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Optional Exercise: From Markov chain to ODE

The expected value of the random variables ΔNSI(t)=NSI(t+δ)NSI(t), which take values on the finite set {0,,N}, is E[ΔNSI(t)|S(t),I(t),R(t)]=Nx=0xP[ΔNSI(t)=x]=P[ΔNSI(t)=1|S(t),I(t),R(t)]+o(δ)=μSI(t)S(t)δ+o(δ) Taking the limit as δ0 gives back the original ODE rate lim

Optional Exercise: SDE version of the SIR model

The Euler-Maruyama method for an SDE representation of the closed-population SIR model results in the following equations The so-called Euler-Maruyama approximation \tilde X is generated by \begin{aligned} \tilde S(\tilde t_{k+1}) &= \tilde S(\tilde t_k) - \delta\,\mu_{SI}(t)\,\tilde S(\tilde t_k) + \sigma_{SI} \sqrt{\delta} \, Z_{SI} \\ \tilde I(\tilde t_{k+1}) &= \tilde I(\tilde t_k) - \delta\,\big(\mu_{SI}(t)\,\tilde S(\tilde t_k) - \mu_{IR}(t)\,\tilde I(\tilde t_k)\big) + \sigma_{IR} \sqrt{\delta} \, Z_{IR}\\ \tilde R(\tilde t_{k+1}) &= N - \tilde S(\tilde t_{k+1}) - \tilde I(\tilde t_{k+1}) \end{aligned} where Z_{SI} and Z_{IR} are independent standard normal random variables. The values of \sigma_{SI} and \sigma_{IR} can be chosen to reflect different sources of stochasticity. For example, setting them to the square root of the preceeding term, e.g., \sigma_{SI}=\sqrt{\mu_{SI}(t)\,\tilde S(\tilde t_k) - \mu_{IR}(t)\,\tilde I(\tilde t_k)}, reflects demographic stochasticity.

Difficulties that might arise with non-negativity constraints can occurr when the left hand side of the equations become negative, e.g., because of large intervals \delta, high rates \mu, large values of the normal random variables Z or because of large values of \tilde S(\tilde t_{k+1}) + \tilde I(\tilde t_{k+1}). One practical way to deal with this issue is to truncate negative numbers to zero.


References