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<title>Nicolas Franz-Pattillo</title>
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<description>Economist at the Central Bank of Chile. Ph.D., Vancouver School of Economics, UBC.</description>
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  <title>COVID-19 Pandexit and the effects on economic activity</title>
  <dc:creator>Nicolas Franz-Pattillo</dc:creator>
  <link>https://www.nicolasfranzpattillo.com/posts/covid19-vaccines/</link>
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<div class="page-hero column-screen"><img src="https://www.nicolasfranzpattillo.com/posts/covid19-vaccines/src/cover.webp" alt="A vial of the Pfizer-BioNTech COVID-19 vaccine. Photo: Lisa Ferdinando, U.S. Department of Defense — public domain."><div class="page-hero__credit">Photo: Lisa Ferdinando, U.S. Department of Defense — public domain.</div></div>
<section id="introduction" class="level2">
<h2 class="anchored" data-anchor-id="introduction">Introduction</h2>
<p>Chile’s vaccination campaign is progressing rapidly, raising a natural question: <strong>when can we expect mobility—and economic activity—to return to normal?</strong> The answer depends not only on how quickly the population is vaccinated, but also on how effective the vaccines are at preventing contagion.</p>
<p>Using the methodology proposed by <span class="citation" data-cites="Rungcharoenkitkul2021">Rungcharoenkitkul (2021)</span>, we project the evolution of the pandemic and the associated optimal mobility response under a vaccination process similar to Chile’s. We consider three scenarios in which vaccines prevent contagion 95%, 60%, and 5% of the time. The exercise allows us to ask how vaccine efficacy affects the expected timing of the return to normality—and what a slower <em>pandexit</em> could cost in terms of economic activity.</p>
<p>Figure 5 shows a sizable difference across scenarios. <strong>With 95% efficacy, mobility returns to normal roughly a month and a half earlier than with 60% efficacy.</strong> Over the following six months, the average mobility gap between the two scenarios is approximately 4.7%.</p>
<p>That delay has an economic cost. We estimate that a 10% reduction in mobility is associated with GDP growth that is roughly 1.7 to 3 percentage points lower. Combining this estimate with the projected mobility paths suggests that <strong>a slower return to normality could subtract around 0.4 to 0.6 percentage points from annual GDP growth</strong>. For an economy expected to grow 8.5% under the 60%-efficacy scenario, this implies growth could instead reach roughly 8.9% to 9.1% if vaccines prevented contagion 95% of the time.</p>
<p>The effectiveness of vaccination therefore matters for more than the evolution of the pandemic. <strong>It also determines how quickly we can return to normal—and how much economic activity we lose while we wait.</strong></p>
</section>
<section id="epidemiological-model-mobility-and-activity" class="level2">
<h2 class="anchored" data-anchor-id="epidemiological-model-mobility-and-activity">Epidemiological model, mobility, and activity</h2>
<p>The mechanism is a feedback loop: <strong>mobility supports economic activity, but also creates opportunities for infection</strong>. Rising health risks make restrictions more attractive. Effective vaccination reduces the susceptible population, allowing mobility to recover with a smaller health cost.</p>
<p>The framework follows the 12-compartment model of <span class="citation" data-cites="Rungcharoenkitkul2021">Rungcharoenkitkul (2021)</span>. People move from susceptible to exposed and then infectious. They subsequently isolate at home, enter quarantine, or go to hospital, before recovering or dying. The six isolated groups distinguish these three settings and the two eventual outcomes; only the infectious group transmits in the model. Vaccination provides a route from susceptibility to immunity without passing through infection.</p>
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<p><em>Population flows in the model. Dashed arrows allow for loss of immunity; the six isolated groups do not transmit. UR, QR, and HR eventually recover; UD, QD, and HD eventually die.</em></p>
<p>The economic block chooses mobility by balancing a quadratic cost of restrictions against a mortality-loss measure. A separate empirical relationship translates mobility into activity. The vaccine percentages below are <strong>scenario assumptions about protection against infection</strong>, rather than estimates of protection against hospitalisation or death.</p>
<p><a href="../pandexit-model/index.html" class="post-modal-link" data-post-modal="true">Explore the model: states, equations, and mobility choice</a></p>
</div>
</section>
