Spreading of an infectious disease between different locations
SSpreading of an infectious disease betweendifferent locations ∗ Alessio Muscillo a , Paolo Pin b , and Tiziano Razzolini a,ca Department of Economics and Statistics, University of Siena b Department of Decision Sciences, IGIER and BIDSA, Bocconi University c IZA and LABORThis version: December 2018[here you find an updated version]
Abstract
The endogenous adaptation of agents, that may adjust their local contactnetwork in response to the risk of being infected, can have the perverse effectof increasing the overall systemic infectiveness of a disease. We study adynamical model over two geographically distinct but interacting locations,to better understand theoretically the mechanism at play. Moreover, weprovide empirical motivation from the Italian National Bovine Database, forthe period 2006-2013.
JEL classification codes : C32, D83, I12
Keywords : infectious disease, Italian National Bovine Database, endogenousspreading, exogenous shocks, islands model
Connections between individuals are beneficial because they enable the exchangeof goods and resources. However, they are also the means through which diseasesand shocks may spread in a society, making it vulnerable to hazards and menaces.Due to the advances in virtual and physical communications, understanding thistradeoff has become increasingly more necessary as well as complicated.We study how the spread of an infection evolves in a population adoptingself-protecting behavioral responses which, in turn, affect the evolution of the epi-demics. We focus on two interplaying mechanisms: First, how does the contactnetwork influence the evolution of the disease by only allowing contagion via ex-isting contacts? Second, how is the network itself endogenously modified by thebehavioral response triggered by the risk perception?In the context of a simple two-location model, we obtain analytical resultsfrom a system of ordinary differential equations. This very stylized cross-locationinteraction may generate complex dynamics in terms of the co-evolution of thecoupled mechanism constituted by the contact network and the infection spread. ∗ We acknowledge funding from the Italian Ministry of Education Progetti di Rilevante In-teresse Nazionale (PRIN) grant 2015592CTH. We are grateful to the Italian Ministry of Health,Istituto Zooprofilattico Sperimentale dell’Abruzzo e del Molise “G. Caporale” and, in particular,to Luigi Possenti and Diana Palma for their help with the data. For their helpful commentswe would like to thank Alberto Alesina (and the participants to his reading group at BocconiUniversity), Leonardo Boncinelli, Simone D’Alessandro, Jakob Grazzini, Kenan Huremovi´c andRoberto Patuelli. a r X i v : . [ ec on . GN ] D ec he main question concerns how this stylized globalization affects the 2-locationsystemic resistance with respect to shocks in the infection rates. Is the coupledsystem more resistant to shocks than 2 separated and autarkic single locations?What happen when shocks simultaneously hit both locations?We assume that each location has a limited recovery ability from the disease(e.g. limited hospitalization capacity for quarantine) and that sudden outbreaksin the infection rate (also called infection shocks hereafter) occur exogenouslyand abruptly. After being hit by such an initial infection shock, the evolutionof the disease (and the effectiveness of the recovery measures) is observed overtime. Small shocks are better controlled when the two locations are connectedtogether than when they are isolated and autarkic: in fact, infected individualswho (out)flow from the most infected location to the least one, are treated in bothlocations and this contributes to diluting and reducing the epidemic overall. Onthe contrary, a large shock, even if initially concentrated in only one location,may end up infecting completely both locations, thus putting at risk the entiresystemic resistance to contagion.In terms of policy implications, given the characteristics of the disease un-der study (e.g. contagiousness, type of infection shocks) and given the ease ofconnection between the locations (e.g. autarky or globalization), the resistanceof the whole system to infections depends on the resources allocated for recov-ery measures. As the system becomes more and more globalized by facilitatingconnections between distant locations, it becomes also more resistant to smallinfection shocks but, conversely, it also becomes more exposed to large shocks.Moreover, the relative advantage (in terms of systemic resistance) of a globalizedworld with respect to a system of autarkic or isolated locations becomes higheras the amount of resources dedicated to recovery measures increases, because ofcomplementarities established between the two locations. Related literature
Epidemiological models have been studied for decades, starting from the seminalKermack-McKendrick compartmental models that go back to the 1920s and 1930s(Allen et al., 2008). In recent years, more attention has been devoted to incor-porating agents’ behavioral response and awareness to the concurrent evolutionof the disease in the population (Funk et al., 2010; Fenichel et al., 2011; Polettiet al., 2012). Moreover, because of the facility through which interconnectionsand interdependencies are established in a globalized world, better models needalso to account for different individuals’ mobility patterns (Brauer and van denDriessche, 2001; Wang and Zhao, 2004; Manfredi and D’Onofrio, 2013).In a literature closer to economics and to the social sciences, some theoreticalworks deal with strategic vaccination or with the adoption of different defensivemechanisms which may depend on the connections of the individuals (Galeotti andRogers, 2013, 2015; Goyal and Vigier, 2015). However, with respect to this paper,other works share the same motivation dealing with diseases spreading throughtrade connections (Horan et al., 2015) or approach a somehow similar problemwith different methodologies (Reluga, 2009).
Roadmap
The paper continues as follows. Firstly, we provide novel and fundamental empir-ical motivations for our analysis (Section 2). We then move to our model, withthe introductory case of a single location (Section 3), the description and the re-sults of the main model (Section 4) and its comparative statics with respect toexogenous shocks (Section 5). Section 6 concludes the main part. Additionally,2he appendix includes an extension of the empirical exercise done in the motiva-tion section (Appendix A), the proofs omitted in Sections 3, 4 and 5 (AppendixB) and the mathematical analysis of the linear case (Appendix C) which endswith the approximation of the basins of attraction of the equilibria used for thecomparative statics analysis (Appendix D).
Before going on to the description of the model and of its contribution to theexisting literature, we provide two strong motivations for our work. The first onecomes from a novel dataset, while the other is a relevant application.
Livestock trading: infections and long-range connections
We perform an empirical analysis of trade flows of bovines in the Italian territoryusing the Italian National Bovine Database (
Anagrafe Nazionale Bovina ). Thisdataset has been created by the Italian Ministry of Health after the outbreak ofthe Bovine Spongiform Encephalopathy (BSE) in accordance with the EuropeanEconomic Community Council Directive 92/102/EEC of 1992. The Directive im-posed to all member states to identify each bovine using ear tags and to followall its movements, from birth until death, through all holdings (farms, assemblycenters, slaughterhouses, markets, staging points, pastures, foreign countries oforigin) in the national territory.For each movement of bovines, we have information on the location (e.g. mu-nicipality) of approximately 220,000 origin and destination premises, 90% of whichare farms. The dataset records the exact date of all these movements between2006-2013 and contains information on the stock of animals in each holding. In-formation on trade flows have been merged with the SIMAN database (
SistemaInformatico Malattie Animali ) which registers the diseases occurred in each hold-ing (Iannetti et al., 2014; Calistri et al., 2013). We refer also to Muscillo et al.(2018) for more detailed information.In our analysis, we will focus on outflows from farms in each quarter, from2007Q1 to 2013Q4. We are mainly interested in determining whether the oc-currence of a disease in farm i at time t − i at time t .Figure 1 shows the histogram of distances for all the flows under analysis.The median value is about 17 kilometers whereas the average value and the 75thpercentile are respectively 43 and 41 kilometers. About 10% of the movementsoccur between farms in the same municipality, thus producing a point mass atzero in the distribution of distance.Since some staging points or assembly centers can be misclassified as farms,we retain in the sample only the holdings with a value of the stock smaller thanthe 99th percentile, which is equal to 954. Our final sample consists of 117,758farms originating 1,541,370 trade flows towards other farms.We thus estimate different regressions with trade flow distance as the depen-dent variable using the following specification:Distance it = β + β Positive i,t − + β Stock it + Region g + θ t + α i + (cid:15) it (1) Information on the stock is recorded on the same date of the movement, before any inflowsor outflows have occurred. With the latter information and the data on movements we computethe stock of animals at the beginning of each quarter. Since 2006 was the first year of introduction of the tracking system we start using the datafrom the first quarter of 2007, when the running-in phase was over. The 99th percentile is twice the size of the 95th percentile, the triple of the 90th percentileand about 14 times the size of the median. The empirical estimates from the unrestricted sampleare qualitatively and quantitatively similar.
Distances of trade flows s i t y The figure displays the histogram of distances (in kilometers) shorter than 100km of the tradeflows occurred in the period 2007-2013. Descriptive statistics of distances are shown in Table 1. t = 2007 Q , . . . , Q g = 1 , . . . , i,t − is equal to one if in the previous quarter farm i has registered at least one disease; Stock it measures the number of bovines infarm i at the beginning of quarter t . The variables Region g are regional dummiesand θ t are quarter dummies, while α i are farm-specific time invariant effects usedonly in fixed effect regressions.Table 1 reports descriptive statistics of the dependent variable and the mainregressors. In the 2007-2013 period we observe 265 diseases which represent 0.02%of the observations (i.e. all movements) used in the empirical analysis.Table 1: Descriptive statisticsMean St. Dev. Median Min MaxDistance 42.80 92.18 17.08 0 1,291Positive 0.0002 0.0131 0 0 1Stock 119.36 142.87 66 1 954Observations 1,541,370N. farms 117,758 The sample excludes farms with a value of the stock of bovines greater than 954.
