The COVID-19 shock and its far-reaching repercussions have not only gripped the personal and professional lives of billions of people, but have also transformed the economic profession: epidemiology has become of general interest to economists. As a result, the profession as a whole has begun to intensively analyze how epidemiological dynamics interact with economic choices and government interventions.

By far the most important epidemiological framework underlying this recent wave of work has been the “SIR” framework in the tradition of Kermack and McKendrick (1927).^{1}The model specifies laws of motion for the population shares of three groups, which differ in their health status: “susceptible”, “infected” or “infectious” and “eliminated” (i.e. deceased or fully recovered and immune). The laws of motion describe how, over time, sensitive households become infected and eventually recover or die.

Modeling the interaction between epidemiological dynamics, economic choices and government interventions requires an economic layer in addition to the differential equations that describe the SIR model. Since the analysis of such a two-level framework is far from trivial, researchers are faced with methodological questions. In our recent article (Gonzalez-Eiras and Niepelt 2020b), we address some of these questions.

## Canonical SIR model

Figure 1 illustrates the evolution of susceptible (variable x

**Figure 1** Dynamics in the canonical SIR model: x

Analysis of a two-layer framework with economic choices built into the SIR model generally requires numerical resolution methods. The absence of closed solutions can obscure key channels for the main outcomes of interest. Researchers are therefore faced with a methodological question: can the epidemiological framework be modified or simplified without substantial loss in order to make the economic analysis more manageable?

## Modified SIR model

In Gonzalez-Eiras and Niepelt (2020b), we argue that the answer to this question is yes. First, we compare the canonical SIR model with a modified framework taken from Bailey (1975) which differs only very little from the canonical SIR model and yet offers advantages.

On the one hand, it is more manageable. For the exogenous pathways of the parameters governing infection, healing, and death rates (which in turn could be functions of policy), the system of differential equations admits a closed-form solution (see Bohner et al. 2019).^{3} On the other hand, Bailey’s (1975) framework offers different (and as some epidemiologists argue, more plausible) implications for scale effects in the infection process (e.g. Hethcote 2000: 602). By implication, the modified framework is less prone to the risk of overestimating the effect of policy interventions that affect both susceptible and infected people (e.g. Alvarez et al. 2020, Gonzalez-Eiras and Niepelt 2020a, Acemoglu et al. 2020).

The modified SIR model does not require recalibration. FIG. 2 illustrates the dynamics (in red) under the same assumptions on the values of the parameters as before. The blue diagrams representing the dynamics in the canonical SIR model are identical to the blue diagrams in FIG. 1. It is noted that during the early phase of the epidemic, the dynamics predicted in the two SIR models are very similar. Indeed, given the uncertainty surrounding the epidemiological parameters, the two predictions are indeed indistinguishable.

**Figure 2** Dynamics in canonical (blue) and modified (red) SIR models: x

Likewise, infections (indicated by the solid line), which determine stress in the health care system and are therefore a key variable from the decision-maker’s perspective, peak at almost the same time in both models, albeit at low levels. different levels. The two models therefore make the same prediction as to when hospital capacities are most in demand.

However, there is also a significant difference between the two models, which relates to the number of sensitive and distant people in the long term. In the canonical model, the share of the sensitive population is strictly positive, while in the modified SIR model it converges to zero. The two models therefore have different implications for optimal policy when the government’s goal depends on these long-term population shares, which seems plausible.

## Hybrid SIR model

In this context, we offer a hybrid model that combines the main advantages of the two previous specifications. The hybrid model simply introduces another group of people, the “lucky ones,” who start out healthy and, for exogenous reasons, are never affected by the epidemic. The group of susceptible or lucky people in the hybrid model should be interpreted as the counterpart of the susceptible group in the canonical model. With this minimal addition (which does not increase the complexity of the model), the only remaining downside of the hybrid model is that, unlike the canonical model, the long-term share of the lucky ones is exogenous.

The hybrid SIR model does not require recalibration. Figure 3 illustrates the dynamics (in black) when we stipulate a 12% share of “lucky” people. The blue and red timetables represent the dynamics in the canonical and modified SIR models, respectively. During the early phase of the epidemic, the dynamics predicted in the three models are very similar. Towards the end, by construction, the dynamics of the hybrid model resemble that of the canonical model. Based on the propositions we discuss in our article, we can exactly date the peaks of infection in the hybrid model.

