Infection models on dense dynamic random graphs

INFECTION MODELS ON DENSE D YNAMIC RANDOM GRAPHS SIMONE BALD ASSARRI, PETER BRAUNSTEINS, FRANK DEN HOLLANDER, AND MICHEL MANDJES Abstra ct. W e consider Susceptible-Infected-Reco vered (SIR) models on dense dynamic ran- dom graphs, in whic h the join t dynamics of v ertices and edges are c o-evolutionary , i.e., they influence eac h other bidirectionally . In particular, edges app ear and disapp ear o ver time dep ending on the states of the t wo connected vertices, on how long they ha ve b een infected, and on the total density of susceptible and infected vertices. Our main results establish functional la ws of large num bers for the densities of susceptible, infected, and recov ered vertices, jointly with the underlying evolving random graphs in the graphon space. Our results are supp orted b y simulations, which characterize the limiting size of the epidemics, i.e., the limiting density of susceptible vertices, and ho w the p e ak of the epidemics dep ends on the rate of the ev olution of the underlying graph. The proofs of our main results rely on the careful construction of a mimicking pr o c ess , obtained by appro ximating the tw o-w a y feedback interaction betw een vertex and edge dynamics with a mean-field type interaction, acting only as one-wa y feedback, that remains sufficiently close to the original co-ev olutionary pro cess. T o treat the more general setting in which edge dynamics are affected by the prop ortions of susceptible and infected individuals, we in tro duce a metho dological extension of existing tec hniques. W e thus show that our mo del exhibits m ultiple epidemic p eaks – a phenomenon observed in real-world epidemics – which can emerge in mo dels that incorp orate mutual feedback b etw een vertex and edge dynamics. MSC 2020 subje ct classific ations. 60F17, 60K35, 60K37. Key wor ds and phr ases. SIR dynamics, graph dynamics, co-ev olution. A cknow le dgment. The w ork in this pap er w as supp orted by the Europ ean Union’s Horizon 2020 researc h and innov ation programme under the Marie Skło do wsk a-Curie grant agreement no. 101034253, and by the NWO Gravitation pro ject NETWORKS under grant no. 024.002.003. SB was further supp orted through “Grupp o Nazionale p er l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMP A-INdAM). Date : F ebruary 17, 2026. 1 Contents 1. In tro duction 2 2. Mo del and main results 5 2.1. Mo del definition 5 2.2. Main results 9 3. Illustrations 10 3.1. Dynamics without global feedback 10 3.2. Multiple p eaks with global feedb ac k 13 4. Road map 16 5. Pro ofs 19 5.1. Pro of of Theorem 2.1 19 5.2. Coupling 21 5.3. Pro of of Theorem 2.2 26 App endix A. Graphons, Lévy metric and an additional lemma 26 References 27 1. Intr oduction Mathematical mo dels pla y a crucial role in understanding and managing the spread of infectious diseases. These mo dels offer insigh t into ho w v arious parameters influence the dynamics of an epidemic, whic h can b e instrumen tal in forecasting outbreaks and designing effectiv e con trol and mitigation strategies. The base epidemic mo del is the Susc eptible-Infe cte d-R e c over e d (SIR) compartmental mo del. The deterministic version of this mo del is characterized by a system of differential equations that captures the evolution of the prop ortion of susceptible, infected, and recov ered individuals in the p opulation o v er time. The same differen tial equations also app ear in the sto chastic v ersion of the mo del, as a large p opulation limit established through a functional law of lar ge numb ers (FLLN). In the presen t pap er, we establish suc h a FLLN for a considerably more general v ersion of the base SIR mo del. The basic v ersion of the SIR mo del relies on simplifying assumptions that are typically not satisfied in practice. One k ey assumption is homo gene ous mixing , which implies that ev ery individual in the p opulation is equally lik ely to come in to contact with every other individual. If w e represen t eac h individual in the p opulation as a vertex in a graph and eac h so cial connection through which the epidemic can spread as an edge, then the assumption of homogeneous mixing 2 is equiv alent to assuming that every e dge is pr esent in the gr aph . How ev er, this assumption o v erlo oks several crucial asp ects of real-world so cial netw orks during a pandemic. In realit y , the net w ork is heter o gene ous , reflecting that some individuals hav e more so cial connections than others. It is also dynamic , as connections can app ear and disappear ov er time. Moreov er, these dynamics are c o-evolutionary : the individual epidemiological states (i.e., susceptible, infected, or recov ered) are influenced b y the structure of the net w ork (since the epidemic spreads through existing connections) and the net w ork itself ev olves in resp onse to the epidemic, as individuals ma y c ho ose to alter their connections to preven t further spread of the disease. A t a more precise level, a realistic mo del should incorp orate tw o types of co-ev olution: (a) First, the dep endence of the net work on the individual epidemiological states can b e lo c al . F or example, susceptible individuals ma y choose to break ties with other individuals they susp ect to be infected. (b) Second, there can b e glob al feedbac k mec hanisms. F or instance, quarantine measures ma y b e in tro duced once the prop ortion of infected individuals exceeds a certain threshold. Ideally , we would work with a mo del in which the underlying graph is heterogeneous (i.e., not complete), is dynamic (i.e., evolv es o v er time), and exhibits co-ev olution (i.e., the net w ork influences the epidemic dynamics and is, in turn, shap ed b y them – b oth at local and global scales). In this pap er w e introduce a stochastic SIR model that explicitly captures the realistic net w ork features discussed ab ov e. W e analyze this mo del in the dense r e gime in whic h the n um b er of edges scales roughly as n 2 , with n denoting the n um b er of individuals. Our first goal is to establish a FLLN for the prop ortion of susceptible, infected, and reco v ered individuals o v er time. While our mo del is inherently more complex, this result is in a similar spirit as the FLLN for the base SIR mo del. In view of the general co-evolutionary dynamics incorp orated in the mo del, we are also in terested in ho w the netw ork evolv es during the pandemic. Our second ob jective is therefore to establish a FLLN for the dynamic netw ork. The prop osed mo del is capable of repro ducing complex real-world phenomena, suc h as the emergence of multiple infection p eaks — patterns that cannot b e captured in SIR mo dels without incorp orating the co-ev olutionary feedbac k b etw een individual states and net work structure. W e use our results to gain insigh t in to the evolution of the epidemic and the state of the net w ork as the epidemic ev olv es. Our results should b e viewed within the con text of SIR mo dels on random graphs. W e b egin b y pro viding a brief o verview of this subarea. Broadly , the literature can b e categorized in to four main groups: ˝ A FLLN for the SIR mo del on a sp arse static configuration mo del was proposed in [ 37 ]. This result was subsequently rigorously established in [ 8 ], [ 10 ], [ 19 ], and [ 26 ] under progressiv ely w eaker assumptions on the distribution of the vertex degrees in the 3 configuration mo del. A similar result for an SIR mo del on a sparse static sto chastic blo c k mo del app eared more recen tly in [13]. ˝ FLLNs for SIR mo dels on dense static random graphs hav e b een considered in [ 20 , 28 , 34 ]. In these pap ers, the random graph through whic h the epidemic spreads is constructed by sampling from a reference graphon. The results in [ 34 ] apply when the pro cess exhibits non-Mark o vian dynamics but hold only when then num ber of edges scale as n 2 , whereas the results in [ 20 , 28 ] consider Mark o vian dynamics but also hold when the num ber of edges scale as n 1 ` a for a P p 0 , 1 s . ˝ FLLNs for the SIR mo del on sp arse dynamic random graphs with co-evolutionary feedbac k are established in [ 5 , 6 , 16 , 27 ] (see also [ 21 ] for the SI model). In these pap ers, susceptible individuals can break edges with infected neigh b ors, potentially rewiring the edge to another individual. A primary fo cus of these pap ers is to establish a discontin uit y in the final prop ortion of susceptible