The empirical results are shown in Table 2. Column 1 reports estimates from aProbit model where the dependent variable is set equal to one when the distance islarger than 41 kilometers (i.e the 75th percentile.) The estimated coefficient of thedummy Positive is statistically significant at 5% level. The marginal effect is 0.05which means that presence at t − t to farms distant more than 41 kilometers.The estimates from standard OLS regressions are shown in column 2. Thecoefficient of the dummy Positive indicates that a disease in the past increases thedistance of trade by about 19 kilometers. In column 3 we report the estimatedeffect from a Tobit to take into account the censored nature of distance which hasa point mass at zero. The estimated effect in this case is equal to 19 kilometers.Finally, in column 4, we show the estimates from a panel regression to controlfor farm-specific fixed effects. The estimated coefficient of the dummy Positive4ndicates that a registered disease at t − t − t . Nevertheless, theestimated effect of the dummy Positive on distance is again equal to 19 kilometers.Estimation results and the identification strategy adopted are shown in AppendixA. Table 2: Empirical Results(1) (2) (3) (4)Probit OLS Tobit Panel FEPositive i,t − i,t σ ∂ Pr( Y i = 1) /∂ Positive
All regressions control for time dummies. Estimations in columns 1,2 and 3 control for regionalfixed effects. Panel fixed effect estimation in column 4 controls for farm-specific effects. Standarderrors clustered at the farm level are shown in parenthesis. Asterisks mean: *** significant at1%, ** significant at 5%,* significant at 10%.
The 2014 Ebola outbreak
The theoretical study of infection dynamics when the (endogenous) behavior ofpatients increases infections has potentially enormous applications. Such a mech-anism has been at play (and possibly at the origin of) some disastrous epidemicevents, like the complex case of the
Zaire ebolavirus epidemic that affected WestAfrica since approximately December 2013 (see Figure 2). A particularly dan-gerous situation can occur when contagious individuals are expelled from theirvillages and are able to reach big towns, or even other countries; or if they vol-untary travel to other countries when they are sick, to avoid social stigma or to See Chowell and Nishiura (2014) for a detailed review; see also Thomas et al. (2015) andthe website of the World Bank for some estimates of the damages done to the economiesof some African countries: and also . “. . . as the situation in one country began to improve, it attractedpatients from neighboring countries seeking unoccupied treatment beds,thus reigniting transmission chains. In other words, as long as onecountry experienced intense transmission other countries remained atrisk, no matter how strong their own response measures had been.”Directly quoted from the web site of the WHO: “Countries in equatorial Africa have experienced Ebola outbreaks fornearly four decades. [...] In those outbreaks, geography aided con-tainment. [...] In West Africa [which had never experienced an Ebolaoutbreak], entire villages have been abandoned after community-widespread killed or infected many residents and fear caused others to flee.[...] West Africa is characterized by a high degree of population move-ment across exceptionally porous borders. Recent studies estimatethat population mobility in these countries is seven times higher thanelsewhere in the world. [...] Population mobility created two significantimpediments to control. [...] [C]ross-border contact tracing is difficult.Populations readily cross porous borders but outbreak responders donot.The importation of Ebola into Lagos, Nigeria on 20 July and Dallas,Texas on 30 September [2014] marked the first times that the virus en-tered a new country via air travelers. These events theoretically placedevery city with an international airport at risk of an imported case.The imported cases, which provoked intense media coverage and publicanxiety, brought home the reality that all countries are at some degreeof risk as long as intense virus transmission is occurring anywhere inthe world - especially given the radically increased interdependenceand interconnectedness that characterize this century.”Figure 2: Air traffic connections from West African countries to the rest of the world.Source Gomes et al. (2014). See also Halloran et al. (2014). Ibidem. The single-location model: the building block
As a warm-up exercise, in this section we develop the building block of the model.We define a system constituted by a single location and describe how the infectionevolves in it, as time passes. The dynamics is kept explicitly abstract and simpleon purpose, and this has one main reason: the so-defined 1-location system is ableto recover from small shocks, but unable to do so in case of large shocks (to bedefined shortly). This, in turn, will allow us in the next Section 4 to considertwo such systems interacting with each other and, then, evaluate what will be theeffects on this whole 2-location system, in terms of resistance to shocks.Consider a population of agents living in one location and susceptible to theinfection from a transmittable disease, which can spread through personal contactswith other agents. Following the motivations from Section 2, the intuitive idea isthat agents trade with each other and meet in pair. These meetings, however, arealso the mean through which the disease may spread.Let x ( t ) denote the fraction of infected individuals at time t . The evolutionof this fraction is ruled by the following differential equation, used for example inecological economics as a development from the classic Bass model (Bass, 1969;D’Alessandro, 2007): dd t x ( t ) = νx ( t )(1 − x ( t ))( x ( t ) − q ) , (2)where ν ∈ (0 ,
1) is a parameter representing the contagiousness of the disease and q ∈ (0 ,
1) is a parameter which measures the capacity of the system to controlthe disease, which we may call quarantine . More specifically, we think of q asthe quantity of resources allocated to hospitalize infected individuals as well as toother disease-control measures. We consider these resources fixed and exogenous,meaning that they can change over a longer time-scale with respect to that of theevolution of the disease. Remark.
Equation (2) can be seen as modified susceptible-infected model: wetake the probability that an infected individual meets a susceptible one, i.e. x (1 − x ), and that this meeting results in a new infection with probability ν . We thenmultiply this by a factor ( x − q ) which modifies the sign of the flow of infecteddepending on whether the fraction of infectives exceeds or not the quarantinethreshold q . PROPOSITION 3.1.
The dynamical system (2) has 3 critical points: • the asymptotically stable, disease-free equilibrium x = 0 ; • an unstable equilibrium x = q ; • the asymptotically stable, endemic equilibrium x = 1 .Consequently, the interval [0 , q ) ⊂ R is the basin of attraction of x = 0 , while ( q, is the basin of attraction of x = 1 .Proof. See Appendix B.
Remark.
In ecology, this dynamics sometimes describes the evolution of a speciesover time (D’Alessandro, 2007). In this context, the analogy is that the specieswe are considering is a bacterium causing the infective disease. The threshold q represents the critical mass of infections that the species has to reach and exceedin order to survive: when there are not enough infected individuals, the speciescannot proliferate and propagate any more and, eventually, the epidemic diesout. Lastly, it is worth noticing that underlying assumption in the susceptible-infected model is the so-called homogeneous mixing : meetings among individualsare random, according to their relative proportion in the whole population. Thisis an assumption that we maintain here.7 esistance to shocks & policy We define a shock as follows: suppose thatat time t = 0 there is a sudden and exogenous variation in the infection rate suchthat x (0) = x ∈ [0 , x of infected is what we will callshock.If the shock is x < q , i.e. below the threshold, then the system will (asymptot-ically) return to the disease-free equilibrium, whereas if the shock is larger than q ,then the dynamic will converge toward 0, where the whole population is infected.If the shocks are assumed uniformly distributed over [0 , q quantifies theability of the system to recover from a shock.The policy implication here is then straightforward: the more resources canbe allocated to control the disease (i.e. the larger q is), the more the system willable to recover from larger shocks in the infection rate (i.e. the larger will be thebasin of attraction of 0).Figure 3: Single-location dynamics x Dynamics defined by equation (2), where the parameters are set at q = 0 . ν = 0 . q and 1. The unstable equilibrium x = q acts as a threshold separating thebasins of attraction of the two asymptotically stable equilibria x = 0 (in green) and x = 1(in red). Small exogenous shocks, i.e. below q , are absorbed, whereas shocks larger than q lead to a fully infected system. Starting from the conclusion of the previous section, we now extend our analysis:trades and meetings will take place both within and across two geographicallydistinct locations (also called islands or countries hereafter) and, consequently,the same will happen to the spread of the disease. To express the incentivesof economic agents, who can choose whether to interact with other agents withintheir own location or in the other location, we stick to the interpretation of tradingand, therefore, speak also of prices. We think that the economic intuition remainsthe same also in other situations where prices are less explicit (as for the Ebolaexample of Section 2), since things could still be modeled in terms of higher andheterogeneous costs for interaction with distant locations.Specifically, we consider 2 locations both populated by interacting agents, e.g.farmers who are trading cattle. Agents benefit from interacting with each otherbut, since there may be a (latent) disease spreading, this potential benefit de-creases as the infection prevalence increases. This accounts for the risk of becom-ing infected and the reduction in performance that diseased cattle experience (e.g.slower growth, death). In the attempt to avoid contagion and risks, the agents ofone location may be willing to interact with other agents in the other location,even if to do so they have to pay a higher cost related to this long-range interaction(e.g. export costs, trade barriers).We restrict our attention to two identical and symmetric locations, where8gents are homogeneous and identical in all aspects but in the export cost. Inparticular, different agents of the same location are assigned different costs toexport to the other location and, intuitively, this may be reflecting different ge-ographical proximity, facility in the contacts with a foreign country, etc. Onekey aspect is that using identical locations and identical agents is a normalizationthat can help guarantee that any variation in the fragility of the coupled system isdue to the cross-country connection structure rather than to differences in othercharacteristics.
