Eplosive Contagion in Netwoks (Supplementay Infomation) Jesús Gómez-Gadeñes,, Laua Loteo, Segei N. Taaskin, and Fancisco J. Péez-Reche Institute fo Biocomputation and Physics of Comple Systems (BIFI), Univesity of Zaagoza, E-8 Zaagoza, Spain Depatment of Condensed Matte Physics, Univesity of Zaagoza, E-9 Zaagoza, Spain Depatamento de Ciencias de la Computación y de la Decisión, Univesidad Nacional de Colombia, Medellín, Colombia. St. Cathaine s College and Depatment of Chemisty, Univesity of Cambidge, Cambidge, UK Institute fo Comple Systems and Mathematical Biology, SUPA, King s College, Univesity of Abedeen, Abedeen, UK SIS SYNERGY MODEL WITH LINEAR SYNERGY RATE. In this section, we solve the SIS model eactly fo a population of individuals having the netwok of contacts with the topology of andom z-egula gaph with linea dependence of tansmission ate on the numbe of ignoant/healthy neighbous and demonstate that this solution is analogous to that fo the eponential dependence of σ z (n h (i)) discussed in the main tet. The equilibium states coespond to the solutions of Eq. (6) in the main tet which can be ecast in the following fom: F (y) y( z λ z ( y)) =. () In the synegy-fee case, λ z = and the stable solution of Eq. () is y = fo c = /z and y = /(z) fo > c. In the model with linea synegy, the tansmission ate is given by λ z = ( + βz( y)) and F (y) is a thid ode polynomial in y which can have fom one to thee eal oots. The equilibium solution, y sf = (speadefee egime), is always pesent while the two othe oots, y ± = + βz ( ± ae eal only fo (β), whee + β ), () (β) = β. () The values of y ± epesent equilibium concentations of speades and thus must be in the ange, y ± [, ]. An equilibium concentation, y eq, coesponding to a oot of F (y) can efe to eithe stable (if F (y eq ) < ) o unstable (if F (y eq ) > ) equilibium. In the β paamete space, thee is a special (ticitical) point, ( tp, β tp ) = (/z, /(z)) (see the point labelled by TP in Fig. ), at which all thee oots of F (y) coincide, i.e. y sf = y = y + =. This point sepaates the egimes of eplosive and continuous tansitions between non-invasive (speade-fee) and invasive (endemic) epidemics. Fig. shows the dependence of the equilibium concentation of speades, y eq, on fo fied value of β above (panel (a)) and below (panel (b)) the ticitical point. Fo fied β above the ticitical point, β > β tp, and values of smalle than citical value, c (β) = z + βz, () both oots y ± ae outside the physical ange [, ] and the only stable equilibium at y sf = coesponds to the speade-fee state (cf. Fig. (a)). Fo = c (β) (see the solid line in Fig. ), the oot y intesects the allowed ange [, ] at a point whee y = y sf =. With inceasing value of > c (β), the equilibium concentation y continuously inceases in the inteval [, ] and it coesponds to the stable equilibium (F (y ) < ) while the speade-fee equilibium, y sf =, is unstable (F (y sf ) > ) fo these values of. This means that an incease in the inheent tansmission ate at fied β > β tp dives the system continuously fom speadefee ( c (β)) to endemic ( > c (β)) state (the egion above continuous line in Fig. ). Fo values of β below the ticitical point, β < β tp, the scenaio is vey diffeent fom that descibed above (see Fig. (b)). Indeed, if <, the only acceptable oot of F is y sf which coesponds to the stable speade-fee state (the egion below the dashed line in Fig. ). At = (β), the oots y ± become eal and take values in the ange (, ), i.e. < y + = y <. With inceasing in the inteval ( (β), c (β)) (the egion between dashed and dot-dashed lines in Fig. ) at fied β, these two oots split in such a way that < y + < y <. The concentation y coesponds to the stable equilibium while y + to the unstable one. Oveall, thee ae two stable equilibia descibing the speade-fee state with concentation of speades y sf = and endemic state with concentation of speades equal to y. The finite gap between these two equilibium states is a signatue of discontinuous eplosive tansition between non-invasive and invasive epidemics. With futhe incease of fo fied value of β, the oot y + leaves the physical ange [, ] when = c (β) (and y + = ), and the only stable equi-