individuals at the critical v alue of the contact rate. In these pap ers, the co-ev olutionary dynamics are lo cal, in the sense that susceptible individuals break the connection with their infected neighbors in a manner that do es not dep end on the o v erall state of the epidemic (i.e., the total n um b er of infected individuals). ˝ A FLLN for the SIR mo del on dense dynamic random graphs is established in [ 24 ]. In this pap er the underlying graph is a dynamic stochastic blo c k mo del that allo ws for graph degrees that scale at least as log n (so at least n log n edges). Ho wev er, there is no co-ev olutionary feedbac k in the graph dynamics considered, i.e., the state of the graph ev olv es indep enden tly of the epidemic states of the individuals. Curren tly , the literature lac ks a FLLN for an SIR mo del on a dense dynamic random graph that incorp orates b oth lo cal and global co-ev olutionary feedbac k. Moreov er, no efforts hav e b een made to establish suc h a la w for the ev olution of the net w ork itself (i.e., join tly with the prop ortion of susceptible, infected, and recov ered individuals). The present pap er aims to fill these gaps. More broadly , our contributions b elong to the gro wing literature for related mo dels on dynamic random graphs. See for instance [ 2 , 9 , 18 , 23 , 25 , 29 , 33 , 36 ] and references therein. Metho dologically , our pap er builds on tw o earlier w orks [ 15 , 3 ] concerning graph-v alued sto c hastic pro cesses. A cen tral elemen t of these pap ers is the concept of a gr aphon (developed in [ 11 , 12 , 30 , 31 , 32 ]), whic h is used to describ e the limit of the netw ork. Graphons hav e b een extensiv ely used to characterize the limit of other dense graph-v alued pro cesses; see for instance [ 1 , 4 , 14 , 17 , 22 , 35 ]. Ho w ever, to the b est of our kno wledge, the present pap er is the first to establish a limit in the space of graphons for a co-ev olutionary mo del with a global feedbac k mec hanism. In [ 15 ] limits are established for a graph-v alued pro cesses with one-w ay dep endence (i.e., not co-ev olutionary), in the setting where the edge dynamics can dep end on the collectiv e states of all v ertices across the net w ork (i.e., global dep endence). On the other hand, in [ 3 ] limits are established for graph-v alued pro cesses that are co-ev olutionary , but where the co-evolutionary 4 feedbac k is lo cal only (i.e., no global feedbac k). In the presen t pap er, we use the mechanism in [ 15 ] to allo w the edge dynamics to dep end, for example, on the total num ber of infected individuals (i.e., global feedback), and w e adapt the pro of technique in [ 3 ] to establish our results in this more general setting. This technique builds on the notion of a mimicking pr o c ess in tro duced in [ 3 ], whic h approximates the bidirectional in teraction b etw een vertex and edge dynamics b y a mean-field-t yp e, single-dir e ctional in teraction that remains sufficien tly close to the original co-ev olutionary process. Our analysis suggests that incorporating global feedback mec hanisms can b e ac hiev ed with surprising ease within the framework established in [ 3 ]. W e exp ect that this approac h is applicable more broadly to other dense co-ev olutionary processes. The pap er is organized as follows. In Section 2 we detail the system dynamics, present our main theoretical findings, and provide some discussion, concluding remarks and outlo ok. The sim ulations rep orted in Section 3 rev eal that our theory is capable of repro ducing patterns observ ed in practice, most notably tra jectories with multiple p eaks. A road map of the pro of is pro vided in Section 4, while the actual pro ofs are pro vided in Section 5. 2. Model and main resul ts In this section w e first define the stochastic dynamics underlying our SIR mo del on a dense dynamic random graph. This description sp ecifies the w a y that we incorp orate b oth lo cal and global co-ev olutionary feedbac k. W e then state and briefly discuss our main results. 2.1. Mo del definition. In this pap er we analyze an SIR mo del on a dynamic random graph. Giv en a time horizon T ą 0 , w e denote b y p G n p t qq t Pr 0 ,T s an ev olving random graph with vertex set r n s : “ t 1 , 2 , ..., n u . F or i, j P r n s , w e let e ij p t q “ 1 if edge ij is active at time t and 0 otherwise. The graph G n p t q is undirected, so e ij p t q “ e j i p t q , and contains no self-lo ops, so e ii p t q “ 0 . Let x i p t q P t S , I , R u b e the state of v ertex i P r n s at time t P r 0 , T s , where ‘ S ’ means that it is susceptible, ‘ I ’ means that it is infected, and ‘ R ’ means that it is reco v ered. Initialization. A t time t “ 0 w e initialize the graph as an Erdős-Rén yi random graph with connection probability p 0 P p 0 , 1 q . Eac h vertex is infected with probability q 0 P p 0 , 1 q and susceptible with probabilit y 1 ´ q 0 , indep enden tly of the states of the other vertices and of the initial random graph. Th us, no v ertex is in the recov ered state at t “ 0 . Definition of vertex typ es. F or any infected vertex i P r n s , we define its infe ction time as t I i : “ inf t s P r 0 , T s : x i p s q “ I u . W e define typ e of v ertex i at time t as y i p t q : “ $ ’ ’ & ’ ’ % ´ 1 if x i p t q “ S , t ´ t I i if x i p t q “ I , T ` 1 if x i p t q “ R , (2.1) 5 where t ´ t I i is the length of time that v ertex i has b een infected for. W e then define the empiric al typ e pr o c ess , p F n p t ; ¨qq t Pr 0 ,T s , through F n p t ; y q : “ 1 n n ÿ i “ 1 1 t y i p t q ď y u , y P R . (2.2) V ertex and e dge dynamics. F or each infected vertex i , w e denote b y I p¨q its infe ctivity , where I : r 0 , 8q Ñ r 0 , 1 s is a con tin uous deterministic function. ˝ V ertex dynamics. Let N I i p t q b e the set of infected neigh b ors of v ertex i at time t . A t an y time t P r 0 , T s , x i p t q transitions from S to I at rate λ n ÿ j P N I i p t q I p y j p t qq , and x i p t q mak es a transition from I to R at a (normalised) rate of 1 . ˝ Edge dynamics. A t any time t P r 0 , T s , e ij p t q transitions from 0 to 1 (i.e., from inactive to activ e) at rate $ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ % γ π SS p F n p t ; ¨qq , if x i p t q “ x j p t q “ S , γ π SI p y i p t q , F n p t ; ¨qq , if x i p t q “ I , x j p t q “ S , γ π SI p y j p t q , F n p t ; ¨qq , if x i p t q “ S , x j p t q “ I , γ π II p y i p t q , y j p t q , F n p t ; ¨qq , if x i p t q “ x j p t q “ I , and e ij p t q transitions from 1 to 0 (i.e., from activ e to inactive) at the same rate with γ π s p¨q replaced b y γ p 1 ´ π s p¨qq , with s P t SS , SI , I I u . When x i p t q transitions from I to R w e supp ose that v ertex i resamples adjacen t edges with the initial probability p 0 , after whic h p oin t these edges are static (i.e., they no longer turn on and off ). Note that in our setup the functions π s p¨q can dep end on F n p t ; ¨q , and therefore on the prop ortion of susceptible, infected and recov ered v ertices, and also on the distribution of the times whic h infected vertices ha v e been infected for. W e imp ose a Lipsc hitz contin uit y condition on the functions π II p¨q . Sp ecifically , w e supp ose that there exists L II ă 8 suc h that | π II p a 1 , b 1 , F 1 q ´ π II p a 2 , b 2 , F 2 q| ď L II r| a 1 ´ a 2 | ` | b 1 ´ b 2 | ` d L p F 1 , F 2 qs for all a 1 , a 2 , b 1 , b 2 ą 0 and all distribution functions F 1 and F 2 , where d L is the Lévy metric defined in (A.4). W e also imp ose analogous Lipschitz contin uit y assumptions on π SS and π SI . Gr aphons. F or an y t P r 0 , T s , let h G n p t q b e a function from r 0 , 1 s 2 to r 0 , 1 s whic h is characterized b y h G n p t q p x, y q : “ e r nx s , r ny s p t q . (2.3) The function h G n p t q is referred to as empiric al gr aphon asso ciated with G n p t q and encodes the adjacency matrix of G n p t q (see Figure 2.1). 6 1 2 5 4 3 Graph A djacency matrix Graphon , h blac k: h p x, y q “ 1 white: h p x, y q “ 0 . 