<section id="vaccine-effectiveness-and-alternative-scenarios" class="level2">
<h2 class="anchored" data-anchor-id="vaccine-effectiveness-and-alternative-scenarios">Vaccine effectiveness and alternative scenarios</h2>
<p>In this model, effective vaccination removes people from the susceptible group without them having to get sick. Protection does not arrive immediately. For this 2021 exercise, the vaccination chart assumes two doses 30 days apart, with full effectiveness reached 14 days after the second dose. Figure 1 shows first and second doses alongside the population effectively protected under 100%, 75%, and 50% effectiveness assumptions. The first dose contributes half of eventual effectiveness by day 30, building linearly. These illustrative percentages differ from the 95%, 60%, and 5% assumptions used in the mobility scenarios.</p>
<figure class="paper-plot figure" style="width: 100%;"><div class="paper-plot__header"><div class="paper-plot__title">Vaccination process</div><div class="paper-plot__units">(people)</div></div><div class="paper-plot__body"><div class="paper-plotly" id="vaccines" data-paper-plotly-spec="src/graphs/vaccines.json" data-preview-image="src/previews/src_graphs_vaccines.png" style="width: 100%; height: 360px;"></div></div><div class="paper-plot__footer"><div class="paper-plot__notes">Note: this figure assumes that the first dose delivers half of its effectiveness by the end of day 30, and that the effectiveness gain from the day of injection is linear. The second dose delivers full effectiveness once 14 days have passed.</div></div></figure>
<p>The effects of mobility restrictions on economic activity have not been homogeneous over time.</p>
<div class="paper-plot-row" title="Mobility: economic activity and uncertainty">
<figure class="paper-plot figure" style="width: 100%;"><div class="paper-plot__header"><div class="paper-plot__title">Mobility versus economic activity</div><div class="paper-plot__units">(percentages)</div></div><div class="paper-plot__body"><div class="paper-plotly" id="mobility-activity" data-paper-plotly-spec="src/graphs/mobility2gdp.json" data-preview-image="src/previews/src_graphs_mobility2gdp.png" style="width: 100%; height: 360px;"></div></div><div class="paper-plot__footer"><div class="paper-plot__notes">Note: observed mobility corresponds to the monthly average of the index compiled by Google. IMACEC is the percentage change, relative to the same month of the previous year, of the monthly economic activity index reported by the Central Bank of Chile.</div></div></figure>
<figure class="paper-plot figure" style="width: 100%;"><div class="paper-plot__header"><div class="paper-plot__title">Mobility versus uncertainty</div><div class="paper-plot__units">(percentages, index)</div></div><div class="paper-plot__body"><div class="paper-plotly" id="mobility-depuc" data-paper-plotly-spec="src/graphs/mobility2uncertainty.json" data-preview-image="src/previews/src_graphs_mobility2uncertainty.png" style="width: 100%; height: 360px;"></div></div><div class="paper-plot__footer"><div class="paper-plot__notes">Note: observed mobility corresponds to the monthly moving average of the index compiled by Google. DEPUC from Becerra &amp; Sagner (2020).</div></div></figure>
</div>
<figure class="paper-plot figure" style="width: 100%;"><div class="paper-plot__header"><div class="paper-plot__title">Relationship between activity and mobility</div><div class="paper-plot__units">(percentages)</div></div><div class="paper-plot__body"><div class="paper-plotly" id="mobility-activity-coefficient" data-paper-plotly-spec="src/graphs/mobility2gdpCoefficient.json" data-preview-image="src/previews/src_graphs_mobility2gdpCoefficient.png" style="width: 100%; height: 360px;"></div></div><div class="paper-plot__footer"><div class="paper-plot__notes">Note: the chart shows the coefficient from a regression of activity on mobility plus a constant. This regression was estimated using 8-observation rolling windows.</div></div></figure>
<figure class="paper-plot figure" style="width: 100%;"><div class="paper-plot__header"><div class="paper-plot__title">Return to pre-pandemic mobility</div><div class="paper-plot__units">(percentage deviations from normal)</div></div><div class="paper-plot__body"><div class="paper-plotly" id="mobility" data-paper-plotly-spec="src/graphs/efficacy.json" data-preview-image="src/previews/src_graphs_efficacy.png" style="width: 100%; height: 360px;"></div></div><div class="paper-plot__footer"><div class="paper-plot__notes">Note: observed mobility corresponds to the moving average of the index compiled by Google. The 95% line corresponds to the mobility projection using information through December 20, 2020 and assuming vaccine effectiveness of around 95%. The 60% and 5% calculations are analogous to the 95% one.</div></div></figure>
