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Optional Exercise: From Markov chain to ODE

The expected value of the random variables \(\Delta{N}_{SI}(t) = N_{SI}(t+\delta)-N_{SI}(t)\), which take values on the finite set \(\{0,\ldots,N\}\), is \[ \begin{aligned} \mathbb{E}\big[{\Delta{N}_{SI}(t) \,|\, S(t), I(t), R(t)}\big] &= \sum_{x=0}^{N} x\mathbb{P}\big[{\Delta{N}_{SI}(t)=x}\big] \\ &= \mathbb{P}\big[{\Delta{N}_{SI}(t)=1 \,|\, S(t), I(t), R(t)}\big] + o(\delta) \\ &= \mu_{SI}(t)\,S(t)\,\delta + o(\delta)\\ \end{aligned} \] Taking the limit as \(\delta\to 0\) gives back the original ODE rate \[ \begin{aligned} \lim_{\delta\to 0}\frac{\mathbb{E}\big[{\Delta{N}_{SI}(t) \,|\, S(t), I(t), R(t)}\big]}{\delta} &= \lim_{\delta\to 0}\frac{\mu_{SI}(t)\,S(t)\,\delta}{\delta} + \lim_{\delta\to 0}\frac{o(\delta)}{\delta} = \mu_{SI}(t)\,S(t) \\ \end{aligned} \]

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