions for extinction and persistence for this model. Afterwards, we take a further step of incorporating both types of the aforementioned noise in the SIS epidemic model. We not only show the existence of a unique global positive solution but also examine the asymptotic properties, including extinction and persistence. The results are illustrated by computer simulations, including examples based on real life diseases for the rst and second stochastic models. Computer simulations based on the explicit solution and the Euler{Maruyama scheme are compared for the SIS model with telegraph noise. Furthermore, statistical inference is always essential in disease analysis. That is the motivation for us to conduct parameter estimation for the SDE SIS model with white noise introduced. Three estimation methods, least squares estimation, the pseudo-Maximum Likelihood Estimation (pseudo-MLE) method and the Bayesian approach are applied to the SDE SIS model. Our main contribution in least squares estimation and pseudo-MLE is variance estimation. We obtain not only the point estimators but also the interval estimators and the joint con dence regions for both estimation techniques. Additionally we investigate the factors which in uence variance in estimation. As for the Bayesian approach, although strong results have been obtained by using the MCMC technique, we use a di erent method where analytic results are obtained without the need to deal with the signi cant computational cost. Computer simulations are performed to illustrate our theory. The three estimation methods are compared both analytically and in the simulation examples