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 5 2 5 3 5 4 5 1 0 1 5 2 5 3 5 4 5 1 Figure 2.1: Illustration of the relation b et ween graph, adjacency matrix, and graphon. Figure 2.2: T w o graphons with differen t lab ellings (left and middle panels), plus the limiting graphon when the lab elling of the middle panel has b een used (right panel). Empirical graphons b elong to the space of gr aphons W , con taining all functions h : r 0 , 1 s 2 ÞÑ r 0 , 1 s suc h that h p x, y q “ h p y , x q for all p x, y q P r 0 , 1 s 2 . W e endo w W with the cut distanc e : d ˝ p h 1 , h 2 q : “ sup S,T Ďr 0 , 1 s ˇ ˇ ˇ ˇ ż S ˆ T d x d y r h 1 p x, y q ´ h 2 p x, y qs ˇ ˇ ˇ ˇ , h 1 , h 2 P W . The space W is used to establish limiting results for random graphs; we establish our main result in the space of W -v alued paths. W e need to carefully c hoose ho w to lab el the v ertices, since d ˝ p h 1 , h 2 q can dep end on the lab eling conv en tion. F or example, in Figure 2.2 the left and center empirical graphons represent the same graph up to a relabeling of vertices, but only the center graphon is close to the right limiting graphon in the cut distance. Throughout this paper we choose to label the v ertices suc h that y 1 p t q ď y 2 p t q ď ¨ ¨ ¨ ď y n p t q , for all t P r 0 , T s , with ties brok en arbitrarily . This means that susceptible vertices hav e a lo w er lab el than infected vertices, infected vertices hav e a low er lab el than recov ered v ertices, and infected v ertices that ha v e b een infected for a short time ha v e a lo w er lab el than those that ha v e b een infected for a long time. This also implies that the lab els of the vertices change ov er time. 7 Candidate limit. In FLLNs, the limiting ob ject can b e informally understo o d as capturing the system’s dynamics in the limit of large system size. This deterministic limit, often referred to as the fluid limit , is t ypically c haracterized b y a system of differential equations. Belo w, w e outline the procedure for identifying this candidate limit in the con text of our system. W e let p F p t ; ¨qq t Pr 0 ,T s b e a time-v arying distribution function which can b e though t of as a candidate limit of p F n p t ; ¨qq t Pr 0 ,T s as n Ñ 8 . F or an y t P r 0 , T s , F p t ; y q is absolutely con tin uous on y P r 0 , t q and has p oin t masses at ´ 1 , t , and T ` 1 , with no probabilit y mass elsewhere. W e c haracterize F through differen tial equations; ho w ever, in order to write them in a compact form w e require additional notation. First, let f I p t ; y q : “ d d y F p t ; y q for y P r 0 , t q , whic h can b e in terpreted as the limiting densit y of the amount of time that infected individuals ha v e b een infected for at time t , and p S p t q : “ F p t ; ´ 1 q ´ lim s Ò´ 1 F p t ; s q , p I p t q : “ q 0 e ´ t ` ż t ´ 0 d y f I p t ; y q , p R p t q : “ F p t ; T ` 1 q ´ lim s Ò T ` 1 F p t ; s q , (2.4) whic h can b e interpreted as the limiting prop ortion of susceptible, infected, and recov ered individuals at time t . Next, let H b e an edge connection probabilit y function, defined b y H p t ; u, v , p F p s ; ¨qq s Pr 0 ,t s q : “ P p e ij p t q “ 1 | y i p t q “ u, y j p t q “ v , p F n p s ; ¨qq s Pr 0 ,t s “ p F p s ; ¨qq s Pr 0 ,t s q ; w e remark that this probability is defined more rigorously as an in tegral of the ob jects π SS , π SI , and π II that w e in tro duced in Equation (4.2). Finally let J p t q : “ I p t ; F p t ; ¨qq : “ ż t 0 d F p t ; u q H p t ; ´ 1 , u, F p t ; ¨qq I p u q , (2.5) where λ J p t q can b e in terpreted as the limiting rate (as n Ñ 8 ) at which susceptible individuals b ecome infected at time t . The function F is characterized by the initial conditions p S p 0 q “ 1 ´ q 0 , p I p 0 q “ q 0 , (2.6) with the initial infection-age distribution concentrated at u “ 0 (that is, f I p 0; u q “ q 0 δ 0 p u q ), while the updating equations are B B t p S p t q “ ´ λ J p t q p S p t q , B B t f I p t ; u q ` B B u f I p t ; u q “ ´ f I p t ; u q , u ą 0 , (2.7) f I p t ; 0 q “ λ J p t q p S p t q , B B t p R p t q “ p I p t q , 8 where f I p t ; u q denotes the densit y of infected individuals with infection age u at time t , and p I p t q “ ş 8 0 f I p t ; u q d u . Due to the initially infected individuals, there is a probability mass at t yp e t for all t P r 0 , T s , and hence f I p t ; u q do es not exist when u “ t . This technical issue can b e resolv ed in a standard w a y b y in tro ducing the measure-v alued notation F I p t ; d u q and treating the point mass at t separately . Since this mo dification is entirely straightforw ard and do es not affect any of the subsequen t argumen ts, we suppress it in the notation to streamline the presen tation. Note that (2.7) can b e written (in a more inv olv ed form) in terms of F and the mo del parameters defined ab ov e, i.e., p 0 , q 0 , λ , γ , I p¨q , π SS p¨q , π SI p¨ , ¨q , and π II p¨ , ¨ , ¨q . Moreov er, F can b e reco v ered from the solution to (2.7) by letting, for t P r 0 , T s , F p t ; y q “ $ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ % 0 , y ă ´ 1 , p S p t q , y P r´ 1 , 0 q , p S p t q ` ş y 0 d u f I p t ; u q , y P r 0 , t q , p S p t q ` p I p t q , y P r t, T ` 1 q , 1 , y ě T ` 1 . (2.8) W e also need to construct a candidate limit for the graphon v alued pro cess p h G n p t q q t Pr 0 ,T s as n Ñ 8 . T o this end, we let ¯ F denote the generalized in verse of F , whic h is formally defined as ¯ F p t ; x q : “ inf t u : F p t ; u q ą x u . F or t P r 0 , T s , let the candidate limit g r F s p t ; ¨q P W b e giv en via g r F s p t ; x, y q : “ H p t ; ¯ F p t ; x q , ¯ F p t ; y q , p F p t ; ¨q t Pr 0 ,T s qq . (2.9) 2.2. Main results. First we establish that the candidate limit p F p t ; ¨qq t Pr 0 ,T s is well-defined. Note that, giv en (2.9), this also implies that p g r F s p t ; ¨qq t Pr 0 ,T s is w ell-defined. Theorem 2.1. Ther e is a unique solution p F p t ; ¨qq t Pr 0 ,T s to the system of differ ential e quations char acterize d by (2.7) with initial c ondition (2.6) . In p articular, for al l t P r 0 , T s , F p t ; ¨q in (2.8) fol lows fr om (i) p S p t q “ p 1 ´ q 0 q exp ˆ ´ λ ż t 0 d s J p s q ˙ and (ii) f I p t ; u q , with u P r 0 , t q , b eing the unique solution of the fixe d-p oint e quation f I p t ; u q “ e ´ t f I p 0; u ´ t q ` λ ż t 0 d s e ´p t ´ s q J p s q p S p s q . Next w e establish conv ergence of the random pro cesses p F n p t ; ¨qq t Pr 0 ,T s and p h G n p t q q t Pr 0 ,T s to their resp ectiv e candidate limits p F p t ; ¨qq t Pr 0 ,T s and p g r F s p t ; ¨ , ¨qq t Pr 0 ,T s , as n Ñ 8 . T o state this result, we let p M , d L q denote the space of probability distributions equipp ed with the Lévy metric (whic h induces the weak top ology; see (A.4) ). In the sequel, denote by D pp M , d L q , r 0 , T sq 9 (resp. D pp W , d ˝ q , r 0 , T sq ) the Skorokhod space of M -v alued (resp. W -v alued) paths on r 0 , T s . Here and in the sequel, ‘ ñ ’ denotes con vergence in distribution. Theorem 2.2. As n Ñ 8 , on D pp M , d L q , r 0 , T sq , p F n p t ; ¨qq t Pr 0 ,T s ñ p F p t ; ¨qq t Pr 0 ,T s and, on D pp W , d ˝ q , r 0 , T sq , p h G n p t q p¨ , ¨qq t Pr 0 ,T s ñ p g r F s p t ; ¨ , ¨qq t Pr 0 ,T s . 3. Illustra tions In this section, we illustrate our findings. In the first subsection, w e fo cus on the case with only lo cal feedbac k, presen ting expressions for the basic repro duction num ber R 0 and the final fraction of susceptible individuals, and discussing the epidemic peak. The second subsection presen ts a mo del with global feedback, which shows that multiple infection p eaks can arise. 3.1. Dynamics without global feedback. In this subsection, we consider a simpler setting in whic h global feedbac k is absent. More sp ecifically , we make the assumption that π SS p¨q “ p 0 , and π SI p u, F q “ π SI p u q . (3.1) Note that π SS p¨q “ p 0 implies that susceptible individuals are connected with probabilit y p 0 at any time t , and π SI p u, F q “ π SI p u q en tails that there is no global co-ev olutionary feedbac k mec hanism. Under these assumptions, w e can follo w standard intuitiv e calculations to derive compact expressions for the basic repro duction n umber R 0 and the final prop ortion of susceptible individuals p S p8q as n b ecomes large (cf. [8, Section 1]). Note that, under (3.1) , an individual who w as infected u time units ago is connected to an y susceptible individual with probability p SI p u q : “ p 0 e ´ γ u ` ż u 0 d s γ e ´ γ s π SI p u ´ s q . The probabilit y that the infected individual infects any susceptible individual throughout their lifetime is then ¯ p SI ,n p u q : “ ż 8 0 d u p SI p u q λ n I p u q e ´ u . Considering the exp ected n um b er of infections caused b y a single infected individual among a large n um b er (i.e., n Ñ 8 ) of susceptible individuals, w e then obtain the basic repro duction n um b er R 0 “ ż 8 0 d u p