EPIDEMIC DYNAMICS AND ENDEMIC STATES IN PHYSICAL REVIEW E 63 066117 =F(1,1+y,2+y,-[mA6(A)) (18) That is, we obtain a power-law behavior with exponent B 1/(y-1)>l, but now we observe the presence of a non- In order to solve eqs.(17)and(18)in the limit p-0 zero threshold (which obviously corresponds also to 0-0, we must per form a Taylor expansion of the hypergeometric function. The expansion for Eq.(17)has the form [33] (27) F(1,y,1+y,-[mA6(A)]) In this case, a critical threshold reappears in the model. How- ever, the epidemic threshold is approached smoothly without (m6)y+y∑( (m入e) any sign of the singular behavior associated with critical point. (iii)y>2: The relevant terms in the e expansion are now where T(x)is the standard gamma function. An analogous 6(A)=-,mA6 (28) expression holds for Eq (18). The expansion(19) is valid for -2(mA6)2 any y*+ 1, 2, 3,... Integer values of y must be analyzed in a case by case basis. (The particular value y=I was consid- and the relevant expression for e is ered in the previous section. For all values of y, the leading behavior of Eq (18)is the same 6(A)= (29) 1+ nO which yields the behavior for The leading behavior in the rhs of Eq. (19), on the other P-x-Ac hand, depends on the particular value of y (0<y<1: In this case, one has with the same threshold Ac as in Eq. (27)and an exponent B=1. In other words, we recover the usual critical frame- work in networks with connectivity distribution that decays (m入) (21) faster than k to the fourth power. Obviously, an exponen- tially bounded network is included in this last case, recover- from which we obtain ing the results obtained with the homogeneous approxima- tion of sec. Ill. It is worth remarking that the above results 6(A) yT 1-y (m)%(1-y) (22) are obtained by neglecting connectivity correlations in the network, i.e., the probability that a link points to an infected node is considered independent of the connectivity of the Combining this with Eq(20), we obtain node from which the link is emanated [see Eq. (7)]. This approximation appears to be irrelevant in the BA network. In (23) different SF networks with more complex topological prop- Here we have again the total absence of any epidemic thres erties, however, connectivity correlations could play a more old and the associated critical behavior, as we have already important role and modify the analytic forms obtained in this shown for the case y= 1. In this case, however, the relation section, Eqs. (23)and(26). On the other hand, conclusions concerning the epidemic threshold absence for connectivity between p and A is given by a power law with exponent p distributions decaying more slowly than a cubic power can =1/1-y),ie,B>1 be considered of general validity. Indeed, for connectivities 1i)1<y<2: In this case, to obtain a nontrivial informa- decaying faster than the cubic power, the connectivity fluc- tion for 0, we must keep the first two most relevant terms in tuations are bounded, and one would expect to obtain the Eq.(19), same qualitative behavior as in exponential distribution In summary, for all SF networks with 0<ysl, we re 6(A)=-(m入6)y+ no (24) cover the absence of an epidemic threshold and critical be- SIn( yTT From here we get when A-0. In the interval 1<y<2, an epidemic threshold reappears(p→0ifA→入), but it is also approached with sin(y丌)m vanishing slope, i.e., no singular behavior. Eventually, for 6(A)=m(y=1)(mA)y (25) y>2 the usual MF critical behavior is recovered and the SF rk is indistinguishable from an exponential The expression for p is finally VL CONCLUSIONS The emerging picture for disease spreading in complex networks emphasizes the role of topology in epidem 661177r5F„1,11g,21g,2@mlQ~l!# 21…. ~18! In order to solve Eqs. ~17! and ~18! in the limit r→0 ~which obviously corresponds also to Q→0, we must per￾form a Taylor expansion of the hypergeometric function. The expansion for Eq. ~17! has the form @33# F„1,g,11g,2@mlQ~l!# 21… . gp sin~gp! ~mlQ! g 1g(n51 ` ~21! n ~mlQ! n n2g , ~19! where G(x) is the standard gamma function. An analogous expression holds for Eq. ~18!. The expansion ~19! is valid for any gÞ1,2,3,... . Integer values of g must be analyzed in a case by case basis. ~The particular value g51 was consid￾ered in the previous section.! For all values of g, the leading behavior of Eq. ~18! is the same, r. 11g g mlQ. ~20! The leading behavior in the rhs of Eq. ~19!, on the other hand, depends on the particular value of g. ~i! 0,g,1: In this case, one has Q~l!. gp sin~gp! ~mlQ! g , ~21! from which we obtain Q~l!.F gp sin~gp! G 1/(12g) ~ml! g/(12g) . ~22! Combining this with Eq. ~20!, we obtain r;l1/(12g) . ~23! Here we have again the total absence of any epidemic thresh￾old and the associated critical behavior, as we have already shown for the case g51. In this case, however, the relation between r and l is given by a power law with exponent b 51/(12g), i.e., b.1. ~ii! 1,g,2: In this case, to obtain a nontrivial informa￾tion for Q, we must keep the first two most relevant terms in Eq. ~19!, Q~l!. gp sin~gp! ~mlQ! g 1 g g21 mlQ. ~24! From here we get Q~l!.F 2sin~gp! p~g21! m ~ml! g S l2 g21 mg D G 1/(g21) . ~25! The expression for r is finally r.S l2 g21 mg D 1/(g21) ;~l2lc ! 1/(g21). ~26! That is, we obtain a power-law behavior with exponent b 51/(g21).1, but now we observe the presence of a non￾zero threshold lc5g21 mg . ~27! In this case, a critical threshold reappears in the model. How￾ever, the epidemic threshold is approached smoothly without any sign of the singular behavior associated with critical point. ~iii! g.2: The relevant terms in the Q expansion are now Q~l!. g g21 mlQ2 g g22 ~mlQ! 2, ~28! and the relevant expression for Q is Q~l!. g22 g21 1 l2m S l2 g21 mg D , ~29! which yields the behavior for r r;l2lc ~30! with the same threshold lc as in Eq. ~27! and an exponent b51. In other words, we recover the usual critical frame￾work in networks with connectivity distribution that decays faster than k to the fourth power. Obviously, an exponen￾tially bounded network is included in this last case, recover￾ing the results obtained with the homogeneous approxima￾tion of Sec. III. It is worth remarking that the above results are obtained by neglecting connectivity correlations in the network, i.e., the probability that a link points to an infected node is considered independent of the connectivity of the node from which the link is emanated @see Eq. ~7!#. This approximation appears to be irrelevant in the BA network. In different SF networks with more complex topological prop￾erties, however, connectivity correlations could play a more important role and modify the analytic forms obtained in this section, Eqs. ~23! and ~26!. On the other hand, conclusions concerning the epidemic threshold absence for connectivity distributions decaying more slowly than a cubic power can be considered of general validity. Indeed, for connectivities decaying faster than the cubic power, the connectivity fluc￾tuations are bounded, and one would expect to obtain the same qualitative behavior as in exponential distribution. In summary, for all SF networks with 0,g<1, we re￾cover the absence of an epidemic threshold and critical be￾havior, i.e., r50 only if l50, and r has a vanishing slope when l→0. In the interval 1,g,2, an epidemic threshold reappears (r→0 if l→lc), but it is also approached with vanishing slope, i.e., no singular behavior. Eventually, for g.2 the usual MF critical behavior is recovered and the SF network is indistinguishable from an exponential network. VI. CONCLUSIONS The emerging picture for disease spreading in complex networks emphasizes the role of topology in epidemic mod￾EPIDEMIC DYNAMICS AND ENDEMIC STATES IN . . . PHYSICAL REVIEW E 63 066117 066117-7