Let A and B denote two populations of agents living for an infinite time horizonand let a and b denote one of their generic agent, respectively. Benefits from interaction and costs
Agents benefit from trading/interactingwith other agents and, in particular, any agent a ∈ A receives a gross utility of p A , when trading in her home country A , and a possibly higher gross utility p B ,when instead exporting to the other country B . Benefits are assumed to be equalacross agents and to be decreasing functions of the current infection prevalencerates x A ( t ) , x B ( t ) ∈ [0 , This last assumption reflects the fact that tradingbecomes riskier as contagion spreads. Formally: p A = p A ( x A ( t ) , x B ( t )) , p B = p B ( x A ( t ) , x B ( t )) , for any time t ∈ R .Any generic agent a ∈ A chooses between two (mutually exclusive) actions,respectively labeled as A and B , which are either “trading in her home country” or“exporting to the other country”. However, to export to the other country eachagent a ∈ A has to pay an exporting cost c a >
0, which is assumed to be randomlydistributed across agents according to a cumulative distribution function F A . Thecost of trading in the home country, instead, is normalized to 0. Depending onthe chosen action A or B , agent a ’s utility at time t is then given by: u a ( t ) = (cid:26) p A ( t ) , if trading in A,p B ( t ) − c a , if exporting to B, so that agent a ∈ A decides to export to B at time t if and only if p A < p B − c a . Symmetrically, analogous definitions and notations hold for all agents b ∈ B . Thesame happens in the rest of this section. Remark.
Notice that in our formulation agents make decisions only based on theprices p A and p B that they are able to observe in the two markets. In particular,they are not able to observe neither their status (as susceptible or infected) northe status of the others. In the case of the cattle trade mentioned in the motiva-tion section, this assumption is economically justified as follows. Movements arestressful for bovines, which can result in the development of diseases and reducedgrowth (or even death) of the animals. This also implies that latent diseases canbe masked as stress and go undetected. For these reasons, farmers are compelledto report to local health institutions any situation that may be related to a disease– they would otherwise incur in high fines. These can be interpreted as prices at which trade happens. According to our notation, then, agents a ∈ A can export to B and, conversely, agents b ∈ B to A . c a ∼ F A , the above expression implies that the fraction of A ’s agentswilling to export to B at time t is given by P { a ∈ A : p A ( t ) < p B ( t ) − c a } = P { a ∈ A : c a < p B ( t ) − p A ( t ) } = F A { p B ( t ) − p A ( t ) } , (3)or, equivalently, that the fraction of A ’s agents trading in A and not exporting is1 − F A { p B ( t ) − p A ( t ) } . Cross-country meetings and flows of infected individuals
Let us proceedwith the analysis: of the fraction of agents that are exporting from A to B ,a subfraction of them given by x A · F A { p B − p A } is of currently infected agents.Consequently, when these exporting and infected agents meet the fraction of thosesusceptible in B that remain in B for trade, which is (1 − x B )(1 − F B { p A − p B } ),this will give rise to an additional source of infected individuals for country B : x A · F A (cid:124) (cid:123)(cid:122) (cid:125) A ’s infected exporting to B · (1 − x B ) · (1 − F B ) (cid:124) (cid:123)(cid:122) (cid:125) B ’s susceptible remaining in B . Still another source of infection for B comes from the meetings between B ’s in-fected individuals remaining in B with A ’s susceptible exporting to B : x B · (1 − F B ) (cid:124) (cid:123)(cid:122) (cid:125) B ’s infected remaining · (1 − x A ) · F A (cid:124) (cid:123)(cid:122) (cid:125) A ’s susceptible exporting to B . However, this additional infective activity due to cross-country interactions issomehow compensated with a reduction in the home country. In particular, now B ’s within-country spreading cannot follow the single-location equation (2) givenin Section 3: not only because the meetings only happen between B ’s susceptibleand infected agents that are not exporting, but also because we have to subtractthe fraction of B ’s infected agents that are exporting, as an outflow. ν B x B (1 − F B )(1 − x B )(1 − F B )( x B − q ) (cid:124) (cid:123)(cid:122) (cid:125) meetings among B ’s remaining agents resulting in infections − x B F B (cid:124) (cid:123)(cid:122) (cid:125) outflow of infected , where ν B ∈ (0 ,
1) is the contagiousness parameter for B . Analogous reasoningshold symmetrically for A .By putting all these elements together, we can build a system of coupled dif-ferential equations ruling the evolution over time of the infection rates in the twocountries. The first line of each equation accounts for the possibly reduced within-country epidemic spreading, whereas the second line accounts for the additionalinflow of infection due to cross-country interactions just described above: dd t x A = ν A (cid:104) x A (1 − F A )(1 − x A )(1 − F A )( x A − q A ) ++ x A (1 − F A )(1 − x B ) F B + (1 − x A )(1 − F A ) x B F B (cid:105) − x A F A dd t x B = ν B (cid:104) x B (1 − F B )(1 − x B )(1 − F B )( x B − q B ) ++ x B (1 − F B )(1 − x A ) F A + (1 − x B )(1 − F B ) x A F A (cid:105) − x B F B , (4)where ν A , ν B ∈ (0 ,
1) and q A , q B ∈ (0 ,
1) are the contagiousness and quarantineparameters respectively of location A and B . The system can be algebraically For ease of notation, throughout we will write F A { p B − p A } = F A and F B { p A − p B } = F B . For ease of notation, we omit the time t . However, it is worth remembering that F A and F B depend on p A and p B which, in turn, depend on x A ( t ) and x B ( t ). dd t x A = ν A (1 − F A ) (cid:104) x A (1 − x A )( x A − q A )(1 − F A ) + ( x A + x B − x A x B ) F B (cid:105) − x A F A dd t x B = ν B (1 − F B ) (cid:104) x B (1 − x B )( x B − q B )(1 − F B ) + ( x A + x B − x A x B ) F A (cid:105) − x B F B . (5) Remark.
If cross-country export is not allowed, i.e. when F A = F B = 0, thensystem (4) is reduced to two uncoupled equations, corresponding to two single-location models of the form of equation (2), both evolving separately. Remark.
In this model, export at any instant only occurs in one direction attime, either from A to B or vice versa. Indeed, suppose that p A ( t ) < p B ( t ) ata certain time t ∈ R . Since F A and F B are cumulative distributions which arepositive only for positive costs, then in such a case F B ( p A ( t ) − p B ( t )) = 0 while F A ( p A ( t ) − p B ( t )) >
0. So, there is an outflow of infection in the first equationfor x A and an inflow in the second for x B . However, as the following analysiswill show, the infection rates x A ( t ) and x B ( t ) (as well as p A ( t ) , p B ( t )) are notnecessarily monotone functions of time.In the following, we will make the assumption that F A = 1 when x A = 1. Thisis intuitive: whenever in A the rate of infection is the maximum, i.e. x A = 1, thenall A ’s agents would be facing the minimum home benefit p A and thus be willingto export, so F A = 1. PROPOSITION 4.1.
System (3) is well defined in the unit square describingany ( x A , x B ) ∈ [0 , .Proof. See Appendix B.
