Numeric methods for stochastic disease spread models

Disease spread models are important in controlling the new infectious diseases that suddenly threaten the public health. Mathematical tools are especially important to understand the spread of the disease since experiments are not possible in the area. This study focusses on a stochastic SIR (susceptible-infected-recovered) model for a finite population. We first assume a homogeneous population and study Markov modelling of disease spread for an exponential infectious period. We present the algorithms that compute the expected duration of an epidemic, the final outbreak size distribution and the maximum number of simultaneously infected individuals distribution. After stating the problems with exponential infectious period, we assume an Erlang distributed infectious period allowing us to use Markov chains. The Markov disease spread model proposed for it uses the remaining stages as state variables and treats the Erlang distributed infectious period as simply exponential. This enables us to compute the exact final outbreak size distribution for large populations efficiently. Moreover, we propose an approximation for the distribution of the maximum epidemic size using the exact distribution of the remaining stages. We also consider a mixture of Erlangs so that by using the first two moments of an infectious period one can fit a corresponding mixture. Furthermore, by considering a mixture of Erlangs distribution for the infectious period and assuming two types of infected individuals as symptomatic and asymptomatic, our proposed models are implemented with the parameters similar to those reported for COVID-19 spread. Finally, a stochastic SIR model for a non homogeneous population is considered. The notion of R0 for heterogeneous populations is discussed and individual R0 as the expected number of secondary cases produced by a unique given initially infected individual. We propose a general formula for individual R0 and use it for the assessment and development of the intervention methods.