SI p u q λ I p u q e ´ u . (3.2) T o write down a formula for the final v alue p S p8q , note that an initially susceptible individual do es not ev er get infected with probability p S p8q{p 1 ´ q 0 q , and if it remains susceptible, then it 10 do es not get infected by an y of the n p 1 ´ s p8qq individuals that get infected at some point. Consequen tly p S p8q 1 ´ q 0 “ lim n Ñ8 p 1 ´ ¯ p SI ,n p u qq n p 1 ´ p s S p8qq “ lim n Ñ8 ˆ 1 ´ ż 8 0 d u p SI p u q λ n I p u q e ´ u ˙ n p 1 ´ p S p8qq “ lim n Ñ8 ˆ 1 ´ R 0 p 1 ´ p S p8qq n p 1 ´ p S p8qq ˙ n p 1 ´ p S p8qq “ e ´ R 0 p 1 ´ p S p8qq , whic h implies that p S p8q is the unique solution in r 0 , 1 s of the fixed-p oint equation p S p8q “ p 1 ´ q 0 q e ´ R 0 p 1 ´ p S p8qq . (3.3) Let p S ,n p t q denote the prop ortion of susceptible individuals at time t in a system with n individuals. As a consequence of Theorem 2.2, it holds that if p S p8q “ lim t Ñ8 lim n Ñ8 p S ,n p t q , then p S p8q satisfies (3.3) . It should b e noted, how ev er, that the limiting proportion of susceptible individuals is most naturally expressed as p S p8q “ lim n Ñ8 lim t Ñ8 p S ,n p t q . The required interc hange of limits is generally justified b y dominating the epidemic pro cess with a sub critical branc hing pro cess after some large time t (see, for instance, [ 13 , Section 6.3]). Here, w e encounter tw o challenges: (1) the v ertex types are con tin uous, whic h would likely require w orking with a dominating branching pro cess with a con tin uous t yp e space, and (2) the dynamic nature of the edges complicates the construction of suc h a dominating branc hing pro cess. Addressing these technical challenges is b eyond the scop e of the presen t w ork. W e are also in terested in the p e ak of the epidemic, defined as i max : “ max t ě 0 p I p t q . A classical result for the conv en tional SIR mo del is that i max “ 1 ´ R ´ 1 0 ` R ´ 1 0 log R ´ 1 0 . (3.4) In our setting, deriving an analogous formula is c hallenging (see also [ 24 , Section 3.2] for a related analysis). Nevertheless, the findings b elo w illustrate how γ , whic h defines the timescale of the edge dynamics, influences i max . T o this end, w e first analyze the effect of γ on the basic repro duction num ber R 0 ” R 0 p γ q , assuming I p¨q ” 1 . F rom (3.2), b y direct computations we find R 0 p γ q “ λ γ ` 1 p 0 ` λγ ż 8 0 d u e ´ u ˆ ż u 0 d s e ´ γ s π SI p u ´ s q ˙ “ : A p γ q ` B p γ q ; observ e that in the static setting (in which γ “ 0 ) this reduces to R 0 p 0 q “ λp 0 . After applying F ubini–T onelli and the change of v ariables v “ u ´ s in the inner in tegral, w e can write B p γ q “ λγ ż 8 0 d s e ´ γ s ˆ ż 8 0 d v e ´p s ` v q π SI p v q ˙ “ λγ C ż 8 0 d s e ´p γ ` 1 q s “ λγ C γ ` 1 , 11 Figure 3.1: Eac h solid curve represents the n umerical solution of the system of PDEs for π SS “ p 0 , λ “ 4 , q 0 “ 0 . 1 , and v arying p 0 , π SI , γ . where C : “ ż 8 0 d v e ´ v π SI p v q . Note that C ă 8 b ecause π SI p¨q w as assumed to b e Lipsc hitz con tin uous. It thus follows that R 0 p γ q “ λ p p 0 ` γ C q γ ` 1 , and as a consequence R 1 0 p γ q “ λ p C ´ p 0 q p γ ` 1 q 2 . W e therefore conclude that R 0 is increasing in γ if and only if C ą p 0 . Note that through Equation (3.3) w e see that p S p8q is monotonically increasing in R 0 , and hence p S p8q is monotonically increasing in γ if and only if C ą p 0 . The condition C ą p 0 is simplest to understand when π SI p¨q is constant (say π SI ), where it reduces to π SI ą p 0 . In this case, once an individual b ecomes infected, the probability that they are connected to a susceptible individual, p SI p u q , increases with the time since infection u . If the edge re-sampling rate γ increases, then the increase in p SI p u q is sp ed up, leading to an increase in R 0 (see (3.2) ). Equiv alen t reasoning also explains wh y R 0 is decreasing in γ when C ” π SI ă p 0 . W e no w turn to the p eak of the epidemic, i max . First observe that if the classical result (3.4) w ere to hold for our model, then i max w ould be increasing in R 0 . Second, since the n um b er of infected individuals at any given time cannot exceed the total fraction that will ev en tually b e infected, w e obtain the upp er b ound i max ď 1 ´ p S p8q , where the right-hand side is lik ewise increasing in R 0 . T aken together with the preceding discussion, these observ ations suggest that i max is monotonically increasing in γ if and only if C ą p 0 . As w e are unable to deriv e a counterpart of (3.4) for the present mo del, we cannot establish this claim rigorously . Nev ertheless, the conclusion is strongly supp orted b y the numerical results sho wn in Figure 3.1. 12 3.2. Multiple p eaks with global feedbac k. In this subsection, we present an example that illustrates ho w global feedback b et w een the epidemic state and netw ork connectivity can pro duce multiple infection p eaks. T o capture this b ehavior, we consider a mo del in which net w ork connectivit y ev olv es in resp onse to the o v erall infection lev el. F or a ą 0 , we define ϕ p t q : “ ż a 0 d s f I p t ; s q , whic h can b e equiv alen tly written as ϕ p t q “ F n p t ; a q ´ F n p t ; 0 q . Here, ϕ p t q denotes the prop ortion of individuals who w ere infected at most a time units ago and who are still infectious at time t . It may b e interpreted as the p opulation’s p erceiv ed ov erall ‘threat level’. Since ϕ p t q dep ends on infections that o ccurred in the past (namely , within the preceding a time units), the p erceived threat level generally lags b ehind the actual threat level. The latter is most naturally quan tified b y the curren t rate of infection (see (2.5) for the limiting rate as n Ñ 8 ). F or 0 ă ϕ 1 ă ϕ 2 ă 8 , define the piecewise linear function d p ϕ q “ $ ’ ’ ’ & ’ ’ ’ % 0 . 1 if ϕ ď ϕ 1 , 0 . 1 ` 0 . 8 ϕ ´ ϕ 1 ϕ 2 ´ ϕ 1 if ϕ 1 ă ϕ ă ϕ 2 , 0 . 9 if ϕ ě ϕ 2 , whic h is a b eha vioral con trol function, represen ting ho w strongly the p opulation is distancing: d p ϕ q « 0 . 9 (resp. d p ϕ q « 0 . 1 ) means that the system is in ‘distancing mo de’ (resp. ‘normal mo de’). W e then put π SS p ϕ q : “ p 1 ´ d p ϕ qq p norm SS ` d p ϕ q p dist SS , π SI p ϕ q : “ p 1 ´ d p ϕ qq p norm SI ` d p ϕ q p dist SI , (3.5) for given n umbers p norm SS , p dist SS , p norm SI , and p dist SI , where it is en visaged that p norm SS ą p dist SS and p norm SI ą p dist SI . W e hav e thus constructed a simplified model of ‘b ehavioural resp onse’ (e.g., b y go v ernment in terv en tions or volun tary distancing), enforcing that contact rates drop when p eople p erceiv e high infection. In the n umerical illustration in Figures 3.2 and 3.3, w e ha v e c hosen the parameters: p 0 “ 0 . 1 , q 0 “ 0 . 05 , ϕ 1 “ 0 . 24 , ϕ 2 “ 0 . 28 , γ “ 20 , λ “ 10 , p norm SS “ 0 . 9 , p dist SS “ 0 . 3 , p norm SI “ 0 . 6 , p dist SI “ 0 . 01 , π II “ 0 . 3 , a “ 1 . These figures display the functional la w of large num bers for the epidemic states and netw ork that was established in Theorem 2.2, and simulations with finite n . The explanation b elow fo cuses on the large p opulation l imit, and describ es how to interpret the figures. The following phases can be distinguished: ˝ Initial rise: In the top ro w of Figure 3.2 we observe that the prop ortion of infected individuals is increasing for t P r 0 , 0 . 69 s . As displa y ed in Figure 3.3, the pro cess is in normal mo de for the v ast ma jority of this time in terv al. Consequen tly , there is a high proportion of activ e SI edges through whic h the epidemic can spread, and still a relativ ely high prop ortion of susceptible individuals which can b e infected. Indeed, 13 Figure 3.2: The grey tra jectories in the top row, from left to righ t, corresp ond to simulations of 100 sto c hastic SIR tra jectories of the fraction of infected individuals for n “ 200 , 500 , 1000 , resp ectiv ely . The solid blue curv es are the n umerical solution of the system of PDEs with the same parameters. Observ e that the ‘cloud’ of simu lated tra jectories shrinks with n , reflecting the con v ergence to the deterministic limiting path. The second ro w displa ys the empirical graphons corresponding to the blac k tra jectory ( n “ 200 ) when t “ 0 . 69 (first p eak), t “ 1 . 4 (v alley b etw een the first tw o p eaks) and t “ 1 . 