Abstract
Motivated by the complex control issues raised by COVID-19, this article investigates the optimal control of an epidemic of a Susceptible-Infective-Removed-Susceptible (SIRS) infection when two main instruments – social distancing and vaccination – are available to the social planner. The resulting two-control optimal problem is set within a parsimonious economic model in which the planner minimizes an objective function that weights epidemiological and economic costs by choosing the extent of social distancing in a first stage of the epidemic and both social distancing and the tax rate to finance vaccination during a subsequent stage. The article shows (i) how to combine the two policy tools depending on the planner’s degree of rationality; (ii) the importance of the planner’s expectation on the date of vaccine arrival, and how the expected efficacy of the vaccine can affect the optimal social distancing trajectory in the pre-vaccination period, and (iii) the use of social distancing as the only instrument to optimally control the epidemic when the vaccine supply is rationed.
Data Availability
No datasets were generated or analyzed during the current study.
Notes
In our model, consumption has been introduced because output represents goods and services produced and entirely distributed. Since there is no saving, households can only consume their disposable income. Consequently, the loss of output due to the lockdown and/or the taxation of labor income – which reduces disposable income – implies a reduction in consumption. Hence, in the objective function, including the output loss is equivalent to including the loss of consumption.
In what follows, u(t) can be determined through an optimization process or follow a non-optimal trajectory (naive behavior).
See the discussion below for the economic reasons of using quadratic costs.
There may be other costs associated with infection, such as hospitalization costs paid primarily (in Italy) from public funds, which, however, in the interest of a parsimonious model, we do not consider. Alternatively, this part of the objective may reflect a proxy for the evaluation of worsening health due to the infection.
For the sake of clarity, \(T_{SD}\), \(T_{V}^{e}\), \(T_{V}\) and T respectively represent the beginning of Stage 1, the planner’s expected date of vaccine arrival (end of Stage 1), the actual date of vaccine arrival (beginning of Stage 2) and the end of Stage 2.
See Buratto et al. (2022) for an approach in which this date is influenced by R&D expenditure and is treated by the social planner as a random variable with a given distribution.
The scrap value function of the first period (Stage 1) has been removed from the social planner’s objective, as it is only meaningful in the disjoint case.
The ensuing optimality conditions for Stage 1 remain unchanged except for the last two lines in (19), as do all the optimality conditions in Stage 2 except for those in the second-to-last line in (17). The transversality conditions for Stage 1 and initial conditions for Stage 2 given in the last two lines of (19) and in the second last line of (17) must be replaced with appropriate continuity conditions.
The trajectories of the sub-optimal tax rate used to fund the organization and administration of vaccines (associated with varying levels of vaccine efficacy) are similar, but the required level of taxation increases as the vaccine’s efficacy decreases.
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Acknowledgements
The authors acknowledge Raouf Boucekkine and an anonymous journal reviewer for insightful comments. The authors are also indebted to participants of the (1) 11th MDEF conference held at the University of Urbino, Italy (June 2023), and (2) 64th Riunione Scientifica Annuale (RSA) of the Italian Economic Association (SIE) held at the Gran Sasso Science Institute, GSSI, L’Aquila, Italy (October 2023). The usual disclaimer applies.
Funding
Luca Gori’s research was supported by the University of Pisa under the “PRA – Progetti di Ricerca di Ateneo” (Institutional Research Grants) – Project No. PRA_2020_64 “Infectious diseases, health and development: economic and legal effects.” Simone Marsiglio’s research was supported by the University of Pisa under the “PRA – Progetti di Ricerca di Ateneo” (Institutional Research Grants) – Project No. PRA_2020_79 “Sustainable development: economic, environmental and social issues.” Luca Gori and Simone Marsiglio’s research was supported by the Italian Ministry of University and Research as part of the PRIN 2022 (Grant Protocol Number 20227339KC) program. Mauro Sodini also acknowledges the support by VSB-TUO project SP2021/15 and the Science without Borders 2.0 project, CZ.02.2.69/0.0/0.0/18_053/0016985, within the Operational Programme Research, Development and Education. Piero Manfredi carried out this work when he was the main advisor of the ERC project IMMUNE.
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Appendix
Appendix
This Appendix presents a simulation exercise related to the naïve control during Stage 1 and the optimal control during Stage 2. The social distancing effort u(t) during the initial phase of the epidemic (Stage 1) is modeled as a reactive feedback law rather than a globally optimized trajectory. This approach represents the behavior of a policy-maker who adjusts restrictions based on immediate observations of epidemiological indicators, without long-term predictive information about vaccine availability. The lockdown intensity under the naïve control during Stage 1 is defined by the following state-dependent rule:
The control logic expression in (24) integrates two distinct signals that characterize the evolution of the contagion: (1) the term \(\alpha ( \mathcal {R}_{t}-1)\) acts as a proportional correction based on the effective reproduction number, where \(\alpha >0\) is a transmission stability parameter. It increases the control effort when the virus is in an expansive phase \(\mathcal {R}_{t}>1\)) and reduces it when the transmission falls below the critical threshold, seeking to stabilize the system at a sub-critical equilibrium; the term \(\beta _{g}\frac{\dot{I}(t)}{I(t)}\) accounts for the relative growth rate of the infected population, where \(\beta _{g}>0\) is a contagion momentum. This serves as a precautionary mechanism that allows the policy-maker to react to the "speed" of the outbreak. It ensures that restrictions are intensified during periods of exponential acceleration, providing a buffer to prevent the healthcare system from reaching capacity.
To ensure the physical and socio-economic consistency of the model, the control variable is nested within \(\max \) and \(\min \) operators. This mathematical structure guarantees that u(t) remains bounded within the interval [0, 1]. Specifically, it prevents the model from generating negative restrictions – which would lack physical meaning – and caps the distancing effort at its theoretical maximum, representing the complete ban of non-essential social interactions. This Stage 1 naïve policy serves as a behavioral baseline, capturing the period of incremental governance and high uncertainty that typically precedes the transition to the optimal control framework triggered by the arrival of pharmaceutical interventions in Stage 2. The total cost of the policy increases dramatically compared to the joint and disjoint optimal control problems studied in the main text, but the number of people infected during the first phase is greatly reduced. This makes clear the trade-off between protecting lives and the economy. In the absence of a plan allowing the authority to rationally choose an optimal trajectory for the control variables, protecting human lives through lockdowns significantly increases the economic cost of the policy implemented in the two stages, although social distancing measures can be almost entirely relaxed thanks to the activation of the vaccine campaign (Fig. 9).
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Gori, L., Manfredi, P., Marsiglio, S. et al. A parsimonious model of optimal social distancing and vaccination during an outbreak. J Evol Econ 36, 65 (2026). https://doi.org/10.1007/s00191-026-00984-3
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DOI: https://doi.org/10.1007/s00191-026-00984-3