To keep the analysis tractable, we restrict our model to a linear specification ofsystem (4): • the two locations A and B are assumed to be identical, from the point ofview of the epidemic parameters, so ν A = ν B = ν ∈ (0 ,
1) and q A = q B = q ∈ (0 , • the agents’ exporting costs c a >
0, for a ∈ A , are uniformly distributed overthe interval [0 ,
1] (analogously for b ∈ B ), so that the cumulative distribu-tions are identical and of the form F A = F B = U (0 , F A ( c ) = F B ( c ) = , for c ≤ c, for c ∈ [0 , , for c ≥ . In particular, the maximum and minimum cost are respectively 1 and 0. • The gross utilities from trading, p A and p B , are assumed to depend linearlyon the infection rate of the own location: p A ( x A ( t ) , x B ( t )) := 1 − x A ( t ) , p A ( x A ( t ) , x B ( t )) := 1 − x B ( t ) . Then, maximum and minimum gross utility attainable are thus normalizedto 1 and 0, respectively. 11ith these assumptions in place, equation (3) becomes: F A { p B − p A } = F A { x A − x B } = , if x A − x B < x A − x B , if 0 ≤ x A − x B ≤ , if 1 < x A − x B = max { , x A − x B } , and, analogously, F B = max { , x B − x A } . We can then rewrite system (4) asfollows: dd t x A = ν (1 − max { , x A − x B } ) (cid:104) x A (1 − x A )( x A − q )(1 − max { , x A − x B } )+ ( x A + x B − x A x B ) max { , x B − x A } (cid:105) − x A max { , x A − x B } dd t x B = ν (1 − max { , x B − x A } ) (cid:104) x B (1 − x B )( x B − q )(1 − max { , x B − x A } )+ ( x A + x B − x A x B ) max { , x A − x B } (cid:105) − x B max { , x B − x A } . (6)In Appendix C we derive the properties of this system, which can be summa-rized as follows. The system is well defined in the unit square [0 , ⊂ R , whichis invariant under its dynamics, and it is symmetric with respect to the diagonalin R (Propositions C.3 and C.4). This system has three equilibria (PropositionC.6): • ( x A , x B ) = (0 ,
0) and (1 , • ( x A , x B ) = ( q, q ), which is an unstable saddle point.What becomes interesting is to study the basins of attractions of the two stableequilibria, and to characterize the basins’ border, which we call separatrix C .As shown in Figure 4, depending on the parameters ν and q , as time t passes,the solution enters the unit square either crossing its border along the segment[ q, × { } or along { } × [0 , q ] and, eventually, converges toward { q, q } as t → ∞ (Proposition C.9).In Appendix D we show that, although the separatrix C cannot be describedanalytically, it can be very well approximated linearly. In Appendix D we alsoshow the good accuracy of this approximation, done by mean of an extensive gridof numerical simulations. This will allow us to make a comparative statics analysisthat will be then used in the following Section 5. More precisely, in PropositionD.1 and Lemma D.2 we give an explicit approximation (cid:101) C of C and of its point ofintersection with the boundaries of the unit square [0 , .Figure 5 shows similarities and differences between C and (cid:101) C . Depending on theparameters ν and q , the area under the curve (cid:101) C is either a trapezoid or a triangleand is easily computed analytically. By considering this area as an approximationof the area under the curve C , which is instead impossible to compute analyti-cally. This area obtained with this linear approximation will also be used for acomparative statics analysis in Section 5. The results are also shown in Figure 6. Remember that x A , x B ∈ [0 , Basins of attraction of the disease-free and fully-endemic equi-libria, (0 , and (1 , , for changing epidemic parameters ν, q . x A x B Parameters ν = 0 . q = 0 . x A x B Parameters ν = 0 . q = 0 . x A x B Parameters ν = 0 . q = 0 . x A x B Parameters ν = 0 . q = 0 . Mathematica ® to plot the vector field defining system (6) and the basinsof attraction of the two asymptotically stable states ( x A , x B ) = (0 ,
0) and (1 ,
1) (respec-tively colored in white and shaded gray). The arrows depict the vector field definingsystem (6) in each point ( x A , x B ) ∈ [0 , and confirm that the unit square is invariantand that the same is true for the diagonal and for the super-diagonal and sub-diagonal“triangles”. Moreover, from the saddle point ( q, q ) one can identify the separatrix curves,the unstable separatrix coinciding with the diagonal, while the stable one, i.e. C , consti-tutes part of the border of the basins of attraction, thus separating them. Lastly, noticethat as quarantine q increases, the system exhibits a larger and larger basin of attractionof the disease-free equilibrium (0 , η,
0) and (0 , η ), i.e. the intersection pointsof the separatrix C with the horizontal and vertical axis mentioned in Proposition C.9.Analogously, the triangular dots are (1 , ζ ) and ( ζ, Separatrix C and comparison with its linear approximation (cid:101) C x A x B Parameters ν = 0 . q = 0 . x A x B Parameters ν = 0 . q = 0 . (cid:101) C (dashed straight line) and comparison with the ac-tual separatrix C (continuous black curve). (cid:101) C is a first-order approximation of C in aneighborhood of the saddle ( q, q ). Figure 6:
Approximated area of the triangle/trapezoid
Parameters ν = 0 . q = 0 .
2. Parameters ν = 0 . q = 0 . (cid:101) C (dashed lines), in the rectangles of interest. Comparative statics with respect to exogenous shocks
We now focus our attention on the conclusions that can be drawn from the analysisof system (6) performed in Section 4.2, Appendix C and Appendix D. We willcompare the two following situations: • first, no cross-country trade between the two locations is allowed, i.e. theyare considered separated and autarkic; • second, cross-country trading is instead allowed, as described in the previoussection.By comparing these two situations we are thus able to analyze the effects of a very“stylized globalization” (the second case) on the systemic resistance to potentialshocks in the infection rates. Depending on the “intensity” and “dimensionality”of the shock, being “autarkic” or “globalized” may or may not be advantageous.In particular, small shocks are better absorbed by an interconnected system,independently of their dimensionality: intuitively, the shock is more easily dilutedin a larger system. On the contrary, somehow surprisingly, large shocks may or maynot have worse consequences when the locations are interconnected, depending onthe amount of resources dedicated to recovering (formalized by the parameter q ). Let us consider two autarkic locations, where no trade is possible between themand where each location is subjected to a disease-spread dynamic described by thesingle-location model of Section 3. The evolution over time of the two infectionrates x A ( t ) and x B ( t ) of these two locations A and B can be written as a systemof two (uncoupled) differential equations: dd t x A = νx A (1 − x A )( x A − q )dd t x B = νx B (1 − x B )( x B − q ) . (7)The dynamics and results are shown in Figure 7 (left) and summarized in thefollowing proposition. PROPOSITION 5.1.
Given two autarkic locations A and B , system (7) hasthe following properties: • it is symmetric with respect to the diagonal, which is then an invariant set.The super-diagonal and sub-diagonal sets in R , { ( x A , x B ) ∈ R : x A < x B } and { ( x A , x B ) ∈ R : x A > x B } , are also invariant; • the unit square [0 , is invariant; • the critical points where d x A d t = d x B d t = 0 are: – (0 , q ) , (1 , q ) , ( q, , ( q, , which all are (unstable) saddle points; – ( q, q ) , which is an unstable point (source); – (0 , , (0 , , (1 , , (1 , , which all are asymptotically stable equilibria.Moreover, the separatrix curves of the saddle points are the lines x A = 0 , x B = 0 , x A = q , x B = q , x A = 1 and x B = 1 , and this also makes possible characterizingthe basins of attraction of the stable points in [0 , : For a reasonable comparison with an analogous globalized 2-location model, here we assumethe same symmetric epidemic parameters ν A = ν B = ν and q A = q B = q . Of course, we only consider their intersection with the unit square, which is the sets in whichthe fraction of infected x A and x B make sense. Comparison between autarkic and interconnected locations x A x B Autarky, parameters ν = 0 . q = 0 . x A x B Globalized, parameters ν = 0 . q = 0 . , ,
0) is left in white. Only the autarkic case(left) exhibits two partially-endemic asymptotically stable states, (0 ,
1) and (1 , • [0 , q ) is the basin of attraction of (0 , ; • [0 , q ) × ( q, is the basin of attraction of (0 , ; • ( q, × [0 , q ) is the basin of attraction of (1 , ; • ( q, is the basin of attraction of (1 , .Proof. See Appendix B.As shown in Figure 7 (left), the points (1 ,
0) and (0 ,
1) play a peculiar role:they represent a situation in which only one of the two locations is fully infected,while the other is disease free. In case of autarky, this may happen when theinitial point of infection at time t = 0 belongs to [0 , q ) × ( q,
1] or ( q, × [0 , q ),which will cause the dynamics to convergence toward (0 ,
1) or (1 , In line with what done in Section 3, shocks are assumed to be uniform at randomover the unit square [0 , and are represented by a vector of initial conditions: s = ( s A , s B ) := ( x A (0) , x B (0)) . Figures 7 and 8 show the comparison of the basin of attraction of the point (1 , We assume a uniform distribution of shocks just to simplify the exposition. What is importantfor our analysis, is just that the support of our random shocks is the unit square [0 , , so thatwe can compare regions of this support in the two regimes of atarky and globalization . With Systemic resistance to small vs. large shocks
Comparison by juxtaposition of the areas ob-tained in Figure 7, with same parameters ν = 0 . q = 0 .