<p>The observed mobility figures show that there are factors that have changed drastically relative to the base scenario.<sup>1</sup> These factors cannot be identified without assuming a vaccine efficacy and, therefore, are not estimated in our analysis. The contagion rate estimated when assuming 95% efficacy is 6 times higher than the maximum recorded before the vaccination process began. We interpret this as the model rejecting the 95%-efficacy hypothesis.</p>
<p>The vaccination chart illustrates a timing issue that the mobility counterfactuals do not separately model: the delay between inoculation and effective protection. Under the two-dose schedule assumed here, full protection takes about a month and a half from the first injection. The vaccination illustration should therefore not be read as a separate dose-by-dose model underlying the mobility projections.</p>
</section>
<section id="conclusion" class="level2">
<h2 class="anchored" data-anchor-id="conclusion">Conclusion</h2>
<p>Chile’s rapid vaccination campaign means that a return to normality may finally be getting close. But <strong>how close depends critically on how effective vaccines are at preventing contagion</strong>. If vaccination substantially weakens the link between mobility and infections, restrictions can be relaxed sooner and economic activity can recover faster. If it does not, the <em>pandexit</em> will take longer.</p>
<p>Our estimates suggest that the difference could be economically meaningful. Under the 95%-efficacy scenario, mobility returns to normal roughly a month and a half earlier than under the 60% scenario. Over the following six months, this translates into an average mobility gap of about 4.7 percentage points and, using our estimated relationship between mobility and activity, <strong>around 0.4–0.6 percentage points of annual GDP growth</strong>.</p>
<p>That last calculation should be taken with a grain of salt. The regressions relating mobility to economic activity are deliberately simple and leave out many factors that changed simultaneously during the pandemic—fiscal support, uncertainty, firms’ and households’ adaptation, external conditions, and the composition of restrictions, among others. These omitted variables may bias the estimated relationship between mobility and activity. <strong>The GDP figures should therefore be read as an illustration of the possible economic magnitude of a delayed <em>pandexit</em>, rather than as a causal estimate.</strong></p>
<p>The data arriving as the vaccination campaign progresses also give us reasons to be cautious about the most optimistic epidemiological scenario. Matching observed mobility under the assumption of 95% effectiveness against contagion requires a contagion rate around six times higher than anything estimated before vaccination began. Within the model, this makes the 95% scenario increasingly difficult to reconcile with what we are seeing.</p>
<p>The next few months will therefore be particularly informative. As vaccination advances and more data become available, we should learn much more about how strongly vaccines break the connection between mobility and contagion. <strong>That connection will help determine not only when Chile can return to normal, but also how much economic activity is lost on the way there.</strong></p>
<section id="our-world-in-data" class="level3">
<h3 class="anchored" data-anchor-id="our-world-in-data">Our World in Data</h3>
<iframe src="https://ourworldindata.org/explorers/coronavirus-data-explorer?zoomToSelection=true&amp;pickerSort=desc&amp;pickerMetric=population&amp;hideControls=true&amp;Metric=People+vaccinated+%28by+dose%29&amp;Interval=Cumulative&amp;Relative+to+Population=true&amp;Align+outbreaks=true&amp;country=BRA~CHL~FRA~DEU~ISR~GBR~USA~URY~ESP~ITA~ARE" loading="lazy" style="width: 100%; height: 600px; border: 0px none;">
</iframe>
<iframe src="https://ourworldindata.org/explorers/coronavirus-data-explorer?zoomToSelection=true&amp;pickerSort=desc&amp;pickerMetric=total_deaths&amp;hideControls=true&amp;Metric=Confirmed+deaths&amp;Interval=7-day+rolling+average&amp;Relative+to+Population=true&amp;Align+outbreaks=false&amp;country=BRA~CHL~FRA~DEU~ISR~GBR~USA~URY~ESP~ITA~ARE" loading="lazy" style="width: 100%; height: 600px; border: 0px none;">
</iframe>
</section>
</section>
<section id="references" class="level2">
<h2 class="anchored" data-anchor-id="references">References</h2>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-Rungcharoenkitkul2021" class="csl-entry">
<span class="smallcaps">Rungcharoenkitkul, P.</span> (2021): <a href="https://www.bis.org/publ/work932.htm">Macroeconomic Consequences of Pandexit</a>, BIS Working Papers, Bank for International Settlements.