Bistability -/z * c TP (-/(z),/z) Speade-fee Endemic spead -. -... β FIG.. Contagion diagam fo the SIS model with linea synegy. The solid line epesents the theshold c(β) fo continuous tansitions between the speade-fee and invasive endemic egimes. The cicle labelled by TP indicates the ticitical point. The dot-dashed line coesponds to c(β) in the egion with eplosive tansitions. The bi-stability egion is bounded fom below by the dashed line coesponding to the function (β). Numeical values along the aes coespond to andom z-egula gaphs with z = and =. The hoizontal ais shows only meaningful values of β > /z. y eq y eq.8.6.. (b).8 y -.6. (a) y c y -. y + y * c libium at y coesponds to the endemic state (the egion above the dot-dashed line in Fig. ). In the bi-stable egime with ( (β), c (β))), the mean-field system, depending on initial conditions, eaches the speade-fee egime, y sf, o the endemic egime, y. The dotted lines in Fig. (b) indicate the eplosive tansitions obseved by inceasing fom < (up aow) o deceasing fom > c (down aow). A hysteesis loop of width c becomes wide as β becomes moe negative. MODELS WITH REMOVAL OF SPREADERS ON z-random REGULAR GRAPHS In this section, we deive the geneal solution (Eq. () of the main tet) fo the mean-field models with emoval of speades and illustate its popeties using the SIR model with linea synegistic tansmission ate as a benchmak. Fom Eqs. ()-() of the main tet and the definition of λ z () = σ z (), one obtains, d y = zσ z () dt = d γ() dt. () Integating the second equation in Eq. () ove time in the inteval [, t] leads to the following epession: (t) γ() σ z () d = z (t) d. (6) FIG.. Equilibium concentation, y eq (y eq [, ]), of speades in SIS epidemics on andom z-egula gaphs with z = and = vs inheent tansmission ate,. Linea synegy ates with (a) β =. > β tp and (b) β =. < β tp, illustate continuous and eplosive tansitions, espectively. In (b), the equilibium concentation y + coesponds to unstable states (dashed line). The dotted vetical lines at c and indicate eplosive tansitions with inceasing and deceasing, espectively. Hee, we have assumed a population which initially consists of only ignoants and speades, i.e., () =, () = and y() =. Fom Eq. (6), the concentation of emoved individuals ove time, (t), can be epessed as a function of the concentation of ignoants as follows: (t) = z [F ( ) F ((t))]. (7) The function F () is defined in Eq. (6) of the main tet. The fied points of the system given by Eqs. ()-() in the main tet coespond to states without speades, y =. In geneal, any finite system with an initially positive concentation of speades, y >, and positive emoval ate, γ() >, evolves towads a fied point with y =, = and =. The condition y = points out the end of the epidemic. Eamples of the evolution of and ae shown in Figs. and fo the
.8 =, β =. = =. =. =..8 =.9; β =....6.6.7.8.9......6.8.8.6.....6.8 =.9; β =. = =. =. =. =. =. FIG.. Tajectoies of the concentation of emoveds and ignoants duing SIR epidemics with = and linea synegy ate fo β =. speading on andom z-egula gaphs with z =. The initial concentation of ignoants is = and =.9 in the uppe and lowe panel, espectively. Epidemics stop (cicles) when the tajectoies each the dashed line coesponding to =. The final concentation of emoveds,, inceases smoothly with in both panels. SIR model with linea synegy fo seveal values of, and β. The value of the final concentation of ignoants, (o emoveds, = ), depends in geneal on the initial concentation of ignoants, = y, the inheent tansmission ate,, as well as on the synegistic and ecovey mechanisms encoded by the functions σ z and γ, espectively. Such dependence can be ecast fom Eq. (7) in the implicit fom given by Eq. () of the main tet which we epeat hee fo convenience: = f( ; ) z( ) [F ( ) F ( )]. (8) It is clea fom Eq. (8) that systems chaacteised by a function f( ; ) that deceases monotonically with will ehibit continuous tansitions fom smalle to lage (fom