4. The white area, [0 , q ) ,and the grid-shaded area, ( q, , depict wherea hitting shock would produce the same out-come, independently of locations being autar-kic or globalized. Light-gray areas measurewhere shocks result in a partial endemic state,in case of autarkic locations, or where they areinstead totally recovered, in case of globalizedlocations. On the contrary, dark-gray areasare those where shocks result fully infectedsystem, if globalized, whereas only partial in-fection, if autarkic. Given a shock s = ( s A , s B ), if it is large enough in both components or smallenough in both components, then the resulting outcome is the same for an autarkicsystem and for a globalized system. In particular: • if s A < q and s B < q , then both the autarkic system and the globalizedsystem will be able to fully recover (white areas in Figure 8); • if s A > q and s B > q , then both systems will converge to a fully infectedendemic state (grid-shaded areas in Figure 8).On the contrary, the outcome resulting from a shock hitting mainly one lo-cation is completely different when the two locations are autarkic or connected.Indeed, consider an “almost” 1-dimensional shock s targeting mainly location A ,that is s = ( s A , ε ) , with ε < q < s A . In the autarkic case, the dynamics will converge to a partial epidemic equilibrium:Proposition 5.1 and Figure 7 (left) show that A would converge to fully infectionwhile, independently, B would recover.Instead, what happens when A and B are connected while facing the sameshock s = ( s A , ε ) as before? Two different situations may arise: • if s A is large enough and such that s = ( s A , ε ) belongs (1 , • if, instead, s A is still greater than q but not large enough, then ( s A , ε ) belongsto (0 , uniform shocks areas in the support region are proportional to probabilities, but all followingresults could be adapted also to any other distribution of shocks, remembering that in the moregeneral case areas should be translated into probabilities. Notice that assuming that the shock distribution is uniform or continuous, thus atom-less,guarantees that the probability that one component hits 0, q or 1 is zero. Symmetrically, the argument is the same for shocks mainly concentrated in B . Comparative statics q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . q = 0 . ν = 0 . q , thepolicy parameter, increases in (0 , area, then, exactly measures the weakness of the system with respect to this kindof mainly 1-dimensional shocks and, in addition, it also captures the advantage ofan autarkic system over a globalized one.Analogously, but in an opposite way, the light-gray areas in Figure 8 capturethe advantage of a globalized system over an autarkic one: shocks belonging tothese regions are recovered by a connected system, whereas they result in a partialepidemic equilibrium in the autarkic case. Understanding the relationship between the dark-gray areas and the light-grayareas in Figure 8 becomes necessary, because it gives an indication of the relative(dis)advantage of an autarkic system over a globalized system for systemic resis-tance. Figure 9 also shows that this advantage changes as the recovery parameter q varies: this turns out to be crucial for policy making.One way to address this issue is by analyzing the separatrix curve C of thesaddle ( q, q ), because it separates the basins of attraction of the regions of interest.Unfortunately, apart from Proposition C.9, which relies on the “local” informationprovided by the eigenvector of the linearized system in the neighborhood of thesaddle point ( q, q ) and on the monotonicity of the components of the vector field These results resemble those obtained in the context of financial networks, where agents (e.g.banks) are exposed via financial dependence to others’ default and the goal is to understandhow shocks spread in a financial network. As argued in Acemoglu et al. (2015): “as long as themagnitude of negative shocks affecting financial institutions are sufficiently small, a more denselyconnected financial network [...] enhances financial stability. However, beyond a certain point,dense interconnections serve as a mechanism for the propagation of shocks, leading to a morefragile financial system.” The same kind of results are achieved in Cabrales et al. (2017). C analytically.Specifically, we first numerically approximate the intersection points betweenthe separatrix C and the boundaries of the unit square [0 , and, then, numeri-cally measure the gray areas and determine their relative ratio, which, as alreadyobserved, is key to understanding whether a globalized system is shock-resistancesuperior to an autarkic system, given the same parameters q and ν . (Numerical) comparative statics Let us first deal with the (numerical) com-putation of the intersection point between the separatrix C and the border of theunit square below the diagonal, i.e. the segments [0 , × { } and { } × [0 , Depending on whether C intersects the former or latter segment, we follow thenotation used in Proposition C.9 and Figure 4 respectively denote this point with( x A , x B ) = ( η ( q, ν ) ,
0) or (1 , ζ ( q, ν )).This analysis is shown in Figure 10: • holding fixed ν ∈ (0 , C crosses the segment [ q, × { } in thepoint ( η ( q, ν ) , q (cid:55)→ η ( q, ν ) is increasing in q and spans from 0 to 1.Moreover, η ( q, ν ) > q ; • analogously, when q exceeds a certain threshold , then C crosses the segment { } × [0 , q ] in the point (1 , ζ ( q, ν )); moreover, q (cid:55)→ ζ ( q, ν ) is increasing, goingfrom 0 to 1 and always satisfying ζ ( q, ν ) < q .Let us now turn to the relative advantage/disadvantage of an autarkic sys-tem over a globalized system, especially when subjected to mainly 1-dimensionalshocks. We have already observed that the areas in light gray and dark gray ofFigures 8 and 9 measure the extent to which an autarkic system or a globalizedsystem is relatively more or less able to recover from shocks of this kind.Holding fixed the contagiousness ν , as the recovery parameter q increases,the light-gray areas expand while the dark-gray areas shrink. According toour previous interpretation, this means that it becomes more likely that a 1-dimensional shock lead the autarkic system to a partial endemic equilibrium, whilea corresponding reduction of the dark-gray areas means that a globalized systembecomes more able to recover from shocks. This, in turn, means that the largerit is the available level of quarantine q , the more convenient it becomes to be in aglobalized system relative to an autarkic one. In this respect, Figure 9 shows howthe light-gray and dark-gray areas change, as the quarantine q changes. Thisanalysis is also shown in Figure 11, where we plot the percentage of the rectangle[ q, × [0 , q ] which is occupied by the dark-gray area. By using the shock analysisdone above, as q increases, we observe that having a connected 2-location systembecomes more and more advantageous and resistant overall than an autarkic 2-location system.This conclusion directly translates in terms of policy: if the available quaran-tine level q can be taken large enough, then allowing cross-country import-exportis beneficial and preferable for systemic resistance to infection shocks. On the con-trary, two autarkic countries constitute a more resistant system against infectionshocks when only a small level of quarantine q is available. By symmetry with respect to the diagonal, the same analysis holds also for the border of theunit square above the diagonal. Threshold that corresponds to q = 0 .
39 in Figure 10. Shocks that mainly start from a single location, of the form s = ( ε, s B ) or ( s A , ε ), with ε ≈ We think of contagiousness as a parameter strictly related to the type of disease considered,so not of interest for policy making. Since it corresponds to an expansion of the white recovery area, for a globalized system. While contagiousness ν is kept fixed, because we think of it as a disease-related parameter,not subject to policy making. Intersection between separatrix C and boundaries of [0 , q On the left, intersection points q (cid:55)→ η ( q, .
7) (squares) and q (cid:55)→ ζ ( q, .
7) (triangles),with fixed ν = 0 .
7. As q increases, the separatrix C first crosses the horizontal segment[ q, × { } in ( η ( q, ν ) , q exceeds a certain threshold ( q = 0 .
39 in this case,signaled by the dotted vertical line), C starts crossing the boundary in the vertical segment { } × [0 , q ] in the point (1 , ζ ( q, ν )). The diagonal (dashed) shows that η > q while ζ < q .On the right, intersection η ( q, ν ) as a function of both parameters ( q, ν ) ∈ (0 , . Allsections η ( · , ν ) and η ( q, · ) are increasing. Starting from a very simple model of epidemic diffusion among homogeneousagents, we consider the case in which two identical countries are inhabited bysuch agents. These agents interact and trade with each other in (random) pairsto obtain benefits and, by doing so, they also spread a contagious disease amongthem, which lowers the attainable gain from trade. As a response to the infectionrisk, agents can choose to bear (heterogeneous) costs to interact with the agentspresent in the other country, establishing then a stylized form of cross-countryimport-export trade. By assuming that both countries have (limited and fixed)resources to intervene against the infection, we are also able to introduce thepossibility of recovery, that is, of reducing the infection rate.Given the epidemic parameters, we compare the resistance to exogenous shocksin infection rates of the “autarkic” system, in which the two countries are assumednot to trade with each other, with the resistance of the “globalized” system where,instead, cross-country trade is allowed. Overall, globalized systems result more“extreme” in their reaction to shocks with respect to autarkic systems. Thisis a consequence of the two countries being connected: on the one hand, theglobalized system has a larger “recovery capacity” when facing relatively smallshocks but, on the other hand, it has a larger area where both countries end upbeing completely infected. In particular, the main possibility which is precludedto globalized systems with respect to autarkic ones is a situation in which onlyone country is infected while the other is not. On the contrary, “autarkic” systemsoffer a wider spectrum of possible outcomes resulting from infection shocks and,in particular, they exhibit partial endemic equilibria in which only one location isfully infected while the other is disease free.By comparing how an autarkic system and a globalized system behave in re-sponse to shocks, we are able to understand their similarities and differences. Themain result of this shock-resistance analysis is that the behavior of the two sys-tems is substantially different especially when they are subjected to “1-dimensionallarge shocks”: when infection shocks hit mainly one location (and only slightly theother), a globalized system either fully recovers or becomes fully infected, while anautarkic system could exhibit partial endemic equilibria, if exposed to the same20igure 11:
Ratio between the gray areas q On the left, ratio between the dark-gray area and the sum of the dark-gray plus light-grayareas (i.e. [ q, × [0 , q ] ∪ [0 , q ] × [ q, q ∈ (0 , ν = 0 .
7. On the right, the ratio as a function of both parameters ( q, ν ) ∈ (0 , .As the quarantine q increases, a globalized system becomes more and more convenientrelative to an autarkic one. shock. Depending on the amount of resources allocated to recovery, as measuredby the quarantine level q in our framework, a globalized system may be preferablewhen large resources for quarantine are available, whereas an autarkic system ispreferable in case of low resources. References
Acemoglu, D., A. Ozdaglar, and A. Tahbaz-Salehi (2015, February). Systemicrisk and stability in financial networks.
American Economic Review 105 (2),564–608.Allen, L. J., F. Brauer, P. Van den Driessche, and J. Wu (2008).
Mathematicalepidemiology . Springer.Bass, F. M. (1969). A new product growth for model consumer durables.
Man-agement Science 15 (5), 215–227.Brauer, F. and P. van den Driessche (2001). Models for transmission of diseasewith immigration of infectives.
Mathematical Biosciences 171 (2), 143–154.Cabrales, A., P. Gottardi, and F. Vega-Redondo (2017). Risk sharing and conta-gion in networks.
The Review of Financial Studies 30 (9), 3086–3127.Calistri, P., A. Conte, F. Natale, L. Possenti, L. Savini, M. L. Danzetta, S. Iannetti,A. Giovannini, et al. (2013). Systems for prevention and control of epidemicemergencies.
Veterinaria italiana 49 (3), 255–261.Cavoretto, R., S. Chaudhuri, A. De Rossi, E. Menduni, F. Moretti, M. C. Rodi,E. Venturino, T. E. Simos, G. Psihoyios, C. Tsitouras, et al. (2011). Approxima-tion of dynamical system’s separatrix curves. In
AIP Conference Proceedings-American Institute of Physics , Volume 1389, pp. 1220.Chowell, G. and H. Nishiura (2014). Transmission dynamics and control of ebolavirus disease (evd): a review.