</div>
</div>


</section>


<div id="quarto-appendix" class="default"><section id="footnotes" class="footnotes footnotes-end-of-document"><h2 class="anchored quarto-appendix-heading">Footnotes</h2>

<ol>
<li id="fn1"><p>The base projection scenario uses estimates with data through December 20, 2020. After that, all shock innovations are assumed to disappear.↩︎</p></li>
</ol>
</section></div> ]]></description>
  <category>COVID-19</category>
  <category>Economics</category>
  <guid>https://www.nicolasfranzpattillo.com/posts/covid19-vaccines/</guid>
  <pubDate>Sun, 01 Aug 2021 04:00:00 GMT</pubDate>
  <media:content url="https://www.nicolasfranzpattillo.com/posts/covid19-vaccines/src/cover.webp" medium="image" type="image/webp"/>
</item>
<item>
  <title>The Pandexit model: states, equations, and mobility choice</title>
  <dc:creator>Nicolas Franz-Pattillo</dc:creator>
  <link>https://www.nicolasfranzpattillo.com/posts/pandexit-model/</link>
  <description><![CDATA[ 






<p>This is a companion piece to <a href="../covid19-vaccines/index.html">COVID-19 Pandexit and the effects on economic activity</a>, covering the underlying epidemiological model in full.</p>
<p>This is a restatement of the core framework in <span class="citation" data-cites="Rungcharoenkitkul2021">Rungcharoenkitkul (2021)</span>, sections 2.1–2.2, checked against the author’s <a href="https://github.com/phurichai/covid19macro/blob/b2f89d7d45e5cfffcfb08e11c983f1c67433b845/codes/seir_simple.py">public implementation</a>. The notation below groups the six isolation equations to make the population accounting easier to follow.</p>
<p><strong>States and notation</strong></p>
<p>Each state is a number of people; one period is one day. Write <img src="https://latex.codecogs.com/png.latex?%5CDelta%20X_%7Bt+1%7D=X_%7Bt+1%7D-X_t">.</p>
<table class="caption-top table">
<colgroup>
<col style="width: 50%">
<col style="width: 50%">
</colgroup>
<thead>
<tr class="header">
<th style="text-align: left;">State</th>
<th style="text-align: left;">Interpretation</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?S_t"></td>
<td style="text-align: left;">Susceptible</td>
</tr>
<tr class="even">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?E_t"></td>
<td style="text-align: left;">Exposed, not yet infectious</td>
</tr>
<tr class="odd">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?I_t"></td>
<td style="text-align: left;">Infectious and able to transmit</td>
</tr>
<tr class="even">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?U%5ER_t,%5C%20Q%5ER_t,%5C%20H%5ER_t"></td>
<td style="text-align: left;">Isolated and eventually recovering: undetected, quarantined, hospitalised</td>
</tr>
<tr class="odd">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?U%5ED_t,%5C%20Q%5ED_t,%5C%20H%5ED_t"></td>
<td style="text-align: left;">Isolated and eventually dying, in the same three settings</td>
</tr>
<tr class="even">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?R_t"></td>
<td style="text-align: left;">Recovered and immune</td>
</tr>
<tr class="odd">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?D_t"></td>
<td style="text-align: left;">Cumulative deaths</td>
</tr>
<tr class="even">
<td style="text-align: left;"><img src="https://latex.codecogs.com/png.latex?V_t"></td>
<td style="text-align: left;">Effectively vaccinated and immune</td>
