Mathematical Modeling and Simulation for Epidemic Models

Abstract

Mathematical modeling has been important to explore transmission dynamics and construct effective control strategies to prevent the spread of disease. Most simple mathematical model is deterministic model. However, we need to take account into the stochastic model when the system involves the intrinsic fluctuations or randomness. Moreover, stochastic models can capture exactly the dynamics of individuals in a small population. But generally, it is difficult to solve the stochastic system. Since bio-chemical reaction is described according to the law of mass action which is used to construct epidemic model. We derive the explicit formula of the solution in terms of block matrices. We formulate mathematical models for epidemic disease. Then, we apply the stochastic computational methods such as the stochastic simulation algorithm (SSA) and the moment closure method (MCM) to the model. First, we apply the stochastic methods to an disease transmission model with government$'$s control policies against the 2009 H1N1 influenza in Korea. We investigate the impact of various vaccination and antiviral treatment intervention scenarios to prevent the spread of disease. As the result, it is verified that the earlier vaccination is more effective. Second, we consider the two-strain dengue transmission model with seasonality for sequential infection. Despite of having no autochthonous dengue outbreaks in Korea, the potential risk of dengue transmission in Jeju Island increases. We investigate the possible impacts of the potential outbreak of dengue fever in Jeju Island considering climate change based on Representative Concentration Pathways (RCP) scenarios and the migration of infected international travel. Finally, if there are a small number of cases at the initial stage of the epidemic. Infection processes occur randomly. Transmission dynamics involve the probabilistic properties in the system. Therefore, stochastic model provides more accurate predictions. We compare the dynamics of epidemic outbreaks quantitatively under stochastic and deterministic models. We investigate that as the initial number of infectives increases, the difference between the deterministic and stochastic solutions decreases.