BMC medicine 12 (1), 196.D’Alessandro, S. (2007). Non-linear dynamics of population and natural re-sources: The emergence of different patterns of development.
Ecological Eco-nomics 62 (3), 473–481. 21enichel, E. P., C. Castillo-Chavez, M. Ceddia, G. Chowell, P. A. G. Parra, G. J.Hickling, G. Holloway, R. Horan, B. Morin, C. Perrings, et al. (2011). Adaptivehuman behavior in epidemiological models.
Proceedings of the National Academyof Sciences 108 (15), 6306–6311.Funk, S., M. Salath´e, and V. A. Jansen (2010). Modelling the influence of humanbehaviour on the spread of infectious diseases: a review.
Journal of the RoyalSociety Interface 7 (50), 1247–1256.Galeotti, A. and B. W. Rogers (2013). Strategic immunization and group struc-ture.
American Economic Journal: Microeconomics 5 (2), 1–32.Galeotti, A. and B. W. Rogers (2015). Diffusion and protection across a randomgraph.
Network Science 3 (03), 361–376.Gomes, M. F., A. Piontti, L. Rossi, D. Chao, I. Longini, M. E. Halloran, andA. Vespignani (2014). Assessing the international spreading risk associatedwith the 2014 west african ebola outbreak.
PLOS Currents Outbreaks 1 .Goyal, S. and A. Vigier (2015). Interaction, protection and epidemics.
Journal ofPublic Economics 125 , 64–69.Halloran, M. E., A. Vespignani, N. Bharti, L. R. Feldstein, K. Alexander, M. Fer-rari, J. Shaman, J. M. Drake, T. Porco, J. Eisenberg, et al. (2014). Ebola:mobility data.
Science (New York, NY) 346 (6208), 433–433.Horan, R. D., E. P. Fenichel, D. Finnoff, and C. A. Wolf (2015). Managingdynamic epidemiological risks through trade.
Journal of Economic Dynamicsand Control 53 , 192–207.Iannetti, S., L. Savini, D. Palma, P. Calistri, F. Natale, A. Di Lorenzo, A. Cerella,and A. Giovannini (2014). An integrated web system to support veterinaryactivities in italy for the management of information in epidemic emergencies.
Preventive veterinary medicine 113 (4), 407–416.Manfredi, P. and A. D’Onofrio (2013).
Modeling the interplay between humanbehavior and the spread of infectious diseases . Springer Science & BusinessMedia.Muscillo, A., P. Pin, T. Razzolini, and F. Serti (2018). Does “network closure”beef up import premium? mimeo .Poletti, P., M. Ajelli, and S. Merler (2012). Risk perception and effectivenessof uncoordinated behavioral responses in an emerging epidemic.
MathematicalBiosciences 238 (2), 80–89.Reluga, T. C. (2009). An SIS epidemiology game with two subpopulations.
Journalof Biological Dynamics 3 (5), 515–531.Roodman, D. (2011). Fitting fully observed recursive mixed-process models withcmp.
The Stata Journal 11 (2), 159–206.Thomas, M. R., G. Smith, F. H. Ferreira, D. Evans, M. Maliszewska, M. Cruz,K. Himelein, and M. Over (2015). The economic impact of ebola on sub-saharanafrica: updated estimates for 2015.Wang, W. and X.-Q. Zhao (2004). An epidemic model in a patchy environment.
Mathematical Biosciences 190 (1), 97–112.22 ppendix A More on the econometric analysis
In this section we investigate whether the significant effect of the dummy Positive,observed in Table 2, could be a result of a selection process. To this aim, we haveestimated a bivariate selection model by maximum likelihood estimation wherethe main equation is a Tobit model with distance as the dependent variable. Theparticipation equation is a Probit, and estimates the probability of being active(i.e. sending at least one bovine) in quarter t .Although the dependent variable in the main equation is – when not censored– continuous, the identification of the model could depend only on distributionalassumptions. For this reason, we have added an exclusion restriction in the partic-ipation equation, using data on rainfalls provided by the Italian Air Force ( CentroOperativo Dati per la Meteorologia ). For each municipality, we have imputed thelevel of rainfalls and its deviation from its quarterly mean by averaging the threeclosest meteorological stations. We have thus included as a regressor in theparticipation equation the lagged value of the deviation of rainfalls from quarterlymean. Since reduced rainfalls at t − t .The bivariate model has been estimated using the Stata ® command cmp developed by David Roodman. The coefficient of the dummy Positive i,t − indicates that farms with a sickbovine at t − t −
1. The deviation of rainfallsfrom quarterly mean has the expected positive and statistically significant effecton the probability of sending cattle at time t . The ρ coefficient, which estimatesthe correlation between error terms is negative and significant at 10% , thus sug-gesting the presence of a weak negative selection effect. The estimated effect ofPositive i,t − in the Tobit main equation is, however, very close to the result shownin column 3 of Table 2. The meteo stations are around 115 with daily data covering the entire Italian territory. See Roodman (2011) for details. t Positive i,t − it i,t − σ ρ -0.0077*(0.0047)Observations 2,267,463 2,267,463Log likelihood -10,207,407 The bivariate Tobit/Probit model has been estimated using the
Stata ® command cmp . Theregression includes time and regional effects. Standard errors clustered at the farm level areshown in parenthesis. Asterisks mean: *** significant at 1%, ** significant at 5%,* significant at10%. Appendix B Proofs for Sections 3, 4 and 5
Proof of Proposition 3.1.
The derivative d x d t , which is a cubic function of x , hasonly three roots x = 0, x = q and x = 1, where it becomes equal to 0. Moreover,it is strictly negative when x ∈ (0 , q ) and strictly positive when x ∈ ( q, Proof of Proposition 4.1.
We want to show that the unit square [0 , is an in-variant set under the dynamics defined by system (3). In order to do that, weneed to the vector field defining the system of equation, i.e. the right-hand side of(3) as 2-dimensional function of ( x A , x B ) is “pointing toward the interior” of thesquare, while restricted on the borders of it. More formally: • suppose that x A = 0. Then ˙ x A = ν A (1 − F A ) x B F B ≥
0, for any x B ∈ [0 , • Suppose, instead, that x A = 1. By assumption, we have that F A = 1 when x A = 1, then ˙ x A = ν A (1 − F A )(1 − x B ) F B − x A F A = − < , as we wanted.An analogous and symmetric reasoning shows that ˙ x B ≥
0, when x B = 0, andthat ˙ x B ≤
0, when x B = 1. Proof of Proposition 5.1.
The vector field defining system (7) is of the form F ( x A , x B ) =( F A ( x A ) , F B ( x B )), where F A ( x ) = F B ( x ) = f ( x ) := νx (1 − x )( x − q ). Then,clearly the system is symmetric with respect to the diagonal, that is F ( x B , x A ) =( F B ( x A , x B ) , F A ( x A , x B )).Since f ( x ) = 0 if and only if x = 0 or x = q or x = 1, then the equilibriaof system (7) are: (0 , , , , q, q ), (0 , q ), (1 , q ), ( q,
0) and ( q, x A = 0 when x A = 0, this means that the line x A = 0 in R cannotbe crossed by the trajectories of the system. Analogously, the lines x A = q , x A = 1, x B = 0, x B = q , x B = 1 cannot be crossed, which implies that they are invariantand that the unit square [0 , is also invariant under the dynamics defined bysystem (7).In order to evaluate the stability of such equilibria, it suffices to study theJacobian of the system. Now, since the Jacobian is of the form (cid:18) f (cid:48) ( x A ) 00 f (cid:48) ( x B ) (cid:19) , where f (cid:48) ( x ) = ν [(2 − x ) x + q (2 x − f (cid:48) (0) = − νq < f (cid:48) ( q ) = νq (1 − q ) > f (cid:48) (1) = − ν (1 − q ) when ν, q ∈ (0 , , , ,
0) and (1 ,
1) are asymptoticallystable, because the eigenvalues are both negative. The points (0 , q ), ( q, , q ),( q,
1) are saddle point because they have eigenvalues of different sign. Lastly, ( q, q )is an unstable source point because both its eigenvalues are positive.
Appendix C Analysis of the linear case
We here study system (6), which comes from the assumptions of agents’ linearutility and uniform cost distributions. In principle, the system is well definedin R , but we will restrict our analysis to the unit square ( x A , x B ) ∈ [0 , , inwhich the fractions of infected agents make sense. It is continuously differentiableeverywhere but the diagonal of R , i.e. over R \ { ( x A , x B ) ∈ R : x A = x B } .However, thanks to the symmetry of the system guaranteed by the assumptionsmade, we can separate the analysis focusing on three different parts: the diagonal,the super-diagonal set and the sub-diagonal. This allows us to use an ad hoc strategy to obtain some explicit results. There are two asymptotically stableequilibria, ( x A , x B ) = (1 ,
1) and (0 , q, q ) which is an unstable saddle point. Its separatrix curves separatethe basins of attraction of the asymptotically stable states, as depicted in Figure4. It is worth noting, though, that they are not explicitly characterizable. For ease of exposition, let us re-write system (6) in vector notation as follows:dd t ( x A , x B ) = V ( x A , x B ) , (8)where V ( x A , x B ) := ( V A ( x A , x B ) , V B ( x A , x B )) for all ( x A , x B ) ∈ R and V A , V B are the 2-variable, real-valued functions defined respectively by the first and secondrow of (6). Let us also denote the diagonal by D := { ( x A , x B ) ∈ R : x A = x B } ,and the sets above and below the diagonal respectively by ∆ + := { ( x A , x B ) ∈ R : x A < x B } and ∆ − := { ( x A , x B ) ∈ R : x A > x B } . LEMMA C.1.