</tr>
</tbody>
</table>
<p>The twelve compartments sum to the initial population <img src="https://latex.codecogs.com/png.latex?N">, including deaths. The public code also tracks auxiliary series, such as cumulative detected cases; these are not additional mutually exclusive population compartments.</p>
<p><strong>Infection and vaccination</strong></p>
<p>Let <img src="https://latex.codecogs.com/png.latex?F_t"> be new exposures and <img src="https://latex.codecogs.com/png.latex?%5Cnu_t"> the daily flow of susceptible people gaining vaccine protection. With <img src="https://latex.codecogs.com/png.latex?%5Csigma"> the incubation-transition rate, <img src="https://latex.codecogs.com/png.latex?%5Cdelta"> the isolation-transition rate, and <img src="https://latex.codecogs.com/png.latex?%5Comega"> the immunity-loss rate:</p>
<p><img src="https://latex.codecogs.com/png.latex?%0AF_t=%5Cgamma_t%5Cfrac%7BS_tI_t%7D%7BN%7D,%5Cqquad%0A%5Cbegin%7Baligned%7D%0A%5CDelta%20S_%7Bt+1%7D&amp;=-F_t-%5Cnu_t+%5Comega(R_t+V_t),%5C%5C%0A%5CDelta%20E_%7Bt+1%7D&amp;=F_t-%5Csigma%20E_t,%5C%5C%0A%5CDelta%20I_%7Bt+1%7D&amp;=%5Csigma%20E_t-%5Cdelta%20I_t,%5C%5C%0A%5CDelta%20V_%7Bt+1%7D&amp;=%5Cnu_t-%5Comega%20V_t.%0A%5Cend%7Baligned%7D%0A"></p>
<p>Here <img src="https://latex.codecogs.com/png.latex?%5Cnu_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 <img src="https://latex.codecogs.com/png.latex?%5Comega=0"> rules out waning immunity. Flows must be bounded so no compartment becomes negative.</p>
<p><strong>Isolation, recovery, and deaths</strong></p>
<p>Let <img src="https://latex.codecogs.com/png.latex?q"> be detection probability, <img src="https://latex.codecogs.com/png.latex?h"> hospitalisation probability conditional on detection, and <img src="https://latex.codecogs.com/png.latex?p_t"> eventual death probability after infection. Define the allocation weights</p>
<p><img src="https://latex.codecogs.com/png.latex?%0Aw_U=1-q,%5Cqquad%20w_Q=q(1-h),%5Cqquad%20w_H=qh.%0A"></p>
<p>For each <img src="https://latex.codecogs.com/png.latex?j%5Cin%5C%7BU,Q,H%5C%7D">, the two equations</p>
<p><img src="https://latex.codecogs.com/png.latex?%0A%5Cbegin%7Baligned%7D%0A%5CDelta%20j%5ER_%7Bt+1%7D&amp;=%5Cdelta(1-p_t)w_jI_t-%5Crho_j%20j%5ER_t,%5C%5C%0A%5CDelta%20j%5ED_%7Bt+1%7D&amp;=%5Cdelta%20p_tw_jI_t-%5Cmu%20j%5ED_t%0A%5Cend%7Baligned%7D%0A"></p>
<p>represent six state transitions. The recovery rates satisfy <img src="https://latex.codecogs.com/png.latex?%5Crho_U=%5Crho_Q=%5Crho">, while hospital recovery uses <img src="https://latex.codecogs.com/png.latex?%5Crho_H">; <img src="https://latex.codecogs.com/png.latex?%5Cmu"> is the death-transition rate. The remaining two equations are</p>
<p><img src="https://latex.codecogs.com/png.latex?%0A%5Cbegin%7Baligned%7D%0A%5CDelta%20R_%7Bt+1%7D&amp;=%5Crho(U%5ER_t+Q%5ER_t)+%5Crho_H%20H%5ER_t-%5Comega%20R_t,%5C%5C%0A%5CDelta%20D_%7Bt+1%7D&amp;=%5Cmu(U%5ED_t+Q%5ED_t+H%5ED_t).%0A%5Cend%7Baligned%7D%0A"></p>
<p>Summing all twelve changes gives zero. The outcome superscripts are accounting categories, not assumptions that an individual’s outcome is observed in advance.</p>
<p><strong>Mobility and the policy rule</strong></p>
<p>Mobility <img src="https://latex.codecogs.com/png.latex?m_t%5Cin%5B-1,0%5D"> is a fractional deviation from normal: <img src="https://latex.codecogs.com/png.latex?-0.1"> means 10% below normal. Its contribution to transmission is exponential:</p>