lage to smalle ) with inceasing. Eamples of this type of behaviou of f( ; ) ae shown by the continuous lines in Fig. fo the SIR model. =. (invasive) =. (invasive) =. (citical) =. ( non-invasive )...6.8 FIG.. Tajectoies of the concentation of emoveds and ignoants duing SIR epidemics on andom z-egula gaphs with z = fo =.9, β =., = and linea synegy ate with β =.. The dashed line shows the function = giving the locus of concentations at the end of epidemics. Cicles indicate the final state of each tajectoy at =. The final concentation of emoveds,, changes abuptly with inceasing, thus indicating an eplosive tansition to lage contagion. The eplosion occus at the citical value of c.. Fo this citical value, the blue dotted line shows solutions of Eq. (8) that ae not eached duing the epidemic because the epidemic (solid blue line) teminates at the point denoted by the blue cicle which coesponds to the lagest value of. with linea synegy ate. In contast, discontinuous tansitions can occu when f( ; ) is not monotonic and it inceases with in some sub-inteval of (, ). In this case, Eq. (8) can have seveal solutions fo coesponding to seveal fied points (cf. dashed lines in Fig. ). The evolution given by Eqs. ()-() in the main tet is such that deceases with time fom and the system evolves towads the solution coesponding to the lagest value of ; the est of solutions ae not accessible to the system. The tajectoies of the SIR model with linea synegy shown in Fig. illustate this behaviou. In paticula, the tajectoy fo c shows both the eachable (continuous line) and uneachable (dotted line) solutions of Eq. (8). As mentioned in the main tet, the egimes with continuous and eplosive tansitions ae sepaated by a citical egime fo which f( ; ) displays an inflection point at some value of = tp (, ). This situation coesponds to the ticitical point discussed in the main tet. At the inflection point, f(; ) = f(; ) tp =. (9) tp These conditions and definition of f( ; ) given by Eq. (8) esult in Eqs. (7) and (8) given in the main
(a) = (b) =.9 the following elations at the ticitical point: f(; ) c β = β = -. b a tp =, zβ tp () = zβ tp ( + e (zβtp+) ), () tp = β tp. ()...6.8...6.8 FIG.. Function f(; ) defined by Eq. (8) with eplaced by, fo (a) = and (b) =.9 coesponding to SIR epidemics with linea synegy speading on andom egula gaphs with z =. In both panels, the continuous and dashed lines coespond to β = and β =., espectively. The hoizontal dot-dashed line in (b) illustates the solutions (cicles) of Eq. (8) fo =. The system evolves towads the lagest solution and eaches the final concentation of ignoants = a. Fig. 6 shows the phase diagam fo the SIR model with linea synegistic tansmission fo two initial conditions: = (i.e. a negligible initial concentation of infecteds, y ) and =.9. Fo =, one obtains β tp = /(z) fom Eq. () which leads to tp = and tp = /z. The value of β tp deceases with (see Fig. 7). This implies that social phenomena stating with a elatively lage initial concentation of speades, y =, will equie lage synegistic effects of the contet in ode fo them to be eplosive. Howeve, eplosive tansitions eist fo any initial conditions with > since β tp is finite fo any > (fom Eq. (), it is clea that β tp only fo.). tet fo the ticitical point. Fo the SIR model with linea synegy, Eqs. (7)-(9) in the main tet lead to
Continuous tansition, = Eplosive tansition, = Eplosive tansition, =.9 tp = -β tp invasive TP (-/z,/z) non-invasive -. -... β FIG. 6. Contagion phase diagam fo the SIR model on andom z-egula gaphs. The continuous black line indicates the invasion theshold fo continuous tansitions obseved fo an initial concentation of ignoants, =. The solid staight line displays the locus of ticitical points given by Eq. (). The blue and geen dashed lines give the eplosive invasion theshold fo epidemics with = and =.9, espectively. Numeical values along the aes coespond to andom z-egula gaphs with z = and =..8.6.. - -8-6 - - zβ tp FIG. 7. Gaphical epesentation of the dependence of β tp on fo the SIR model with linea synegy ate (cf. Eq. ()).