The vector field V is symmetric with respect to the diagonal D ,that is, for all ( x A , x B ) ∈ R : V ( x B , x A ) ≡ ( V B ( x A , x B ) , V A ( x A , x B )) . Proof.
The proof follows directly from the definition of V . There is no known way to analytically determine these curves, even in simple dynamicalsystems. Progresses have been made with their numerical approximations (Cavoretto et al.,2011). V can be seen as constituted by three basic pieces, all ofwhich are defined over the entire R but such that they coincide with V itselfwhen appropriately restricted on the sets D , ∆ + and ∆ − . The following lemmaformalizes this idea. LEMMA C.2.
1. The vector field V when restricted on the diagonal D coincides with V D ( x A , x B ) := (cid:18) νx A (1 − x A )( x A − q ) νx B (1 − x B )( x B − q ) (cid:19) , which, in turn, is well defined over R .2. The vector field V when restricted on ∆ − coincides with V − ( x A , x B ) := ν (1 − x A + x B ) (cid:104) x A (1 − x A )( x A − q )(1 − x A + x B ) (cid:105) − x A ( x A − x B ) ν (cid:104) x B (1 − x B )( x B − q ) + ( x A + x B − x A x B )( x A − x B ) (cid:105) .
3. The vector field V when restricted on ∆ + coincides with V + ( x A , x B ) := ν (cid:104) x A (1 − x A )( x A − q ) + ( x A + x B − x A x B )( x B − x A ) (cid:105) ν (1 − x B + x A ) (cid:104) x B (1 − x B )( x B − q )(1 − x B + x A ) (cid:105) − x B ( x B − x A ) . Proof.
When ( x A , x B ) ∈ D , then max { , x A − x B } = max { , x B − x A } = 0. Fromthis, the first point follows from the computation of V as defined by (6).The second point follows because when ( x A , X B ) ∈ ∆ − , then max { , x A − x B } = x A − x B while max { , x B − x A } = 0. Analogously for the third point. PROPOSITION C.3.
System (6) is symmetric with respect to the diagonal andit is well defined in R . The diagonal D , the sets ∆ + and ∆ − are all invariantwith respect to the dynamics defined by system (6) .Proof. The symmetry of system (6) in R follows from that of V . This impliesthat the diagonal D has to be invariant and, consequently, also ∆ + , ∆ − have tobe invariant.Because of the invariance, it suffices to show that the system is well definedwhen restricted on each of D , ∆ + and ∆ − . From the previous Lemma C.2, itfollows that the system is well defined because V D , V − and V + are smooth on R and, in particular, on D , ∆ − and ∆ + respectively. PROPOSITION C.4.
The unit square [0 , is invariant with respect to thedynamics defined by system (6) .Proof. The following Lemma C.5 implies that, on the borders of the unit square,the vector field V points towards the interior. LEMMA C.5. V B ( · , · ) > on the segment (0 , ×{ } and V A < on the segment { } × (0 , . Consequently, by symmetry, V B < on (0 , × { } and V A > on { } × (0 , .Proof. From the definition, it follows that for all x A ∈ (0 , V B ( x A ,
0) = νx A >
0. Analogously, for all x B ∈ (0 , V A (1 , x B ) = − (1 − x B ) <
0. Now we can show that system (6) has 3 critical points, specifically with ( q, q )being a saddle point, whose separatrix curves naturally form the boundaries ofthe basins of attraction of the asymptotically stable points (0 ,
0) and (1 , ROPOSITION C.6.
System (6) has (only) three equilibria: • ( x A , x B ) = (0 , and (1 , , which are asymptotically stable states; • ( x A , x B ) = ( q, q ) , which is an unstable saddle point.Moreover, the two separatrix curves of the saddle ( q, q ) are such that the unstableone coincide with the diagonal of the square { ( x A , x B ) ∈ [0 , : x A = x B } , whilethe stable separatrix are part of the boundary of the basins of attraction of thestable equilibria.Proof. The proof follows directly by using Lemma C.2 and then applying LemmaC.8.
LEMMA C.7.
Consider V − , as defined in Lemma C.2. Then its Jacobian J ac − ( x A , x B ) := (cid:16) ∂ V − ( x A ,x B ) ∂x A | ∂ V − ( x A ,x B ) ∂x B (cid:17) when evaluated: • in (0 , , it has both eigenvalues equal to − qν < , for all q, ν ∈ (0 , ; • in (1 , , it has eigenvalues equal to − (1 − q ) ν and − − (1 − q ) ν , which areboth negative for all q, ν ∈ (0 , ; • in ( q, q ) , it has eigenvalues equal to qν (1 − q ) > , − q (1 + ν (1 − q )) < andcorresponding eigenvectors equal to (1 , and (cid:16) − − q ) ν , (cid:17) .Consider V + . Analogously, its Jacobian has negative eigenvalues when evaluatedin the points (0 , and (1 , . Whereas, when evaluated in ( q, q ) , the eigenval-ues are the same as above, qν (1 − q ) > and − q (1 + ν (1 − q )) < , but theircorresponding eigenvectors are (1 , and ( − − q ) ν, .Proof. By definition of V − in Lemma C.2, the computation of the derivatives inthe point ( q, q ) yields: J ac − ( q, q ) = (cid:18) − q (1 − ν (1 − q )) q − q ) qν − (1 − q ) qν (cid:19) . The eigenvalues and eigenvectors of this matrix are easily computed.Analogously, the computation in the point (0 ,
0) and (1 ,
1) gives:
J ac − (0 ,
0) = (cid:18) − qν − qν (cid:19) , J ac − (1 ,
1) = (cid:18) − − (1 − q ) ν − (1 − q ) ν (cid:19) , from which one obtains the eigenvalues and eigenvectors as claimed above.The last part follows from the same computations done symmetrically for V + . LEMMA C.8.
Consider the two vector fields V − and V + defined in Lemma C.2.Then:1. The points (0 , , (1 , and ( q, q ) are the equilibria of both V − and V + respectively in the region D ∪ ∆ − and D ∪ ∆ + .2. (0 , and (1 , are asymptotically stable for both V − and V + .3. ( q, q ) is a saddle for both V − and V + .Proof. For the first point, consider V − . It suffices to verify that, in the region D ∪ ∆ − , V − ( x A , x B ) = if and only if ( x A , x B ) is one of the points considered inthe claim. Analogously, for V + .The second and third points follow directly from Lemma C.7.27astly, we focus on the crucial role played by the stable separatrix curve of thesaddle ( q, q ), hereafter denoted by C . From the theory of dynamical systems, itfollows that C is partitioned as the image of three distinct trajectories/solutionsof system (6): C = C − ∪ { q, q } ∪ C + . In our case, C − is the piece obtained as the separatrix of the saddle ( q, q ) withrespect to the vector field V − , while C + is the piece obtained as the separatrix of( q, q ) with respect to V + .The following result formalizes what is shown in Figure 4: depending on theparameters ν and q , as time t increases, the solution C − enters the unit squareeither crossing its border along the segment [ q, × { } or along { } × [0 , q ] and,eventually, converges toward { q, q } as t → ∞ . Symmetrically, the same occurs for C + . PROPOSITION C.9.
Let C denote the (unique) stable separatrix of the saddlepoint ( q, q ) of system (6) . The curve C can be naturally partitioned according tothe following three distinct trajectories that compose it: C = C − ∪ { q, q } ∪ C + . Then C − ∩ [0 , is included in [ q, × [0 , q ] and, depending on the parameters q , ν ,it either crosses the segment [ q, × { } in a point ( η, or the segment { } × [0 , q ] in a point (1 , ζ ) . A symmetric result holds for C + .Proof. The result is based on the following lemmas.
LEMMA C.10.
Let C − denote the part of the (stable) separatrix curve of ( q, q ) with respect to V − that belongs to ∆ − . Symmetrically, let C + denote the (stable)separatrix of ( q, q ) for V + belonging to ∆ + . Then: C − ⊂ [ q, × [0 , q ] and C + ⊂ [0 , q ] × [ q, for times t large enough.Proof. Consider the case of C − (the other case of C + is symmetrical). The point( q, q ) is a saddle so its stable separatrix converges to ( q, q ) as t → ∞ and, addition-ally, it is locally linearly approximated by the vector (cid:16) − − q ) ν , (cid:17) , which is theeigenvector corresponding of the negative eigenvalue of J ac − ( q, q ), from LemmaC.7. LEMMA C.11.