<p><img src="https://latex.codecogs.com/png.latex?%0A%5Cgamma_t(m_t)=%5Cbeta_0%20e%5E%7B%5Cbeta_1m_t%7D+z_t,%0A%5Cqquad%20%5Cbeta_0,%5Cbeta_1%3E0.%0A"></p>
<p>The residual <img src="https://latex.codecogs.com/png.latex?z_t"> captures transmission changes not explained by mobility. The paper’s baseline projection uses a mean-reverting residual, <img src="https://latex.codecogs.com/png.latex?z_t=%5Cvarrho%20z_%7Bt-1%7D+%5Cvarepsilon_t">, with future innovations set to zero. This still allows the inherited residual to decay gradually.</p>
<p>Writing <img src="https://latex.codecogs.com/png.latex?s_t=S_t/N"> and <img src="https://latex.codecogs.com/png.latex?i_t=I_t/N">, the reduced policy block uses the mortality-loss measure</p>
<p><img src="https://latex.codecogs.com/png.latex?%0A%5Cwidetilde%20d_t(m)=p_t%5Cdelta%20i_t%0A%5Cleft%5B1-%5Cdelta+%5Cgamma_t(m)s_t%5Cright%5D.%0A"></p>
<p>This is the policy rule’s mortality measure, <strong>not</strong> observed daily deaths <img src="https://latex.codecogs.com/png.latex?%5CDelta%20D_%7Bt+1%7D/N">. The mobility decision balances it against the cost of restrictions:</p>
<p><img src="https://latex.codecogs.com/png.latex?%0Am_t%5E*%5Cin%5Cunderset%7B-1%5Cle%20m%5Cle0%7D%7B%5Coperatorname%7Bargmin%7D%7D%0A%5Cleft%5C%7B%5Cwidetilde%20d_t(m)%5E2+%5Cvarphi%20m%5E2%5Cright%5C%7D,%0A%5Cqquad%20%5Cvarphi%3E0.%0A"></p>
<p>An interior choice satisfies</p>
<p><img src="https://latex.codecogs.com/png.latex?%0A-%5Cvarphi%20m_t=%0A%5Cwidetilde%20d_t(m_t)%5C,%0A%5Cunderbrace%7Bp_t%5Cdelta%20i_t%20s_t%5Cbeta_0%5Cbeta_1e%5E%7B%5Cbeta_1m_t%7D%7D_%7B%5Cpartial%5Cwidetilde%20d_t/%5Cpartial%20m_t%7D.%0A"></p>
<p>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.</p>
<p>A useful summary of epidemic pressure is</p>
<p><img src="https://latex.codecogs.com/png.latex?%0A%5Cmathcal%20R_t%5E%7B%5Cmathrm%7Beff%7D%7D=%0A%5Cfrac%7B%5Cgamma_t%7D%7B%5Cdelta%7D%5Cfrac%7BS_t%7D%7BN%7D.%0A"></p>
<p>Reducing mobility lowers <img src="https://latex.codecogs.com/png.latex?%5Cgamma_t">; effective vaccination lowers <img src="https://latex.codecogs.com/png.latex?S_t">. Both can push this quantity below one, when the combined exposed-and-infectious pool starts shrinking.</p>
<p><strong>From mobility to activity</strong></p>
<p>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 <img src="https://latex.codecogs.com/png.latex?%5CDelta%20y_t%5Csimeq%5Cwidehat%20b_t%5CDelta%20m_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.</p>
<p>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.</p>
<section id="references" class="level2">
<h2 class="anchored" data-anchor-id="references">References</h2>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-Rungcharoenkitkul2021" class="csl-entry">
<span class="smallcaps">Rungcharoenkitkul, P.</span> (2021): <a href="https://www.bis.org/publ/work932.htm">Macroeconomic Consequences of Pandexit</a>, BIS Working Papers, Bank for International Settlements.
</div>
</div>


</section>

 ]]></description>
  <category>COVID-19</category>
  <category>Economics</category>
  <category>Methodology</category>
  <guid>https://www.nicolasfranzpattillo.com/posts/pandexit-model/</guid>
  <pubDate>Sun, 01 Aug 2021 04:00:00 GMT</pubDate>
</item>
</channel>
</rss>
