Modelling the effect of stochasticity in epidemic and HIV models

An epidemic of an infectious disease can be modelled by using either a deterministic model or a stochastic model. In this thesis, we consider the effect that different types of noise has on the dynamical behaviour of deterministic SIS models and SIR/SIRS models as well as an HIV model.We start off with a literature review giving previous work and the mathematical background to the area. Next, we introduce demographic stochasticity into the well-established deterministic SIS model with births and deaths and derive a stochastic differential equation (SDE). We assume that an infected individual or a susceptible individual who dies is immediately replaced by a susceptible individual and thus the population size is kept constant. In order for our model to make sense, we then prove that the SDE has a strong unique nonnegative solution which is bounded above and establish the conditions needed for the disease to become extinct. Based on the idea of the Feller test, we also calculate the respective probabilities of the solution first hitting zero or the upper limit. Numerical simulations are then produced using the Milstein method with both theoretical and realistic parameter values to confirm our theoretical results. Motivated by the model discussed in the first topic, we then continue our study on the effect of demographic stochasticity on the deterministic SIS model by now assuming that the births and deaths of individuals are independent of each other and thus the population size can vary with respect to time. In this case, the per capita disease contact rate may be dependent on the population size and we have shown that this model allows us to consider the cases when the population size tends to a large number and when the population size tends to a small number. First we look at the SDE model for the total population size and show that there exists a strong unique nonnegative solution. Then we look at the two-dimensional SDE SIS model and show that there also exists a strong unique nonnegative solution which is bounded above given the total population size. We then obtain the conditions needed in order for the disease to become extinct in finite time almost surely. Numerical simulations with both theoretical and realistic parameter values are also produced to confirm our theoretical results.Next we look at a different type of noise, namely the telegraph noise, which is an example of an environmental noise. Telegraph noise could be modelled as changing between two or more regimes of environment which differ by factors such as rainfalls or nutrition.This form of switching can be modelled using a finite-state Markov Chain. We incorporate the telegraph noise into the SIRS epidemic model. First we start with a two-state Markov Chain and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence for the stochastic SIRS model. We then explain how the results can be generalised to a finite-state Markov Chain. Furthermore we also show that the results for the SIR model with Markov switching are a special case of the SIRS model. Numerical simulations are produced using theoretical and realistic parameter values to confirm our theoretical results.Lastly we look at the modified Kaplan HIV model amongst injecting drug users. We introduce environmental stochasticity into the deterministic HIV model by the well-known standard technique of parameter perturbation. We then prove that the resulting SDE has a unique global nonnegative solution. As well as constructing the conditions required for extinction and persistence we also show that there exists a stationary distribution for the persistence case. Simulations using the Euler-Maruyama method with realistic parameter values are then constructed to illustrate and support our theoretical results.A brief discussion and summary section is given at the end to conclude the thesis.An epidemic of an infectious disease can be modelled by using either a deterministic model or a stochastic model. In this thesis, we consider the effect that different types of noise has on the dynamical behaviour of deterministic SIS models and SIR/SIRS models as well as an HIV model.We start off with a literature review giving previous work and the mathematical background to the area. Next, we introduce demographic stochasticity into the well-established deterministic SIS model with births and deaths and derive a stochastic differential equation (SDE). We assume that an infected individual or a susceptible individual who dies is immediately replaced by a susceptible individual and thus the population size is kept constant. In order for our model to make sense, we then prove that the SDE has a strong unique nonnegative solution which is bounded above and establish the conditions needed for the disease to become extinct. Based on the idea of the Feller test, we also calculate the respective probabilities of the solution first hitting zero or the upper limit. Numerical simulations are then produced using the Milstein method with both theoretical and realistic parameter values to confirm our theoretical results. Motivated by the model discussed in the first topic, we then continue our study on the effect of demographic stochasticity on the deterministic SIS model by now assuming that the births and deaths of individuals are independent of each other and thus the population size can vary with respect to time. In this case, the per capita disease contact rate may be dependent on the population size and we have shown that this model allows us to consider the cases when the population size tends to a large number and when the population size tends to a small number. First we look at the SDE model for the total population size and show that there exists a strong unique nonnegative solution. Then we look at the two-dimensional SDE SIS model and show that there also exists a strong unique nonnegative solution which is bounded above given the total population size. We then obtain the conditions needed in order for the disease to become extinct in finite time almost surely. Numerical simulations with both theoretical and realistic parameter values are also produced to confirm our theoretical results.Next we look at a different type of noise, namely the telegraph noise, which is an example of an environmental noise. Telegraph noise could be modelled as changing between two or more regimes of environment which differ by factors such as rainfalls or nutrition.This form of switching can be modelled using a finite-state Markov Chain. We incorporate the telegraph noise into the SIRS epidemic model. First we start with a two-state Markov Chain and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence for the stochastic SIRS model. We then explain how the results can be generalised to a finite-state Markov Chain. Furthermore we also show that the results for the SIR model with Markov switching are a special case of the SIRS model. Numerical simulations are produced using theoretical and realistic parameter values to confirm our theoretical results.Lastly we look at the modified Kaplan HIV model amongst injecting drug users. We introduce environmental stochasticity into the deterministic HIV model by the well-known standard technique of parameter perturbation. We then prove that the resulting SDE has a unique global nonnegative solution. As well as constructing the conditions required for extinction and persistence we also show that there exists a stationary distribution for the persistence case. Simulations using the Euler-Maruyama method with realistic parameter values are then constructed to illustrate and support our theoretical results.A brief discussion and summary section is given at the end to conclude the thesis