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), agentbased models, internetbased
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 EulerMaruyama 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 wellknown
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.