71 (second p eak); a dot represen ts an edge. The lab els of the verti ces are up dated dynamically so that they are ordered lexicographically , i.e., the vertices with state S ha ve low er lab els than the vertices with state I , which in turn hav e low er lab els than the vertices with state R , and then by increasing t yp e. The third row displays the corresp onding FLLN. The b ottom row displays ¯ F 200 p t ; x q (dashed line) corresp onding to the black tra jectory and ¯ F p t ; x q (solid line) when t “ 0 . 69 , 1 . 4 , 1 . 71 for x P r 0 , 1 s . 14 Figure 3.3: An illustration of ϕ p t q for t P r 0 , 5 s : the black and the blue lines corresp ond to the empirical black tra jectory and the smo oth blue curve, resp ectively , of the top left panel in Figure 3.2. the initial proportion of susceptible individuals is 1 ´ p 0 “ 0 . 9 and, b y observing the b ottom left plot of Figure 3.2, w e see that at t “ 0 . 69 the prop ortion of susceptible individuals is just ov er 0.4 (i.e., the prop ortion of type ´ 1 v ertices). ˝ First p e ak: Just b efore t “ 0 . 69 the pro cess en ters distancing mo de, and after t “ 0 . 69 the proportion of infected individuals starts to decline. In distancing mo de, when resampled, the edge probability b et w een SI individuals is π SI p ϕ q “ 0 . 069 , compared to π SI p ϕ q “ 0 . 541 in normal mo de (see Equation (3.5) ). The adjustmen t to distancing mo de is not instantaneous, how ev er, since edges are resampled at the finite rate γ “ 20 . Consequen tly , in the left middle tw o panels in Figure 3.2, where we observe the empirical graphon and limiting graphon at t “ 0 . 69 , the graph is losing SI edges rapidly . F rom the b ottom left plot of Figure 3.2, w e see that the prop ortions of susceptible and infected individuals are appro ximately 0.43 and 0 . 4 “ 0 . 83 ´ 0 . 43 resp ectiv ely (refer to x -axis); hence these SI edges corresp ond to p x, y q P r 0 , 0 . 43 s ˆ r 0 . 43 , 0 . 83 s Y r 0 . 43 , 0 . 83 s ˆ r 0 , 0 . 43 s in the left second and third row of Figure 3.2. ˝ De cline fr om first p e ak and first dip: Figure 3.3 demonstrates that even once the prop ortion of infected individuals starts to decline, the perceived danger lev el ϕ p t q is increasing (i.e., for t P r 0 . 69 , 1 s ). This is b ecause ϕ p t q is calculated using past infections, and hence the p erceived danger level lags b ehind the actual threat level, i.e., the current rate of infection. This can also b e observed after the first dip ( t “ 1 . 4 ), where the p erceiv ed danger lev el con tin ues to decline even though the prop ortion of infected individuals is increasing, and after the second p eak, where the p erceiv ed threat lev el con tin ues to increase even though the prop ortion of infected individuals is declining. It 15 is these delay ed dynamics that lead to the tw o infection p eaks that w e observ e in the top ro w of Figure 3.2. ˝ Se c ond p e ak: At t “ 1 . 71 , the prop ortion of infected individuals reaches its second p eak. In the b ottom righ t of Figure 3.2, for x P r 0 . 23 , 0 . 56 s , we can observ e the t yp es (i.e., the time since infection) of the currently infected individuals. Here we see that there ha ve b een t w o wa v es of infection: of those curren tly infected, a proportion of approximately 0 . 1 “ 0 . 33 ´ 0 . 23 (refer to x -axis) of the total p opulation w ere infected in the second w a v e whic h o ccurred b et w een 0 and 0 . 31 time units ago (refer to y -axis), appro ximately 0 . 04 w ere infected during the dip which o ccurred b et w een 0 . 31 and 1 . 02 time units ago, and the remaining 0 . 19 “ 0 . 56 ´ 0 . 37 were infected during the first wa v e whic h o ccurred b et ween 1 . 02 and 1 . 71 time units ago. ˝ Final de cline: After t “ 1 . 71 the prop ortion of infected individuals declines monotonically . This is b ecause there are now very few susceptible individuals left to infect. Indeed, the b ottom righ t of Figure 3.2 shows that the prop ortion of susceptible individuals is just o v er 0.2 at t “ 1 . 71 . Moreov er, at t “ 1 . 71 , the prop ortion of infected individuals is no longer increasing despite the relatively high prop ortion of active SI edges in comparison to t “ 0 . 69 and t “ 1 . 4 , whic h can be observed by comparing the plots in the third ro w of Figure 3.2. The contin ued decline in the prop ortion of infected individuals after t “ 1 . 71 is therefore b ecause the densit y of SI edges is nev er high enough to comp ensate for the lo w n um b er of susceptible individuals to infect. 4. R o ad map In this section, we outline a four-step pro cedure leading to the proof of our main results. Eac h step is presented systematically , as this approac h may also serve as a useful framework for analyzing related co-evolutionary systems. The main idea is that w e construct a pro cess, referred to as the mimicking pr o c ess , that exhibits one-way dep endenc e : the edge dynamics dep end on the states of the vertices, but not vice versa. This structural simplification makes the mimic king pro cess significan tly more tractable than the original co-ev olutionary mod el, whic h features t w o-w a y dep endence. In the first three steps of the pro cedure outlined b elo w w e pro v e our FLLN claims for the mimic king pro cess. In the fourth step we show that the tw o pro cesses remain ‘sufficien tly close’, allo wing the FLLN that we establish for the mimic king pro cess to carry o v er to the co-evolutionary pro cess. Step 1: Define the mimicking pr o c ess, and establish c onver genc e of its empiric al typ e pr o c ess. A k ey role in our pro ofs is pla y ed b y the mimic king pro cess, a pro cess with one-wa y dep endence that shares the same edge dynamics as the co-evolutionary mo del but differs in its vertex dynamics. It is expressed via the evolving random graph pro cess p G ˚ n p t qq t Pr 0 ,T s that corresp onds to the empirical type pro cess p F ˚ n p t ; ¨qq t Pr 0 ,T s . F or this pro cess, w e define the generalized type analogously to the original model, no w denoted by X ˚ i p t q “ p x ˚ i p t q , y ˚ i p t qq . In the mimic king 16 pro cess at an y time t P r 0 , T s , x ˚ i p t q transitions from state S to state I at rate λ J p t q , as giv en in (2.5) . The infected and recov ered vertices b ehav e as in the original mo del. W e define the mimic king pro cess in full detail in Section 5.2. W e then fo cus on establishing the con v ergence of the empirical type pro cess of the mimic king pro cess, p F ˚ n p t ; ¨qq t Pr 0 ,T s as n Ñ 8 . The following prop ert y pla ys a k ey role. Prop ert y 4.1. The empiric al typ e pr o c ess satisfies a sto chastic-pr o c ess limit: F ˚ n ñ F as n Ñ 8 on D p M pr 0 , T s Y t T ` 1 uq , r 0 , T sq . T o establish this prop erty in our setting, we characterize F via the generator of the pro cess p x ˚ i p t q , y ˚ i p t qq t Pr 0 ,T s . It tak es the form p L t f qp x, y q “ ` λ J p t q 1 t x “ S u ` 1 t x “ I u ˘ r f p x 1 , y q ´ f p x, y qs ` b p x, y q B B y f p x, y q , (4.1) where x 1 is the state of the selected vertex after changing its state x at time t . Here b p x, y q is c haracterized as follo ws: (i) b p x, y q “ 0 for x P t S , R u , because the t yp e of a susceptible or reco v ered v ertex do es not c hange in time if its state do es not, and (ii) b p I , y q “ 1 . Indeed, if v ertex i has state I during the entire time in terv al p t, t ` d t q , then y ˚ i p t ` d t q “ y ˚ i p 0 q ` t ` d t ´ t I i “ y ˚ i p t q ` d t, so that the time deriv ativ e of y ˚ i p t q equals 1. This giv es rise to the K olmogoro v forw ard equations in (2.7) that characterizes F , and that v erify Prop erty 4.1 ab ov e. Step 2: Expr ess the e dge-c onne ction pr ob ability, c orr esp onding to the mimicking pr o c ess, in terms of the typ es. The goal is develop expressions (i) for the probability that in the mimicking pro cess there is an active edge at time t , in terms of the t yp es of v ertices i and j , and (ii) for the path of the empirical distribution p F ˚ n p s ; ¨qq s Pr 0 ,t s up to time t . In this context, the key prop ert y is the follo wing. Prop ert y 4.2. A t any time t , e dge ij is active with pr ob ability H p t ; y ˚ i p t q , y ˚ j p t q , p F ˚ n p s ; ¨qq s Pr 0 ,t s q , c onditional ly indep endently of al l the other e dges given p y ˚ i p s q , y ˚ j p s q , F ˚ n p s ; ¨qq s Pr 0 ,t s . Giv en the paths of x ˚ i p¨q and x ˚ j p¨q , the probabilit y that edge ij is active at time t is p ij p t q “ e ´ γ t p 0 ` γ ż t 0 d s e ´ γ s π x ˚ i p t ´ s q ,x ˚ j p t ´ s q p¨q “ e ´ γ t p 0 ` γ ż t 0 d s e ´ γ p t ´ s q π x ˚ i p s q ,x ˚ j p s q p¨q . T o verify Prop erty 4.2, we need to show that this probability can b e expressed only in terms of the t yp es of vertices i and j at time t (i.e., x ˚ i p t q and x ˚ j p t q “ S ) and p F ˚ n p s ; ¨qq s Pr 0 ,t s . T o this end, w e first observ e that if x ˚ i p t q “ x ˚ j p t q “ S , then p ij p t q “ e ´ γ t p 0 ` γ ż t 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq . 17 Denote y ˚ ij, _ p t q : “ y ˚ i p t q _ y ˚ j p t q and y ˚ ij, ^ p t q : “ y ˚ i p t q ^ y ˚ j p t q . If x ˚ i p t q “ x ˚ j p t q “ I , then, using the iden tities t I i ^ t I j “ t ´ y ˚ ij, _ p t q and t I i _ t I j “ t ´ y ˚ ij, ^ p t q , w e can write p ij p t q “ e ´ γ t p 0 ` γ ż t I i ^ t I j 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq ` γ ż t I i _ t I j t I i ^ t I j d s e ´ γ p t ´ s q π SI p s ´ p t I i ^ t I j q , F ˚ n p s ; ¨qq ` γ ż t t I i _ t I j d s e ´ γ p t ´ s q π II p s ´ p t I i ^ t I j q , s ´ p t I i _ t I j q , F ˚ n p s ; ¨qq “ e ´ γ t p 0 ` γ ż t ´ y ˚ ij, _ p t q 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq ` γ ż t ´ y ˚ ij, ^ p t q t ´ y ˚ ij, _ p t q d s e ´ γ p t ´ s q π SI p s ´ p t ´ y ˚ ij, _ p t qq , F ˚ n p s ; ¨qq ` γ ż t t ´ t ´ y ˚ ij, ^ p t q d s e ´ γ p t ´ s q π II p s ´ p t ´ y ˚ ij, _ p t qq , s ´ p t ´ y ˚ ij, ^ p t qq , F ˚ n p s ; ¨qq . Similarly , if x ˚ i p t q “ I and x ˚ j p t q “ S , then p ij p t q “ e ´ γ t p 0 ` γ ż t ´ y ˚ i p t q 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq ` γ ż t t ´ y ˚ i p t q d s e ´ γ p t ´ s q π SI p s ´ p t ´ y ˚ i p t qq , F ˚ n p s ; ¨qq . The case x ˚ i p t q “ S and x ˚ j p t q “ I is fully analogous. Finally , if x i p t q “ R or x j p t q “ R , then eviden tly p ij p t q “ p 0 . In view of the ab o ve computations, we can write the probability p ij p t q , in terms of y ˚ i p t q , y ˚ j p t q and p F ˚ n p s ; ¨qq s Pr 0 ,t s only: p ij p t q “ 1 t y ˚ ij, _ p t q “ T ` 1 u p 0 ` 1 t y ˚ ij, _ p t q ‰ T ` 1 u e ´ γ t p 0 ` 1 t y ˚ ij, _ p t q “ ´ 1 u γ ż t 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq ` 1 t y ˚ ij, _ p t q P r 0 , T ` 1 qu γ ż t ´ y ˚ ij, _ p t q 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq ` 1 t y ˚ ij, ^ p t q “ ´ 1 u 1 t y ˚ ij, _ p t q P r 0 , T ` 1 qu ˆ γ ż t t ´ y ˚ ij, _ p t q d s e ´ γ p t ´ s q π SI p s ´ p t ´ y ˚ ij, _ p t qq , F ˚ n p s ; ¨qq ` 1 t y ˚ ij, ^ p t q P r 0 , T ` 1 qu 1 t y ˚ ij, _ p t q P r 0 , T ` 1 qu ˆ γ ˜ ż t ´p y ˚ ij, ^ p t qq t ´p y ˚ ij, _ p t qq d s e ´ γ p t ´ s q π SI p s ´ p t ´ y ˚ ij, _ p t qq , F ˚ n p s ; ¨qq ` 18 ż t t ´ y ˚ ij, ^ p t q d s e ´ γ p t ´ s q π II p s ´ p t ´ y ˚ ij, _ p t qq , s ´ p t ´ y ˚ ij, ^ p t qq , F ˚ n p s ; ¨qq ¸ . Consequen tly , the probability of having an activ e edge b et w een tw o vertices with type u and v at time t is given by , with p u, v q ´ : “ u ^ v and p u, v q ` : “ u _ v , H p t ; u, v , F ˚ n p t ; ¨qq “ 1 tp u, v q ` “ T ` 1 u p 0 ` 1 tp u, v q ` ‰ T ` 1 u e ´ γ t p 0 ` 1 tp u, v q ` “ ´ 1 u γ ż t 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq ` 1 tp u, v q ` q P r 0 , T ` 1 qu γ ż t ´p u,v q ` 0 d s e ´ γ p t ´ s q π SS p F ˚ n p s ; ¨qq 1 tp u, v q ´ “ ´ 1 u 1 tp u, v q ` P r 0 , T ` 1 qu ` ˆ γ ż t t ´p u,v q ` d s e ´ γ p t ´ s q π SI p s ´ t ` p u, v q ` , F ˚ n p s ; ¨qq ` 1 tp u, v q ´ P r 0 , T ` 1 qu 1 tp u, v q ` P r 0 , T ` 1 qu ˆ γ ˜ ż t ´p u,v q ´ t ´p u,v q ` d s e ´ γ p t ´ s q π SI p s ´ t ` p u, v q ` , F ˚ n p s ; ¨qq ` ż t t ´p u,v q ´ d s e ´ γ p t ´ s q π II p s ´ t ` p u, v q ` , s ´ t ` p u, v q ´ , F ˚ n p s ; ¨qq ¸ . (4.2) W e hav e thus verified Prop erty 4.2. Step 3: Establish c onver genc e of the gr aphon-value d pr o c ess that c orr esp onds to the mimicking pr o c ess. If F is the limit describ ed in Prop ert y 4.1, then the induced reference graphon pro cess (see (2.9) ) is our candidate limit for the graph-v alued pro cess G ˚ n p t q in the space of graphons. T o sho w this, w e use the contin uous mapping theorem, for which we need the follo wing prop ert y . Prop ert y 4.3. The map F ÞÑ g r F s fr om D p M pr 0 , T s Y t T ` 1 u , r 0 , T sq to D pp W , d ˝ q , r 0 , T sq is c ontinuous, wher e W and d ˝ ar e define d in App endix A. Step 4: Extend to the original c o-evolutionary mo del. W e apply the framew ork ab o v e to the mimic king pro cess, and couple the mimic king pro cess with the co-ev olutionary pro cess in suc h a w a y that discrepancies during the time in terv al r 0 , T s ha v e a sufficien tly small probability . This will imply that h G n p t q p¨ , ¨q ù ñ g r F s p t ; ¨ , ¨q as n Ñ 8 in the space D pp W , d ˝ q , r 0 , T sq 5. Pr oofs 5.1. Pro of of Theorem 2.1. The expression for p S p t q directly follo ws from the integration of the first equation in (2.7) . W e just need to pro ve that, for any t P r 0 , T s and u P r´ 1 , T ` 1 s , there exists a unique solution f I p t, u q of the second equation in (2.7) . T o this end, we first deriv e the fixed-p oin t equation f I p t, u q m ust satisfy b y using the metho d of c haracteristics, and then we use the Banac h fixed-p oin t theorem to prov e existence and uniqueness of the solution. 19 Metho d of char acteristics. The c haracteristics satisfy d u { d t “ 1 , meaning that u p t q “ u p 0 q ` t . Along this curv e the PDE b ecomes an ODE for the profile F p t q “ f I p t, u p t qq , whic h reads as F 1 p t q ` F p t q “ λ J p t q p S p t q , whic h is equiv alent to d d t ` e t F p t q ˘ “ λ e t J p t q p S p t q . This thereb y leads to F p t q “ e ´ t F p 0 q ` λ ż t 0 d s e ´p t ´ s q J p s q p S p s q , whic h reads as f I p t, u q “ e ´ t f I p 0 , u ´ t q ` λ ż t 0 d s e ´p t ´ s q J p s q p S p s q . Note that the solution is not explicit, but it dep ends on solving a coupled system along all the c haracteristics. Indeed, the term J p s q dep ends on the function f I p s, ¨q itself. Contr action pr op erty. Define the space X “ C pr 0 , T s ˆ r´ 1 , T ` 1 s , R q with the norm } f } 8 : “ sup t Pr 0 ,T s , u Pr´ 1 ,T ` 1 s | f p t, u q| , f P X . Define no w the operator T : X Ñ X as p T f qp t, u q : “ e ´ t f p 0 , u ´ t q ` λ ż t 0 d s e ´p t ´ s q J f p s q p S p s q , where J f p s q : “ ş s 0 d F f p s ; u q H p s ; ´ 1 , u, F f p s ; ¨qq I p u q , with F f defined as in (2.8) after replacing f I b y f (note that also p I p t q is obtained after this replacemen t in (2.4) ). A solution f I of the starting PDE corresp onds then to a fixed p oin t of f “ T f . Letting f , g P X , w e find, by using that J p¨q is Lipschitz contin uous with Lipschitz constant L , }p T f qp t, u q ´ p T g qp t, u q} 8 “ sup t Pr 0 ,T s ˇ ˇ ˇ ˇ λ ż t 0 d s e ´p t ´ s q p S p s q p J f p s q ´ J g p s qq ˇ ˇ ˇ ˇ ď λL } f ´ g } 8 sup t Pr 0 ,T s ż t 0 d s s e ´p t ´ s q ď λL } f ´ g } 8 sup t Pr 0 ,T s ż t 0 d s s, “ K p T q} f ´ g } 8 , with K p T q : “ 1 2 λLT 2 , sho wing that T is a contraction for small enough T . L o c al existenc e and uniqueness. First, note that K p T q ą 0 . Moreov er, if K p T q ă 1 , then T is a contraction for an y t P r 0 , T s , therefore leading to global existence and uniqueness b y the Banac h fixed-p oin t theorem. Otherwise, by the same theorem we deduce that there exists a unique solution on r 0 , T 0 s ˆ r´ 1 , T ` 1 s , with T 0 the unique p ositive v alue such that K p T 0 q “ 1 . 20 Glob al existenc e. After obtaining the solution on r 0 , T 0 s , w e use the v alue at t “ T 0 as new initial data and repeat the same argument. Since T ă 8 and the norm of the solution can b e b ounded uniformly in t and u as sup t Pr 0 ,T s sup u Pr´ 1 ,T ` 1 s | f I p t, u q| ď sup u Pr´ 1 ,T ` 1 s | f I p 0 , u q| ` λK p T q ă 8 , w e can iterate finitely many times to cov er all t P r 0 , T s . This yields global existence and uniqueness for p t, u q in the whole rectangle r 0 , T s ˆ r´ 1 , T ` 1 s . 