# Covid-19 briefing: extensions to the SIR model

*John Lewis*

The SIR model, first developed by Kermack and McKendrick (1927), remains the canonical epidemiological model. It a natural choice for economists seeking to model the interplay between economic and epidemiological dynamics. This briefing surveys some of the many adaptations to the basic SIR setup which have emerged in the epi-macro literature over the past six months. These have all been used to analyse issues such as lockdown policies, super-spreaders, herd immunity, hospital capacity and ‘test- and-trace’.

**The canonical SIR model**

The SIR model divides the population into three categories (or ‘compartments’ in epidemiology jargon): **S**usceptible, **I**nfected and **R**ecovered. It models individuals transitions between those states using a set of exogenously given transition rates which are related to the relative sizes of each group. And it assumes that the transition can only ever occur in ‘one direction’ (eg susceptible people may become infected, infected people may recover, but no other moves between groups are possible).

SIR based epi-macro models modify this basic framework by assuming these transitions between compartments are endogenous and related to individuals’ economic decisions (eg consumption or labour supply). It then bolts on a simple macro model to allow for feedback between the pandemic dynamics and the macroeconomy and *vice versa*.

Many epi-macro papers use this basic SIR model, either because they want to generate a simple benchmark (eg Eichenbaum Rebelo and Trabandt* *(2020)), and/or because the main modelling focus of the paper lies elsewhere. Over the past six months, other authors have added more epidemiological richness to epi-macro models by extending the SIR framework in various ways.

**Uncertainty over infection status**

Forsyth (2020) introduces a plausible informational friction: some infected people are asymptomatic, and some susceptible people display Covid-like symptoms though they are not infected. This setup allows for a comparison of policies targeted at those with symptoms against the alternative of a uniform lockdown. The former are less costly in terms of output, because fewer agents have to isolate, but result in some transmission via asymptomatic agents. Calibration on UK data in the paper suggests mitigation measures reduce fatalities by 39.1% over 18 months. Under uniform lockdown, the GDP hit in 2020 is estimated at 21.4%, but if policies are conditional on symptoms, GDP costs can be reduced without increasing fatalities.

**Variation in ‘R’**

Other papers have relaxed the assumption that the probability of a susceptible individual becoming infected is uniform across the whole population. This can be done by introducing variation in either the contact rates between people, or in the probability of transmission when two individuals meet.

Ellison (2020) uses epidemiological modelling advances of the 80s and 90s to relax the ‘uniform contact rates’ assumption in two ways. First, agents are split into sub-groups with different activity levels, implying different probabilities of encountering others. Second, agents are more likely to encounter those within their own group than outsiders. He shows that key population parameters, like herd immunity thresholds and composite R, are not merely averages of the underlying groups, but depend on the range and distribution of activity levels across the population. He shows that R can decline faster when meeting is more heterogeneous, and so models based on uniform meeting assumptions may overstate the initial impact of lockdown measures and understate how fast the virus would have spread absent any lockdown measures.

Holden and Thornton (2020) allow for variation in the probability of transmission across individual pairs, by making the pair-specific reproduction rate a random variable. This is a way of modelling ‘super-spreaders’ who are much more infectious than average. If by ‘bad luck’ more of the initially infected are super-spreaders, this raises the initial case count, and has long-lasting effects as cases compound from that higher base. The role of ‘luck’ is much greater early on when fewer people are infected, whereas later on the ‘law of averages’ tends to kick in and R converges to its population average. As a result, otherwise identical populations can exhibit very different paths simply due to chance. In this model, optimal policy depends not just on the average R, but crucially on the proportion of its distribution that lies above one. The authors show that the relative efficacy of measures to reduce the transmissibility of a virus (eg facemasks) vs. lowering contact rates (eg shelter-in-place orders) is affected by the distribution of R.

**Waning immunity**

In the canonical SIR model, immunity is permanent once acquired, because recovered agents are no longer susceptible. This assumption can be relaxed in the so-called ‘SIRS’ model, where recovered individuals can lose immunity and move back into the susceptible category. This effectively assumes that immunity can wane to some extent: consistent with the documented cases of Covid reinfection (eg Tillett* et al* (2020)).

The SIRS model of Çenesiz and Guimarães (2020) implies that the shorter-lived immunity is, the harder it is to achieve herd immunity, and the greater and longer the period of social distancing required. Waning immunity has relatively little role early on in pandemics, because policy prescriptions are similar regardless of assumptions about longevity of immunity. But over time the difference between the two models’ results grows larger, because if immunity is permanent, the stock of immune agents steadily cumulates. At longer horizons, therefore, even relatively small changes in lockdown tightness can have large implications.

They then also allow for richer determinants of immunity, include the possibility that previously infected agents are less likely to have severe forms of Covid, and consider optimal policy if a future vaccine is expected. In short, the faster immunity wanes, and the more distant a vaccine is, the greater the impact of waning immunity on optimal policy, because the pandemic becomes longer-lived.

**Additional compartments**

Many papers have extended the number of compartments the population is divided into, (well) beyond the three in the classic SIR model. And, by allowing for a richer set of transition equations, they have removed the requirement that individuals can only go through the groups in a set sequence. Favero (2020) develops a model with a compartment for hospitalised patients. This allows the model to make direct predictions about the numbers hospitalised and also allows for explicit consideration of the role of ICU capacity. When admissions exceed capacity, those additional patients are termed as constrained, and face a higher probability of death because they cannot be treated to the same standard as others.

Favero (2020) shows the addition of this constraint can explain the much higher case fatality rate (CFR) in the Lombardy region of Italy. More broadly, he analyses the role of the capacity constraint, and the risk of ICU saturation, and shows that strategies that involve large numbers of simultaneous infections are associated with much higher death rates.

Giordano *et al *(2020) develop a model which adds five extra compartments. Some capture different possibilities for those who have the disease, in terms of having symptoms or not, and being detected or not. Other compartments are for agents who are acutely ill, and for two final states where agents either end up dead or fully healed.

This richer model allows for more parameters, including different fatality rates and differential transmission of symptomatic cases. It also helps explain misperceptions of the CFR and speed of spread. The authors conclude that lockdown measures can only be lifted when widespread testing and contact tracing is available, and that a combination of both tools is needed to reduce cases.

But more complexity may not always be preferable. Roda *et al* (2020) compare the SIR model with a SEIR model, which has an extra category capturing exposed agents, who have the disease but are not yet infectious. This creates two additional parameters, governing the latent period (how long before those who get the disease become infectious) and the initial share of the population in the exposed category. In practice, it is difficult to identify the parameters empirically, because the model cannot distinguish between a case with a long latent period and a low initial share, or a short latent period and a high initial share. Using Akaike information criteria, the authors show that the small increase in model fit for the SEIR model versus the SIR model does not compensate for the additional complexity introduced by these two extra parameters.

**Concluding remarks**

The original SIR model has stood the test of time remarkably well. In its simplest form it is able to capture key features of pandemics, and this makes it a natural choice in epi-macro for modellers seeking to bolt on an economic model while keeping their model simple. In addition, the compartmental structure allows for relatively easy incorporation of further complexity. These extensions yield important insights about the relative merits of different lockdown policies, the role of super-spreaders in determining the path of a pandemic, the additional difficulty of achieving herd immunity, the constraint posed by hospital capacity and the role of test and trace.

*John Lewis works in the Bank’s Research Hub.*

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