This is a companion piece to COVID-19 Pandexit and the effects on economic activity, covering the underlying epidemiological model in full.
This is a restatement of the core framework in Rungcharoenkitkul (2021), sections 2.1–2.2, checked against the author’s public implementation. The notation below groups the six isolation equations to make the population accounting easier to follow.
States and notation
Each state is a number of people; one period is one day. Write \(\Delta X_{t+1}=X_{t+1}-X_t\).
| State | Interpretation |
|---|---|
| \(S_t\) | Susceptible |
| \(E_t\) | Exposed, not yet infectious |
| \(I_t\) | Infectious and able to transmit |
| \(U^R_t,\ Q^R_t,\ H^R_t\) | Isolated and eventually recovering: undetected, quarantined, hospitalised |
| \(U^D_t,\ Q^D_t,\ H^D_t\) | Isolated and eventually dying, in the same three settings |
| \(R_t\) | Recovered and immune |
| \(D_t\) | Cumulative deaths |
| \(V_t\) | Effectively vaccinated and immune |
The twelve compartments sum to the initial population \(N\), including deaths. The public code also tracks auxiliary series, such as cumulative detected cases; these are not additional mutually exclusive population compartments.
Infection and vaccination
Let \(F_t\) be new exposures and \(\nu_t\) the daily flow of susceptible people gaining vaccine protection. With \(\sigma\) the incubation-transition rate, \(\delta\) the isolation-transition rate, and \(\omega\) the immunity-loss rate:
\[ F_t=\gamma_t\frac{S_tI_t}{N},\qquad \begin{aligned} \Delta S_{t+1}&=-F_t-\nu_t+\omega(R_t+V_t),\\ \Delta E_{t+1}&=F_t-\sigma E_t,\\ \Delta I_{t+1}&=\sigma E_t-\delta I_t,\\ \Delta V_{t+1}&=\nu_t-\omega V_t. \end{aligned} \]
Here \(\nu_t\) measures effective protection, not injections: an efficacy assumption scales the vaccination input. A delay can be represented in that input without adding compartments. The displayed core equations allocate vaccination to susceptible people; the upstream code also offers allocation across susceptible and recovered people. Setting \(\omega=0\) rules out waning immunity. Flows must be bounded so no compartment becomes negative.
Isolation, recovery, and deaths
Let \(q\) be detection probability, \(h\) hospitalisation probability conditional on detection, and \(p_t\) eventual death probability after infection. Define the allocation weights
\[ w_U=1-q,\qquad w_Q=q(1-h),\qquad w_H=qh. \]
For each \(j\in\{U,Q,H\}\), the two equations
\[ \begin{aligned} \Delta j^R_{t+1}&=\delta(1-p_t)w_jI_t-\rho_j j^R_t,\\ \Delta j^D_{t+1}&=\delta p_tw_jI_t-\mu j^D_t \end{aligned} \]
represent six state transitions. The recovery rates satisfy \(\rho_U=\rho_Q=\rho\), while hospital recovery uses \(\rho_H\); \(\mu\) is the death-transition rate. The remaining two equations are
\[ \begin{aligned} \Delta R_{t+1}&=\rho(U^R_t+Q^R_t)+\rho_H H^R_t-\omega R_t,\\ \Delta D_{t+1}&=\mu(U^D_t+Q^D_t+H^D_t). \end{aligned} \]
Summing all twelve changes gives zero. The outcome superscripts are accounting categories, not assumptions that an individual’s outcome is observed in advance.
Mobility and the policy rule
Mobility \(m_t\in[-1,0]\) is a fractional deviation from normal: \(-0.1\) means 10% below normal. Its contribution to transmission is exponential:
\[ \gamma_t(m_t)=\beta_0 e^{\beta_1m_t}+z_t, \qquad \beta_0,\beta_1>0. \]
The residual \(z_t\) captures transmission changes not explained by mobility. The paper’s baseline projection uses a mean-reverting residual, \(z_t=\varrho z_{t-1}+\varepsilon_t\), with future innovations set to zero. This still allows the inherited residual to decay gradually.
Writing \(s_t=S_t/N\) and \(i_t=I_t/N\), the reduced policy block uses the mortality-loss measure
\[ \widetilde d_t(m)=p_t\delta i_t \left[1-\delta+\gamma_t(m)s_t\right]. \]
This is the policy rule’s mortality measure, not observed daily deaths \(\Delta D_{t+1}/N\). The mobility decision balances it against the cost of restrictions:
\[ m_t^*\in\underset{-1\le m\le0}{\operatorname{argmin}} \left\{\widetilde d_t(m)^2+\varphi m^2\right\}, \qquad \varphi>0. \]
An interior choice satisfies
\[ -\varphi m_t= \widetilde d_t(m_t)\, \underbrace{p_t\delta i_t s_t\beta_0\beta_1e^{\beta_1m_t}}_{\partial\widetilde d_t/\partial m_t}. \]
Boundary choices must also be considered. The upstream numerical routine searches a mobility grid for the smallest first-order-condition residual. This is the source’s reduced policy rule, rather than a newly solved full dynamic-planning problem.
A useful summary of epidemic pressure is
\[ \mathcal R_t^{\mathrm{eff}}= \frac{\gamma_t}{\delta}\frac{S_t}{N}. \]
Reducing mobility lowers \(\gamma_t\); effective vaccination lowers \(S_t\). Both can push this quantity below one, when the combined exposed-and-infectious pool starts shrinking.
From mobility to activity
The activity calculation is a separate empirical step. The rolling regressions below estimate an intercept and a mobility coefficient using eight monthly observations. For a scenario comparison, the approximation is \(\Delta y_t\simeq\widehat b_t\Delta m_t\), with mobility and activity changes expressed in consistent percentage-point units. It is an estimated association, not an additional epidemiological equation or a causal identification result.
The 95%, 60%, and 5% mobility paths keep the vaccination scenario comparable while varying assumed protection against infection. This section documents their model framework; it does not supply a new calibration or re-estimate the historical charts.