The signs of the components of the vector field defining system (6) , when computed in ( x A , x B ) ∈ ( q, × (0 , q ) , are such that V A ( x A , x B ) ≤ and V B ( x A , x B ) ≥ .Symmetrically, V A ≥ and V B ≤ in (0 , q ) × ( q, .Proof. Consider V − A ( x A , x B ), the first component of V − (the other cases are anal-ogous), and let us show that V − A ( x A , x B ) < < x B < q < x A <
1. Bydefinition in Lemma C.2: V − A ( x A , x B ) = ν (1 − x A + x B ) [ x A (1 − x A )( x A − q )] − x A ( x A − x B ) . It has to be shown that for 0 < x B < q < x A < q, ν ∈ (0 , ν (1 − x A + x B ) x A (1 − x A )( x A − q ) ? < x A ( x A − x B ) , that is ν (1 − x A + x B ) (1 − x A )( x A − q ) ? < x A − x B . The right-hand side is always greater than x A − q , so that the inequality becomes: ν (1 − x A + x B ) (1 − x A )( x A − q ) ? < x A − q, ν (1 − x A + x B ) (1 − x A ) ? < . Taking the supremum of the left-hand side for x B ∈ (0 , q ), which is attained at x B = q gives ν (1 − x A + q ) (1 − x A ) ? < . Then the supremum for x A ∈ ( q, x A = q gives ν (1 − q + q ) (1 − q ) ≡ ν (1 − q ) < , which implies that all the inequalities above have to hold, as wanted. Appendix D Linearization of the separatrix curve C and approximation of the basins of at-traction Since that we have already observed that the separatrix C cannot be describedanalytically, we first compute a linear approximation of it and, then, confirm theresults by numerical analysis. This allows us to approximate the area of the basinsof attraction which is key for the comparative statics analysis done in Section 5.We linearize system (6) in a neighborhood of the saddle ( q, q ) using Lemma C.7.Figure 5 shows similarities and differences between C and its linear approximation (cid:101) C . PROPOSITION D.1.
The separatrix C is linearly approximated in ( q, q ) by thetwo-piece line (cid:101) C , which we respectively call (cid:101) C + and (cid:101) C − , defined by (cid:101) C = (cid:101) C + : x B = 1 − − q ) ν ( x A − q ) + q, defined for x A ≤ q, (cid:101) C − : x B = − − q ) ν ( x A − q ) + q, defined for x A ≥ q. Proof.
Let us consider (cid:101) C − (the case of (cid:101) C + is analogous). From Lemma C.7 itfollows that the approximation of the (stable) separatrix of ( q, q ) with respectto V − is the line for ( q, q ) with tangent given by the eigenvector correspondingto the negative eigenvalue, that is the vector (cid:16) − − q ) ν , (cid:17) . Such line in R isparametrically described by (cid:18) x A x B (cid:19) = t (cid:18) − − q ) ν (cid:19) + (cid:18) qq (cid:19) , for t ≥ x B = − − q ) ν ( x A − q ) + q , for x A ≥ q . LEMMA D.2.
The intersection between (cid:101) C − and the sub-diagonal boundaries ofthe unit square [0 , , that is, the segments { } × [0 , q ] and [0 , q ] × { } , is the point P − = ( P − A , P − B ) given by P − = (cid:40) (cid:0) , − ν (1 − q ) + q (cid:1) , if ν < q − q ) , (cid:16) q ν (1 − q ) + q, (cid:17) , if ν ≥ q − q ) , Symmetrically, a point P + can be found as the intersection of (cid:101) C + and the segments { } × [ q, and [0 , q ] × { } . emark. Notice that, provided q ∈ (0 ,
1) and ν ∈ (0 , ν < q − q ) ⇐⇒ ν − √ ν + 14 ν < q. Moreover, when q > / ν < q − q ) for all ν ∈ (0 , q, ν ) ∈ (0 , where this condition is satisfied.Figure 12: Condition on the parameters q and ν q ν Subregions of the square ( q, ν ) ∈ (0 , separated by the curve ν = q − q ) . Thewhite area is where ν < q − q ) , whereasthe gray area is where the opposite in-equality holds. In particular, in the whitearea (respectively, gray) P − belongs to thevertical segment { } × [0 , q ] (resp. hori-zontal segment [ q, × { } ) and the areaunder the curve (cid:101) C − is the trapezoid Q − (resp. triangle T − ). Lastly, the dashedline is at q = 1 / Depending on the parameters ν and q , the area under the curve (cid:101) C is either atrapezoid or a triangle and is easily computed in the following result. By consid-ering this area as an approximation of the area under the curve C , this will alsoallow us to make a comparative statics analysis. The results are also shown inFigure 6. LEMMA D.3. If ν ≥ q − q ) , consider the triangle T − ⊂ { ( x A , x B ) ∈ [0 , : x A ≥ x B } defined as the convex hull in R of the following set of vertexes T − = Conv (cid:0) { ( q, q ) , ( q, , P − } (cid:1) . If, instead, ν < q − q ) , consider the trapezoid Q − ⊂ { ( x A , x B ) ∈ [0 , : x A ≥ x B } defined by Q − = Conv (cid:0) { ( q, q ) , ( q, , (1 , , P − } (cid:1) . The measure of their area is: A( T − ) = q × ( P − A − q )2 = q ν (1 − q ) , defined whenever ν ≥ q − q ) , A( Q − ) = (1 − q ) × ( q + P − B )2 = (1 − q ) (cid:0) q − ν (1 − q ) (cid:1) , when ν < q − q ) . Whenever defined, q (cid:55)→ [A( T − )] ( q, ν ) is always increasing for all ν . Moreover, itsderivative with respect to q is: ∂ A( T − ) ∂q = q (2 − q )4 ν (1 − q ) > , ∀ q, ν ∈ (0 ,
1) : ν ≥ q − q ) . The derivative of A( Q − ) is ∂ A( Q − ) ∂q = 1 − q + 3 ν (1 − q ) , defined whenever ν < q − q ) , nd it is positive if and only if the following condition holds (cid:26) < q ∧ − q − q ) < ν < q − q ) (cid:27) ∨ (cid:26) q < ∧ (cid:20) ν < q − q ) ∨ ν > − q − q ) (cid:21)(cid:27) . Analogously, for P + and correspondingly T + , Q + and their areas and derivatives.Proof. The measure of the areas of the triangle T − or trapezoid Q − are easilycomputed by using the coordinates of P − obtained in Lemma D.2. Computingthe derivatives is then straightforward.Now let us compute the ratio between the area under the curve (cid:101) C − and theentire rectangle [ q, − q ] × [0 , q ], as in Figure 6. LEMMA D.4.
Let (cid:101) R − ( q, ν ) be the ratio between the area under the curve (cid:101) C − and the rectangle [ q, × [0 , q ] ⊂ [0 , . Then (cid:101) R − ( q, ν ) is given by (cid:101) R − ( q, ν ) := [A( T − )] ( q, ν ) q (1 − q ) = q ν (1 − q ) , if ν ≥ q − q ) [A( T − )] ( q, ν ) q (1 − q ) ≡ [A( Q − )] ( q, ν ) q (1 − q ) = , if ν = q − q ) [A( Q − )] ( q, ν ) q (1 − q ) = q − ν (1 − q ) q , if ν ≤ q − q ) . In an analogous fashion, the ratio above the line (cid:101) C + is denoted by (cid:101) R + ( q, ν ) and isequal to (cid:101) R − ( q, ν ) by symmetry.Proof. The numerator of (cid:101) R − is given by the area of the triangle or trapezoidgiven by the previous Lemma D.3, that is, respectively A ( T − ) or A ( Q − ). Thedenominator, instead, is simply the area of the rectangle [ q, × [0 , q ] in R .The behavior of (cid:101) R ( q, ν ), as a function of the parameters q and ν , is describedby the following result. LEMMA D.5 ( Comparative statics on the approximated ratio (cid:101) R ( q, ν )) . Consider (cid:101) R − ( q, ν ) defined above. It is bounded in [0 , and its sections q (cid:55)→ (cid:101) R − ( q, ν ) are increasing for all ν ∈ (0 , , whereas ν (cid:55)→ (cid:101) R − ( q, ν ) are decreasing forall q ∈ (0 , . Furthermore, whenever defined, its derivatives are: ∂∂q (cid:101) R − ( q, ν ) = q ν (1 − q ) > , if ν > q − q ) (cid:18) q − (cid:19) ν > , if ν < q − q ) ∂∂ν (cid:101) R − ( q, ν ) = − q − q ) ν < , if ν > q − q ) − (1 − q ) q < , if ν < q − q ) . Proof.
The computation of the derivatives follows directly from the formulas defin-ing (cid:101) R − in the previous Lemma. Moreover, it is straightforward to check thatthe denominator is always greater than the numerator, thus guaranteeing that (cid:101) R − ( q, ν ) ≤ The complicated condition derives from the fact that − q − q ) is decreasing while q − q ) isincreasing, when q ∈ (0 , q = 2 / Area of basin of attraction and its approximation
The plot shows: the area of the basin of attraction of (0 ,
0) as a function of q and ν computed numerically (in blue), the area computed using the approximation (cid:101) C (in yellow,almost indistinguishable from that in blue) and, finally, their differences (the almost-plainsurface, in red). Remark.
Given q and ν , R − ( q, ν ) just represents the part of the basin of attrac-tion of (0 ,
0) within the rectangle [ q, × [0 , q ]. So, the exact total area of the basinof attraction of (0 ,
0) is readily computed as: 2 · R − ( q, ν ) + q . Analogously for itsapproximation obtained with (cid:101) R −−