**figure 3** Dynamic in canonical (blue), modified (red) and hybrid (black) SIR models: x

## Logistics model

For the purposes of economic analysis, a major drawback of the three SIR models is that they have two state variables. Reducing the number of state variables to one promises to simplify the model, making closed solutions of integrated economic programs more likely. We therefore consider a simplified model with one state variable – the logistic model – which captures many essential aspects of SIR models but offers more flexibility and similar degrees of “realism” and flexibility.^{4}

To get the logistic model as a special case of the SIR models, we set the death and cure rates to zero. It is much less restrictive than it seems at first glance. In particular, this does not imply that the logistics model cannot capture the costs of deaths or infections. On the contrary, to represent such costs, it is not necessary to explicitly account for the deceased population or the stock of people currently infected. Rather, it suffices to take into account the flows induced by the infection from the susceptible population and to associate costs with these flows.

Due to the reduced number of state variables, the logistic model only includes two groups in addition to the “lucky” group, denoted by f

Figure 4 illustrates the dynamics in the logistics model (in green). The blue and black schedules (representing the dynamics in the canonical and hybrid SIR models, respectively) are identical to the schedules in the previous figures. Obviously, the paths predicted by the logistic model closely mimic the corresponding paths in the SIR models.

**Figure 4** Dynamics in the canonical (blue) and hybrid (black) SIR models and in the logistic model (green). SIR models: x

## Conclusion

Researchers interested in the intersection of epidemiology and economics could usefully use variations of the canonical SIR model. The logistics model can be particularly promising for simplifying epidemiological and economic analyzes, without substantial loss in terms of “realism” or flexibility.

## The references

Acemoglu, D, V Chernozhukov, I Werning and MD Whinston (2020), “A multi-risk SIR model with optimally target lockdown”, NBER working paper 27102.

Alvarez, F, D Argente and F Lippi (2020), “A simple planning problem for COVID-19 Lockdown”, NBER Working Paper 26981.

Atkeson, A (2020), “What will be the economic impact of COVID-19 in the United States? Rough Estimates of Disease Scenarios ”, NBER Working Paper 26867.

Bailey, NTJ (1975), *The mathematical theory of infectious diseases and its applications*, New York: Hafner Press.

Bohner, M, S Streipert and DFM Torres (2019), “Exact solution to a dynamic SIR model”, *Nonlinear analysis: hybrid systems* 32: 228-238.

Eichenbaum, MS, S Rebelo and M Trabandt (2020), “The macroeconomics of epidemics”, NBER Working Paper 26882.

Gonzalez-Eiras, M and D Niepelt (2020a), “On optimal ‘containment’ during an epidemic”, *Economy of Covid 7*: 68-87.

Gonzalez-Eiras, M and D Niepelt (2020b), “Treatable epidemiological models for economic analysis”, CEPR Discussion Paper 14791.

Harko, T, FS Lobo and M Mak (2014), “Exact analytical solutions of the susceptible-infected-recovered (SIR) epidemic model and the SIR model with equal death and birth rates”, *Applied Mathematics and Calculus* 236: 184-194.

Hethcote, HW (2000), “The Mathematics of Infectious Diseases”, *SIAM review* 42 (4): 599-653.

Kermack, WO and AG McKendrick (1927), “A contribution to the mathematical theory of epidemics”, *Acts of the Royal Society*, Series A 115 (772): 700–721.

## End Notes

^{1} The first articles in recent literature merging epidemiology and economics include Atkeson (2020) and Eichenbaum et al. (2020). In a few days, the number of articles increased considerably, with many contributions being published in the CEPR* Covid economy: articles verified and in real time* initiative. For epidemic simulation tools based on variants of the SIR model, see for example see here.

^{2} For a detailed description of our calibration strategy, see Gonzalez-Eiras and Niepelt (2020b).

^{3} See Harko et al. (2014) for a solution strategy for the canonical SIR model.

^{4} See Gonzalez-Eiras and Niepelt (2020a) for an application. Another model treatable with a state variable is the “SIS” model which is used to represent the dynamics of infection without immunity (eg Hethcote 2000).