Pandemic simulation is considered to be crucial as a scenario simulation and it is performed by many kinds of methods; the classical ordinary differential models (SIR model), agent-based models, internet-based models, and etc are among them. The SIR model is one of the fundamental methods to see the behavior of the pandemic with easy computation, where S, I, and R denote susceptible, infected and removed populations respectively, and it computes the number of people infected with a contagious disease in a closed population over time. The model can quickly deal with simulations of infectious disease spread among homogeneous populations using simple simultaneous ordinary differential equations and a few parameters. However, there are no stochastic variation terms in the equations. The objective of our study is to obtain the confidence intervals for stochastic variations for the predicted values using real world cases.
The stochastic differential equations (SDE) can provide such kind of variations. Although the SDE are applied to many fields such as economics, less attention has been paid to the SIR simulations. In this paper, we propose a SDE version of the SIR simulation model by using Euler-Maruyama method as a simple numerical method. The diffusion is added to the dI(t) term, which corresponds to infected population derivative. The SIR mean parameters were obtained by using the difference equations, and the parameters of the SDE was obtained by using a well-known property of the quadratic variation associated with the stochastic process for I(t).
The proposed method was applied to SARS (Severe Acute Respiratory Syndrome) case in 2003 in Hong Kong. In that case, we pursued the appropriate number of runs to to obtain the confidence intervals for the estimates, resulting in 10000 runs in the simulations. We have found that the SIR model gives us the final value around 2300 at the time of day 40. This estimated value and the observed value of 1755 are close to each other. However, the confidence interval show a possibility that the number of infected people would be twice as many as the actually observed number. As time goes on, the highest value in 95% confidence intervals, which we can interpret the possible worst case, is becoming lower.