5.2. Coupling. T o prov e con v ergence of the pro cess in the space of graphons, w e construct a mimic king pro cess follo wing Steps 1–3 in Section 4 and that is close to the original pro cess in L 1 (Step 4). Mimicking pr o c ess. Supp ose that the pro cess p G ˚ n p t qq t Pr 0 ,T s is c haracterised by the following dynamics: ‚ G ˚ n p 0 q is an ERRG with connection probability p 0 . ‚ Eac h v ertex i b eing susceptible (resp. infected) is equipp ed with an indep enden t rate- λ J ˚ p t q (resp. rate- 1 ) Poisson clo ck, with J ˚ p t q : “ I p t ; f ˚ I p t, ¨q , p F ˚ p t, ¨qq t Pr 0 ,T s q , where f ˚ I p t, ¨q “ P p X ˚ i p t q P p I , ¨qq and p F ˚ p t, ¨qq t Pr 0 ,T s is the asso ciated limiting empirical t ype pro cess. When the clo ck asso ciated to vertex i rings, the follo wing happ ens: – if x i p t q “ S , then v ertex i gets infected; – if x i p t q “ I , then vertex i reco v ers and all the adjacent edges to vertex i are resampled according to the initial edge probability p 0 . ‚ Eac h edge (not inv olving any vertex having state R ) is equipp ed with an indep enden t rate- γ P oisson clo ck. When the clo c k asso ciated to edge ij rings, then the edge is active with a probability that dep ends on the state of the connecting v ertices. This probability is defined as follows: – π SS p F ˚ p t, ¨qq if b oth adjacen t v ertices are susceptible in G ˚ n p t q ; – π SI p y ˚ i p t q , F ˚ p t, ¨qq (resp. π SI p y ˚ j p t q , F ˚ p t, ¨qq ) if vertex i (resp. v ertex j ) is infected and vertex j (resp. vertex i ) is susceptible, where y ˚ i p t q denotes the t yp e of vertex i in G ˚ n p t q for an y i P r n s ; – π II p y ˚ i p t q , y ˚ j p t q , F ˚ p t, ¨qq if b oth adjacen t v ertices are infected in G ˚ n p t q . Lemma 5.1. The pr o c ess t G ˚ n p t qq t ě 0 u n P N satisfies the assumptions of [ 15 , Theorem 3.10] , and ther efor e h G ˚ n ñ g r F ˚ s as n Ñ 8 in the sp ac e D pp W , d ˝ q , r 0 , T sq . Pr o of. T o pro ve the claim, it suffices to verify that [ 15 , Assumptions 3.1, 3.6–3.7, 3.9] hold. The dynamics of eac h v ertex in the mimic king process are indep enden t of those of the other v ertices. Consequently , the first three prop erties discussed in Section 4 are satisfied. W e ma y therefore follo w the same reasoning as in [ 3 , Section 2.4.1] to establish a FLLN for the empirical t yp e pro cess as n Ñ 8 . □ 21 Lemma 5.2. Ther e exists a c oupling of tp G n p t qq t Pr 0 ,T s u n P N and tp G ˚ n p t qq t Pr 0 ,T s u n P N such that, for any δ ą 0 , lim n Ñ8 1 n log P ´ || ˜ h G n p t q ´ ˜ h G ˚ n p t q || L 1 ą δ for some t P r 0 , T s ¯ “ 0 . (5.1) Pr o of. The claim is prov ed in three steps. In Step I, w e construct a coupling b etw een the original and the mimicking pro cesses. The key idea is to asso ciate susceptible vertices with Poisson clo c ks whose rates differ b etw een the t w o pro cesses, and to couple these clo cks b y sim ulating a P oisson pro cess of intensit y λ up to time t and retaining each p oint in G n p t q and G ˚ n p t q with appropriately c hosen probabilities, sp ecified b elo w. The first retained p oint determines the infection time of the v ertex in the corresp onding pro cess; if no p oint is retained up to time t , the v ertex remains susceptible. The same construction applies to the edge dynamics. A differ enc e arises when a vertex (resp. an edge) is infected (resp. activ e) in one process but susceptible (resp. inactiv e) in the other. Step I I c haracterizes the probability of such differences, and Step I I I pro vides precise estimates. Step I: description of the c oupling. Supp ose that p G n p t qq t ě 0 and p G ˚ n p t qq t ě 0 are generated in the follo wing manner. Recall that N I i p t q is the set of infected neighbors of v ertex i at time t . ‚ Eac h initially infected v ertex i P r n s is assigned the same (coupled) rate-1 P oisson clo c k in b oth pro cesses. When the clo c k asso ciated with vertex i rings, v ertex i reco v ers and all the adjacent edges to v ertex i are resampled according to the initial edge probability p 0 in b oth G n p t q and G ˚ n p t q . ‚ Eac h initially susceptible vertex i P r n s is assigned the same (coupled) rate- λ P oisson clo c k in b oth pro cesses. When the clock asso ciated to v ertex i rings, generate an outcome U dra wn from Unif p 0 , 1 q distribution. – If U ď ř j P N I i p t q I p y j q{ n , then vertex i b ecomes infected, otherwise remains suscep- tible, in G n p t q . – If U ď J ˚ p t q , then v ertex i b ecomes infected, otherwise remains susceptible, in G ˚ n p t q . ‚ Assign eac h edge the same (coupled) rate- γ P oisson clo ck in both pro cesses. When the Poisson clo ck asso ciated with edge ij rings, generate an outcome U dra wn from Unif p 0 , 1 q distribution. – When x i p t q “ x j p t q “ S , if U ď π SS p F n p t, ¨qq , then edge ij is activ e, otherwise it is inactiv e, in G n p t q . – When either x i p t q “ S and x j p t q “ I , or x i p t q “ I and x j p t q “ S , if U ď π SI p y j p t q , F n p t, ¨qq or U ď π SI p y i p t q , F n p t, ¨qq , resp ectiv ely , then edge ij is active, otherwise it is inactive, in G n p t q . – When x i p t q “ x j p t q “ I , if U ď π II p y i p t q , y j p t q , F n p t, ¨qq , then edge ij is activ e, otherwise it is inactive, in G n p t q . – When x ˚ i p t q “ x ˚ j p t q “ S , if U ď π SS p F ˚ p t, ¨qq , then edge ij is activ e, otherwise it is inactiv e, in G ˚ n p t q . 22 – When either x ˚ i p t q “ S and x ˚ j p t q “ I , or x ˚ i p t q “ I and x ˚ j p t q “ S , if U ď π SI p y ˚ j p t q , F ˚ p t, ¨qq or U ď π SI p y ˚ i p t q , F ˚ p t, ¨qq , resp ectively , then edge ij is activ e, otherwise it is inactive, in G ˚ n p t q . – When x ˚ i p t q “ x ˚ j p t q “ I , if U ď π II p y ˚ i p t q , y ˚ j p t q , F ˚ p t, ¨qq , then edge ij is activ e, otherwise it is inactive, in G ˚ n p t q . Step II: majorization of the L 1 distanc e. If vertex i is susceptible in b oth mo dels and the clock asso ciated with vertex i rings, then a differ enc e is formed when vertex i is susceptible in one pro cess and infected in the other. This happ ens with probabilit y ˇ ˇ ˇ ˇ ˇ ÿ j P N I i p t q I p y j q n ´ J ˚ p t q ˇ ˇ ˇ ˇ ˇ . If edge ij has the same state (active or inactiv e) in b oth mo dels and the clock asso ciated with edge ij rings, then a difference is formed with probability dep ending on the generalised t yp es X i p t q , X j p t q , X ˚ i p t q , X ˚ j p t q , and on F n p t, ¨q , F ˚ p t, ¨q . F or instance, if x i p t q “ x ˚ i p t q “ S and x j p t q “ x ˚ j p t q “ I , this probabilit y is equal to | π SI p y j p t q , F n p t, ¨qq ´ π SI p y ˚ j p t q , F ˚ p t, ¨qq| . Step III: b ounding the dominating pr o c ess. Our approach is to sho w that, ov er a small time windo w r t, t ` ∆ q with ∆ ą 0 , the n um b er of differences is sto chastically dominated by a random v ariable with suitable prop erties. Let ‚ t ∆ “ r t, t ` ∆ q b e the time windo w, ‚ N E „ Bin p ` n 2 ˘ , 1 ´ e ´ γ ∆ q b e the total num b er of edge clo c ks that ring in t ∆ , ‚ N V „ Bin p N S p t q , 1 ´ e ´ λ ∆ q b e the total n um b er of clo cks asso ciated with susceptible v ertices in both pro cesses at time t that ring in t ∆ , where N S p t q is the n um ber of susceptible v ertices at time t , ‚ d E p t q b e the total num b er of differences in the edges b etw een G n p t q and G ˚ n p t q , ‚ d V p t q b e the total num ber of differences in vertex states betw een F n p t ; ¨q and F ˚ n p t ; ¨q , where F n is defined in (2.2) and F ˚ n is defined analogously using the graph G ˚ n p t q . Here, b y F n p t ; ¨q and F ˚ n p t ; ¨q w e mean the complete information on vertex types and states. Giv en Ω ” p N E , N V , d E p t q , d V p t q , F n p t ; ¨q , F ˚ n p t ; ¨qq , we wan t to b ound the probability Π n p ∆ q that, when an edge clo ck (of an edge c hosen uniformly at random) rings, a difference is formed during an y time s P t ∆ . Note that if the clo c k of a vertex pair ij rings, then a difference can only b e formed when either x i p s q ‰ x ˚ i p s q or x j p s q ‰ x ˚ j p s q , or there is a difference in the t ypes or in the empirical t yp e distributions F n p s ; ¨q and F ˚ p s ; ¨q . At any time t , we can then partition the set of all the v ertex pairs as those connecting at least one vertex with a difference of state b et ween the t w o mo dels, and those whic h connect b oth vertices having the same state in the t w o mo dels, which we refer to as E 1 p t q and E 2 p t q , resp ectively . Since each vertex i is part of n v ertex pairs, note that | E 1 p t q| ď n p d V p t q ` N V q . 