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Social interactions and the prophylaxis of SI epidemics on networks Géraldine Bouveret, Antoine Mandel To cite this version: Géraldine Bouveret, Antoine Mandel. Social interactions and the prophylaxis of SI epi- demics on networks. Journal of Mathematical Economics, Elsevier, 2021, 93, pp.102486. 10.1016/j.jmateco.2021.102486. halshs-03165772 HAL Id: halshs-03165772 https://halshs.archives-ouvertes.fr/halshs-03165772 Submitted on 17 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Social interactions and the prophylaxis of SI epidemics on networkssa G´eraldineBouveretb Antoine Mandel c March 17, 2021 Abstract We investigate the containment of epidemic spreading in networks from a nor- mative point of view. We consider a susceptible/infected model in which agents can invest in order to reduce the contagiousness of network links. In this setting, we study the relationships between social efficiency, individual behaviours and network structure. First, we characterise individual and socially efficient behaviour using the notions of communicability and exponential centrality. Second, we show, by computing the Price of Anarchy, that the level of inefficiency can scale up to lin- early with the number of agents. Third, we prove that policies of uniform reduction of interactions satisfy some optimality conditions in a vast range of networks. In setting where no central authority can enforce such stringent policies, we consider as a type of second-best policy the implementation of cooperation frameworks that allow agents to subsidise prophylactic investments in the global rather than in the local network. We then characterise the scope for Pareto improvement opened by such policies through a notion of Price of Autarky, measuring the ratio between social welfare at a global and a local equilibrium. Overall, our results show that individual behaviours can be extremely inefficient in the face of epidemic propaga- tion but that policy can take advantage of the network structure to design welfare improving containment policies. Keywords: Network, Epidemic Spreading, Public Good, Price of Anarchy JEL codes: D85, D62, I18 aAntoine Mandel acknowledges support from ITN ExSIDE European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 721846. Declarations of interest: none bNanyang Technological University, School of Physical and Mathematical Sciences, 21 Nanyang Link, 637371 Singapore, Email: [email protected] (corresponding author) cParis School of Economics-Universit´eParis 1 Panth´eon-Sorbonne, Centre d'Economie´ de la Sor- bonne, 106 Boulevard de l'hopital, 75013, Paris, France, Email: [email protected] 1 1 Introduction In the context of the 2020 COVID-19 pandemic, very strong policy measures have been implemented to contain epidemic diffusion. State of emergency has been declared in certain countries and certain civil liberties (e.g. freedom of assembly) have been suspended. The implementation of such stringent policies, labelled as social distancing measures, has been justified by the role of social interactions in epidemic diffusion. In economic terms, the premise is that individual behaviour is extremely inefficient in the presence of disease/network externalities. Yet, there is, to our knowledge, no normative analysis of the challenges posed by the containment of epidemic spreading in a network. This is the issue we address in this paper. The containment of epidemic processes defines a specific class of externality prob- lems: through prophylactic investment, agents can reduce their own contamination risk but also reduce the risk of contagion of their peers in the network. The external effect hence created has certain features of a public good as the investment of each agent benefits to all the agents to whom it is connected. However, the magnitude of the effect depends on the specific connectivity between each pair of agents and thus on the structure of the network. In this setting, our first aim is to characterise, as a function of the network structure, individual and socially efficient behaviours. Second, we measure, using the notion of Price of Anarchy (PoA), the inefficiency induced by individual behaviours. Third, we investigate policy measures that can be implemented to overcome these inefficiencies. We place ourselves in a setting where the network structure is given, each agent can be initially contaminated with a certain probability, and contagion spreads through network links proportionally to their contagiousness. Once infected, agents remain so permanently, i.e. we consider a susceptible/infected type of model according to the epidemiological terminology. In this context, agents aim at minimising their probability of contagion before a given date. In a narrow interpretation, this date can be seen as the expected date at which a treatment will be available. In a broader sense, the objective of each individual is to reduce the speed of incoming epidemic propagation. We assume that agents can invest in the network to reduce the speed of contagion. More precisely, they can decrease the contagiousness of links at a fixed linear cost. As the impact of individual investments depends on global contagiousness, and hence on the investment 2 of other players, the situation defines a non-cooperative game. We consider two variants of the game. The local game in which an agent can only invest in the links through which it is connected. The global game in which an agent can invest in each link of the network. The local game naturally applies to settings where agents are individuals that can take individual and costly measures to limit their social interactions. The global game corresponds to a more complex setting where agents are usually organisations (regions, countries) that are involved in a scheme that allows one agent to subsidise, directly or indirectly, the investment of other agents in the reduction of contagiousness. Our main results characterise the relationships between social efficiency, individual behaviours and network structure. First, we show that individually rational and so- cially efficient behaviours can be characterised using the notions of communicability and exponential centrality (Estrada and Hatano, 2008; Estrada and Higham, 2010). It is individually rational to invest in a link proportionally to the communicability be- tween the investor and the edges of the link while it is socially efficient to invest in a link proportionally to the total communicability/exponential centrality of its edges. Second, we derive a quantitative measure of the inefficiency induced by individual be- haviours using the notion of PoA. We show that in worst cases the level of inefficiency can scale up to linearly with the number of agents. This strongly calls for public policy interventions. In this respect, we show the "-optimality of a policy of uniform reduction of interactions in a wide range of networks. This latter result provides normative foun- dations for the social distancing policies implemented during the COVID-19 pandemic. The implementation of such policies nevertheless requires the existence of an authority with sufficient legitimacy to implement such coercive measures. It can be thus imple- mented in a domestic context but is much harder to implement at the global scale, unless all agents/countries have individual incentives to do so. If this is not the case, we regard the shift from a local to a global game as a type of second-best policy. In the latter game, agents can subsidise investments towards contagiousness reduction in the global rather than in the local network. The scope for Pareto improvement gener- ated by such policies is then characterised through a notion of Price of Autarky (PoK), which assesses the ratio between social welfare at a global and a local equilibrium. We derive a lower bound on this PoK, as a function of the network structure and thus give sufficient conditions under which a shift to the global game actually improves social welfare. Overall, our results underline the possible extreme inefficiency of individual 3 behaviours to limit epidemic propagation, but also the possibility to design efficient containment policies taking into account the network structure. The remaining of this paper is organised as follows. Section2 reviews the related literature. Section3 introduces epidemic dynamics as well as our behavioural model of the containment of epidemic spreading. Section4 provides our main results on the relationship between individual behaviours, social efficiency and network structure. Sec- tion5 investigates the social efficiency of policy measures aiming at reducing epidemic diffusion. Section6 concludes. All proofs are given in the appendix. 2 Related literature The paper builds on the very large literature on the optimal design and defense of net- works (see, e.g., Bravard et al.(2017)) and on epidemic spreading in networks. The latter literature has been extensively reviewed in Pastor-Satorras et al.(2015) and gen- erally combines an epidemiological
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