The COVID-19 shock and its broad repercussions has not only gripped the personal and professional lives of billions of people but has also transformed the economics profession: Epidemiology has become of general interest for economists. As a consequence, the wider profession has started to intensively analyse how epidemiological dynamics interact with economic choices and government interventions.

By far the most prominent 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 with respect to their health status: ‘Susceptible’, ‘infected’ or ‘infectives’, and ‘removed’ (that is, either deceased or fully recovered and immune). The laws of motion describe how, over time, susceptible households are infected and eventually recover or die.

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

## Canonical SIR model

Figure 1 exemplifies the evolution of susceptible (variable x(t), indicated by the dotted line), infected (variable y(t), indicated by the solid line), and removed (variable z(t), indicated by the dashed line) persons in the canonical SIR model. To generate the depicted paths, we stipulate commonly assumed values for the epidemiological parameters governing the infection, death, and recovery rates in the case of COVID-19.^{2} Starting from a tiny share in mid-March 2020 (t=0), the model predicts the stock of infected persons (indicated by the solid line) to peak in July 2020. Only 12% of the population are predicted to escape infection in the long run.

**Figure 1** Dynamics in the canonical SIR model: x(t) (dotted), y(t) (solid), and z(t) (dashed).

The analysis of a two-layered framework with economic choices embedded in the SIR model typically requires numerical solution methods. The lack of closed-form solutions may obscure which channels are key for the central results of interest. Researchers are therefore confronted with a methodological question: Can the epidemiological framework be modified or simplified without substantive loss in order to render the economic analysis more tractable?

## Modified SIR model

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

On the one hand, it is more tractable. For exogenous paths of the parameters governing infection, recovery, and death rates (which could in turn 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 process of infections (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 persons (e.g. Alvarez et al. 2020, Gonzalez-Eiras and Niepelt 2020a, Acemoglu et al. 2020).

The modified SIR model does not require new calibration. Figure 2 illustrates the dynamics (in red) under the same assumptions about parameter values as before. The blue schedules representing the dynamics in the canonical SIR model are identical to the blue schedules in figure 1. We note that during the early phase of the epidemic, the predicted dynamics in the two SIR models are very similar. In fact, given the uncertainty surrounding the epidemiological parameters, the two predictions are effectively indistinguishable.

**Figure 2** Dynamics in the canonical (blue) and modified (red) SIR models: x(t) (dotted), y(t) (solid), and z(t) (dashed).

Similarly, infections (indicated by the solid line), which determine the stress in the healthcare system and thus constitute a key variable from a policymaker's perspective, peak at nearly the same time in both models, although at different levels. The two models therefore make the same prediction as to when hospital capacities are in highest demand.

However, there is also a significant difference between the two models, which concerns the number of susceptible and removed persons in the long run. In the canonical model, the share of the susceptible 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 objective depends on these population shares in the long run, which seems plausible.

## Hybrid SIR model

Against this background, we propose a hybrid model which combines the key advantages of the previous two specifications. The hybrid model simply introduces another population group, the ‘lucky’, who start out healthy and, for exogenous reasons, are never affected by the epidemic. The group of susceptible or lucky persons 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 drawback of the hybrid model is that, unlike in the canonical model, the long-run share of lucky persons is exogenous.

The hybrid SIR model does not require new calibration. Figure 3 illustrates the dynamics (in black) when we stipulate a share of 12% ‘lucky’ persons. The blue and red schedules represent the dynamics in the canonical and modified SIR models, respectively. During the early phase of the epidemic, the predicted dynamics in the three models are very similar. Towards the end, by construction, the dynamics in the hybrid model resemble those in the canonical model. Based on propositions that we discuss in our paper, we can exactly date peak infections in the hybrid model.

**Figure 3** Dynamics in the canonical (blue), modified (red), and hybrid (black) SIR models: x(t) (dotted, including “lucky”), y(t) (solid), and z(t) (dashed).

## Logistic model

For the purpose of economic analysis, a main drawback of the three SIR models lies in the fact that they feature two state variables. Reducing the number of state variables to one holds the promise of simplifying the model, rendering closed-form solutions of embedded economic programs more likely. We therefore consider a streamlined model with one state variable—the logistic model—which captures many of the essential aspects of SIR models but offers more tractability and similar degrees of ‘realism’ and flexibility.^{4}

To obtain the logistic model as a special case of the SIR models, we set the death and recovery rates equal to zero. This is much less restrictive than it appears at first sight. In particular, it does not imply that the logistic model cannot capture costs of deaths or infections. On the contrary, to represent such costs, there is no need to explicitly account for the deceased population or the stock of currently infected persons. Rather, it suffices to account for infection-induced flows 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, indicated by f(t) and s(t). The role of the stock of infected persons in the SIR models is now played by the flow of persons from the first to the second group, indicated by the time derivative s’(t). Using propositions discussed in our paper, one can easily relate the logistic and SIR models, and calibrate the former in order to match peak infection rates (or other statistics) implied by the latter.

Figure 4 illustrates the dynamics in the logistic 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. Evidently, 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(t) (dotted, including “lucky”), y(t) (solid), and z(t) (dashed). Logistic model: f(t) (dotted, including “lucky”), s’(t) (solid, scaled), and s(t) (dashed).

## Conclusion

Researchers interested in the intersection of epidemiology and economics might usefully employ variants of the canonical SIR model. The logistic model may hold particular promise for simplifying epidemiologic-economic analyses, without substantive loss in terms of ‘realism’ or flexibility.

## References

Acemoglu, D, V Chernozhukov, I Werning and M D Whinston (2020), “A multi-risk SIR model with optimally targeted 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 US? Rough estimates of disease scenarios”, NBER Working Paper 26867.

Bailey, N T J (1975), *The Mathematical Theory of Infectious Diseases and its Applications*, New York: Hafner Press.

Bohner, M, S Streipert and D F M Torres (2019), “Exact solution to a dynamic SIR model”, *Nonlinear Analysis: Hybrid Systems* 32: 228–238.

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

Gonzalez-Eiras, M and D Niepelt (2020a), “On the optimal `lockdown' during an epidemic”, *Covid Economics 7* [4]: 68–87.

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

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

Hethcote, H W (2000), “The mathematics of infectious diseases”, *SIAM Review* 42(4): 599–653.

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

## Endnotes

^{1} Early papers in the recent literature merging epidemiology and economics include Atkeson (2020) and Eichenbaum et al. (2020). Within days, the number of papers has grown substantially, with many contributions being published in CEPR's* Covid Economics: Vetted and Real-Time Papers [5]* initiative. For epidemic simulation tools based on variants of the SIR model, see for instance see here [6].

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

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

^{4} See Gonzalez-Eiras and Niepelt (2020a) for an application. Another tractable model with one state variable is the ‘SIS’ model that is used to represent infection dynamics without immunity (e.g. Hethcote 2000).