23 Observing that d V p t q “ ř n i “ 1 1 t x i p t q ‰ x ˚ i p t qu , considering the w orst case scenario that ev ery time a v ertex clo c k rings during t ∆ a new difference is formed, and observing that the type of t w o infected v ertices can differ of at most ∆ during t ∆ , w e ha v e the follo wing upper b ound: Π n p ∆ q ď | E 1 p t q| ` n 2 ˘ ` L ` 2∆ ` d L p F n p s ; ¨q , F ˚ p s ; ¨qq ˘ ď n p d V p t q ` N V q ` n 2 ˘ ` L ` 2∆ ` d L p F n p s ; ¨q , F ˚ p s ; ¨qq ˘ . (5.2) where w e ha v e used the Lipsc hitz con tin uit y of the functions π SS , π SI , π II , with the constant L defined as max t L SS , L SI , L II u ă 8 . T o control this probability , using the triangular inequality for the Lévy metric (see App endix A), we can write d L p F n p s ; ¨q , F ˚ p s ; ¨qq ď d L p F n p s ; ¨q , F ˚ n p s ; ¨qq lo oooooooooomo ooooooooo on p a q ` d L p F ˚ n p s ; ¨q , F ˚ p s ; ¨qq looooooooooomooooooooooon p b q . Considering the worst case scenario as explained abov e, for any s P t ∆ , the term (a) can b e b ounded as d L p F n p s, ¨q , F ˚ n p s, ¨qq ď d V p t q ` N V n , while the term (b) tends to zero as n Ñ 8 directly from the Glivenk o-Can telli theorem, b ecause in the mimicking process vertices b eha v e indep endently and therefore F ˚ n p s ; ¨q con v erges in distribution to F ˚ p s ; ¨q . Thus, (5.2) b ecomes Π n p ∆ q ď n p d V p t q ` N V q ` n 2 ˘ ` L ˆ 2∆ ` d V p t q ` N V n ` o p n q ˙ ď ˆ 1 ` L 2 ˙ n p d V p t q ` N V q ` n 2 ˘ ` 2 L ∆ ` o p n q . Because this bound is uniform in t ∆ and do es not depend on N E , giv en Ω , and the term o p n q can b e ignored in the limit as n Ñ 8 , we therefore ha v e d E p t ` ∆ q ´ d E p t q st ď Bin ˜ N E , ˆ 1 ` L 2 ˙ n p d V p t q ` N V q ` n 2 ˘ ` 2 L ∆ ¸ , where st ď means ‘stochastically dominated by’. Giv en Ω , w e next w an t to b ound the probability that, when a vertex clo ck (of a susceptible v ertex in b oth pro cesses at time t , c hosen uniformly at random, that is) rings, a difference is formed during an y time s P t ∆ . In other w ords, w e wish to establish an upp er b ound on 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N I i p s q I p y j q n ´ J ˚ p s q ˇ ˇ ˇ ˇ ˇ , 24 whic h applies for all s P t ∆ . Applying the triangular inequalit y , we hav e 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N I i p s q I p y j q n ´ J ˚ p s q ˇ ˇ ˇ ˇ ˇ ď 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N I i p s q I p y j q n ´ ÿ j P N ˚ I i p s q I p y ˚ j q n ˇ ˇ ˇ ˇ ˇ looooooooooooooooooooooomooooooooooooooooooooooon p i q ` 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N ˚ I i p s q I p y ˚ j q n ´ J ˚ p s q ˇ ˇ ˇ ˇ ˇ loooooooooooooooooomoooooooooooooooooon p ii q , where N ˚ I i p s q (eviden tly) denotes the set of infected neigh b ours of v ertex i in G ˚ n p s q . W e first b ound (i) as follo ws: 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N I i p s q I p y j q n ´ ÿ j P N ˚ I i p s q I p y ˚ j q n ˇ ˇ ˇ ˇ ˇ “ 1 n 2 n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N I i p s q I p y j q ´ ÿ j P N ˚ I i p s q I p y ˚ j q ˇ ˇ ˇ ˇ ˇ ď 1 n 2 p d E p t q ` N E q , where w e used that sup u Pr 0 ,t ` ∆ s I p u q “ 1 . W e next bound (ii). T o this end, w e use the conv ergence of the mimicking pro cess in the space of graphons. Let g ˚ n p s ; u, v q “ h G ˚ n p s ; u, v q denote the empirical graphon at time s . In addition, let r S ` I n p s q and r S ` I p s q denote the sum of the prop ortions of susceptible and infected v ertices in the empirical and limiting graphon, resp ectively , at time s . Similarly , r S n p s q and r S p s q denote the corresp onding quan tities for susceptible v ertices. Th us, w e can write 1 n ÿ j P N ˚ I i p s q I p y ˚ j q “ ż r S ` I n p s q r S n p s q d u g ˚ n ´ s ; i n , u ¯ I p F ˚ p s ; u qq , using the definition of the empirical graphon. In addition, by Lemma A.1 w e ha v e J ˚ p s q “ ż r S ` I p s q r S p s q d y g r F ˚ s p s ; y , 0 q I p F ˚ p s ; y qq . W e can therefore write 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ÿ j P N ˚ I i p s q I p y ˚ j q n ´ J ˚ p s q ˇ ˇ ˇ ˇ ˇ “ 1 n n ÿ i “ 1 ˇ ˇ ˇ ˇ ˇ ż r S ` I n p s q r S n p s q d u g ˚ n ´ s ; i n , u ¯ I p F ˚ p s ; u qq ´ ż r S ` I p s q r S p s q d y g r F ˚ s p s ; y , 0 q I p F ˚ p s ; y qq ˇ ˇ ˇ ˇ ˇ “ ˇ ˇ ˇ ˇ ˇ ż r S ` I n p s q r S n p s q d u g ˚ n p s ; 0 , u q I p F ˚ p s ; u qq ´ ż r S ` I p s q r S p s q d y g r F ˚ s p s ; y , 0 q I p F ˚ p s ; y qq ˇ ˇ ˇ ˇ ˇ ď ˇ ˇ p r S ` I n p s q ´ r S ` I p s qq ` p r S n p s q ´ r S p s qq ˇ ˇ 25 ` d ˝ ´ g ˚ n p s ; 0 , ¨q , g r F ˚ s p s ; 0 , ¨q ¯ . Note that, by the FLLN claimed in Lemma 5.1), as n Ñ 8 eac h of these terms tends to zero uniformly in s with high probability . Th us, w e conclude that d V p t ` ∆ q ´ d V p t q st ď Bin ˆ N V , 1 n 2 p d E p t q ` N E q ` ϵ p n q ˙ , where ϵ p n q is a function that can be c hosen arbitrarily small with high probability , and can therefore effectiv ely ignored in the limit as n Ñ 8 . Our plan is now to b ound the n um b er of differences formed in the vertices and edges b et w een the tw o mo dels b y considering the time in terv als r 0 , ∆ s , r ∆ , 2∆ s , ..., r T ´ ∆ , T s in sequence, and then use the union b ound. Without loss of generality , w e are tacitly assuming that T is a m ultiple of ∆ . W e use the results ab o ve which, for n large, can be summarised as follows: N E st ď Bin ˆˆ n 2 ˙ , λ ∆ ˙ , N V st ď Bin p n, γ ∆ q , d E p t ` ∆ q ´ d E p t q st ď Bin ˜ N E , ˆ 1 ` L 2 ˙ n p d V p t q ` N V q ` n 2 ˘ ` 2 L ∆ ¸ , d V p t ` ∆ q ´ d V p t q st ď Bin ˆ N V , d E p t q ` N E n 2 ˙ . The pro of is then completed by follo wing argumen ts analogous to those used in the pro of of [ 3 , Lemma 3.6]. □ 5.3. Pro of of Theorem 2.2. Theorem 2.2 no w follo ws from Lemmas 5.1 – 5.2 and [ 15 , Theorem 3.10]. Appendix A. Graphons, Lévy metric and an additional lemma Graphons. Let W b e the space of functions h : r 0 , 1 s 2 Ñ r 0 , 1 s suc h that h p x, y q “ h p y , x q for all p x, y q P r 0 , 1 s 2 , endo w ed with the cut distanc e d ˝ p h 1 , h 2 q : “ sup S,T Ďr 0 , 1 s ˇ ˇ ˇ ˇ ż S ˆ T d x d y r h 1 p x, y q ´ h 2 p x, y qs ˇ ˇ ˇ ˇ , h 1 , h 2 P W . (A.1) On W , called the space of graphons, there is a natural equiv alence relation „ . Let Σ b e the space of measure-preserving bijections σ : r 0 , 1 s Ñ r 0 , 1 s . Then h 1 p x, y q „ h 2 p x, y q if δ ˝ p h 1 , h 2 q “ 0 , where δ ˝ is the cut metric defined by δ ˝ p ˜ h 1 , ˜ h 2 q : “ inf σ 1 ,σ 2 P Σ d ˝ p h σ 1 1 , h σ 2 2 q , ˜ h 1 , ˜ h 2 P Ă W , (A.2) with h σ p x, y q “ h p σ x, σ y q . This equiv alence relation yields the quotient space p Ă W , δ ˝ q , whic h is compact. 26 A finite simple undirected graph G on n v ertices can b e represen ted as a graphon h G P W b y setting h G p x, y q : “ # 1 if there is an edge b etw een vertex r nx s and vertex r ny s , 0 otherwise , (A.3) whic h referred to as the empiric al gr aphon associated with G , and has a blo c k structure. Lévy metric. W e equip the space M p R ` q with the L évy metric , so that for tw o distribution functions F , G w e ha v e d L p F , G q : “ inf t ϵ ą 0 : F p x ´ ϵ q ´ ϵ ď G p x q ď F p x ` ϵ q ` ϵ, @ x P R u (A.4) An imp ortan t prop ert y of the Lévy metric is that it satisfies the triangle inequalit y . Lemma A.1. F or any s P t ∆ , J ˚ p s q “ ż r S ` I p s q r S p s q d y g r F ˚ s p s ; y , 0 q I p F ˚ p s ; y qq . Pr o of. By definition, we hav e J ˚ p s q “ I p s ; f ˚ I p s, ¨q , F ˚ p s, ¨qq “ ż T 0 d u f ˚ I p s, u q H p s ; ´ 1 , u, F ˚ p s, ¨qq I p u q . By using the change of v ariable u “ F ˚ p s ; y q , we obtain J ˚ p s q “ ż r S ` I p s q r S p s q d y H ` s ; ´ 1 , F ˚ p s ; y q , F ˚ p s, ¨q ˘ I p F ˚ p s ; y qq “ ż r S ` I p s q r S p s q d y g r F ˚ s p s ; y , 0 q I p F ˚ p s ; y qq . where in the first equalit y w e used the fact that d y “ f ˚ I p s, F ˚ p s ; y qq d u . This follows from the deriv ative of the inv erse function, whic h is w ell defined b ecause the function F ˚ p s ; ¨q is righ t-con tin uous and non-decreasing. Thus, the deriv ativ e is zero only in the in terv als in which the function is constan t, meaning that there are no infected individuals with those types, and therefore w e ma y think the in tegral not running on these v alues. Finally , note that the in tegral go es from r S p s q to r S ` I p s q b ecause otherwise I p¨q ” 0 and therefore the in tegrand is zero. □ References [1] S. A threya, F. den Hollander, A. Röllin. Co-evolving v ertex and edge dynamics in dense graphs. 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Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100 L’Aquila, It al y Email addr ess : simone.baldassarri@gssi.it School of Ma thema tics and St a tistics, Anit a B. La wrence Centre, UNSW Sydney, Sydney NSW 2052, A ustralia Email addr ess : p.braunsteins@unsw.edu.au Ma thema tisch Instituut, Universiteit Leiden, Einsteinweg 55, 2333 CC Leiden, The Nether- lands Email addr ess : denholla@math.leidenuniv.nl Ma thema tisch Instituut, Universiteit Leiden, Einsteinweg 55, 2333 CC Leiden, The Nether- lands Email addr ess : m.r.h.mandjes@math.leidenuniv.nl 29